DevelopmentandValidationofanObject … · 2017. 1. 3. · Abstract...

Facoltà di IngegneriaDipartimento di Energetica Sergio Stecco

DOTTORATO DI RICERCA IN ENERGETICA E TECNOLOGIEINDUSTRIALI INNOVATIVE

XX CICLO

Development and Validation of an ObjectOriented CFD Solver for Heat Transfer andCombustion Modeling in Turbomachinery

Applications

Candidato:Ing. Luca Mangani

Correlatore:Hrvoje Jasak

Relatore:Prof. Ing. Bruno Facchini

Coordinatore:Prof. Ing. Francesco Martelli

Abstract

A new object oriented code named OpenFOAM, released by OpenCFD Ltdand aimed at solving continuum mechanics problems, has been provided withsuitable tools for predicting ows involving heat transfer and combustion in con-ditions typical of turbomachinery applications. OpenFOAM supports operator-based implicit and explicit second and fourth-order Finite Volume (FV) dis-cretization in three dimensional space and on curved surfaces.

Expansions have been added in this directions: Development of an All-Machpressure based solver for co-located arrangement, development of RANS eddyviscosity isotropic and anisotropic models with particular interest of wall treat-ment, development of combustion models for premixed, partially premixed anddiusion ames and development of radiative heat transfer models.

Tests for validation here presented concern geometries for impingement aswell as lm and internal duct cooling turbine blade. While for combustion andradiation typical testcases for the validation was simulated. All the simulationsresults were compared against in-house and from literature experimental data.

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To the rst Ing. PhD Luca Mangani

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Heisenberg reportedly said on his deathbed that he would ask God twoquestions: why relativity and why turbulence? "I really think he may have an

answer to the rst question."

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Acknowledgements

The author would like to express his gratitude to Prof. Bruno Facchini foragreeing to coordinate and to supervise this thesis and to Prof. Hrvoje Jasak foragreeing to act as a co-supervisor.I also would like to thank Stefano Traverso for his encouragement.As well I would like to thank all the guys in our group for their numerousdiscussions over the course of the code development.An heartfelt thankyou to all those people that trust my capabilities, to my familyfor everything they have done during these long years of studying and to my halfsweetheart.Financial support for this study was provided by ANSALDO ENERGIA and isgreatfully acknowledge.

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Contents

Nomenclature xxiii

1 Basic CFD 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Physical principles and Mathematical model . . . . . . . . . . . . 4

1.2.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . 41.2.2 Momentum equation . . . . . . . . . . . . . . . . . . . . . 51.2.3 Energy equation . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 A model for the uid . . . . . . . . . . . . . . . . . . . . . 8

1.3 Discretization approaches . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . 11

1.3.1.1 Approximation of Volume Integrals: Source Terms 111.3.1.2 Approximation of Surface Integrals . . . . . . . 121.3.1.3 Diusive ux . . . . . . . . . . . . . . . . . . . . 131.3.1.4 Convective ux . . . . . . . . . . . . . . . . . . . 13

1.4 Finite Approximation: Interpolation and Dierentiation Schemes 131.4.1 Upwind Interpolation UDS . . . . . . . . . . . . . . . . . 141.4.2 Linear Interpolation CDS . . . . . . . . . . . . . . . . . . 141.4.3 Self Filtered Centered Interpolation SFCD . . . . . . . . . 151.4.4 Algebraic Equation . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 OpenFOAM 212.1 Introduction to C++ . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 OpenFOAM software . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Matrix structure in OpenFOAM . . . . . . . . . . . . . . . . . . 252.4 Working with main classes in OF . . . . . . . . . . . . . . . . . . 26

3 All Mach Number Pressure Based Solver 293.1 Segregated solvers . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Pressure velocity coupling . . . . . . . . . . . . . . . . . . . . . . 303.3 Problems associated with density based solvers . . . . . . . . . . 313.4 SIMPLE Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Pressure checkerboarding . . . . . . . . . . . . . . . . . . . . . . 36

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3.6 Co-located grid: Rhie - Chow interpolation . . . . . . . . . . . . 373.7 Compressibility eects . . . . . . . . . . . . . . . . . . . . . . . . 393.8 Critical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.9 Solver validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9.1 GAMM tests . . . . . . . . . . . . . . . . . . . . . . . . . 423.9.2 Goldman test . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Turbulence Modelling 454.1 The physics of turbulence . . . . . . . . . . . . . . . . . . . . . . 474.2 Models for Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 484.3 RANS approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Standard k − ε turbulence model . . . . . . . . . . . . . . . . . . 53

4.4.1 Equation for turbulent kinetic energy . . . . . . . . . . . 534.4.2 Equation for turbulent kinetic energy dissipation . . . . . 544.4.3 Drawbacks of standard two equation model . . . . . . . . 554.4.4 Wall functions . . . . . . . . . . . . . . . . . . . . . . . . 554.4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Failures in predictions for heat transfer . . . . . . . . . . . . . . . 574.6 Low Reynolds k − ε . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6.1 Abe Kondoh Nagano . . . . . . . . . . . . . . . . . . . . . 594.6.2 Chien . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6.3 Chen Lien Leschziner . . . . . . . . . . . . . . . . . . . . 614.6.4 Hwang Lin . . . . . . . . . . . . . . . . . . . . . . . . . . 624.6.5 Lam Bremhorst . . . . . . . . . . . . . . . . . . . . . . . . 634.6.6 Lien Leschziner . . . . . . . . . . . . . . . . . . . . . . . . 634.6.7 Realizability . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Two Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.8 Anisotropic Turbulence Model . . . . . . . . . . . . . . . . . . . . 66

4.8.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . 674.9 The k − ω class . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.9.1 Original k − ω . . . . . . . . . . . . . . . . . . . . . . . . 704.9.2 Baseline model . . . . . . . . . . . . . . . . . . . . . . . . 714.9.3 Shear Stress Transport model . . . . . . . . . . . . . . . . 724.9.4 Automatic Near Wall Treatment . . . . . . . . . . . . . . 73

5 Combustion and Flame Modelling 755.1 Reynolds Averaged Navier-Stokes equations of mass fraction . . . 775.2 Perfect gases law and thermodynamical properties . . . . . . . . 775.3 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.1 Arrhenius model . . . . . . . . . . . . . . . . . . . . . . . 785.3.2 Reaction mechanisms . . . . . . . . . . . . . . . . . . . . 795.3.3 Chemical kinetics in turbulent ow . . . . . . . . . . . . . 79

5.4 Flame Stretch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.5 Diusion Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.5.1 Soot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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5.5.2 Laminar diusion ames modelling . . . . . . . . . . . . . 845.5.3 Mixture fraction . . . . . . . . . . . . . . . . . . . . . . . 845.5.4 Diusive amelets . . . . . . . . . . . . . . . . . . . . . . 865.5.5 Turbulent combustion diagram . . . . . . . . . . . . . . . 875.5.6 Turbulent diusion ames modelling . . . . . . . . . . . . 895.5.7 Presumed pdf amelets models . . . . . . . . . . . . . . . 90

5.6 Premixed Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.6.1 Combustion regimes and types of ame . . . . . . . . . . 945.6.2 Turbulence and ame scales . . . . . . . . . . . . . . . . . 945.6.3 Spectral Diagram . . . . . . . . . . . . . . . . . . . . . . . 965.6.4 Premixed turbulent combustion models . . . . . . . . . . 975.6.5 Bray-Moss-Libby model . . . . . . . . . . . . . . . . . . . 985.6.6 Level-Set Model: G-Equation . . . . . . . . . . . . . . . . 101

5.7 Partially Premixed Flames . . . . . . . . . . . . . . . . . . . . . . 1025.7.1 Partially Premixed Flames Modelling . . . . . . . . . . . 102

6 Radiative Heat Transfer 1056.1 Radiation physic . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 Radiative Intensity . . . . . . . . . . . . . . . . . . . . . . 1076.2.2 Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.2.3 Radiant energy ux vector . . . . . . . . . . . . . . . . . . 1086.2.4 Black Body Emissive Power . . . . . . . . . . . . . . . . . 108

6.3 Radiative Properties of Surfaces . . . . . . . . . . . . . . . . . . . 1096.3.1 Emittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.3.2 Absorbance . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3.3 Reectance . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . 1106.4.1 Interaction with Participating Media . . . . . . . . . . . . 110

6.4.1.1 Absorption . . . . . . . . . . . . . . . . . . . . . 1116.4.1.2 Emission . . . . . . . . . . . . . . . . . . . . . . 1116.4.1.3 Scattering . . . . . . . . . . . . . . . . . . . . . . 111

6.4.2 Derivation of RTE . . . . . . . . . . . . . . . . . . . . . . 1126.4.3 Boundary conditions for RTE . . . . . . . . . . . . . . . . 112

6.5 Solution Methods For RTE . . . . . . . . . . . . . . . . . . . . . 1136.5.1 Optical Thickness . . . . . . . . . . . . . . . . . . . . . . 1136.5.2 Spectral Nature Modelling . . . . . . . . . . . . . . . . . . 113

6.5.2.1 Modelling of Radiative Properties Gaseous Com-bustion Product . . . . . . . . . . . . . . . . . . 113

6.5.2.2 Radiative properties of soot . . . . . . . . . . . . 1156.5.3 Directional Nature Modelling . . . . . . . . . . . . . . . . 1166.5.4 Spherical Armonics Method . . . . . . . . . . . . . . . . . 1176.5.5 Finite Volume Method . . . . . . . . . . . . . . . . . . . . 119

6.6 Radiative Heat Transfer Eect On Overall System Energy Balance121

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7 Results: Heat Transfer 1237.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.2 Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3 Impingement Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.3.1 ERCOFTAC C25 Axial-symmetric Impingement . . . . . 1277.3.2 Single hole case . . . . . . . . . . . . . . . . . . . . . . . . 1317.3.3 Multirow Impingement Cooling . . . . . . . . . . . . . . . 134

7.4 Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.4.1 Sinha Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.4.2 Multirow Eusion Cooling . . . . . . . . . . . . . . . . . . 1417.4.3 Fan-Shaped Cooling Holes . . . . . . . . . . . . . . . . . . 143

7.4.3.1 SAA Geometry . . . . . . . . . . . . . . . . . . . 1457.4.3.2 PAA Geometry . . . . . . . . . . . . . . . . . . . 150

7.5 Blade internal cooling - Trailing Edge . . . . . . . . . . . . . . . 1537.5.1 Trailing Edge Cooling System: AITEB2 . . . . . . . . . . 153

7.5.1.1 Geometry G1.0 . . . . . . . . . . . . . . . . . . . 1587.5.1.2 Geometry G1.2A and geometry comparison . . . 159

8 Results: Turbulent Combustion Modeling 1618.1 Results: Diusion Flames . . . . . . . . . . . . . . . . . . . . . . 162

8.1.1 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . 1628.1.1.1 Libraries generation . . . . . . . . . . . . . . . . 1628.1.1.2 Coupling with CFD code . . . . . . . . . . . . . 163

8.1.2 Model Validation: Blu Body SANDIA Burner . . . . . . 1648.2 Results: Premixed Flames . . . . . . . . . . . . . . . . . . . . . . 176

8.2.1 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . 1768.2.2 Model Validation: Lean Premixed Test Rig . . . . . . . . 177

8.3 Results: Partially Premixed Flames Model . . . . . . . . . . . . 1808.3.1 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . 1808.3.2 Model Validation: GE10 Burner . . . . . . . . . . . . . . 181

9 Results: Radiative Heat Transfer 1899.1 Results: P1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.1.1 RTE Numerical Modeling . . . . . . . . . . . . . . . . . . 1909.1.2 Model Validation: Concentrical Cylinders Testcase . . . . 191

9.2 Results: Finite Volume . . . . . . . . . . . . . . . . . . . . . . . 1949.2.1 RTE Numerical Modeling . . . . . . . . . . . . . . . . . . 1949.2.2 False Scattering . . . . . . . . . . . . . . . . . . . . . . . . 1969.2.3 Model Validation: RADIARE . . . . . . . . . . . . . . . . 197

9.2.3.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . 1979.2.3.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 199

9.2.4 Model Validation: Spherical Isothermal Domain . . . . . 202

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10 Results: Combustion & RTE Models Coupling 20510.1 Numerical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.1.1 Participating Media Absorption Coecient Model: GaseousMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

10.1.2 Soot absorption coecient . . . . . . . . . . . . . . . . . . 20610.1.3 Soot modelling . . . . . . . . . . . . . . . . . . . . . . . . 20610.1.4 Coupling with CFD code . . . . . . . . . . . . . . . . . . 207

10.2 Model Validation: Craneld Jet-k Burner . . . . . . . . . . . . . 208

Conclusions 217

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List of Figures

1.1 Fluid element for conservation laws. . . . . . . . . . . . . . . . . 51.2 Stress component in the x direction. . . . . . . . . . . . . . . . . 61.3 Heat ux vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Typical bi-dimensional quadrilateral control volume. . . . . . . . 111.5 Typical structured C grid example. . . . . . . . . . . . . . . . . . 161.6 Typical hybrid tetrahedral hexaedral unstructured mesh example. 17

2.1 Polyhedral mesh example. . . . . . . . . . . . . . . . . . . . . . . 252.2 UML diagram for GeometricField class. . . . . . . . . . . . . . . 26

3.1 Bidimensional control volume. . . . . . . . . . . . . . . . . . . . . 333.2 Prole Mach number in upper and lower wall. . . . . . . . . . . . 423.3 Density gradient contour plots. . . . . . . . . . . . . . . . . . . . 433.4 Stator Blade analysis. . . . . . . . . . . . . . . . . . . . . . . . . 44

4.1 Da Vinci sketch of turbulent ow. . . . . . . . . . . . . . . . . . . 474.2 LES predictions of shear stress (top row), rate of strain (middle

row), and eddy viscosity (bottom row) for the xy component (leftcolumn), yz component (middle column) and xz component(rightcolumn). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 Comparison of analytical correlation and DNS data for the anisotropyfactor γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Premixed ame propagation speed . . . . . . . . . . . . . . . . . 805.2 Examples of strained ames: a) steady strained one-dimensional

diusion ame and b) counterow premixed ames . . . . . . . . 815.3 Diusion ame structure . . . . . . . . . . . . . . . . . . . . . . . 825.4 Example of ame structure in Z-space, methane-air equilibrium

ame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.5 Regimes in nonpremixed turbulent combustion . . . . . . . . . . 875.6 Shapes of the β-function for dierent value of Z and γ . . . . . . 915.7 Schematic representation of a premixed ame . . . . . . . . . . . 935.8 Schematic representation of both amelet regimes. If Ka < 1

amelets are corrugated (a) while for 1 < Ka < 100 amelets arethickened (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9 Peters' combustion regimes diagram . . . . . . . . . . . . . . . . 96

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5.10 Schematic representation of a lifted jet diusion ame . . . . . . 103

6.1 Symbols for the denition of radiative intensity . . . . . . . . . . 1076.2 Scattering due to a small particles . . . . . . . . . . . . . . . . . 1116.3 Solution methods for the dierential radiative equation . . . . . . 1166.4 Coordinate system for FVM angular discretization . . . . . . . . 120

7.1 Turbulence quantities proles. . . . . . . . . . . . . . . . . . . . . 1267.2 Stanton number prole. . . . . . . . . . . . . . . . . . . . . . . . 1277.3 ERCOFTAC - Schematic view of experimental set up. . . . . . . 1287.4 ERCOFTAC - Geometry detail of the grid around the stagnation

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.5 ERCOFTAC - Nusselt number distribution along radius. . . . . . 1297.6 Turbulent kinetic energy distribution on symmetry plane [m2/s2]. 1307.7 Impingement single hole - Grid. . . . . . . . . . . . . . . . . . . . 1317.8 Impingement single hole - Heat transfer coecient on impinged

wall along symmetry line. . . . . . . . . . . . . . . . . . . . . . . 1327.9 Impingement single hole - Heat transfer coecient distribution

on impinged wall [W/(m2K)]. . . . . . . . . . . . . . . . . . . . . 1337.10 Impingement single hole - Velocity magnitude on cutting hole

symmetry plane [m/s]. . . . . . . . . . . . . . . . . . . . . . . . . 1347.11 Impingement single hole - Turbulent kinetic energy on cutting

hole symmetry plane [m2/s2]. . . . . . . . . . . . . . . . . . . . . 1357.12 Impingement ve holes - Grid. . . . . . . . . . . . . . . . . . . . 1357.13 Impingement ve holes - Heat transfer coecient along center lines.1367.14 Impingement ve holes - Velocity magnitude on symmetry plane

[m/s]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.15 Impingement ve holes - Temperature distribution on symmetry

plane [K]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.16 Impingement ve holes - Heat transfer coecient distribution on

impinged wall [W/(m2K)]. . . . . . . . . . . . . . . . . . . . . . . 1377.17 Sinha - Calculation domain. . . . . . . . . . . . . . . . . . . . . . 1387.18 Sinha - Near wall mesh details. . . . . . . . . . . . . . . . . . . . 1397.19 Spanwise distribution of eectiveness. . . . . . . . . . . . . . . . 1407.20 Laterally averaged and local center line eectiveness. . . . . . . . 1407.21 Eectiveness distribution over the wall. . . . . . . . . . . . . . . 1417.22 Mesh details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.23 Spanwise averaged adiabatic eectiveness. . . . . . . . . . . . . . 1437.24 Comparison between laterally averaged and local center line lm

cooling eectiveness. . . . . . . . . . . . . . . . . . . . . . . . . . 1437.25 SAA Computational grids: coarser 7.25(a) and ner 7.25(b) . . 1447.26 PAA Computational domain . . . . . . . . . . . . . . . . . . . . 1457.27 Dierences between numerical and experimental domain . . . . . 1457.28 Local lateral η for three location downstream for SAA smooth

conguration BR=0.73 . . . . . . . . . . . . . . . . . . . . . . . . 146

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7.29 Local lateral η for three location downstream for SAA smoothconguration BR=1.31 . . . . . . . . . . . . . . . . . . . . . . . . 146

7.30 Local lateral η for three location downstream for SAA sharp con-guration BR=0.74 . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.31 Local lateral η for three location downstream for SAA sharp con-guration BR=1.29 . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.32 Numerical adiabatic eectiveness map for SAA smooth congu-ration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.33 Experimental adiabatic eectiveness map for SAA smooth con-guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.34 Numerical adiabatic eectiveness map for SAA sharp congura-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.35 Experimental adiabatic eectiveness map for SAA sharp cong-uration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.36 Spanwise averaged eectiveness < η > for SAA geometry forsmooth (7.36(a), 7.36(b)) and sharp (7.36(c), 7.36(d)) congura-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.37 Local lateral η for three location downstream for PAA smoothconguration BR=0.75 . . . . . . . . . . . . . . . . . . . . . . . . 150

7.38 Local lateral η for three location downstream for PAA smoothconguration BR=1.36 . . . . . . . . . . . . . . . . . . . . . . . . 150

7.39 Numerical adiabatic eectiveness map for PAA smooth congu-ration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.40 Experimental adiabatic eectiveness map for PAA smooth con-guration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.41 Spanwise averaged eectiveness < η > for PAA geometry andsmooth conguration . . . . . . . . . . . . . . . . . . . . . . . . 151

7.42 Typical compound cooling system of a blade. . . . . . . . . . . . 1537.43 Computationl domain and boundaries . . . . . . . . . . . . . . . 1547.44 Computational grids: pedestal and rib details for unstructured

and structured meshes . . . . . . . . . . . . . . . . . . . . . . . . 1557.45 Geometry G1.1A: heat transfer coecient map . . . . . . . . . . 1567.46 Geometry G1.1A: heat transfer coecient distribution . . . . . . 1567.47 Geometry G1.0: heat transfer coecient map . . . . . . . . . . . 1577.48 Geometry G1.0: heat transfer coecient distribution . . . . . . . 1587.49 Geometry G1.0: contour of x vorticity component on six sections

normal to x axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.50 Geometry G1.2a: heat transfer coecient map . . . . . . . . . . 1597.51 Geometry comparison: heat transfer coecient distribution . . . 1597.52 Geometry comparison: velocity vector plot . . . . . . . . . . . . . 1607.53 Geometry comparison: static pressure distribution . . . . . . . . 160

8.1 Diusion ame model owchart . . . . . . . . . . . . . . . . . . . 1648.2 BBody burner: geometry and ame images . . . . . . . . . . . . 165

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8.3 BBody burner: schematic representation of the ame zones withsome sampling sections . . . . . . . . . . . . . . . . . . . . . . . . 165

8.4 BBody mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1668.5 BBody burner: mixture fraction (left) and temperature contours

(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1678.6 BBody burner: OH and CO mass fraction contours . . . . . . . 1688.7 BBody burner: details of velocity eld contour in the recircula-

tion zone with streamlines . . . . . . . . . . . . . . . . . . . . . . 1698.8 BBody burner: mixture fraction variance contour near blu body. 1698.9 BBody burner: mixture fraction average proles. . . . . . . . . . 1708.10 BBody burner: mixture fraction variance proles. . . . . . . . . . 1718.11 BBody burner: temperature proles. . . . . . . . . . . . . . . . . 1728.12 BBody burner: species proles at Y=13 mm. . . . . . . . . . . . 1738.13 BBody burner: species proles at Y=30 mm. . . . . . . . . . . . 1738.14 BBody burner: species proles at Y=45 mm. . . . . . . . . . . . 1748.15 BBody burner: species proles at Y=65 mm. . . . . . . . . . . . 1748.16 BBody burner: species proles at Y=90 mm. . . . . . . . . . . . 1758.17 BBody burner: species proles at Y=120 mm. . . . . . . . . . . . 1758.18 Localization of mean ame front by linear interpolation. . . . . . 1768.19 Vanderbilt combustor geometry and boundary conditions. . . . . 1788.20 Contour plots of temperature and progress variable source for tfc

model (left) and of temperature and iso-g for level-set model (right).1788.21 Averaged temperature proles at three dierent locations x/d =

0.1, x/d = 0.8 and x/d = 1.0. . . . . . . . . . . . . . . . . . . . . 1798.22 CO Mole fraction. . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.23 Flowchart of BML for partial premixed ame. . . . . . . . . . . . 1818.24 GE 10: schematic representation . . . . . . . . . . . . . . . . . . 1828.25 GE 10: schematic representation of the burner, property of GE-NP1828.26 GE 10: used mesh, property of GE-NP . . . . . . . . . . . . . . . 1848.27 GE 10: schematic representation of geometrical model and ame 1848.28 GE 10: plot of mixture fraction near combustor inlet . . . . . . . 1858.29 GE 10: velocity contours, left STAR− CDTM right FOAM . . 1868.30 GE 10: vector plot of velocity eld near combustor cap by (An-

dreini, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.31 GE 10: vector plot of velocity eld near combustor cap by the

implement model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.32 GE 10: progress variable contours from right to left: KPP closure,

TFC closure and STAR− CDTM KPP closure . . . . . . . . . . 1878.33 GE 10: temperature contours from right to left: KPP closure,

TFC closure and STAR− CDTM KPP closure . . . . . . . . . . 1878.34 GE 10: ame lift, STAR− CDTM . . . . . . . . . . . . . . . . . 1878.35 GE 10: ame lift, FOAM-KPP . . . . . . . . . . . . . . . . . . . 1878.36 GE 10: ame lift, FOAM-TFC . . . . . . . . . . . . . . . . . . . 187

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8.37 GE 10: progress variable source term from right to left: KPPclosure, TFC closure and STAR− CDTM KPP closure . . . . . 188

8.38 GE 10: progress variable source term from right to left: KPPclosure, TFC closure and STAR− CDTM KPP closure . . . . . 188

9.1 Flowchart of P1 solving algorithm . . . . . . . . . . . . . . . . . 1919.2 Flowchart of FMV-RTE solution . . . . . . . . . . . . . . . . . . 1959.3 Bidimensional control volume for RTE . . . . . . . . . . . . . . . 1969.4 RADIARE case 1: well-stirred combustion chamber . . . . . . . 1989.5 RADIARE perfectly stirred combustor wall radiative heat ux

contourt. Emissivity=0.5 . . . . . . . . . . . . . . . . . . . . . . 1989.6 RADIARE perfectly stirred combustor wall radiative heat ux

vs z coordinate. Emissivity=0.5 . . . . . . . . . . . . . . . . . . . 1999.7 RADIARE perfectly stirred combustor wall radiative heat ux

vs z coordinate. Emissivity=0.8 . . . . . . . . . . . . . . . . . . . 1999.8 RADIARE case 2: cylindrical combustion chamber. . . . . . . . 2009.9 RADIARE case 2 temperature distribution for short-ame (up-

per) and long-ame (lower) . . . . . . . . . . . . . . . . . . . . . 2009.10 RADIARE case 2 quarter mesh . . . . . . . . . . . . . . . . . . . 2019.11 RADIARE case 2 wall irradiance by short-ame . . . . . . . . . 2029.12 RADIARE case 2 wall irradiance by long-ame . . . . . . . . . . 2039.13 Spherical domain unstructured mesh . . . . . . . . . . . . . . . . 2039.14 Spherical domain %-error on enthalpy sink vs absorption coecient204

10.1 Diusion ame combined with soot and radiative heat exchangemodelling owchart . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.2 Jet-k: schematically representation . . . . . . . . . . . . . . . . . 20910.3 Jet-K burner : axial plots . . . . . . . . . . . . . . . . . . . . . . 21110.4 Jet-K burner : radial plots at 100 mm from burner . . . . . . . . 21210.5 Jet-K burner : radial plots at 300 mm from burner . . . . . . . . 21310.6 Jet-K burner : absorption coecient plots . . . . . . . . . . . . . 21410.7 Jet-K burner : temperature contours left P1, right FVM . . . . . 21510.8 Jet-K burner : enthalpy radiative sink contours left P1, right FVM 21510.9 Jet-K. From left to right contours of: mixture fraction Favre's

average, hydroxide mass fraction, soot source term [s−1] and sootmass fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

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List of Tables

4.1 Standard coecients values for standard k − ε. . . . . . . . . . . 544.2 Standard coecients values for Abe et al. (1994) Low Reynolds

k − ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Standard coecients values for Chien Low Reynolds k − ε. . . . 614.4 Standard coecients values for Chen et al. Low Reynolds k − ε. 624.5 Standard coecients values for Hwang and Lin Low Reynolds k−ε. 634.6 Standard values for Lam and Bremhorst Low Reynolds k − ε

coecients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.7 Standard coecients values for Lien and Leschziner Low Reynolds

k − ε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.8 Standard coecients values for Two Layer k − ε. . . . . . . . . . 664.9 Standard values for k − ω coecients. . . . . . . . . . . . . . . . 704.10 Standard values for k − ω BSL coecients. . . . . . . . . . . . . 724.11 Standard values for k − ω SST coecients. . . . . . . . . . . . . 73

7.1 Acronyms for the various turbulence models. . . . . . . . . . . . 1247.2 Typical values for the underrelaxation factors used. . . . . . . . . 1247.3 Flat plate - Computational boundary conditions. . . . . . . . . . 1257.4 ERCOFTAC - Flow conditions. . . . . . . . . . . . . . . . . . . . 1297.5 Impingement single hole - Flow conditions. . . . . . . . . . . . . 1327.6 Impingement single hole - Computational boundary conditions. . 1327.7 Sinha - Flow conditions. . . . . . . . . . . . . . . . . . . . . . . . 1397.8 Sinha - Computational boundary conditions. . . . . . . . . . . . 1407.9 Eusion - Flow conditions. . . . . . . . . . . . . . . . . . . . . . . 1427.10 Eusion - Computational boundary conditions. . . . . . . . . . . 1427.11 Blowing Ratio and Reynolds number for hot tests . . . . . . . . . 1447.12 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.13 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1557.14 Grid characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.1 BBody INLETS boundary conditions . . . . . . . . . . . . . . . 1668.2 GE10: inlet air condition, TCD and PCD stand for Temperature

and Pressure at Compressor Discharge . . . . . . . . . . . . . . . 183

9.1 P1 cylinders: temperatures of inner T1 and outer T2 cylinders. . 192

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9.2 P1 cylinders: %−Error on nondimensional heat ux Psi versusoptical thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

9.3 Constants value for RADIARE case 2 temperature distribution . 200

10.1 Jet-K: Sandia absorption coecient constants . . . . . . . . . . . 20610.2 Jet-K: boundary conditions. . . . . . . . . . . . . . . . . . . . . . 209

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Nomenclature

Symbols

A generic coecient matrix, generic diagonal coecientin algebraic equation

[ ],[A]

a generic o diagonal coecient in algebraic equation [A]b mass imbalance [kg/s]Cµ coecient for eddy viscosity [ ]Cp constant pressure specic heat [J/(kgK)]Cω1, Cω2 coecients in specic turbulent dissipation equation [ ]Cε1, Cε2 coecients in turbulent dissipation equation [ ]c speed of light in vacuum [m/s]c speed of sound [m/s]d ∆x

a [m/A]E specic energy [J/kg]fµ damping function for eddy viscosity [ ]f1, f2 damping functions for ε equation [ ]f external force per unit volume [N/m3]F external force, surface mass ux [N ], [kg/s]G explicit part of turbulent kinetic energy production [kg/(m3s)]h0 specic total enthalpy [J/kg]H remainder [ ]h specic static enthalpy [J/kg]h specic thermochemical static enthalpy [J/kg]HTC heat transfer coecient [W/(m2K)]i, j, k generic counters [ ]i specic internal energy [J/kg]~J generic ux vector [Φ/(m2s)]j generic face ux [Φ/(m2s)]k+ non-dimensional turbulent kinetic energy = k

u2τ

[ ]

k absorption coecient, turbulent kinetic energy [m−1], [m2/s2]k thermal conductivity [W/(mK)], [m2/s2]lµ turbulent length scale for eddy viscosity [m]lε turbulent length scale for dissipation [m]lt turbulent length scale [m]

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m mass [kg]~n normal versor [ ]n number of cell [ ]Pk turbulent kinetic energy production term [kg/(m3s)]P pitch [m]p pressure [Pa]Prt turbulent Prandtl number [ ]Pr Prandtl number [ ]Q rate of heat [W ]Q source term [Φ/s]q source term per unit volume [Φ/(m3s)]~q heat ux vector [W/m2]R gas constant [J/(kg K)]R perfect gases universal constant [J/(kg K)]Ret Reynolds turbulent number = k2

εν [ ]

Rey Reynolds y number =√

kyν [ ]

Re Reynolds' number [ ]~s light beam versor [ ]Sij rate of strain tensor [s−1]S work done by body forces per unit volume, surface [W/m3], [m2]Sc Schmidt's number [ ]T+ non dimensional temperature [ ]Ts turbulent time scale [s]T temperature [K]t time [s]

Tu turbulent intensity = u′

U≈√

23 k

U[%]

T Reynolds stress tensor [Pa]~U velocity vector [m/s]u0 pseudo velocity [m/s]uτ friction velocity [m/s]u0 equivalent friction velocity [m/s]Ui generic velocity component [m/s]u, v, w velocity components in x,y,z direction [m/s]W rate of work [W ]xi generic cartesian coordinate [m]x, y, z cartesian coordinates [m]y+ non-dimensional wall distance yuτ

ν [ ]y wall distance [m]Z mixture fraction []Z ′′2 mixture fraction variance []

Greek symbols

α thermal diusivity, relaxation factor [m2/s], [ ]

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α absorbance []β∗ constant for turbulent kinetic energy production term

k − ω class[ ]

γ specic heat ratio, blending factor, v′w′

[ ], [ ], [ ]Γ diusivity coecient [1/sm]δij Kronecker delta [ ]∆x,∆y dimensions of bidimensional control volume [m]ε turbulent kinetic energy dissipation [m2/s3]ε+ non-dimensional turbulent kinetic energy dissipation

= νεu4

τ

[ ]

ε emittance []η adiabatic eectiveness [ ]< η > averaged adiabatic eectiveness [ ]κ Von Karman's constant = 0.41 [ ]λ distance weighting factor, bulk viscosity [ ], [kg /(m s)]µ viscosity [kg /(m s)]µt eddy viscosity [kg /(m s)]ν kinematic viscosity [m2/s]ρ density [kg/m3]σk, σε, σω coecients for eddy viscosity in k, ε and ω equation [ ]τ shear stress [Pa]Φ generic scalar, viscous dissipation [Φ], [W/m3]Ψ 1

RT - compressibility factor [kg/J ]χ net ux through surface [Φ/s]ω specic turbulent kinetic energy dissipation [s−1]Ω volume [m3]Ω solid angle [strad]

Subscripts

b bodyb blackbody, or burnt state (BML model)c coolantE,W,N, S east, west, north, south cell neighbore, w, n, s east, west, north, south faceEE right (east) neighbor of east neighborg hot gasλ wavelenght [nm]m counter for inner iterationn normal directionnb neighborP cell of reference nodeR referred to radiationref reference

xxv

s surfaceT referred to turbulenceu unburnt state (BML model)w wall

Superscripts

n outer iteration counter′ correction, Reynolds uctuation′′ Favre uctuation∗ guessed

Reynolds averaged˜ Favre averaged

xxvi

Introduction

One of the most demanding task in gas turbine design is the proper evalu-ation of all phenomena involved with heat transfer. Of particular interest arethe compound cooling systems for the hot components of the engine. Preciseinvestigations for the zones in which heat transfer to solid walls is higher isfundamental as long as actual design criteria force cooling system to more andmore limiting conditions. Cooling system is extremely important in order toraise the gases temperature at turbine inlet that results in an improvement ofthe machine performances and consequently in fuel economy. Thinking aboutblade and combustion chamber cooling, the need for improvements in overallperformances tend to increase temperature in turbine, at the same time theamount of available cooling air is wanted to be as low as possible to keep worklosses at the minimum. For combustors the situation is not so dierent with thenew lean premixed ames design criteria reducing the quantity of air for linercooling. The combustor is the device that gives the power to the machineand by doing this it burns fuel and produces pollutants, it is then clear that aproper combustor design is fundamental in order to meet either performancesor pollution reduction goals. So the combustor design has two main tasks, bothaecting each other: the reaction zone design and the combustor cooling systemdesign.

Accurate heat transfer measurements are however very expensive to supportdue to the complexity of geometries, the high costs of measuring apparatus andthe long set up necessary to collect reliable data. This is why CFD analysis isbecoming more and more popular in each phase of the design process.

Nevertheless, CFD simulations for evaluation of thermal loads and eective-ness of the cooling devices in gas turbine engines are one of the most complexto face. Complexity arises especially because of the need for advanced physicalmodelling and exibility in geometrical mesh handling. Typical example for thenecessity of particular models is the treatment of turbulence both in combustionand heat transfer analysis. Complexity grows because the good functioning ofthese models is quite case dependent. Viceversa, other kinds of cooling devicesare very demanding in terms of mesh handling: internal ducts are usually im-proved with non aerodynamic turbulator such as ribs or pins, where a very negrid is required to resolve the complex structure of the local ow.

The capability to handle hybrid unstructured meshes and the availability of

xxvii

a quite large set of turbulence models become a must for a CFD code aimed atsolving heat transfer related problems. Unfortunately these requirements resultto be quite strict and not many softwares worldwide can satisfy them. Thatis why, in the last decade, both industrial and academic researchers have beenwidely using only few well known commercial solvers.

Commercial solvers aspire to resolve many kinds of ow and so they areoften provided with a big amount of dierent packages for various models. Thisgenerality often results in poor performances in terms of calculation times dueto a waste of system resources for a large part of packages that are not used instandard simulations.

Apart from this, the main drawback, according to expert users, stands inhow commercial CFD codes behave like a black box solution maker. Especiallyin case of disagreements with experiments, it is not of secondary importanceto perfectly understand under which hypothesis numerical predictions couldbe generated. Information on the process used to obtain convergence can befound only on often poor of details user manuals: source codes are obviouslyinaccessible.

Advanced users in heat transfer and combustion modelling applications some-times need the use of ad hoc models or modications suitable for specic cases.User subroutine features provided by commercial packages become quickly inad-equate as the complexity of modications grows. Furthermore R&D departmentof big companies usually need to tune built-in models in order to feed calcula-tions tools with their design practice frequently based on detailed and expensiveexperimental tests.

This work is a project aimed to build a software package containing manyof the CFD tools needed for turbomachines cooling and combustion systemsdesign.

The objective of this thesis is to show the capabilities of a new open-sourcesoftware environment where it is possible to implement new models, renew theexisting ones and experiment with model combinations. The OpenFOAM pack-age (Field Operation And Manipulation) is an object-oriented numerical simu-lation toolkit written in C++ language. Besides its advanced basic native CFDfeatures its essential characteristic, in opposition to commercial solvers, is theopportunity to build new models and solvers with high simplicity and in lesstime than with standard FORTRAN based codes. Object-oriented program-ming of C++ drastically reduces the probability of bugs introduction with aconsequent reduction in debugging time.

This thesis describes the attempt to build a CFD package suitable for typicalsteady state heat transfer analysis and to add the combustion and radiativeheat exchange modelling. As will be described later, to reach such goal it wasnecessary to introduce specic features in the package in order to overcomethe limitations of built-in approaches: rst of all a compressible steady statesolver capable of handling transonic ows, then a set of turbulence two-equationsclosures with particular reference to a detailed near wall treatment. Concerning

xxviii

combustion modelling the software is now capable of simulating diusion andpartially premixed ames aected by soot formation and radiative exchange.

As conrmation of the work done a set of validation applications were per-formed. These capabilities have been tested by proper testcases and we willfocus our attention on the validation of the code with some complex congura-tions typical of heat transfer and combustion problems.

Thesis OutlineChapter 1 is based on a review of classical arguments concerned with Com-

putational Fluid Dynamics. All the dierent parts numerical solution methodsare composed by, are briey presented with no aim of completeness but as hintof the approach used in this thesis.

Chapter 2 presents main features of the base code OpenFOAM with anoverview on the C++ philosophy of programming, a description of how thelinear systems are built and an example of the structure of one of the mostcommon classes. Moreover an example on how to write a postprocessing utilityis given.

Chapter 3 describes the solver used. First general themes regarding solutionalgorithms such as pressure velocity coupling and segregation are introduced.Thus the ancestor of the solver used is described and nally expansion to co-located approach and compressibility eects are presented.

Chapter 4 introduces the problem of modeling turbulence as rst thing.Eddy viscosity RANS models are introduced rst deriving standard k−ε model,then presenting a number of dierent models for Low Reynolds grids. Finally thespecic OpenFOAM class for turbulence models is described and commented.

Chapter 5 describes the RANS equations set for a combustion system to-gether with perfect gases equations of state and chemical kinetics.

Chapter 5.5 presents the diusion ame modeling starting from laminarames models that are further used in turbulent combustion modelling after abrief description of combustion regimes. It also deals with the soot formationwhich is peculiar of diusion ames.

Chapter 5.6 describes those models suitable to simulate premixed amesmodelling and provides details on combustion regimes.

Chapter 5.7 shows how the merge of diusion and premixed ames modelsis performed in order to analyse partially premixed combustion systems.

Chapter 6 briey presents the radiative heat exchange phenomenon thatleads to the denition of the radiative transfer equation. Then are reported thesolution methods for this equation that represent the radiative transfer models.

Chapter 7 reports the results for the simulations done to validate the modelsproposed. After an initial test over a at plate, typical geometries for turboma-chinery cooling devices are presented.

Chapter 8 reports the implementation procedure of the diusion, premixedand partially premixed ames models appliyed to reference test case and indus-

xxix

trial combustor .Chapter 9 shows how the Finite Volume Method and P1 models for radiative

transfer has been implemented. The Finite Volume is validated by three test-cases, the rst two by RADIARE program, the last one, representing a sphericalblackwalled domain, is intended to test the capability to work with unstructuredmeshes as well as to analyse the behaviour of the model in troublesome condi-tions. While the P1 is validated by comparing obtained results to an analyticalsolution by Modest (2003).

Chapter 10 describes how the radiative transfer modelling has been coupledto the diusion ames one. Moreover has been implemented a soot formationmodel, being this specie heavily produced in jet engines combustors and stronglyaecting the radiative transfer. The model has been validated by the CraneldUniversity Jet-K testcase representing a diusion Jet-A ame.

Chapter 10.2 gives the conclusions of this research work.

xxx

Chapter 1

Basic CFD

This chapter introduces and discusses the basilar aspects of ComputationalFluid Dynamics starting from a general overview, with the aim of organizingsuch branch into the more complex environment of uid ow predictions, andgoing through the structural parts in which numerical methods can besubdivided. In particular all basic aspects of numerical prediction techniquesare touched, starting from the governing equations of the mathematical model,giving a brief introduction to discretization methods and nite approximationand mentioning the methods for solving the resulting system. In the followingparticular interest will be placed on the Finite Volume Method.

Contents1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Physical principles and Mathematical model . . . 4

1.2.1 Continuity equation . . . . . . . . . . . . . . . . . . 41.2.2 Momentum equation . . . . . . . . . . . . . . . . . . 51.2.3 Energy equation . . . . . . . . . . . . . . . . . . . . 71.2.4 A model for the uid . . . . . . . . . . . . . . . . . . 8

1.3 Discretization approaches . . . . . . . . . . . . . . 91.3.1 Finite Volume Method . . . . . . . . . . . . . . . . . 11

1.3.1.1 Approximation of Volume Integrals: SourceTerms . . . . . . . . . . . . . . . . . . . . . 11

1.3.1.2 Approximation of Surface Integrals . . . . 121.3.1.3 Diusive ux . . . . . . . . . . . . . . . . . 131.3.1.4 Convective ux . . . . . . . . . . . . . . . . 13

1.4 Finite Approximation: Interpolation and Dier-entiation Schemes . . . . . . . . . . . . . . . . . . . 13

1.4.1 Upwind Interpolation UDS . . . . . . . . . . . . . . 141.4.2 Linear Interpolation CDS . . . . . . . . . . . . . . . 141.4.3 Self Filtered Centered Interpolation SFCD . . . . . . 151.4.4 Algebraic Equation . . . . . . . . . . . . . . . . . . . 15

1.5 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Solution method . . . . . . . . . . . . . . . . . . . . 16

1

Chapter 1. Basic CFD

1.1 OverviewPrediction of heat transfer and uid ow processes can be obtained by two

main methods: experimental investigation and theoretical calculation.Even if reliable information about physical processes could only be obtained

by actual measurement, experimental investigations involving full scale equip-ment often result to be too expensive. The alternative is so to use scaled modelsand conditions, at the end extrapolate the results to full scale. This scaling,however, is not completely free from errors: general rules for correct scaling areoften unavailable, phenomena may not be scalable (i.e. combustion, turbulence,etc ...), measuring instruments errors may weight more.

Such problems can be avoided with the help of a mathematical model ableto suitably represent the physical process. Theoretical predictions consist inworking out the consequences of such model. For uid dynamics problems, themathematical model basically consists in a set of partial dierential equation.If classical mathematics techniques were to be used for solving such equations,there would be little hope of predicting many cases of practical interest withclosed form solution. Hopefully, with the development of numerical methodsand the availability of large digital computers, closure can be found almost forany practical problem.

Furthermore, computer analysis oers several advantages with respect toexperimental investigation: low cost, speed, complete and detailed information,capability in simulating both realistic and ideal conditions. At the same time,numerical calculation are not free from disadvantages: it can happen that asuitable mathematical model for describing the physical conditions cannot befound, it is possible for prediction with a very limited objective not to be cheaperthan experiments, problems involving complex geometries, strong non linearitymay be harder, longer and again more expensive to solve.

It appears clearly now how experiments and computations must coexist andinteract to have wide and reliable predictions (Patankar, 1980).

Let's now analyze in detail what a numerical solution method is composedby. As above written, the starting point of a numerical method is the Math-ematical Model, i.e. the set of partial dierential equations and boundaryconditions. Fluid dynamic science teaches that exact conservation laws describethe behavior of all ows: no matter the type of ow, it will respect the generalgoverning equations. General purpose methods however are often impractical, ifnot impossible, to solve so it is more convenient to include simplications in themathematical model and develop a solution method designed for that particularset of equations. Then a suitable Discretization Method, approximating theset of dierential equations by a system of algebraic equations for the variablesat a number of discrete points in space and time, is necessary. The most im-portant discretization methods are: Finite Dierence Method (FDM), FiniteVolume Method (FVM) and Finite Element Method (FEM). The discrete loca-tions where the variables want to be calculated, are dened by the NumericalGrid. The numerical grid is a discrete representation of the ow domain (both

2

Basic CFD Chapter 1.

in space and time) through the use of a nite number of subdomains such aselements, control volumes etc...

Then a Finite Approximation technique has to be selected taking in con-sideration the choice for the discretization method and the numerical grid. Thischoice inuence a lot the accuracy of the solution as well as the development,coding, debugging and the speed of the solution method. More accurate approx-imations involve, in fact, more nodes and give usually a fuller coecient matrix.A compromise between accuracy and eciency is always necessary. Once thislarge system of non-linear algebraic equations has been built by discretizationtechniques, it must be solved using a Solution Method. Such methods usesuccessive linearization of the equations and the resulting linear systems arealmost always solved by iterative techniques. Usually there are two levels ofiterations: inner iterations, within which the linear equation are solved, and theouter iterations, that deal with the non linearity and coupling of the equations.As last point, it is important to determine suitable Convergence Criteria. Itis fundamental to well set stopping conditions for both the inner and the outercycles in order to obtain accurate solution in an ecient way.

Once dened, numerical methods must be checked to posses certain prop-erties in order to establish whether a method is appropriate or not. The mostimportant properties are:

• Consistency: discretization should become exact as the grid spacingtends to zero. In other words truncation error, i.e. the dierence betweenexact and discretized equation, must go to zero as 4t→ 0 and 4x→ 0.

• Stability: errors appearing in the course of numerical solution process donot magnify. For iterative methods, stable methods are the ones that donot diverge.

• Convergence: the solution of the discretized equation tends towards theexact solution as the grid spacing tends to zero. It is a very dicultproperty to demonstrate, it is usually accepted to test grid-independencefor a solution.

• Conservation: solution must respect conservation of physical quantitiesboth on local and global scale. It is a very important property becauselimits solution error. Even if on ne grids non-conservative schemes canalso lead to correct solutions, conservative ones are usually preferred.

• Boundedness: solution should lie within proper bounds. Boundednessis dicult to guarantee and often unbounded schemes have stability andconvergence problems too.

• Realizability: guarantee a model to give physically realistic solutions forthe phenomena it is representing. Usually in connection with phenomenatoo complex to be directly simulated.

3


• Accuracy: is the property of well approximating the exact solution, inother words limiting modeling, discretization and iteration errors.

1.2 Physical principles and Mathematical model

In this section conservation equations for mass, momentum and energy fornon reacting mono-phase compressible ows, will be described.

Firstly equations are derived in the most general form as possible. Thesame approach has been used to obtain all the basic equation for uid motion:apply the appropriate fundamental physical principle to a suitable model of theow and then extract the mathematical equations which embody such physicalprinciples. The uid ow has been modeled with innitesimal control volumexed in space with the uid moving through it. As a consequence equationsare proposed in the dierential conservation form. To switch from one form toanother it must be remembered the concept of the substantial derivative:

D

Dt=

∂

∂t+ (~U · ∇) , (1.1)

meaning that the rate of increase of a scalar of uid particle, on the left handside of Eq. 1.1, is equal to the rate of increase of uid element plus the net rateof ow of the scalar out of uid element, on the right hand side of the sameequation (Anderson, jr., 1995).

Secondly an appropriate model for the uid is proposed: simplications forthe cases of interest such as ideal gas and Newtonian uid are introduced.

Third passage in manipulating the set of governing equations, i.e. steady-state simplication and turbulence modeling, is explicitly done in chapter 4 buthas been widely used in all the cases of need in the previous chapters too.

1.2.1 Continuity equation

The fundamental physical concept standing behind the continuity equationis that mass is conserved. In other words the rate of increase of mass in uidelement must equal the net rate of ow of mass into uid element or the rate ofchange of mass in particle is equal to zero (Malalasekera and Versteeg, 1995):

D m

Dt= 0 . (1.2)

With reference to Fig. 1.1 the rate of increase of mass in the uid elementis:

∂

∂t(ρ δx δy δz) =

∂ρ

∂tδx δy δz , (1.3)

4


Figure 1.1: Fluid element for conservation laws.

thus the net rate of ow of mass into uid element is:

(ρu− ∂(ρu)

∂x

12δx

)δy δz −

(ρu+

∂(ρu)∂x

12δx

)δy δz

+(ρv − ∂(ρv)

∂y

12δy

)δx δz −

(ρv +

∂(ρv)∂y

12δy

)δx δz

+(ρw − ∂(ρw)

∂z

12δz

)δx δy −

(ρw +

∂(ρw)∂z

12δz

)δx δy

= −(∂(ρu)∂x

+∂(ρv)∂y

+∂(ρw)∂z

)δx δy δz . (1.4)

Eq. 1.3 and Eq. 1.4 together give the well known dierential continuity equa-tion in conservation form:

∂ρ

∂t+∇ · (ρ~U) = 0 . (1.5)

1.2.2 Momentum equation

From Newton's second law directly follows the momentum equation: therate of increase of momentum of uid particle equals the sum of forces on uidparticle:

mD~U

Dt= Fs + Fb . (1.6)

It is easier to separate external forces into surface forces Fs and body forces Fb.The rst ones being dened upon control volume boundaries, the second oneson the volume itself. It is common to separate surface forces into pressure, thehydrostatic part of the stress tensor, and viscous stress, the deviatoric part.

To derive x-momentum equation it is better start with a balance of surface

5


Figure 1.2: Stress component in the x direction.

forces in the x-direction on the (E,W) faces (see Fig. 1.2):[(p− ∂p

∂x

12δx

)−

(τxx − ∂τxx

∂x

12δx

)]δy δz

+[−

(p+

∂p

∂x

12δx

)+

(τxx +

∂τxx

∂x

12δx

)]δy δz

=(−∂p∂x

+∂τxx

∂x

)δx δy δz , (1.7)

then on (N,S) faces:

−(τyx − ∂τyx

∂y

12δy

)δx δz+

(τyx +

∂τyx

∂y

12δy

)δx δz =

∂τyx

∂yδx δy δz, (1.8)

and on (T,B):

−(τzx − ∂τzx

∂z

12δz

)δx δy+

(τzx +

∂τzx

∂z

12δz

)δx δy =

∂τzx

∂zδx δy δz. (1.9)

Putting Eq. 1.7, Eq. 1.8 and Eq. 1.9 together we obtain the x-component ofthe momentum equation:

∂(ρu)∂t

+∇ ·(ρu~U

)=∂(−p+ τxx)

∂x+∂τyx

∂y+∂τzx

∂z+ fbx , (1.10)

similarly can be done for y-direction:

∂(ρv)∂t

+∇ ·(ρv~U

)=∂τxy

∂x+∂(−p+ τyy)

∂y+∂τzy

∂z+ fby , (1.11)

and z-direction:

∂(ρw)∂t

+∇ ·(ρw~U

)=∂τxz

∂x+∂τyz

∂y+∂(−p+ τzz)

∂z+ fbz . (1.12)

6


1.2.3 Energy equationThe energy equation is derived from the rst principle of thermodynamics

stating that the rate of change of energy of a uid particle is equal to the rateof heat addition to the uid particle plus the rate of work done on the particle

mDE

Dt= Q+W . (1.13)

Let's determine the rate of heat addition and the work done on the particlein terms of more suitable physical quantities such as temperature gradients,shear and normal stresses.

Figure 1.3: Heat ux vector.

Referring to Fig. 1.3, the net rate of heat transfer due to heat ow in thex-direction is:

[(qx +

∂qx∂x

12δx

)−

(qx − ∂qx

∂x

12δx

)]δy δz = −∂qx

∂xδx δy δz , (1.14)

and including contributions for the y and z directions, the rate of heat transferadded to the particle per unit volume is:

−∂qx∂x

− ∂qy∂y

− ∂qz∂z

= −∇ · ~q . (1.15)

Furthermore, from Fourier's law:

~q = −k∇T, (1.16)

giving nally:

Q = −∇ · (k∇T ) δx δy δz . (1.17)

7


The net rate of work done on the particle by surface forces acting on x-direction is:

[(pu− ∂pu

∂x

12δx

)−

(pu+

∂pu

∂x

12δx

)]δy δz

+[−

(τxxu− ∂τxx

∂x

12δx

)+

(τxxu+

∂τxx

∂x

12δx

)]δy δz

+[−

(τyxu− ∂τyxu

∂y

12δy

)+

(τyxu+

∂τyxu

∂y

12δy

)]δx δz

+[−

(τzxu− ∂τzxu

∂z

12δz

)+

(τzxu+

∂τzxu

∂z

12δz

)]δx δy

=[∂(u(−p+ τxx))

∂x+∂(uτyx)∂y

+∂(uτzx)∂z

]δx δy δz . (1.18)

Following the same approach for the y and z direction as in Eq. 1.18 andsumming, the total rate of work done on the uid particle by surface stress is:

W =(−∇ · (pU) +

∂τijUi

∂xj

)δx δy δz . (1.19)

Including Eq. 1.19 and Eq. 1.17 into Eq. 1.13, the nal form of the energyequation in obtained, eventually adding the rate of work done by body forcesas a source term SE :

∂ρE

∂t+∇ ·

(ρE~U

)= −∇ ·

(p~U

)+∂τijUi

∂xj+∇ · (k∇T ) + SE . (1.20)

For compressible ows it is however more correct, and surely less complicatedto treat, to express energy equation in terms of another variable: total enthalpyin which pressure energy is directly included is one of the choices:

h0 = E +p

ρ= i+

12

(u2 + v2 + w2

)+p

ρ. (1.21)

Substituting Eq. 1.21 into Eq. 1.20, one may obtain energy equation in termsof total enthalpy:

∂ρh0

∂t+∇ ·

(ρh0

~U)

=∂p

∂t+∂τijUi

∂xj+∇ · (k∇T ) + Sh , (1.22)

sometimes the term ∂τijUi

∂xjis referred to as the viscous dissipation term Φ.

1.2.4 A model for the uidAs above mentioned, a Mathematical Model result in being more accurate

and easier to handle if suitable simplications are introduced in the generalequations to be used. The aim of this section is so to reduce the area of interestto problems concerned with heat transfer in turbomachinery, giving a specicmodel to the uid.

In order to simplify the above derived equations into a simpler set of equationstill capable of well modeling such ows, at least three assumptions can be made.

8


First hypothesis is the assumption of the uid to be a perfect gas, meaningthat it follows the well-known equation of state:

p = ρRT. (1.23)

Second assumption concerns the modeling of the viscous stress. Fluids ofinterest are considered Newtonian, meaning that viscous stresses, in analogywith hookean elasticity, are linearly proportional to the rates of deformation. Arst coecient µ named dynamic viscosity, relates stress to linear deformation,a second coecient λ called second viscosity or bulk viscosity takes into accountvolumetric deformation eect:

τij = µ

(∂Ui

∂xj+∂Uj

∂xi

)+ δijλ

(∇ · ~U

). (1.24)

It may noticed that, following such denition, mechanical and thermodynamicpressure may not coincide. In fact following this denition τij is not implicitlytraceless. Since little is known about λ it is common to avoid this inconsistencysimply assuming that λ = − 2

3µ. It is also true that the term(∇ · ~U

)is usually

very small and some authors proposed to directly neglect the eects due tovolumetric deformation (White, 1991).

Third comes the modeling of thermophysical properties. Even if thevalidation runs have been performed with constant thermophysical properties,uid properties are treated, not to loose generality for the proposed model, asfunction of both temperature and pressure.

All these assumptions and simplications lead to the nal set of governingequations:

∂ρ

∂t+∂(ρUj)∂xj

= 0 , (1.25)

∂(ρUi)∂t

+∂(ρUiUj)∂xj

= (1.26)

− ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj

)+ Fbi ,

∂ρh0

∂t+∂(ρh0Uj)∂xj

=∂p

∂t+

∂

∂xj

(k∂T

∂xj

)+ Φ + Sh , (1.27)

p = ρRT , (1.28)

where Φ is the viscous dissipation:

Φ = τij∂Ui

∂xj= µ

[12

(∂Uj

∂xi+∂Ui

∂xj

)2

− 23

(∇ · ~U

)2]. (1.29)

1.3 Discretization approachesAfter the physical mathematical model has been derived, the goal is to ma-

nipulate it in a form suitable for computer calculations. First step on this pathregards the discretization of the equations. The main task of a discretization

9


approach is to convert a partial dierential equation, valid on the entire do-main, into a set of discrete algebraic equation, one for every node considered.The value at the node is, of course, put in relation with neighbor nodes, thesimultaneous satisfaction of all the equation in the set then give the numer-ical solution. The most popular discretization techniques are presented anddiscussed:

• Finite Dierence Method (FDM) approximates conservation equationsin dierential form substituting partial derivatives via truncated Taylorseries expansions or polynomial tting. Even if, in principle, it can beapplied to all kind of grids, actual applications are limited to structuredgrids where grid lines are used as local coordinate lines. In such a way infact, it is easy to obtain higher-order schemes. The biggest drawback ofFDM is that it does not enforce conservation, consequently it is very hardto get reliable simulations of complex geometries and use is restricted tothe simple ones.

• Finite Volume Method (FVM) works with the integral form of theconservation equations. The domain is divided into nite volumes whosecentroid represent the calculation node. Grid just dene boundaries inbetween dierent volumes and need not to be related to any metrics.Interpolation is used to express variable values at the surface in termsof nodal values. This method is conservative by construction as long assurface integrals for volumes sharing the same face are equal for bothof them. The disadvantage of FVM in comparison with FDM is thatbuilding higher than second order schemes for 3D simulation is much moredicult due to the three levels of approximation introduced: interpolation,dierentiation and integration. Because of its very physical approach,ease to be understood and implemented, FVM is the most widely usedapproach.

• Finite Element Method (FEM) may appear similar to FVM, the distin-guish feature is the weight function: conservation equations are multipliedby a weight function before being integrated. Solution is supposed to ad-here within each volume to a shape function constructed from values atthe corners of the elements. Such an hypothesis is substituted into theconservation equations whose derivative with respect to nodal value is putto zero, selecting so the residual minimizing allowed function. The mainadvantage in using FEM is the ability in dealing with arbitrary geome-tries, while the main drawback, common to all integral methods, is thatthe resulting matrix may result not well structured meaning that ecientsolving method are dicult to implement.

Discussing which of the previously described methods should be used, exactsatisfaction of scalar conservation over control volumes has been focused to bethe most important properties to search for. It is fundamental in fact that all

10


the simplications and discretizations to be introduced in the governing equa-tions, always respect exactly basilar physical principles they are built on. Thephysical laws to be obeyed are all conservation laws so conservation should beperfectly observed. Exact conservation can be imposed externally or intrinsi-cally respected by the discretization method. The second choice seemed themost logic and that is why implicitly conservative methods were selected. Fi-nite Volume Method has so been chosen, as a consequence from now on it willbe the method of reference and will be discussed more in detail, see Sec. 1.3.1.

1.3.1 Finite Volume MethodAs above mentioned, FVM is based on conservation equations in the integral

form. To obtain such equations from the ones derived in Sec. 1.2, integrationover a nite volume must be performed. To better x ideas and not to avoidirrevelant calculation, detailed presentation of this method will be done consid-ering the generic conservation equation for a transported scalar Φ:

∫

S

~JΦ · ~ndS =∫

S

ρΦ(~U · ~n)dS︸︷︷︸

Convection

−∫

S

ΓΦ(∇Φ · ~n)dS︸︷︷︸

Diffusion

=∫

Ω

qΦdΩ︸︷︷︸

Source

. (1.30)

Such an equation applies over each control volume and the entire domain aswell, underlying once again the main feature of FVM: global conservation. Toobtain an algebraic equation the three integral must be approximated by quadra-ture formulae. In the following discussion about approximation techniques thebi-dimensional quadrilateral CV in Fig. 1.4 has been considered.

Figure 1.4: Typical bi-dimensional quadrilateral control volume.

1.3.1.1 Approximation of Volume Integrals: Source Terms

Source terms require integration over the entire CV. Approximations of suchintegrals basically reduce in calculating an averaged value q for the source terms

11


to be multiplied by the volume

QP =∫

Ω

qdΩ = qΩ . (1.31)

The rst and easiest approximation just replaces the mean value with thevalue at the CV center node

QP = qP Ω . (1.32)

Such approximation is obviously exact if q is constant or varies linearly, in anyother case is second order accurate. Higher order of accuracy schemes can beobtained via shape function or interpolation.

1.3.1.2 Approximation of Surface Integrals

The net ux through the CV boundaries is obtained summing over the sixfaces, the surface integral of the total ux normal vector j composed by convec-tive and diusive contributions:

∫

S

~J · ~ndS =∫

S

JndS =∑

k

∫

Sk

jdS =∑

k

χk . (1.33)

To exactly calculate surface integrals, the value of j is needed everywhereon the face Sk. This is of course impossible as long as only nodal values of Φare computed, it is so necessary to introduce some approximations. Usually twolevel of approximation are introduced:

• the integral is approximated in terms of the variable values at one or morelocations on the cell face (face approximation)

• the cell-face values are approximated in terms of nodal values (nodal ap-proximation).

For each type several approximation modes have been proposed. For theface approximation:

• midpoint rule - the value over the face is equal to the value at cell-facecenter χe = jeSe. This approximation is of second-order accuracy.

• trapezoid rule - the value on the face is the mean between face extremes(cell vertex) shared with 'neighbor' cells χe = Se

12 (jne + jse). This ap-

proximation is second order too.

• Simpson's rule - the value on the face is a combination of the value at thecenter of the face and the values at the vertices χe = Se

16 (jne +4je + jse).

This approximation is forth order.

Of course to maintain such levels of accuracy, nodal approximation of, atleast, the same order must be chosen. Extension in three dimensions is quitedirect for both the midpoint and trapezoid rule, but gets more complicated for

12


higher-order approximation. Nodal approximation is discussed later Sec. 1.4being a task requiring deeper treatment.

It has already been cleared that ~J is composed by two dierent contributions:convection and diusion. These two terms indeed behave in a quite dierentmanner as will be shown in the following.

1.3.1.3 Diusive ux

Discretization of the diusive uxes gives, using midpoint rule and assuminglinear variation:∫

S

ΓΦ(∇Φ · ~n)dS =∑

k

Sk(ρΓΦ∇Φ)k =∑

k

Sk(ρΓΦ)k(∇Φ)k . (1.34)

Let's see how it is not that dicult to compute gradients on the boundary facee:

(∇Φ)e =ΦE − ΦP

xE − xP. (1.35)

It is however possible to compute the gradient in other way that is:

(∇Φ)P =1Ω

∑

k

SkΦk , (1.36)

and nd the corresponding value on the face e interpolating (∇Φ)P and (∇Φ)E

with techniques to be proposed later in Sec. 1.4.

1.3.1.4 Convective ux

The discretization of the convective uxes gives using the midpoint rule:∫

S

ρΦ(~U · ~n)dS =∑

k

Sk(ρΦUn)k =∑

k

Sk(ρUn)kΦk =∑

k

FkΦk . (1.37)

Transport of Φ across the boundary faces has been computed in quite a di-rect way, the problem has now shifted in nding an expression for Φk involvingonly nodal values. This pretty much is the most crucial passage of the en-tire discretization process and so the following section is entirely dedicated tointerpolation and dierentiation schemes.

1.4 Finite Approximation: Interpolation and Dif-ferentiation Schemes

It has already been shown how it is not possible, especially for high orderapproximation, to compute the algebraic equations only in terms of cell centervalues but values on the boundary are also needed. The aim of this section isto present some of the most common schemes for interpolation in order to getvalues at locations other than computational nodes. As in previous sections theow eld is supposed to be known, in addition density and transport coecientsare also determined everywhere.

13


1.4.1 Upwind Interpolation UDSThe idea is to approximate Φe with the value at the node upstream. So,

depending on the sign of ~U ·~n on the face of reference, it will either be the rightor the left one:

Φe =

ΦP if (~U · ~n)e > 0

ΦE if (~U · ~n)e < 0 .(1.38)

This approximation is equivalent to using a backward forward dierence approx-imation for rst derivative, hence the name Upwind Dierencing Scheme (UDS).Such interpolation scheme result in being stable, boundedness criterion alwayssatised, but numerically diusive. Consider in fact Taylor series expansion:

Φe = ΦP + (xe − xp)(∂Φ∂x

)

P

+(xe − xp)2

2

(∂2Φ∂x2

)

P

+H . (1.39)

UDS neglect all terms but the rst so it is a rst order scheme. The rstneglected term behave like a diusive ux:

jde = Γe

(∂Φ∂x

)

e

, (1.40)

meaning that scalars are diused normally and parallel to the ow. The coe-cient of numerical diusion is proportional to grid dimension and to mass ux,moreover is magnied for multidimensional oblique to the grid ows. To avoidsuch inaccuracy, fundamental in case of shocks or rapid changes, mesh must berene a lot because the scheme is only rst order accurate.

1.4.2 Linear Interpolation CDSAgain in analogy with nite dierence dierential approximation, linear in-

terpolation can be implemented for FVM: the value at CV-face center is a linearinterpolation, with distance as weighting factor, of the values at the centers;

Φe = ΦEλe + ΦP (1− λe) , (1.41)

with λe = (xe − xP )/(xE − xP ). Taylor series expansion for ΦE at point xP

shows that Eq. 1.41 is second order accurate vanishing all rst order terms inthe truncation error. In fact:

Φe = ΦEλe + ΦP (1− λe)− (xe − xp)(xE − xp)2

(∂2Φ∂x2

)

P

+H . (1.42)

Having eliminated the numerical diusion, the leading term of the truncationerror is quadratic and may produce oscillatory solutions. In spite of such oscil-latory behavior CDS, being the simplest second order, is one the most widelyused dierencing scheme.

14


1.4.3 Self Filtered Centered Interpolation SFCDSFCD is a scheme that try to get the best of the two precedent schemes,

namely the boundedness of the UDS and the accuracy of CDS. It is in facta central dierence scheme with a native lter to remove unphysical estremawhenever they would arise. This lter is basically a local shifting towards anupwind scheme in such zones. This is done blending CDS and UDS together bya blending factor γ, ranging 0− 1, depending on computed face ∇Φ:

ΦSFCDe = γeΦCDS

e + (1− γe)ΦUDSe . (1.43)

The blending factor will be as close as possible to unity in case of steep gradientsin order to get sharp discontinuities, while in other more uniform zones stabilityis reached with γ close to zero. The main drawback of this scheme is theintroduction of additional non linearity due to the dependency of γ on Φ values,resulting for some steady ow in oscillatory solutions (Sta, 2004).

1.4.4 Algebraic EquationUsing the discretization and approximation techniques together, general

transported scalar conservation equation Eq. 1.30 can be nally written as analgebraic equation for each nodal point P :

AP ΦP +∑

nb

AnbΦnb = QP . (1.44)

If both the AP and the Anb are known, this is the case if the ow eld isgiven, the equation is linear and putting together the equation for each node itis possible to built up a matrix representing the dierential equation over theentire domain:

A · Φ = Q , (1.45)

in which matrix A is in general quite sparse, because inuence for ΦP is usuallylimited to a small number of other points, actually for FVM is strictly limitedto the number of neighbor nodes. Φ is the solution vector and Q is the sourceterm both of dimension equal to the number of cells.

1.5 GridAs already mentioned, the solution domain should also be discretized into

a nite number of control volumes. The discrete representation of the spatialdomain is called grid or mesh. Meshes can rstly be subdivided into:

• Structured mesh: uid domain of interest is covered with three families oflines (ξ, η, ζ) never intersecting other lines of the same family and normallyintersecting all lines of the other families in sequence. Nodal point Pi,j,k

is dened by the intersection of the three lines (ξi, ηj , ζk). In this gridarrangement it is always possible to identify neighbor nodes moving onthe three lines intersecting at the nodal point.

15


• Unstructured mesh: uid domain of interest is covered with a number ofnite volumes Pi of arbitrary shape sharing the boundaries. Contrarilyto the structured grid where exibility in gridding complex geometriesis limited by the sti organization of mesh elements, unstructured meshoers the maximum freedom in dening cells both in terms of shape andlocation.

A comparison between a typical example of a structured and an unstructuredmesh can be done looking at Fig. 1.5 and Fig. 1.6.

On the contrary of FDM, FVM can also deal with unstructured grid. This isfundamental for heat transfer involved simulations where typically non-aerodynamicgeometries are encountered. The main dierence between FDM and FVM how-ever is that in FVM grids CV are dened via cell boundaries more than cellcenters, even if this could be done. Once the boundary surfaces are dened thecoordinate of cell center are calculated assuring a higher degree of accuracy insubstituting the mean with the center value. Viceversa if centers are denedand boundary calculated, Central Dierence Scheme for derivatives are moreaccurate at CV faces because the face is midway between the two nodes.

Figure 1.5: Typical structured C grid example.

A new class of methods, the so called meshless methods, has been proposedrecently to avoid handling meshes that sometimes can result in being the mostexpensive step in the entire solution process. The idea is not to create a mesh butjust to store data for the cell centers, actually it is a no-sense to talk about cellswhen there are no more, so they will be referred to as nodes. An area of inuenceis hence dened for each node generating a local mesh. Local control volumescan overlap without loosing conservative properties: conservation is respectedno matter the shape, dimension and orientation of the control volumes.

1.6 Solution methodPrevious sections were dealing with building up the algebraic equations to

be solved for a generic scalar Φ transported by a known ow eld. It is not

16


Figure 1.6: Typical hybrid tetrahedral hexaedral unstructured mesh example.

however of secondary importance to underline that the ow eld cannot beknown a priori and scalar quantities for which conservation equations were de-rived (ρ, ρUi, ρh0, p) are responsible for the development of the ow eld itself.So the equations are not independent one from the other but on the contrarydierent scalar conservation laws result to be tightly coupled. In other wordsthe Navier-Stokes Equations cannot be solved separately but must be treatedlike a real system meaning that in the conservation equation for scalar Φ1, allother Φs cannot be considered known but should be treated as variables too.This results in a strong non-linearity of the system, think about convective orviscous dissipation terms for example.

The mechanism used to eliminate such non-linearity is a guess and correctapproach leading to an iterative cycle at the end of which the solution willsatisfy all the equations at the same time. Starting from an initial guess of allΦs, the system is solved taking the values for Φi as constant in the equationfor Φj . This solution is used to update coecients of the matrix representingthe system and the cycle repeats until solution is reached. As the reader mayeasily believe it is not so straight forward that such a method really leads toconvergence. Many dierent algorithms can be used to improve convergenceproperties for this cycle usually called outer cycle, most of which separatethe system in equations to be solved sequentially and not contemporary. Theseare called segregated solvers, in contrast with simultaneous solvers, and will beanalyzed in detail later in Sec.3.1.

Once the Navier-Stokes Equations are linearized, again they appear in thenal form:

A · Φ = Q (1.46)

where A is, this time, the sparse coecient matrix for all the ve conservationlaws and the equation of state, Φ is the solution vector for all variables and Q is

17


the source vector. Linear systems are not dicult to solve, think about Gausselimination techniques or Cramer rule, but, since typically the number of cellsfor a CFD simulation varies from some thousands to hundreads of millions andCPU time is a very requested and still limited resource, solving in an ecientand fast way it is a must.

Methods for solving linear systems are commonly classied into two maingroups: direct methods and iterative methods. Direct methods handle the ma-trix with linear algebra rules to reduce it in a form from which extraction ofnal solution can be done quite easily. Every well posed linear system can besolved directly but the sparse matrix A (every node depends on the value ofnot so many other nodes) gives decomposition matrices that are not sparse in-creasing the computational cost considerably. Iterative methods instead startsfrom a guessed solution and use the system to successively rene it. If eachiteration is cheap and the number of iteration is small enough, this is usuallythe case in CFD, iterative methods result in being much faster than direct ones.This iterative cycle is called inner cycle. The algorithms used in this the-sis are the Incomplete Cholesky preconditioned Conjugate Gradient (ICCG),Incomplete Cholesky preconditioned Biconjugate Gradient (BICCG) and theAlgebraic Multi Grid (AMG). For details on such methods see Ferziger andPeric (2002); Lacor (2006); Murthy and Mathur (2002).

To avoid instability due to the non-linearity of the system as a consequence ofthe coupling between the equations, underelaxation techniques are widely used(Patankar, 1980). The concept standing behind underrelaxation is to relax thechanges in variable values multiplying the increments from the previous valueby an underrelaxation factor (more commonly referred to as relaxation factor)αΦ with values in the interval [0, 1]:

Φn = Φn−1 + αΦ(Φn − Φn−1) , (1.47)

where Φn on the right hand side in the exact result of the linear system atouter iteration n while Φn on the left hand will the best available value forΦ at iteration n. Eq. 1.47 is sometimes referred to as explicit relaxation,meaning that matrix A is not modied. It is however possible to express thesame relation in an implicit way involving a modication of the matrix itself.Considering the linearized equation for ΦP :

AP ΦnP +

∑

nb

AnbΦnnb = QP , (1.48)

and using the Eq. 1.47 it is possible to write a new expression for ΦP at iterationn:

ΦnP = Φn−1

P + αΦ

(QP −

∑nbAnbΦn

nb

AP− Φn−1

P

)⇒

AP

αΦΦn

P +∑

nb

AnbΦnnb = QP +

1− αΦ

αΦAP Φn−1

P . (1.49)

18


This technique is known as implicit relaxation. Stability is increased becausethe diagonal dominance of the matrix A is higher: AP

αΦΦn

P > AP ΦnP . Unfortu-

nately optimum underrelaxation factors are problem dependant, moving towardslower values to increase stability and to higher values to speed up convergence.A general strategy could be to start with low relaxation factors for early itera-tion and move to higher values reaching convergence. It is now time to clarifywhat really convergence is and how it can be established.

Using iterative methods, the most important parameter to check how exactlya prediction satisfy the equation is the residual dened as:

Res = Q−A · Φ .

Res is a vector, with cell number size, and it is not a very convenient factorto establish whether or not an equation is converged. The global parameterusually used is:

GRes =∑

n

Res =∑

n

(Q−A · Φ) , (1.50)

where n correspond to the number of cells in uid domain. Equations arechecked separately so 5 dierent residuals plus the ones for turbulence will befollowed. Usually, and this is the case, residuals are normalized with respect tosome mean value, trying to account relative errors more than absolute. In thiswork the normalization factor is dened as follows:

Φref = Φ ,

S = A · Φ ,

Sref = A · Φref ,

Norm =∑

(|S − Sref |+ |Q− Sref |) .

As already hinted, two nested iterative cycles are present: the inner oneis directly related to the solution of linear system at iteration n, the outer oneregards the algorithms of segregated solvers going from the initial guess at step 0to actual iteration at time n. Convergence criteria must be set up and controlledfor both inner and outer iteration cycles. Outer iterations control parameter isInitial Residuals, while inner iterations are managed by Final Residual:

InitialResidual =∑

n(Qi0 −Ai0 · Φi0)Normi0

,

FinalResidual =∑

n(Qim −Aim · Φim)Normim

.

19

Chapter 2

OpenFOAM

The aim of this chapter is to introduce the reader into the main features ofOpenFOAM as support code for CFD simulations. In particular focus will beposed more on the way it is born and developed than on its practical using.OpenFOAM in fact is not thought to be a ready-to-use software, even if it canbe used as a standard simulation package, but to oer a backing to CFDprogrammers in building their own codes. Being at the core just a C++library, rst a brief introduction to such programming language is necessary.Then OpenFOAM itself is described in detail giving both a philosophicalintroduction and practical examples on how to work with its classes and itsapplications.

Contents2.1 Introduction to C++ . . . . . . . . . . . . . . . . . 222.2 OpenFOAM software . . . . . . . . . . . . . . . . . 232.3 Matrix structure in OpenFOAM . . . . . . . . . . 252.4 Working with main classes in OF . . . . . . . . . . 26

21

Chapter 2. OpenFOAM

2.1 Introduction to C++Giving such a small introduction to C++ is intended here as describing the

supported methodologies of programming and some of the tools to make themconvenient. Such methodologies are technically called programming paradigms.Supporting a programming paradigm does not mean to merely enable suchprogramming style but to provide facilities to make such paradigm convenient touse in terms of ease, safety, eciency. Support must not be limited to languagefacilities to directly use the paradigm, but must be extended at compile-timeand run-time checks.

The most common programming paradigms are briey introduced:

• Procedural Programming: the focus is on the processing, i.e. thealgorithm needed to perform the desired computation. Languages providesfacilities for passing and returning the most suitable arguments to themost suitable functions in the most suitable way. Discussions regardthe concept of most suitable.

• Modular Programming: also known as the data-hiding principle, isconcerned with subdiving programs into smaller modules data are hiddenwithin. This need reects a constant increase in program size.

• Data Abstraction: this paradigm is dealing with user-dened datatypes. Data types suitable for the problem are newly dened providing afull set of operations for each type. Connected and fundamental for doingit, are the concepts of operator overloading and polymorphism, also calledlate binding.

• Object-Oriented Programming: the concept introduced with such aparadigm is the concept of inheritance in connection with class hierarchy.Base classes are dened as the common structure and then specialized intoderived classes. This can actually be done even with previous paradigmbut now the possibility of sharing (inherit) members from base to derivedclass is introduced.

• Generic Programming: the aim is to parametrize algorithm in sucha way they can work for a variety of suitable types and data structures.Generic Programming introduces the concept of containers: classes thatcan hold a collection of elements of dierent types.

C++ was designed to support data abstraction, object-oriented and genericprogramming in addition to traditional C programming techniques such as pro-cedural and modular programming. It was not meant to force one particularprogramming style upon all users (Stourstroup, 1997).

It would be better to clarify some of the above introduced concepts, inparticular: polymorphism, inheritance and containers.

Connected with data abstraction paradigm is the concept of operator over-loading meaning that the same function can actually have dierent meaning for

22

OpenFOAM Chapter 2.

dierent classes. Polymorphism is the capability of deciding at run-time whichof the several overloaded member functions to call.

Inheritance is the ability of one class to inherit capabilities or propertiesfrom another class. To x ideas an example is given. Consider the class of theHuman-beings, it has some specic properties (two arms walk, intelligence, etc)plus some others (giving birth to live young, warm blood, etc) derived from theclass Mammal which is belonging to. Equally Mammals have inherited some oftheir properties to other base classes such as Vertebrate or Animals. In such away a sort of chain is built to correctly link a property to the right class.

Thinking about containers the best example is a list. Lists are basically acollection of elements stored in a row. It is possible to dene a list of integer aswell as a list of people, meaning that the class denition is not bound by thetype of objects it is listing. It is common to refer to containers as templates.Templates are a compile-time mechanism so that their use result in no run-timeoverhead compared to hand written code (Davis, 2005).

2.2 OpenFOAM softwareThe OpenFOAM (Field Operation And Manipulation) code is an object-

oriented numerical simulation toolkit for continuum mechanics, written in C++language, released by OpenCFD Ltd (http://www.opencfd.co.uk) (OpenCFD,2006c). It is so capable to support all the above discussed features typical ofC++ programming: it enables the construction of new types of data specic forthe problem to be solved (i.e. a virtual class for turbulence Model with virtualfunctions such as ε, k, µt, etc ...), the bundling of data and operations into hi-erarchical classes preventing accidental corruptions (i.e. a base class for storingmesh data and a derived class for acceding them), a natural syntax for userdened classes (i.e. operator overloading) and it easily permits the code re-usefor equivalent operations on dierent types (i.e. templating) (Jasak, 1996; Jasaket al., 2004; Juretic, 2004).

Let's see a little more in detail what are the specicity of OpenFOAM inhelping CFD programmers. First and most important thing is that the toolkitimplements operator-based implicit and explicit second and fourth-order FiniteVolume (FV) discretization in three dimensional space and on curved surface.

Dierential operators can be treated like nite volume calculus (fvc) or nitevolume method (fvm) operators.

The rst approach performs explicit derivatives returning a eld, the secondone is an implicit derivation converting the expression into matrix coecients.The idea standing behind is to think about partial dierential equations in termsof a sum of single dierential operators that can be discretized separately withdierent discretization schemes. At the moment the following, self explanatory,implicit dierential operator are dened:

• fvm::ddt = ∂∂t ,

23

http://www.opencfd.co.uk

Chapter 2. OpenFOAM

• fvm::d2dt2 = ∂2

∂t2 ,

• fvm::div =∑

i∂

∂xi,

• fvm::laplacian =∑

i∂2

∂x2i.

In addition implicit treatment of source terms is done by: fvm:Sp andfvm:SuSp.

Explicit equivalent for the previous operators are dened and furthermoreother common operators such as curl, gradient, etc, are implemented.

Building dierent types of partial dierential equations is now only a matterof combining in a dierent way the same set of basic dierential operators.To give an example of the capability of such a top-level code, let's consider astandard equation like momentum conservation:

∂ρ~U

∂t+∇ ·

(ρ~U ~U

)−∇ ·

(µ∇~U

)= −∇p . (2.1)

It can be implemented in an astonishingly almost natural language which isready to compile source C++ code:

solve(

fvm::ddt(rho, U)+ fvm::div(phi, U)- fvm::laplacian(mu, U)

==- fvc::grad(p)

);

letting programmers concentrate their eorts more on the physics than on pro-gramming.

Such example clearly shows that OpenFOAM programmers do not think interms of cells or faces but in terms of objects (U, rho, phi, etc ...) dened asa eld of values, no matter what dimension, rank or size, over mesh elementssuch as points, edges, faces etc. Just to x ideas for example the velocityeld is dened at every cell centroid and boundary-face centers, with its givendimensions and the calculated values for each direction, and represented by justa single object U of the class GeometricField, see Sec. 2.4.

Important feature allowed by object programming is the dimensional check,physical quantities objects are in fact constructed with a reference to theirdimensions and so only valid dimensional operations can be performed. Avoidingerrors and permitting an easier understanding, come directly as a consequenceof an easier debug.

An important feature of OpenFOAM is the exibility in handling dierenttypes of meshes. Its native grid engine can, in fact, handle every mesh ofarbitrary polyhedral bounded by arbitrary polygons, see Fig. 2.1.

24

OpenFOAM Chapter 2.

(a) (b)

Figure 2.1: Polyhedral mesh example.

In addition, even if never directly tested by the author, automatic meshmotion and runtime topological changes such as attach/detach boundaries, celllayer addition/removal are allowed.

OpenFOAM was thought not really to be only a ready to use code but to beas much exible as possible in dening new models and solvers in the simplestway. Its strength in fact stands in being open not only in terms of sourcecode but, what's more, in its inner structure and hierarchical design, giving theuser the opportunity to fully extend its capability. Moreover, the possibility ofusing top-level libraries containing a set of models for the same purpose whichrefer to the same interface, guarantees programmers for smooth and ecientintegration with the built-in functionality. New models can be added to theappropriate model table at link-time and become available in the same manneras the supplied models.

Most of the selections necessary to set up calculations are done at runtime,meaning that options can change while the code is running. For further in-formation about how to use and how to program OpenFOAM see OpenCFD(2006a,b).

2.3 Matrix structure in OpenFOAMIt is interesting to describe how OpenFOAM is building and storing the

matrix A of coecients for one of the discretized equations. This array is de-composed in three parts: the diagonal coecients, the upper triangular matrixu and the lower triangular matrix l. O diagonal coecients represent the in-uence of neighboring cells. Each face is shared by two dierent cells one ofwhich is the owner and the other is called a sneighbor. The owner cell for ageneral face is the one with lower cell index. The lower matrix is so returningcoecients for all the owner cells while the upper one for the neighbors. To

25

Chapter 2. OpenFOAM

clarify consider line i of array A, it corresponds to equation for cell of indexi. The number of non-zero coecients on this line equals the number of facesplus the diagonal coecient that multiply the solution at cell i itself. On theleft side of diagonal coecient (rows j from 1 to i-1), coecients for faces be-tween cell i and lower index are stored. l array is actually storing coecients forowner cells of faces bounding cell i not owned by cell i itself. At the contrary,coecients for cells who share boundaries owned by cell i, once again the cellswith index higher than i, are stored in the u array. OpenFOAM programmersusually works with FVM, so a special class of matrix fvMatrix in which theinuence is only limited to adjacent cells is dened and normally used.

2.4 Working with main classes in OFJust to give an example of how object oriented programming in OpenFOAM

is working, the class GeometricField is shown and described in its structureand interaction with other classes. First of all, it would be better introducethe class with its Unied Modeling Language UML graphical representation inFig. 2.2.

Figure 2.2: UML diagram for GeometricField class.

To interpret these graphs one should read it in this way: a box representsa class, a solid arrow stands for public inheritance and a dashed arrow indi-cates usage, with the edge of the arrow labeled with the variable responsible forthe relationship. Meaning that GeometricField has a boundaryField and is aField<Type>. As hinted above at, GeometricField could represent any tensoreld; it consists of an internalField and a boundaryField. The rst one, be-longing to Field<Type> class, stores a list of values of appropriate tensor rankfor each computational point, the second, namely a GeometricBoundaryField,stores boundary values subdiving it into one patchField for each boundarysurface. Reference to dimensions is stored in the dimensionSet that basi-cally is the vector of the exponents for the fundamental units. It could beof 5 [kg,m, s,K,mol] or 7 [kg,m, s,K,mol,A, cd] elements.

26

OpenFOAM Chapter 2.

GeometricField is a template and gives a good example of generic pro-gramming: this virtual class in fact contains Fields of scalar, vector and tensortype dened on surfaces and volumes. One can in fact dene objects like asurfaceScalarField as well as a volTensorField.

This completes introduction to one of the most common classes in Open-FOAM has been done.

27

Chapter 3

All Mach Pressure BasedSolver

The problem of developing a suitable solver for heat transfer predictions inturbomachinery is not an easy task. This chapter is aiming at giving a generaloverview on the specic needs for such a solver, then various classes of solversare presented and criticized to explain the whys of the choice of apressure-based solver has been made. Hence a detailed derivation of standardalgorithms for this class of ows. Finally the developed solver is presentedwith particular attention to the aspects that were found of much trouble forthe programmers.

Contents3.1 Segregated solvers . . . . . . . . . . . . . . . . . . . 303.2 Pressure velocity coupling . . . . . . . . . . . . . . 303.3 Problems associated with density based solvers . 313.4 SIMPLE Algorithm . . . . . . . . . . . . . . . . . . 323.5 Pressure checkerboarding . . . . . . . . . . . . . . 363.6 Co-located grid: Rhie - Chow interpolation . . . . 373.7 Compressibility eects . . . . . . . . . . . . . . . . 393.8 Critical aspects . . . . . . . . . . . . . . . . . . . . . 403.9 Solver validation . . . . . . . . . . . . . . . . . . . . 42

3.9.1 GAMM tests . . . . . . . . . . . . . . . . . . . . . . 423.9.2 Goldman test . . . . . . . . . . . . . . . . . . . . . . 43

29

Chapter 3. All Mach Number Pressure Based Solver

3.1 Segregated solversDuring the derivation of the set of conservation equations, it has been re-

peated many times that equations are coupled meaning that coecients of onedepend on solution of another. It has been underlined that such coupling re-sults in non-linear systems and the resolution of this non-linearity is left tothe iterative process. What has not been specied yet is how the system issolved. Considering for example the nal expression derived for the discretizedNavier-Stokes Equations:

A[m×n,m×n]Φ[m×n] = Q[m×n] , (3.1)

where Φ[m×n] is a vector of dimensions equal to the number of cells n timesthe number of unknowns m. For computational costs reasons, this system isusually not solved entirely but sequentially, in other words the system matrixis decomposed into one smaller matrix for each equation and such matrices aresolved in sequence for the scalar of reference:

Aρ[n,n]Φρ[n] = Qρ[n] , (3.2)Au[n,n]Φu[n] = Qu[n] , (3.3)Av[n,n]Φv[n] = Qv[n] , (3.4)Aw[n,n]Φw[n] = Qw[n] , (3.5)Ap[n,n]Φp[n] = Qp[n] , (3.6)Ah[n,n]Φh[n] = Qh[n] . (3.7)

In this way it is clear that the coupling between the dierent equations must betreated with another strategy. An algorithm to iteratively link such discretizedequations is needed in order to reach convergence. It is easy to guess that anumber of dierent algorithm has been proposed to organize the steps in whichthe process should be parted.

3.2 Pressure velocity couplingThe two main classes in which standard segregated solvers can be classied

are: density-based and pressure-based solvers. Each of these classes is naturallyapplied to a certain class of ows: high Mach compressible ows for the density-based and low Mach almost incompressible for the pressure-based.

The dierent nature of conservation equations in case of subsonic or super-sonic ows makes the use of these classes outside the range of Mach number ofreference a dicult task. Unsteady compressible Navier-Stokes Equations arein fact parabolic-hyperbolic in nature, but their incompressible counterparts areof elliptic parabolic type.

However, heat transfer in turbomachinery applications, and other well knowncases of no interest here, involve ows usually covering a wide range of Mach

30

All Mach Number Pressure Based Solver Chapter 3.

regimes. In particular, it usually happens that dierent Mach conditions simul-taneously arise in the same domain. Such situations makes the accurate solutionof viscous ows governing equations a complex task since this kind of ows doesnot fall completely into any of the categories cited before. In order to solve suchows specic solvers should be created trying to enlarge range of applicabilityto the maximum, at the best to All-Mach ows.

The most famous method proposed to change behavior of the incompressibleNavier-Stokes Equations and t them over the compressible shape is the so-called articial compressibility technique: a fake time derivative is added to theincompressible conservation equations rendering them hyperbolic and solvablewith standard time-marching techniques developed for the compressible form ofthe equations. Such methods are applied to steady state solution and requirea dierent set of variables in the compressible and incompressible ow regime.That is why a dierent path to solution has been searched for.

3.3 Problems associated with density based solversConservation equations derived in chapter 1 as well as the other later derived

in chapter 4 can be cast into a general form as:

∂ρΦ∂t

+ (∇ · ~J) = Q , (3.8)

where the total ux ~J is composed by a convective and a diusive term:

~J = ρ~UΦ︸︷︷︸convective

− Γ∇Φ︸︷︷︸diffusive

. (3.9)

In addition an equation of state to describe the relationship between the variousthermodynamic variables is needed. This set of non-linear coupled equationshould be solved for the unknowns ρ, ~U , p, h.

It is so customary to associate each of the equations to one basic variable:density for continuity, velocity components for momentum, enthalpy for energy.In addition the equation of state can be considered as the equation for pressure.At very low Mach number however, the pressure and the density become veryweakly related and in the idealized limit of incompressible ows, the density iscompletely decoupled from the pressure. For such a reason neither pressure nordensity can be directly associated with the incompressible continuity equationand it assumes the role of a compatibility condition on the velocity eld.

For a sequential solution of the equations a mechanism to couple continuityand momentum should be searched, otherwise it is also possible to use thefully compressible equations even in incompressible ows. Indeed all uids arecompressible and the pressure is always related to the density even though therelationship may become very weak. However, because equations are solvednumerically, constraint are imposed by numerical considerations: truncationand round-o errors.

31


The perfect gas law for a nearly incompressible ow in which the speed ofsound can be considered constant, states that:

p = ρRT =c2

γρ⇒ ∆p =

c2

γ∆ρ . (3.10)

Introducing reference values it is possible to rewrite Eq. 3.10 in a non-dimensionalform:

∆pρ0u2

0

=1

γMa20

∆ρρ0

. (3.11)

This expression clearly points out, that even though changes in density maybecome small or innitesimal, as long as Ma number tends to zero too theirratio is nite and changes in pressure may remain large. If sensibility in solvingboth terms is established to be:

∆pρ0u2

0

> 10−n , (3.12)

∆ρρ0

> 10−m , (3.13)

then it is imposed the Mach number to be grater than 10−m+n

2 . In addition forMa = 0 there is a singularity point for the Navier-Stokes Equations, renderingthem improperly scaled in the limit of incompressibility.

The above discussion highlights the problems associated with the use of den-sity as a primary variable for computing low Mach ows or mixed compressibleand incompressible ows (Karki, 1986b).

Development of a computational scheme valid for the whole range of Machnumber, needs a switch to pressure as a primary independent variable in pref-erence to density.

3.4 SIMPLE AlgorithmThis section is written to illustrate in detail one of the segregated solver

algorithm introduced in Sec. 3.1. The algorithm presented here it the originalversion of the Semi-Implicit Method for Pressure Linked Equations (SIMPLE)published for the rst time in Patankar and Spalding (1972) and nowadaysvery well known. This original version was derived for incompressible or atmaximum weakly compressible ows and expansion to fully compressible owswill further be investigated. The main idea is to convert continuity equationinto an equation for pressure or better a pressure corrector and to use a guessand correct procedure to reach solution: pressure corrector becomes equallyzero everywhere. Since continuity contains discrete face velocities a connectionto relate such velocities to a pressure eld is needed: the direct link is themomentum equation.

Consider the discretized x and y momentum equation with notation referredto Fig. 3.1:

32


Figure 3.1: Bidimensional control volume.

aeue =∑

anbunb + (pP − pE)∆y +Qe , (3.14)

anvn =∑

anbvnb + (pS − pN )∆x+Qn . (3.15)

To initialize the SIMPLE calculation a pressure eld p∗ is guessed, hence Eq. 3.14and Eq. 3.15 are solved with such pressure distribution to obtain a rst guessvalue for velocity u∗ and v∗:

aeu∗e =

∑anbu

∗nb + (p∗P − p∗E)∆y +Qe , (3.16)

anv∗n =

∑anbv

∗nb + (p∗S − p∗N )∆x+Qn . (3.17)

A pressure corrector is then dened as the dierence between the correct andthe guessed pressure, and similarly is done for velocity correctors too:

p′ = p− p∗ , (3.18)u′ = u− u∗ , (3.19)v′ = v − v∗ . (3.20)

Subtraction of Eq. 3.16 from Eq. 3.14 and of Eq. 3.17 from Eq. 3.15 gives twoequations for pressure and velocity correctors:

aeu′e =

∑anbu

′nb + (p′P − p′E)∆y , (3.21)

anv′n =

∑anbv

′nb + (p′S − p′N )∆x . (3.22)

Given that correctors tend towards zero as long as solution is converging,dropping terms such as

∑anbu

′nb and

∑anbv

′nb is not giving any error when con-

vergence is reached. This is however the biggest approximation of the SIMPLEalgorithm and its consequences will be discussed later in this section. RewritingEq. 3.21 and Eq. 3.22, one obtains:

u′e = (p′P − p′E)de , (3.23)v′n = (p′S − p′P )dn , (3.24)

33


where de = ∆yae

and dn = ∆xan

. Thus far only momentum equations has beenconsidered, continuity should be used to impose mass conservation:

[Fe − Fw] + [Fn − Fs] = 0 ⇒[(ρuA)e − (ρuA)w] + [(ρvA)n − (ρvA)s] =

[ρeue − ρwuw]∆y + [ρnvn − ρsvs]∆x = 0 . (3.25)

The face mass ux Fi is then split in two parts corresponding to the twocomponents of the exact solution of the velocity u∗ and u′:

[F ∗e − F ∗w] + [F ′e − F ′w] + [F ∗n − F ∗s ] + [F ′n − F ′s] = 0 . (3.26)

Eq. 3.23 and Eq. 3.24 can be inserted in Eq. 3.26 to give:

F ∗e + ρede∆y (p′P − p′E)− F ∗w + ρwdw∆y (p′P − p′W )

+ F ∗n + ρndn∆x (p′P − p′N )− F ∗s + ρsds∆x (p′P − p′S) = 0 , (3.27)

that rearranged gives the equation for the pressure corrector:

AP p′P =

∑

nb

anbp′nb + b , (3.28)

where:

aE = ρede∆y , (3.29)aW = ρwdw∆y , (3.30)aN = ρndn∆x , (3.31)aS = ρsds∆x , (3.32)AP =

∑

nb

anb , (3.33)

b = F ∗w − F ∗e + F ∗s − F ∗n . (3.34)

At this point all the necessary equations have been deduced: they shouldbe included into an organic iterative process to obtain a solution. The overallalgorithm foresees the following steps:

1. Guess a pressure eld p∗.

2. Discretize and solve the momentum equations using the guessed value p∗for the pressure terms. Eq. 3.16 and Eq. 3.17 yield the u∗ and v∗.

3. Find the pressure correction source term b calculating mass imbalancewith Eq. 3.34.

4. Discretize and solve the pressure correction equation: Eq. 3.28.

5. Calculate velocity corrections using Eq. 3.23 and Eq. 3.24.

6. Correct pressure and velocity elds: Eq. 3.18, Eq. 3.19 and Eq. 3.20.

34


7. Solve discrete equations for each scalar Φ using the continuity satisfyingvelocity eld for the convection term.

8. Check convergence: IF satised stop, ELSE use p as a new guess p∗ andstart again from 2.

It is now time to return to Eq. 3.21 to clarify the consequences of droppingthe

∑nb anb. It has already been underlined how at convergence nothing is

changing, so the impact is limited on the rate of convergence.Such dropping in fact places the entire burden of correcting velocities upon

the pressure corrector. This will lead to a velocity eld still satisfying continuitybut also to an overpredicted pressure eld. The pressure correction equation isindeed susceptible to divergence unless some under-relaxation is used during theiterative process. Implicit underrelaxation is usually preferred because it alsoincrease the diagonal dominance of the matrix, but this cannot be case. Being acorrector the history of convergence of p′ is without any particular meaning, inother words values of p′ at previous iteration do not resolve the same, or almostthe same, equation, as a consequence p′n−1 can be very dierent from p′n andimplicit relaxation will violate continuity. What is done to avoid this obstacleis to relax the new values for pressure and not the corrector:

pnew = p∗ + αp p′ . (3.35)

To avoid overpredictions of pressure, velocities are also relaxed. A correctchoice of under-relaxation factors α is essential for cost-eective simulations.Too large a value may lead to oscillatory or even divergent solution while smallvalues comport extremely slow convergence. Unfortunately, the optimum valuesof under-relaxation factors are ow dependent and must be sought on a case-by-case basis (Malalasekera and Versteeg, 1995; Murthy and Mathur, 2002).

Other variants of the same method have been proposed in the course ofyears:

• SIMPLER (SIMPLE-Revised): use a discretized equation for pressureto obtain p∗ from an initial guess of velocity v∗∗ and pressure p∗∗ elds(Patankar, 1980).

• SIMPLEC (SIMPLE-Consistent): use the same algorithm of SIMPLEbut with a better approximation for Eq. 3.21 and Eq. 3.22 (Doormal andRaithby, 1984).

• PISO (Pressure Implicit with Splitting Operators): add a second correc-tor step to the standard SIMPLE algorithm meaning that p∗∗ = p∗ + p′ isonly a starting guess for the second corrector step (Issa, 1986).

At this point the method is completely described, it is not however secondaryto better precise how to treat diculties arising with the use of such a solver.In particular, more attention must be paid on the problem of checkerboardingsolutions and extension to fully compressible ows.

35


3.5 Pressure checkerboardingAs you may already have noticed this method requires values both on cell

(capital letters) and face (lowercase) centers so it can be directly applied only onstaggered grids, uid domains on which velocities and pressures are computed atdierent locations. Staggered approach is however quite heavy in terms of meshgeneration and complicated to organize. If the number of elements increase toactual standard size for CFD simulation of industrial interest, this approachshould be left for a co-located approach.

This is indeed not the case for most of standard CFD codes, so a solutionmust be found in order to be able to use such method. In particular, usingOpenFOAM cell centered approach, all face values such as ρe, ue, Fe are un-known and cell values should be interpolated with some interpolation scheme.With a variety of dierent interpolation schemes available in OpenFOAM (cen-tered, upwinded, TVD, NVD schemes, etc ...), the choice for velocity and densityhas been left to the user. Some words more should be spent on face ux (ad-vective velocities) interpolation scheme. The easiest solution could be to use alinear interpolation to obtain values on faces and still apply the above derivedmethod:

Fe =FP + FE

2. (3.36)

where the interpolated ux Fe is referred to the interpolated velocity Unfortu-nately this approach leads to inaccurate results, in fact it allows for non physicalcheckerboarding solutions.

Checkerboarding proles are due to the fact that linear interpolation bringscontinuity, i.e. Eq. 3.25, to be independent from the value of velocity on thecell of reference, in other words conservation of mass at cell P depends only onneighboring values, as a consequence odd and even nodal values result to bedecoupled:

∑Fi = Fe − Fw + Fn − Fs =

12

(FE − FW + FN − FS) . (3.37)

If such a pattern is satisfying momentum equation too, the checkerboardingwould persist in the nal solution; unfortunately this is the case. To demonstrateit, take the x-direction momentum equation Eq. 3.14 and make a balance overcell P:

uP = dP (pw − pe) + ... (3.38)

= dP

(pW + pP

2− pE + pP

2

)+ ... = dP

(pW − pE

2

)+ ... . (3.39)

The dependency of uP on pP , using linear interpolation, is canceled and ve-locity is only feeling neighboring pressure values. This means that velocityand pressure at the same location do not inuence each other, as a consequencecheckerboarding patterns of velocity and pressure will satisfy momentum equa-tion.

36


Dramatically both momentum and mass conservation equation are solved bythe checkerboarding pattern and so once it is formed it will never smooth. Evenif dicultly perfect checkerboarding proles for velocity or pressure arise, dueto irregularities in the mesh, boundary conditions or physical properties, suchtendency often shows itself in unphysical wiggly elds.

To avoid this unacceptable behavior two methods are proposed in literature:

• use a staggered grid: uid domain discretization on which velocities andpressures are computed at dierent locations,

• use a co-located grid but a dierent interpolation method for face massuxes.

First approach takes advantage of storing values for velocity and pressureprecisely at the points required for the implementation and no interpolation isneeded. Primary drawback for the staggered grid arrangement is the incrementof metric complexity due to the use of dierent grid for dierent variables. Thismethod results as a consequence to be very expensive computationally speakingfor curvilinear coordinates and inapplicable to unstructured mesh, so in practiceit has been abandoned in the course of years.

Second approach prevents checkerboarding by expressing face velocities interms of adjacent cell pressure rather than alternate values. Even if more com-plex to understand, this approach is less dependent on geometries and griddingstrategies and moreover it is the only chance to continue using a co-locatedcode. Advanced interpolation techniques are so computed in order to avoidsuch checkerboarding on co-located grids.

3.6 Co-located grid: Rhie - Chow interpolationThe interpolation technique named after Rhie and Chow (1983), and also

known as momentum interpolation, has been used in the solver. Substantiallyit consists in dening a pseudo velocity eld (on cells) excluding the pressuregradient term to the original formulation of velocity, hence such pseudo velocityis interpolated linearly and nally the pressure gradient is evaluated directly onthe face and summed. A monodimensional domain has been considered in thefollowing derivation. It begins with the denition of the pseudovelocity:

u0P = uP +

(dl

A∇p

)

P

= uP +(pE − pW )

2AP, (3.40)

then interpolate it linearly on face centers

u0e =

u0P + u0

E

2, (3.41)

to nally reconsider pressure contributions with gradients computed preciselyon faces:

ue = u0e −

(dl

A∇p

)

e

= u0e −

(pE − pP )2Ae

. (3.42)

37


Substituting Eq. 3.40 into Eq. 3.41 and into Eq. 3.42, a nal expression for facevelocity is obtained:

ue =(uP + uE)

2+

12

((pE − pW )

2AP+

(pP − pEE)2AE

)− (pE − pP )

2Ae. (3.43)

Similar expressions may be derived for the other faces.Putting together such expressions and summing mass uxes Fi, it is clear

that explicit dependence on cell values do not cancel out as with linear inter-polation. Continuity and momentum equation now do feel pressure at point Pmeaning that wavy gradients are smoothed and so checkerboarding is avoided.

It is interesting to see how the behavior of continuity equation changes withthe introduction of this interpolation also known as added dissipation scheme.Rewrite Eq. 3.43 considering Ae = AE = AP , this happens in case of uniformgrid and uniform ow:

ue =(uP + uE)

2+

14AP

((pE − pW + 2pP ) + (pP − pEE − 2pE)) . (3.44)

Writing down the Taylor series expansion for pressure and reorganizing:(∂2p

∂x2

)

P

=(pW − pE − 2pP )

∆x2+O(∆x2) , (3.45)

so

ue =(uP + uE)

2− 1

4AP

((∂2p

∂x2

)

P

−(∂2p

∂x2

)

E

)∆x2 =

=(uP + uE)

2− 1

4AP

(∂3p

∂x3

)

e

∆x3. (3.46)

Substituting this expression into continuity equation for constant density:

ue − uw =(uE − uW )

2− 1

4AP

(∂3p

∂x3

)

e

∆x3 +1

4AP

(∂3p

∂x3

)

w

∆x3 =

=(uE − uW )

2− 1

4AP

(∂4p

∂x4

)

P

∆x4 = 0 , (3.47)

it comes out that a fourth order derivative is added to continuity. Derivativeterms of even order are known in literature as dissipation terms, as a consequencea more diusive behavior is expected from this equation (Anderson, jr., 1995).More diusion means more inuence of the center value on its direct neighborssweeping away all unphysical wiggles.

With the implementation of such interpolation scheme the problem of checker-boarding patterns is solved and the same overall algorithm can be applied alsoin this case of co-located variables Davidson (2005); Murthy and Mathur (2002);Mangani (2006).

Before going through the derivation of a dierent pressure correction equa-tion valid for compressible uid ows, it is better to specify how to use valuesof pressure and velocity known at step 6 of the SIMPLE algorithm, for solvingconservation equations for the other quantities. Solution of passive scalars are

38


better computed via the use of face ux, on which continuity is respected, morethan cell velocity, on which only momentum is exactly satised. If this adviceis followed in fact, conservation equations are perfectly conservative.

3.7 Compressibility eectsIn turbomachinery and heat transfer applications, involved uid ows may

usually cover a wide range of Mach regimes. In particular, it usually happensthat dierent Mach conditions simultaneously arise in the same domain. Suchsituation makes the accurate solution of viscous ows governing equations acomplex task.

As we said above most widely used algorithms for compressible ows cal-culation use density as one of the main independent variables and pressure isdetermined via an equation of state. As there is very little or no change in den-sity for low subsonic or nearly incompressible ows, these density-based methodsfail in such regimes. Their application in cases of incompressible or low Machnumber ows is questionable, since in that situation the density changes are sosmall that the pressure-density coupling becomes very weak.

To avoid this weakness another class of methods, proposed originally forviscous incompressible ows Patankar (1980); Malalasekera and Versteeg (1995);Ferziger and Peric (2002) and later extended to compressible ows Shyy andBraaten (1988a); Rincon and Elder (1997); Darbandi and Schneider (1999);Moukalled and Darwish (2003b, 1994, 2000, 2003a); Peric (2003); Karki (1986b);Peric et al. (1993); Karki and Patankar (1989); McGuirk and Page (1990); Lienand Leschziner (1993); Moukalled and Darwish (1999); Politis and Giannakoglou(1996); Rhie (1989); Senocak and Shyy (2001); Shyy and Braaten (1988b) usepressure as the main independent variable also with the concept of the `retardeddensity' Lien and Leschziner (1993); Politis and Giannakoglou (1996); McGuirkand Page (1990)

To expand the proposed method to compressible ows dependence of densityon pressure should be explicitly pointed out and included in the equation forthe pressure corrector. This feature is essential in extending applicability of thismethod to ows at high Mach: outside the weak compressibility zone.

Let's begin dening density, on the shape of Eq. 3.18, as the sum of a guessedand a corrector density:

ρ = ρ∗ + ρ′ . (3.48)

Using a compressibility factor Ψ dened as:

Ψ =∂ρ

∂p=

1RT

, (3.49)

it is possible to write:

ρ′ = Ψp′ . (3.50)

39


As a consequence:

Fe = (ρu)eAe = (ρ∗ + ρ′)e(u∗ + u′)eAe =

(ρ∗u∗)eAe + ρ∗eu′eAe + ρ′eu

∗eAe + ρ′eu

′eAe , (3.51)

that, dropping last term because innitesimal of second order and substitutingρ′ with Eq. 3.50 (upwinding is used in the following), gives:

Fe = F ∗e + ρ∗eu′eAe + Ψp′Pu

∗eAe . (3.52)

Repeating the same reasoning for all faces a dierent pressure correction equa-tion is obtained:

F ∗e + ρ∗ede∆y (p′P − p′E) + Ψp′Pu∗e∆y

− F ∗w + ρ∗wdw∆y (p′P − p′W )−Ψp′Wu∗w∆y

+ F ∗n + ρ∗ndn∆x (p′P − p′N ) + Ψp′Pu∗n∆x

− F ∗s + ρ∗sds∆x (p′P − p′S)−Ψp′Su∗s∆x = 0 . (3.53)

Reducing to the same notation

AP p′P =

∑

nb

anbp′nb + b , (3.54)

where:

aE = ρ∗ede∆y, (3.55)aW = ρ∗wdw∆y + Ψu∗w∆y, (3.56)aN = ρ∗ndn∆x, (3.57)aS = ρ∗sds∆x+ Ψu∗s∆x, (3.58)AP =

∑

nb

anb −Ψu∗w∆y −Ψu∗s∆x+ Ψu∗e∆y + Ψu∗n∆x, (3.59)

b = F ∗w − F ∗e + F ∗s − F ∗n . (3.60)

Eq. 3.54 include two dierent types of terms for the pressure correction p′P :rst type involving velocity, a convective term Ψp′Pu

∗e∆y, and second involving

pressure dierence, a diusion like term ρ∗ede∆y (p′P − p′E). The relative impor-tance of the two terms depends on the Mach number of the ow. At low Machnumbers, the diusional part dominates and the equation exhibits an ellipticbehavior. In the supersonic regime, the convective term is much larger thanthe diusional term and the mass ux is governed solely by the upstream pres-sure. This reects the correct hyperbolic behavior of pressure for supersonicows. The nature of the equation is such that the transition from the subsonicto supersonic regime is gentle and the transonic ow calculations are smoothlytreated.

3.8 Critical aspectsDuring the development of the code for the proposed solution algorithm,

critical aspects have been found to be the treatment of the boundary conditions

40


for the pressure corrector and the underrelaxation of the pressure correctorequation.

The starting problem was to understand which was the more suitable bound-ary condition for the pressure corrector. Being a ctitious variable at conver-gence equally zero, every boundary condition satisfying this constraint could beconsidered physical for the problem. However it is true that on boundaries onwhich the pressure is specied there is no need for a correction (p∗ = p ) andso p′ is xed to zero. At the same time boundaries on which the mass owis specied need no mass correction that means no dierence of p′ across thatboundary: zero Neumann condition.

Information about boundary conditions is propagated into the domain alongcharacteristic lines. Characteristics for compressible uid ow are the Machwaves. Theory of characteristics reveals how, at the inlet boundary, one of theconditions should be extrapolated from interior values. Extrapolated boundariesare described mathematics with Neumann type boundary conditions. The nullgradient constraint is standardly applied to pressure, a condition that respectsthe physics, therefore a mathematical enquiry should establish whether or notsuch condition on pressure correction equation results in a well posed problem.In case it results to be untrue and other types of boundary conditions are neededto respect well posedness, a solution to maintain physical constraint is needed.

In the weakly compressible form derived in Sec. 1.2.4, pressure correctorequation is a Poisson type equation that is an elliptic problem. A rst derivativeconstraint, Neumann boundary condition, is suitable for this kind of problem:mathematically the problem results to be well-posed. In the fully compressibleversion however a convective term is added and the behavior of the equationturns hyperbolic. For hyperbolic equations, unless one uses the unsteady form,resulting in an initial boundary value problem, a Dirichlet boundary conditionis needed at the inlet too, to satisfy well-posedness constraint. Physically, ithas already been underlined, specifying pressure corrector means to impose apressure. To avoid inlet pressure not to change, it is updated at every iterationno more via the pressure corrector but via extrapolation from the interior usingboth value and gradient on rst cell.

This passage is somehow the bottleneck of the entire process. To improve thespeed of convergence for inlet pressure better would be to increase the diusivebehavior of the equation near the inlet. This can be done using, instead of thezero xed value boundary condition, a mixed (Robin type) boundary condition:

cφw + (1− c)(∂φ

∂n

)

w

= cφref + (1− c)(∂φ

∂n

)

ref

. (3.61)

This boundary condition imposes a value, depending on previously dened refer-ence quantities φref and

(∂φ∂n

)ref

, both zero in the case, to a linear combinationof wall value and normal gradient weighted on scalar c. Again it is better under-line how the combination of this boundary condition on the pressure correctorand the extrapolation of inlet values for pressure from the internal domain, ismathematically and physically valid for the entire range of Mach number.

41


It has already been shown (see Eq. 3.35) how underrelaxation for the pres-sure corrector is necessary to obtain convergence. Furthermore an additionalweighting factor αmix, here improperly referred to as a relaxation factor, hasbeen included in the pressure correction equation, scaling convective terms, tobalance the relative inuence of the diusion term over convective term:

AP =∑

nb

anb + αmix(−Ψu∗w∆y −Ψu∗s∆x+ Ψu∗e∆y + Ψu∗n∆x) . (3.62)

Just to show how the nature of the two relaxation factors is completely dierent,typical values for them two are indicated: αp = 0.01 and αmix = 0.9.

3.9 Solver validation3.9.1 GAMM tests

The developed calculation procedure has been used to solve a variety ofproblems in heat transfer applications as we see in next sections. Here the em-phasis is on the high compressible ows. The capability of the present method isdemonstrated by computing inviscid ow in a channel with a bump on the lowerwall named GAMM test. This test case has been used by various researchersto test their algorithms Peric et al. (1993); Karki (1986a). Application of themethod to two dierent types of inviscid ow, transonic and supersonic, arepresented below.

The width of the channel is equal to the length of the bump, and the chan-nel length is equal to three lengths of the jump. For transonic calculation, thethickness-to-chord ratio is 10% while for supersonic ow calculations it is 4%. Intransonic and supersonic regime at inlet is assumed that ow has uniform prop-erties and the upstream far eld variable values (except pressure in transoniccase) are specied while at the outlet all variable (except pressure in transoniccase) are extrapolated. At the upper and the lower boundaries wall slip condi-tion is prescribed. First case with imposed inlet Mach number Main = 0.675,

-1.5 -1 -0.5 0 0.5 1 1.5x

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

Mac

h N

umbe

r

Lower-WallUpper-Wall

(a) Mach prole in transonicow.

-1.5 -1 -0.5 0 0.5 1 1.5x

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

Mac

h N

umbe

r


(b) Mach prole in supersonicow.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

Mac

h N

umbe

r


(c) Mach prole in supersonicow with two-bump geome-try.

Figure 3.2: Prole Mach number in upper and lower wall.

gives the Mach number distributions along the walls and density gradient mag-nitude contour plot shown in Fig. 3.2(a) and Fig. 3.3(a). In the supersonic case,

42


Main = 1.65, the ow results supersonic all along the bump: Mach number dis-tributions and density gradient magnitude contour plots are shown in Fig. 3.2(b)and Fig. 3.3(b). These results correspond to reference solutions from literaturePeric et al. (1993); Karki (1986a). Fig. 3.2(c) and Fig. 3.3(c) show the Mach

(a) Transonic ow. (b) Supersonic ow.

(c) Supersonic ow with two-bump geometry.

Figure 3.3: Density gradient contour plots.

number distribution and density gradient magnitude contour plot under thesame condition of supersonic case but with two bumps. As can be seen by com-paring Fig. 3.2(b), Fig. 3.2(c) and Fig. 3.3 the second bump does not inuencethe ow upstream indicating that the solution algorithm correctly reproducesthe hyperbolic behavior of the ow.

3.9.2 Goldman testAs example of highly compressible subsonic, we have reported the simulation

of a test based on the work of Goldman et al.Goldman and McLallin (1977). Itis a 2-D turbulent analysis of a stator blade at the mid-span; the details of thegeometry and the mesh are shown in Fig. 3.4(a). The Reynolds number, basedon the chord length of the blade and the free-stream velocity, is 500000 and theinlet Mach number is approximately 0.2. A comparison of the predictions forblade loading (dened as the ratio of static pressure to the inlet total pressure)with the experimental data is shown in Fig. 3.4(c).

43


(a) Geometry mesh.

(b) Mach contour plot.

(c) Pressure ratio over experimental data.

Figure 3.4: Stator Blade analysis.

44

Chapter 4

Turbulence Modeling

This chapter is focused on the modelling of turbulence with the aim ofnding closure to the set of conservation equations proposed in chapter 1,namely from Eq. 1.25 to Eq. 1.28. After a brief introduction to turbulentphenomena, classical approaches to turbulence modelling are presented withspecial interest in the RANS approach. Main part concerns the presentation ofthe many turbulence models capable of partially avoiding such defects in theprediction of turbulent eects, implemented in OpenFOAM for this thesis.

Contents4.1 The physics of turbulence . . . . . . . . . . . . . . 474.2 Models for Turbulence . . . . . . . . . . . . . . . . 484.3 RANS approach . . . . . . . . . . . . . . . . . . . . 494.4 Standard k − ε turbulence model . . . . . . . . . . 53

4.4.1 Equation for turbulent kinetic energy . . . . . . . . 534.4.2 Equation for turbulent kinetic energy dissipation . . 544.4.3 Drawbacks of standard two equation model . . . . . 554.4.4 Wall functions . . . . . . . . . . . . . . . . . . . . . 554.4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.5 Failures in predictions for heat transfer . . . . . . 574.6 Low Reynolds k − ε . . . . . . . . . . . . . . . . . . 58

4.6.1 Abe Kondoh Nagano . . . . . . . . . . . . . . . . . . 594.6.2 Chien . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6.3 Chen Lien Leschziner . . . . . . . . . . . . . . . . . 614.6.4 Hwang Lin . . . . . . . . . . . . . . . . . . . . . . . 624.6.5 Lam Bremhorst . . . . . . . . . . . . . . . . . . . . . 634.6.6 Lien Leschziner . . . . . . . . . . . . . . . . . . . . . 634.6.7 Realizability . . . . . . . . . . . . . . . . . . . . . . 64

4.7 Two Layer . . . . . . . . . . . . . . . . . . . . . . . . 654.8 Anisotropic Turbulence Model . . . . . . . . . . . 66

4.8.1 Model Formulation . . . . . . . . . . . . . . . . . . . 674.9 The k − ω class . . . . . . . . . . . . . . . . . . . . . 69

4.9.1 Original k − ω . . . . . . . . . . . . . . . . . . . . . 704.9.2 Baseline model . . . . . . . . . . . . . . . . . . . . . 71

45

Chapter 4. Turbulence Modelling

4.9.3 Shear Stress Transport model . . . . . . . . . . . . . 724.9.4 Automatic Near Wall Treatment . . . . . . . . . . . 73

46

Turbulence Modelling Chapter 4.

4.1 The physics of turbulenceIt is known that, for low velocities, wall-bounded ows are smooth and ad-

jacent layers of uid slide past each other in an orderly fashion. When velocityis high instead the ow become intrinsically unsteady and chaotic even withconstant boundary conditions: the ow is said to be turbulent. Turbulence

Figure 4.1: Da Vinci sketch of turbulent ow.

phenomena may be described with the following characteristics (White, 1991;Malalasekera and Versteeg, 1995):

• Fluctuations in time superimposed upon a mean value for each quan-tities: Φ(t) = Φ + Φ′(t). Connected with this is the extension to threedimensions also for bidimensional ows: velocity uctuates always in 3D.

• Eddies or uid packets intermingle and ll the shear layer. Eddies witha wide range of length scales, coexist at the same time and at the sameplace.

• Random variations in uid properties. Such variations are not white-noise shaped, each quantity has a specic continuous energy spectrum.

• Self-sustaining motion. New eddies replace those lost by viscous dissipa-tion. Kinetic energy is handed down from bigger to smaller eddies in whatis termed the energy cascade. Large scales (energy-containing subrange)contain most of the energy while smallest scales (dissipation subrange)dissipate such energy into heat. Scales laying in between belong to the socalled inertial subrange: a transport region for the cascade process.

• Mixing is much stronger than in laminar ows. Turbulent eddies activelyincrease diusion by moving in the three dimensions. Heat transfer andfriction are strongly enhanced.

Transition from laminar and turbulent regime always occurs when a certain,called critical, Reynolds number is reached. Starting from such a description,one cannot hope turbulence to be isotropic: only the smallest eddies in fact arenon-directional, with bigger ones strongly dependent on mean ow direction.

47


4.2 Models for TurbulenceSuch impressive complications do not involve the ow not to respect Navier-

Stokes Equations in the general form (see Eq. 1.25), but certainly do reduceabruptly the chance of being able to solve such equations. It should be markedout, in fact, that a wide range of temporal and length scales for eddy dimensionsmeans a wide range of scales to be computed and solved. Navier-Stokes Equa-tions still exactly model turbulent ows and the typical unsteady turbulencephenomena connected with, but at computational costs that often is prohibitive.The need for simplications in the treatment of turbulence has gained duringthe course of the years to several class of models: each involving dierent rangeof turbulence scale of interest. All these approaches are presented here brieyin decreasing order of computational complexity:

• Direct Numerical Simulation (DNS): it is not a model, Navier-StokesEquations are computed in their most general form meaning that the com-plete spectrum of involved frequencies and length scales are solved. Thisapproach is only feasible at current time for Low Reynolds number ows,due to limitation on computer resources, and used mainly as validation testfor other approaches or as an help in understanding turbulence physics.

• Large Eddy Simulation (LES): remembering that energy is containedmainly in the largest scales, this model reduces the range of interest onlyto the biggest vortices. It employs, in fact, a time dependent three di-mensional computation of the large-eddy structure and a model for thesmall scales. It basically consists in ltering in space the Navier-StokesEquations with a high-pass lter, resolving for scales that actually arethe energy-containing scales and modelling dissipation subrange behavior.The cut-o scale usually lies into the inertial subrange (Davidson, 2006).LES is becoming more and more popular in the CFD community, partic-ular complexities in treating boundary conditions and the need for widecomputer resources still limit the use to simple geometries and specicareas of interest in which turbulence modelling is more than fundamental(turbulent combustion, wake eects, chemical reactions).

• Reynolds Averaged Navier Stokes (RANS): Navier-Stokes Equationsare averaged in time on a period big enough to contain also lowest fre-quency oscillation. The unsteady behavior of the turbulent ows is com-pletely neglected, turbulence become a steady phenomenon simply con-sidering the eects of uctuations onto the mean ow. At the state ofart RANS simulations are the standard for ows involving heat transferof industrial interest. In fact unsteady phenomena result in being deter-minant for such simulations. The ease of implementation and the speedin solving, supported by a good accuracy in modelling mean ows, arereasonably believed to make this approach still practised in middle-term.Since this thesis is treating turbulence with a RANS approach, a deeper

48


introduction to such an approach is needed. This will be the subject ofnext section.

• Hybrid: this class is aiming in taking advantage of low computationalcost of the RANS approach and the good condence of LES simulation.The idea is to use the two approaches in dierent areas where ow condi-tions are diverse. In particular LES is used only in the zone were RANSsimulations are likely to fail, namely separating zone.

4.3 RANS approachIt is already known that this approach simply consists of averaging in time

(Reynolds Averaging) the exact Navier-Stokes Equations, here reported for abetter reference:

∂ρ

∂t+∂(ρUj)∂xj

= 0 , (4.1)

∂(ρUi)∂t

+∂(ρUiUj)∂xj

= (4.2)

− ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj

)+ Fbi ,

∂ρh0


=∂p

∂t+

∂

∂xj

(k∂T

∂xj

)+ Φ + Sh , (4.3)

p = ρRT . (4.4)

Mathematical denition of Reynolds Averaging is given below:

f(x) = f − f ′ = limt→∞

1t

∫ t

0

f(x, τ)dτ . (4.5)

Before applying this operator to the complete set of equations, it would bebetter see what happen if it is applied to instantaneous uid properties:

Φ = Φ + Φ′ = Φ + Φ′ . (4.6)

The average of the mean value is equal to itself, the average of the uctuatingpart is zero:

Φ = Φ , (4.7)Φ′ = 0 , (4.8)

all this comes directly out from the denition of the period t to be big enoughto include all eects of unsteady turbulence. If products appears inside theaveraging operator however, uctuation do not completely cancel out:

ΨΦ = (Φ + Φ′)(Ψ + Ψ′) = ΦΨ + Φ′Ψ + ΦΨ′ + Φ′Ψ′ = ΦΨ + Φ′Ψ′ . (4.9)

It must be underlined that also Favre (density-weighted) average:

f(x) = f − f ′′ = limt→∞

1ρ t

∫ t

0

ρf(x, τ) dτ =ρf(x)ρ

, (4.10)

49


is very important for compressible ows because it exactly respects the followingsimplication (Knight, 1997):

ρUiUj = ρUiUj + ρU ′′i U′′j . (4.11)

Remembering such a clarication, in the following derivation Reynolds andFavre averaging will be distinguished.

After this introduction to averaging, it is better start using this operator onthe Navier-Stokes Equations. First start with continuity:

∂ρ

∂t+∂(ρUj)∂xj

=∂ρ

∂t+∂(ρUj)∂xj

= 0 , (4.12)

followed by momentum equation:

∂ρUi

∂t+∂(ρUiUj)∂xj

= − ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj

)+Fbi ⇒

∂ρUi


= − ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj

)+Fbi ⇒

∂ρUi


+∂(ρU ′′i U

′′j )

∂xj=

− ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj

)+ Fbi , (4.13)

and nally energy:

∂ρh0


=∂

∂xj(k∂T

∂xj) + Φ + Sh ⇒

∂ρh0


+∂(ρh′′0U

′′j )

∂xj=

∂

∂xj(k∂T

∂xj) + Φ + Sh , (4.14)

where Φ is:

Φ =µ

2

(∂Ui

∂xj+∂U ′i∂xj

+∂Uj

∂xi+∂U ′j∂xi

)2

. (4.15)

The RANS equations look pretty much like the Navier-Stokes Equations, Eq. 4.1,except for an extra term in momentum and energy equation, depending on uc-tuating quantities. These terms appear inside the divergence operator and theyare often moved to the right hand side as companion terms of laminar vis-cous stress and conduction terms. That is why they are in literature known asReynolds stresses and turbulent heat uxes.

RANS approach will now used from now on in this thesis and so to clarifynotation all the bars, apart from where specically needed, are dropped withthe implicit notation that a = a, a and a′ = a′, a′′.

After the Reynolds Averaging, temporal dependency is of no interest any-more, so simplication of steady-state can nally be introduced, just drop the

50


time derivative term, and the RANS equations presented:

∂(ρUj)∂xj

= 0 , (4.16)

∂(ρUiUj)∂xj

= (4.17)

− ∂p

∂xi+

∂

∂xj

(µ

(∂Ui

∂xj+∂Uj

∂xi

)− 2

3µδij

∂Uj

∂xj− ρU

′iU

′j

)+ Fbi ,

∂(ρh0Uj)∂xj

=∂

∂xj(k∂T

∂xj− ρh′0U

′j) + Φ + Sh , (4.18)

p = ρRT . (4.19)

It comes out that modelling turbulence is another way of saying for nding away to treat the two terms: ρU ′

iU′j and ρh′0U ′j .

These terms introduce 9 more unknown into the Navier-Stokes Equationssystem, that before was determined but now is undetermined with 9 degreesof freedom. Constraints should be introduced if the system want to be solved:introducing these constraints is modelling turbulence.

Modelling these terms is also known as the closure problem, explicitly point-ing out that the number of unknowns [15: density, velocity (3), pressure, Reynoldsstresses (6), enthalpy, turbulent heat uxes (3)] is larger than the number ofequations (6).

Dierent levels of approximation as been proposed to model in a more com-plex or simpler way those terms, depending on the needed accuracy and theavailable computer resources. Main class of models are quickly introduced againin order of decreasing computational complexity. To better understand energyequation has been neglected (drop o 1 equation and 4 unknowns) and turbulentheat ux is implicitly treated like Reynolds stress.

• Dierential Stress Models, more commonly calledReynolds Stress Trans-port Model (RSTM) or Second Moment Closure models, solve a separatescalar transport equation for each stress component and closure is reached.The biggest drawback of this category of models can be guessed easily: itis computationally very expensive compared with the following methods.Moreover, stability is far from being easy to obtain and, even if they in-clude more turbulence physics than eddy viscosity models with advectionand production terms (energy-in terms) derived exactly, they still requiremodelling for important terms quote: redistribution and dissipation terms(Apsley, 2006).

• Eddy Viscosity Models: the RSTMs are the only models not takingadvantage of Boussinnesq assumption, other models in fact, to preventsolving additional dierential equations, relate Reynolds stress tensor tostrain rate or, in other words, to velocity gradients:

−ρU ′iU ′j = T = µt

(∂Ui

∂xj+∂Uj

∂xi− 2

3δij∂Uk

∂xk

)− 2

3ρkδij (4.20)

51


where last two terms are added to be consistent with continuity and turbu-lent kinetic energy denition, that is to have the proper trace for turbulentstress tensor (Davidson, 2003; Knight, 1997). The problem now shifts toa way of guessing the eddy viscosity µt, that is why the following classesof turbulent models are called Eddy Viscosity Models. The main advan-tage of such methods is the facility of implementation in viscous solvers:just replace molecular viscosity µ with an eective viscosity µeff = µ+µt

and the same for thermal diusivity α with an eective thermal diusivityαeff = α+ αt.

This thesis is proceeding further with eddy viscosity models so the last andreally implemented form of the Navier-Stokes Equationsis presented. Moreovera change in variable for the energy equation has been done solving for staticenthalpy instead of total:

∂(ρUj)∂xj

= 0 , (4.21)

∂(ρUiUj)∂xj

= (4.22)

− ∂p

∂xi+

∂

∂xj

[µeff

((∂Ui

∂xj+∂Uj

∂xi

)− 2

3δij∂Uj

∂xj

)− 2

3ρkδij

]+ Fbi ,

∂(ρhUj)∂xj

=∂

∂xj(αeff

∂h

∂xj) + Ui

∂p

∂xi+ Φ + Sh , (4.23)

p = ρRT . (4.24)

Even if theoretical foundation is limited to simple shear ows, eddy viscositymodels are the most commonly used because they are very cheap in CPU time,well known in their pros and cons and global predictions usually do not fail somuch. At the contrary, the main drawback is the little turbulence physics theyare based on: anisotropy and history eects are completely neglected. Due tothe mathematical structure of the model in fact, in the best case one can hopeto accurately represent only one Reynolds Stress. Applicability of such modelsshould be restricted in cases where stress are almost mono directional. This isthe case almost never but often it is possible to individuate a major stress, theidea is to well predict the main one and neglect if the others are not that wellmodelled.

This category is subdivided again into:

• Algebraic Models also called zero equation models because do not solveany extra partial dierential equation and µt is obtained from mean owproperties and a prescribed, geometry dependent characteristic length.

• One Equation Models, in which a length scale is also prescribed butan extra partial dierential equation is solved for k.

• Two Equations Models where transport equations are specied for bothk and its rate of dissipation (absolute ε or specic ω), and µt is thencomputed locally from the value of these scalars.

52


The fundamental limitation of zero equation models is the assumption thatReynolds Stress can be modelled on mean ow: turbulence does not in factrespond instantly to changes in mean ow but rather adjusts over a time scaletypical of the turbulent structure. One equation models, even if feels the in-uence of uctuating properties via turbulent kinetic energy equation, do stilldepend on geometry via the denition of a characteristic length. First, in orderof complexity, class of model to be independent on the denition of an appro-priate length scale is the two equations class. This property is fundamental inpredicting many type of ows, for example recirculating ows: mixing length infact cannot be used for ows in which diusion and convection are not negligible.

The following will be consistent with the choice of two equations models asa good compromise between area of applicability and computational feasibility.

4.4 Standard k − ε turbulence modelThe landmark model for eddy viscosity two equations models is the k − ε

model by Jones and Launder (1972) which appeared in 1972 and nowadaysknown as the standard k − ε model.

Before building and analyzing the structure of the transport equation forturbulent kinetic energy k and turbulent kinetic energy dissipation ε, a generaloverview common to all k − ε models is needed, in order to understand how kand ε are related to the wanted scalar µt.

First turbulent length scale lt is computed locally as:

lt = Cmuk

32

ε, (4.25)

and second, following the denition of length scale, one obtains:

µt = Cµρk12 lt = Cµρ

k2

ε. (4.26)

4.4.1 Equation for turbulent kinetic energyAn exact equation for k is not dicult to derive from Navier-Stokes Equa-

tions with some manipulations (Wilcox, 1998):

∂(ρk)∂t

+∂(ρkUj)∂xj

=∂

∂xj

(µ∂k

∂xj

)

− ∂

∂xj

(ρ2U ′jU

′iU

′i + U ′jp′

)− ρU ′iU

′j

∂Ui

∂xj− µ

∂U ′i∂xk

∂U ′i∂xk

. (4.27)

Terms on the rst line need no further modelling so eorts should be directedin modelling the last three blocks. First of these three comes the turbulentdiusion of kinetic energy: transport of velocity uctuations by the uctuationsthemselves, almost always modelled by use of gradient-diusion assumption:

−(ρ

2U ′jU

′iU

′i + U ′jp′

)≈

(µt

σk

∂k

∂xj

), (4.28)

53


where σk is the so called Turbulent Prandtl number whose value is usually closeto unity.

Second term represents the rate of production (positive source term) dueto the mean ow and is commonly approximated, with the help of Boussinesqapproximation, by:

Pk = −ρU ′iU ′j∂Ui

∂xj= µt

(∂Ui

∂xj+∂Uj

∂xi− 2

3δij∂Uk

∂xk

)∂Ui

∂xj− 2

3ρk∂Uj

∂xj. (4.29)

In order to make equations the most implicit as possible, in computational uiddynamic it is common to split the implicit and the explicit part of the productionterm:

Pk = G︸︷︷︸explicit

− 23ρk∂Uj

∂xj︸︷︷︸implicit

. (4.30)

Last term is the sink term or dissipation for which a second equation isneeded:

−µ∂U′i

∂xk

∂U ′i∂xk

= ρε . (4.31)

Hence the nal form of the k equation is:

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε . (4.32)

4.4.2 Equation for turbulent kinetic energy dissipationAlthough it is possible to derive an exact equation for dissipation directly

from Navier-Stokes Equations the modelling applied to it is so severe that it isbest to regard the entire equation as a model (Ferziger and Peric, 2002). Themost common form for such an equation is:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]= Cε1Pk

ε

k− ρCε2

ε2

k. (4.33)

In Tab. 4.1 standard values for the introduced constants are given.

Cµ Cε1 Cε2 σk σε

0.09 1.44 1.92 1.0 1.3

Table 4.1: Standard coecients values for standard k − ε.

Such equations should be solved together with the Navier-Stokes Equationsand, as one could guess and see, coupling is pretty strong. But as long as Eq. 4.32and Eq. 4.33 are much stier than the others, they are usually solved segregatedin order to increase convergence properties. In addition it is necessary to useunder-relaxation techniques in iterative methods.

54


4.4.3 Drawbacks of standard two equation modelAfter a so straight forward presentation of this model, it would be better to

enquire the main and well known characteristic drawbacks.The proles of turbulent scalars are typically much more peaked near the

wall than mean velocity prole. Such peaks are dicult to capture if a grid,tuned on mean velocity needs, is used also on turbulent equations, better willbe to use a ner grid for k and ε. This is not usually possible and near walltreatment become a very important task.

At wall complete no-slip condition applies, meaning that both mean anductuating part of velocity vanish. As a consequence k is exactly zero on wallboundary and the same stands for µeff . In near wall layer turbulent stresses,usually much higher, decreases to values of comparable magnitude with theviscous one. At high Reynolds number, this results in very large ow gradientsnear the wall, moreover solid boundaries selectively damp uctuations normalto the wall.

This is usually avoided via wall-functions meaning that the ow is not re-solved in the near-wall region and theoretical proles between boundary surfaceand rst near-wall node are assumed and superimposed. The main disadvan-tage of this approach is that theoretical proles are only known and justied innear-equilibrium boundary layers, in addition grid should be chosen with care,avoiding rst node y+ to be out of the range 15-150 with an optimal range of30-50 (Apsley, 2006).

Another approach could be to model near wall damping directly inside theturbulence model. It is so possible to rene the grid near the wall working withy+ ≈ 1 and include the eects of molecular viscosity in the coecients of theturbulent model. These models, known as Low Reynolds, even if much morecomputationally expensive are fundamental in case near wall behavior is needed.Many models belonging to this class are described in the following sections.

The standard k − ε however is a High Reynolds model meaning that mustbe supported by wall-functions, if at least approximately the near wall behaviorwants to be captured. So let's come back at the concept of wall function to seewhat is the standard formulation for near wall turbulence quantities.

4.4.4 Wall functionsFor both velocity and turbulent kinetic energy no-slip condition stands at

the wall. To force the correct near-wall behavior however it is important to havea good guess for the mean value of turbulent viscosity for ~U and of productionand dissipation term for k.

For velocity, once the right prole for k and ε is imposed, nothing has to bedone and wall functions only permit to correctly calculate the right wall shearstress:

τw = ρνe,wUP

yP, (4.34)

55


where the subscript P stands for rst node from the wall, νe,w is the eectivekinematic viscosity at the wall

νe,w = ν ·

1 for y+P < y+

ν

y+P

y+ν + 1

κ ln(1+κ(y+P−y+

ν ))for y+

P > y+ν

(4.35)

and y+ν is a non-dimensional matching length whose typical value is 11.6 for

smooth walls.In the equation for k, Pk and ε should be replaced at the rst node with an

averaged value:

P avk =

(τw/ρ)2

κu0yP

ln[1 + κ(y+

P − y+ν )]− κ(y+

P − y+ν )

1 + κ(y+P − y+

ν )

, (4.36)

εav =u3

0

κyP

[ln

(yP

yε

)+ 1

], (4.37)

where u0 is the "equivalent" friction velocity:

u0 = C0.25µ k0.5

P , (4.38)

and

yε =2kP ν

C0.5µ u0

. (4.39)

As a consequence dissipation equation is not solved in the rst layer of cellsbeing the near wall node value superimposed (Apsley, 2006).

Furthermore near wall temperature proles should also be corrected whenHigh Reynolds turbulence models are used. The log-law can be adapted toprovide an equation that relates the value of temperature at the near wall nodeTP , to that at the wall Tw and to the value of the wall heat ux rate qw.

T+P = ρCp C

0.25µ k0.5

P

Tw − TP

qw, (4.40)

where

T+P = σt

[1κ

ln

(y+

P exp(κC0.25µ y+

ν )

y+ν

)+ y+

ν C0.25µ

(Pr

σt− 1

)]. (4.41)

This equation can be interpreted in two dierent ways depending on physicalboundary condition for temperature. If xed wall heat ux simulations need tobe performed, the numerical condition for T will be xed value given by:

Tw = TP + T+P

qwρCp C0.25

µ k0.5P

. (4.42)

If instead temperature is imposed on the boundary, the temperature gradientmust be held xed numerically:

qw = k

(∂T

∂n

)

w

= ρCpC0.25µ k0.5

P

Tw − TP

T+P

. (4.43)

56


Here notation can be confusing so it would be better underline that k on theleft hand side is the thermal conductivity while kP on the right hand side isthe turbulent kinetic energy at rst node (Iacovides, 2004; Apsley, 2006). Thevalues of Tw and qw depend on TP so they change during the iteration process.When convergence is reached the value of TP , Tw, qw will however exactly satisfythe wall function.

4.4.5 StabilityAfter this quite long digression, some words more should be spent on the

problems connected with the standard k − ε model. If k − ε is applied onregions of low turbulence, numerical instability may arises. The problem isconnected with the modelling of the destruction term in the ε equation: ε2/k.Even if both quantities goes to zero, the rate of decrease is not always correct(Davidson, 2003). Even if they are known to fail in well predicting adversepressure gradient ows, turbulence model of the k − ε class are still the mostused two equations models world wide (Wilcox, 1998).

4.5 Failures in predictions for heat transferApart from the above mentioned well-known defects of the standard k −

ε, other problems, specically connected with heat transfer simulations, arisemaking it no more suitable for such predictions. These new failures of standardturbulence models are presented and a way to solve them is hinted. Laterimplemented models are described in detail.

The correct modelling of turbulent quantities is fundamental in conductingheat transfer simulations, because of the simultaneous importance of well pre-dicting both the near wall behavior and the complex structures of the mainow. Correct predictions of thermal quantities and gradients inside boundarylayers are necessary to establish whether or not the cooling system is ecient.At the same time such properties strongly depend on the development of thefree stream ow.

Usage of wall function approach has to be avoided because of the unpre-dictability of boundary thermal gradient and the failure in predicting transi-tional ows and low Reynolds as well as adverse pressure gradient ows.

Modelling ows close to solid walls requires integration of the two equationsover a ne grid in order to capture turbulent peaks, see Sec. 4.4, as well ascorrections for Low Reynolds number eects. Several so called Low Reynoldsk−εmodels have been proposed in the course of years, see Patel et al. (1985) andWilcox (1998) for a review. The idea standing behind such models is to dampturbulent viscosity near the wall through a damping function fµ going towardszero as the distance from the wall is reducing. Constants multiplying sourceterms in the turbulent dissipation equation are in some cases also damped. Thebasic structure of the models is the same for all of them diering in the tuning

57


of the damping functions and some extra sources in dissipation equation.It is known from literature that in high strain rate regions eddy viscosity

models overpredicted turbulent kinetic energy: this problem is sometimes re-ferred to as stagnation point anomaly (Durbin, 1996). The cause of such anoverprediction stands in the modeling of the production term Pk supposed to beproportional to the rate of strain Sij . When Sij is too large, and that happensnot only near stagnation points but also in swirling ows and even in turbinepassages, such proportionality fails to be physical or better the errors with thesimplications become unacceptable. To avoid such overprediction of turbulentkinetic energy, turbulent time scale, normally computed as the ratio betweenk and ε, is bounded. Such a bound was derived from a sort of realizabilityconstraint for Reynolds stress tensor (Medic and Durbin, 2002).

Having modeled near wall behavior, now the focus shifts towards well match-ing good near wall predictions with suitable modeling of ow structures far fromthe wall. Being the standard k−ε model still one of the best in predicting HighReynolds zones, Two Layer k−ε models have been implemented. Such methods,in fact, consist in patching together a one equation model in the near wall layerand a two equations High Reynolds model, namely standard k− ε model or therenormalized group variant, in the outer layer (Rodi, 1991).

It has already been mentioned how k−ε models are likely to fail in predictingadverse pressure gradient ows. Another class of model, writing the secondequation in terms of specic dissipation ω, is instead better performing forseparating ows. Several models have been proposed all sharing the same defectin depending from freestream values. Interesting is the k−ω in the SST (ShearStress Transport) variant: it includes the modication of the standard k − ω

to avoid sensitivity to (quite arbitrary) freestream values of ω (Menter, 1994,1993). The basic idea is similar to Two Layer models: two dierent approachesare merged together to model the two dierent ow regions. The sublayer andlogarithmic model is the standard k − ω, chosen because of its robustness, theabsence of damping function and Dirichlet type boundary conditions. From thewake region and outside the boundary layer the standard k−ε, written in termsof ω, has been preferred due to its good compromise in predicting dierent kindof ows.

Summarizing it is kind of easy to obtain turbulence models that well performin wall bounded k−ω or in frees shear ows k− ε but since most complex owsinclude both types of regions, the task is to well match both behaviors.

4.6 Low Reynolds k − ε

As already mentioned, models of this class all share the same basic structure,diering only in tuning of the damping functions namely fµ, f1, f2. First andbiggest change is the introduction of the damping function fµ introduced to

58


mimic the direct eect of molecular viscosity on the shear stress:

µt = Cµfµk2

ε. (4.44)

Little is known about such function but to maintain agreement with standardk − ε in the High Reynolds zone it must tend to unity in the fully turbulentlogarithmic layer. In addition an experimental curve can be obtained calcu-lating an expression for fµ for a bidimensional boundary layer and substitutingmeasured data for k+, ε+, uv+:

fµ =−uv+ε+

Cµk+2(dU+/dy+). (4.45)

It exhibits an almost constant value for y+ < 15, a linear increase up to y+ = 60and a gradual approach toward unity.

Equation for turbulent kinetic energy dissipation is also damped in its sourceterms and an extra term is added:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1f1Pkε

k− ρCε2f2

ε2

k+ E , (4.46)

The function f2 is introduced primarily to incorporate Low Reynolds numbereects in the sink term. The eects of such a function is generally limited tothe viscous sublayer and nuances in the shapes do not exert large inuence onthe overall results. E and f1 usually inuence magnitudes at peak or at thewall. Equation for k remains substantially invariant with only the addition ofan extra dissipation terms for some models.

Damping functions are in general expressed in terms of two turbulent nondimensional numbers:

Rey =√ky/ν , (4.47)

Ret = k2/εν , (4.48)

where here, and from now on, y is wall distance if not specied. After thisdescription of the commonalities between such models, each model is analyzedin its own specic formulation.

4.6.1 Abe Kondoh NaganoFirst model to be presented is the model proposed by Abe et al. (1994).

This model will be referred to from now on with the name AKN. Equation ofreference as well as values for the coecients are reported:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1Pkε

k− ρCε2f2

ε2

k, (4.49)

59


∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε . (4.50)

This model uses another non-dimensional number to express the damping func-tions, it is again a sort of Reynolds number calculated via turbulence dissipation:

y∗ =yε0.25

ν0.75. (4.51)

The production term in turbulence dissipation equation is not damped so onlytwo damping functions are present in this model:

fµ =[1− exp

(y∗

14.0

)]2

1 +5.0

Re0.75t

exp

[−

(Ret

200

)2]

, (4.52)

f2 =[1− exp

(y∗

3.1

)]2

1 + 0.3 exp

[−

(Ret

6.5

)2]

. (4.53)

In Tab. 4.2 standard values for the introduced constant are given. Wall bound-


0.09 1.5 1.9 1.4 1.4

Table 4.2: Standard coecients values for Abe et al. (1994) Low Reynolds k − ε.

ary condition for ε are specied in such a way:

εw = 2νk

y2, (4.54)

where this condition is intended imposed on rst near wall node. For k, it isbetter remind the no-slip condition on solid surfaces.

4.6.2 ChienThe following is the model proposed by Chien (1982), known to the author

from (Yoder and Georgiadis, 1999) and referred to as CH later on.

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1Pkε

k− ρCε2f2

ε2

k− 2

µ

y2exp

(−0.5y+)ε , (4.55)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε− 2

µk

y2. (4.56)

The viscosity damping function:

fµ =[1− exp

(−0.0015y+)]

, (4.57)

is expressed this time in terms of:

y+ =yuτ

ν. (4.58)

60


Looking carefully at such an equation, it appears how dicult can be to imple-ment y+ for cells far from the wall. It is not that easy in fact to dene whatreally uτ is for an internal cell. This computation was done with reference tothe friction velocity value of the closest wall cell.

The other damping function is instead modeled with the standard turbulentReynolds number:

f2 =

1− 0.22 exp

[−

(Ret

6

)2]

. (4.59)

In Tab. 4.3 standard values for the introduced constant are given.


0.09 1.35 1.8 1.0 1.3

Table 4.3: Standard coecients values for Chien Low Reynolds k − ε.

Wall boundary value for ε will be held xed to 0, as one may notice in fact anextra term has been introduced into turbulence kinetic energy equation takinginto account the correct near wall behavior of the sink term in that equation. Itis unnecessary to repeat that wall boundary condition for k is a null Dirichletcondition.

4.6.3 Chen Lien LeschzinerThis is the model proposed in (Chen et al., 1996). For brevity, this model

will be referred to from now on with the name CLL. Equation of reference aswell as values for the coecients are reported:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1Gε

k+ (Cε3 − 2

3Cε1)ρε

∂(Uj)∂xj

+ Cε1P′k

ε

k− ρCε2f2

ε2

k, (4.60)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε . (4.61)

In this model damping functions are again only two:

fµ = [1− exp (−0.0198Rey)][1 +

5.29Rey

], (4.62)

f2 =[1− 0.3 exp

(−Re2t)]

, (4.63)

but source term in ε equation does not remain unchanged. An extra productionterm is doping the near wall behavior:

P ′k = 1.33f2

[Pk + 2µ

k

y2exp

(−0.00375Re2y)]

. (4.64)

61


Cµ Cε1 Cε2 Cε3 σk σε

0.09 1.44 1.92 -0.33 1.0 1.219

Table 4.4: Standard coecients values for Chen et al. Low Reynolds k − ε.

In Tab. 4.4 standard values for the introduced constant are given.Wall boundary condition for ε are specied in such a way:

εw =Pk

ρ+ 2

νk

y2. (4.65)

Again k is zero at solid boundaries.

4.6.4 Hwang LinThis is the model proposed by Hwang and Lin (1998). This model will be

called also with the acronym HW. Equation of reference as well as values forthe coecients are reported.

The peculiarity of such a model, even if shared with many other models isthat it is solving for ε instead of ε. The two variables are linked together by thefollowing equations:

ε = ε− ε , (4.66)ε = 2ν|∇ (

k0.5) |2 . (4.67)

An extra term depending on ε will also be added in the kinetic energy equation:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1Gε

k− 2

3Cε1ρε

∂(Uj)∂xj

− ∂

∂xj

[0.5µ

ε

k

∂k

∂xj

]− ρCε2f2

ε2

k, (4.68)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk− ∂

∂xj

[0.5µ

k

ε

∂ε

∂xj

]−ρε . (4.69)

Apart from the denition of this new parameter yλ:

yλ = y

√ε

νk, (4.70)

used in the denition of the viscosity damping function

fµ =[1− exp

(−0.01yλ − 0.008y3λ

)], (4.71)

the main dierence stands in the tuning of σ constants with the change in yλ:

σk =[1.44− 1.1 exp

(−yλ

10

)], (4.72)

σε =[1.3− 1.0 exp

(−yλ

10

)]. (4.73)

The eect of this tuning is to damp the diusivity factor, without constrainingthe production terms. In Tab. 4.5 standard values for the introduced constantare given.

Wall boundary value for ε and k as well are xed to 0.

62


Cµ Cε1 Cε2

0.09 1.44 1.92

Table 4.5: Standard coecients values for Hwang and Lin Low Reynolds k − ε.

4.6.5 Lam BremhorstThis model was proposed by Lam and Bremhorst (1981). It will be referred

to from now on with the name LB. Equation of reference as well as values forthe coecients are reported:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1f1Gε

k− 2

3Cε1f1ρε

∂(Uj)∂xj

− ρCε2f2ε2

k, (4.74)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε . (4.75)

The three damping functions are:

fµ = [1− exp (−0.0165Rey)]2[1 +

20.5Ret

], (4.76)

f1 =

[1−

(0.05fµ

)3.0], (4.77)

f2 =[1− exp

(−Re2t)]

. (4.78)

In Tab. 4.6 standard values for the introduced constant are given. Wall


0.09 1.44 1.92 1.0 1.3

Table 4.6: Standard values for Lam and Bremhorst Low Reynolds k − ε coecients.

boundary condition for ε is of type zero gradient normal to walls. Obviously kis respecting the no slip condition.

4.6.6 Lien LeschzinerThis is the model proposed in (Lien and Leschziner, 1993) to conform with

one equation model by Wolfshtein (1969), a companion model with closure for-mulae according with Norris and Reynolds length scale is presented here after.This model will be referred to from now on with the name LW. Equations ofreference as well as values for the coecients are reported:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]=

Cε1 (Pk + P ′k)ε

k− ρCε2f2

ε2

k, (4.79)

63


∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − ρε . (4.80)

Like in CLL model, two damping functions and a modied production term leadall the modication to the original equations:

fµ = [1− exp (−0.016Rey)] [1− exp (−0.263Rey)]−1, (4.81)

f2 =[1− 0.3 exp

(−Re2t)]

, (4.82)

P ′k =ρCε2f2k

1.5 exp(−0.00222Re2y

)

3.53y [1− exp (−0.263Rey)]. (4.83)

In Tab. 4.7 standard values for the introduced constant are given. Wall


0.09 1.44 1.92 1.0 1.3

Table 4.7: Standard coecients values for Lien and Leschziner Low Reynolds k − ε.

boundary condition for ε are specied in such a way:

εw = Cε1P ′k

ρCε2f2(4.84)

For the companion model, called LNR (Lien, 1992), everything but the followingremain invariant:

fµ = [1− exp (−0.0198Rey)](

1 +5.29Rey

)−1

, (4.85)

Dε =y√k

y√k + 2ν0.45C−0.75

µ

, (4.86)

P ′k =ρCε2f2k

1.5 exp(−0.00057Re2y

)

0.42Cε1yC−0.75µ Dε

. (4.87)

4.6.7 RealizabilityExcessive levels of turbulence kinetic energy are predicted by standard two-

equation models in region of large rate of strain. This was originally recognizedin stagnation point ows, but it will be seen here that it is a more widespreadanomaly. The problem is connected with a big overproduction of k, in case amoderate level of k is subjected to a large of rate of strain. Such over productioncan be due to, an underestimation of the sink term or/and an overestimation ofturbulent viscosity. These ideas could be merged together into a bound for thelocal turbulent time scale Ts (Medic and Durbin, 2002). In detail, rst step isto reformulate the expression for µt:

µt = Cµfµρ u2 Ts , (4.88)

64


where u is the velocity scale and Ts is the local turbulence time scale. For thepreviously proposed models u = k and Ts = k/ε and so ε equation is rewrittenin terms of Ts:

∂(ρε)∂t

+∂(ρεUj)∂xj

− ∂

∂xj

[(µ+

µt

σε)∂ε

∂xj

]= Cε1

Pk

Ts− ρCε2

ε

Ts. (4.89)

To constrain Reynolds Stress Tensor to be positive denite, a limiter is thenapplied to Ts:

Ts = min

[k

ε,

α√6Cµ‖S∗‖

], (4.90)

in which S∗ = 12 G and α = 0.6 are usually used. As a consequence at large rate

of strain Pk is growing like |S| rather than |S|2.The option of selecting the realizability constraint or not has been imple-

mented in all previous models with a switch to be controlled by the user. Thiscorrection will be referred to in future chapters, when brevity is a must withthe postscript Real.

4.7 Two LayerImprovements in well predictions near wall behavior of previous models were

not followed by a similar enhancement in the region far from the wall. UsuallyLow Reynolds models spoils good behavior of standard k− ε in the outer layer.Better would be to join the advantages of the two models. This approach wasproposed by (Rodi, 1991): in which a one equation model (k − l) in the nearwall region match standard k − ε High Reynolds models at a certain patchingpoint. Several variant of this model has been implemented: following Wolfshteinor Norris and Reynolds closure formulae in the wall layer, with standard k − ε

or the RNG version. For simplicity only the Wolfshtein variant is going to bedescribed.

It would be better start with the denition of such a patching point: thelocation at which the damping function fµ reaches the value of 0.95. It is bettershow how fµ is dened:

fµ =[1− exp

(−Rey

Aµ

)]. (4.91)

In order to avoid that non-turbulent zones far from the wall were calculatedwith the one equation model, a user controlled parameter denes the wall layerthickness meaning that above such value fµ was not checked any more and thestandard k − ε directly applied.

The k − l models are based on the following equations, expressing bothdissipation and eddy viscosity via a turbulent length scale:

ε =k

32

lε, (4.92)

µt = Cµρ√klµ . (4.93)

65


For this model two characteristic lengths are dened:

lµ = Cl yfµ , (4.94)

lε = Cl y

[1− exp

(−Rey

Aε

)]. (4.95)

In Tab. 4.8 standard values for the introduced constants are given.

Cµ Cl Aε Aµ

0.09 2.5 5 62.5

Table 4.8: Standard coecients values for Two Layer k − ε.

4.8 Anisotropic Turbulence ModelIt is known Eddy Viscosity Model (EVM) underpredict the turbulence anisotropy

because of the use of a scalar parameter such as the eddy viscosity to computethe turbulence stresses in all space directions. This inaccuracy, coming fromunreliable or incorrect hypotheses in the theoretical development of eddy dif-fusivity models (see (Lecturer, 2002)) becomes a serious drawback in all thosesituations in which the anisotropy of turbulence plays an important role in thedenition of the ow eld or heavily aect the value and distribution of anyquantity of interest. In case of lm cooling, or generally speaking Jet CrossFlows, inaccurate prediction of turbulence anisotropy yields to an underestima-tion of the lateral spreading of the temperature eld and therefore to inaccurateeectiveness distributions on the wall.Improvements in the prediction of turbulence anisotropy may be obtained byusing nonlinear eddy viscosity models. Yet, in the case of Jet Cross Flows,this kind of model enhancement do not leads to fundamental improvements(A.Hoda and S.Acharya, 2000). At an even more rened level, Reynolds stressmodels solving a single equation for each component of the stress tensor, aretheoretically capable to properly capture the turbulence eld. Nonetheless, thisapproach is computationally expensive and, additionally, it was found not to besystematically better then EVM computations (D.J.Ferguson and D.K. Waltersand J.H. Leylek, 1998). Algebraic Reynolds Stress models are to some extentan intermediate solution which, in theory, is able to provide a tradeo betweenthe simpler and relatively easy to use, eddy viscosity models and the computa-tionally expensive and hard dierential RSM models.The present model is introduced in this scenario as a further step towards eddyviscosity models coming from the RSM. Dierently from RSM, the Reynoldsstress tensor components are still assumed to be proportional to the mean defor-mation rates tensor according to eddy viscosity but a tensorial denition of theeddy viscosity is used so that an anisotropic turbulence modelling is achieved.This approach to the problem is supported by the evidence, arising from exper-imental data or from DNS and LES computations, that eddy viscosity, is not

66


isotropic even for relatively simple ows such as a three-dimensional boundarylayer ((Davenport and Simpson, 1982)).For the particular case of lm cooling ows, eddy viscosity anisotropy was exam-ined by Acharya et al. ((S. Acharya and M. Tyagi and A. Hoda and F.Muldoon,1999)). In this work the dierent eddy viscosity components arising in a lmcooling ow were assessed by taking the ratio of the computed shear stress andits respective strain rate component as computed in a LES. The vertical (xy)eddy viscosity can be regarded as the turbulence viscosity used by conventionalEVMs as all of them are tuned on ows, such as two dimensional boundarylayer, backward facing step and so on, where the mean ow is aected mainlyby this component which is therefore the dominating one. Looking at gure4.2, where the results obtained by Acharya et al. are reported, it can be seenthat the xz eddy viscosity is considerably greater than the xy component, thatis the wall-normal component. This suggests that common EVMs are forcedto underestimate the lateral turbulent diusion. Even if Fig. 4.2 shows a verycomplex situation which can hardly be aorded by simple anisotropy correctionsanyway it clearly indicate that directional modications of eddy viscosity arelikely to improve the predictive capabilities of standard EVMs.

Figure 4.2: LES predictions of shear stress (top row), rate of strain (middle row), and eddyviscosity (bottom row) for the xy component (left column), yz component (middle column)and xz component(right column).

4.8.1 Model FormulationThe model introduced in this work is a modication of the standard Two

Layer k− ε model. The modelling approach followed in this work dates back tolate seventies. Bergeles et al. (G. Bergeles and A. D. Gosman and B.E. Launder,1978) proposed to modify standard EVMs in order to take in to account for theanisotropy of the turbulence eld in jet cross ows by using an anisotropic eddydiusivity. The lateral spreading of jet cross ows is underestimated becausethe turbulent transport in the span-wise direction is underpredicted by isotropic

67


EVMs. An ad hoc correction is achieved by multiplying the eddy viscosity as-sociated to the cross-Reynolds stress by an anisotropy factor γ to be computedalgebraically.In the approach of Bergeles et al. the correction was applied just to the mo-mentum equations leaving the remaining of the equations unaltered. A wallfunction approach was used and γ was assumed to be constant in the boundarylayer assuming a value of 4.5. Outside the boundary layer a value of 1 wasassumed. The value of the anisotropy factor in the boundary layer was inferredby experimental data collected by Quarmby and Quirk (A. Quarmby and R.Quirk, 1974).More recently, a more rened approach was used by Lakehal et al. in a seriesof papers on this topic ((A. Azzi and D. Lakehal, 2001), (A. Azzi and B.A.Jubran, 2001), (A.Azzi and D.Lakehal, 2002), (D.Lakehal and C.S. Theodor-idis and W.Rodi, 2001), (D.Lakehal, 2002)). First of all they used a zonal lowReynolds treatment for wall boundaries. Then, an algebraic expression basedon DNS data was used to compute the anisotropy factor as a function of thenon-dimensional wall distance y+ and also, the energy as well as the turbulencekinetic energy equations, in addition to momentum equation, were corrected totake in account for the anisotropy of the turbulence eld.The model derivation is based on physical reasoning. Following the conceptualscheme of Azzi et al., the transport equations for the Reynolds stress tensor maybe written as follows, if homogeneous near-equilibrium turbulence, unaectedby external forces is assumed:

u′iu′j =

23kδij − Cs

k

ε(Pij − 1

3Pii) (4.96)

Where Cs = 0.27 is a model constant and

Pij = −u′iu′kδuj

δxk− u

′ju

′k

δui

δxk(4.97)

Under the addition hypothesis that the ow is homogeneous in the stream-wise(x) direction and that v′w′ is negligible with respect to u′v′ and u′w′ the onlynon zero shear stresses are

u′w′ = −Γ2δu

δz;u′v′ = −Γ3

δu

δy(4.98)

In the previous expressions we assumed a cartesian reference coordinate systemwith the x axis parallel to the stream-wise direction, the y axis in the span-wisedirection and the z axis normal to the plate in a jet cross ow. From Eqs.4.96, 4.97 and 4.98 the anisotropic eddy viscosities Γ2 and Γ3 may be recast asfollows:

Γ2 = Cskw′2

ε; Γ3 = Cs

kv′2

ε(4.99)

which can be rewritten as

Γ2 = Csw′2

k

k2

ε; Γ3 = Cs

w′2

kγk2

ε(4.100)

68


Where γ = v′2/w′2 is the ratio of the span-wise and wall-normal uctuations.Eq. 4.100 shows that eddy viscosity for the cross stresses in the stream-wise(Γ2) and spanwise (Γ3) directions have to be dierent according to an anisotropyfactor γ which is found to depend on the ratio of the normal stresses v′2 andw′2. Actually, approaching the wall, the normal stress w′2 decays at a muchfaster rate with respect to the v′2 because of the damping eect of wall itself.This eect is completely overridden by conventional EVMs which use a scalareddy viscosity.In order to close the anisotropic eddy viscosity model dened by equations4.100 the function γ has to be dened. As said above, Bergeles assumed apeace-wise constant prole for γ posing γ = 4.5 within the boundary layer andγ = 1 outside. Lakehal et al. (D.Lakehal and C.S. Theodoridis and W.Rodi(2001); A.Azzi and D.Lakehal (2002)) used the channel ow DNS data of Kimet al. (J.Kim and P.Moin and R.Moser (1987); W.Rodi and N.N.Mansour andV.Michelassi (1993)) to dene a proper prole for γ as a function of the non-dimensional wall distance y+. The actual prole is shown in gure 4.3. Thisfunction is strictly valid only for channel ow without separation. In order toobtain a more general functional relation the dependence of γ on Y + is relaxedand nally the relation is given the following expression:

Figure 4.3: Comparison of analytical correlation and DNS data for the anisotropy factor γ.

Γ2 = Max[103(y+)0.42

2.682(y+)2 − 5.463, 4.25] (4.101)

4.9 The k − ω classThe rst complete turbulence model was proposed by Kolmogorov in 1942,

in addition to the same equation for k, he developed a second equation for theparameter ω representing the rate of dissipation of energy in unit volume andtime. The reciprocal of ω serves as a local turbulent time scale while the lengthscale is prescribed by k0.5/ω. Due to unavailability of computer resources thismodel went with no application till the 70's when Saman formulated a k − ω

69


model able to predict eects of adverse pressure gradients and to integratethrough the viscous sub-layer (Wilcox, 1998). After that, many variants hasbeen proposed making k−ω models the second most used class of two-equationturbulence models.

As hinted in Sec. 4.5, both k− ε and k−ω class of models have been showntheir own strengths and weaknesses in the course of the years. k−ε in particularis recognized as the best choice for free shear ows (mixing layer, jets and wakes)and reached good accuracy also in wall bounded ows. k−ω certainly performsbetter in wall bounded ows, especially the ones with adverse pressure gradients,although its freestream values dependency is notorious, in addition it is creditedwith a higher numerical stability (Bredberg, 2001).

4.9.1 Original k − ω

The model introduced now is proposed in (Wilcox, 1998) as the evolutionof the well-known k − ω model presented by Wilcox in 1988. New dissipationcoecients, maintaining the high precision for boundary layers and removingoverprediction of free shear layer spreading rates, have been introduced. Thussuch model is applicable to both wall-bounded and free shear ows and theminimum requirement is satised. Denition of eddy viscosity directly followsfrom the k − ε class one but due to main authors dierent notation sometimesthere is ambiguity regarding the old standard k− ε Cµ constant. Therefore itis not superuous to write:

µt = ρk

ω. (4.102)

Then the equation for turbulence frequency is presented together withstandard k equation:

∂(ρω)∂t

+∂(ρωUj)∂xj

− ∂

∂xj

[(µ+

µt

σω)∂ω

∂xj

]=

Cω1Pkω

k− ρCω2ω

2 , (4.103)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+

µt

σk)∂k

∂xj

]= Pk − β∗ρωk . (4.104)

β∗ Cω1 Cω2 σk σω

0.09 0.5555 0.075 2 2

Table 4.9: Standard values for k − ω coecients.

Wall boundary condition for ω is specied in such a way:

ωw = 10.06.0µ

ρCω2y2. (4.105)

70


Again this expression is intended as an imposition on the rst near wall node,noticing that the value for ω is going towards innity as distance from the wallis reducing to zero.

4.9.2 Baseline modelThe idea of the base line model is to retain the robust and accurate formu-

lation of the Wilcox k−ω model in the near wall region, and to take advantageof the free stream independence of the k − ε model in the outer part of theboundary layer. To achieve this a k−ω formulation of standard k− ε is derivedand merged together the previous model via a blending function F1 being onein the near wall region to activate standard k − ω and zero outside activatingk − ε. The blending will take place in the wake region of the boundary layer.The main dierence between the models for the two regions is the appearanceof an extra cross-diusion term (scalar products of gradients) in the ω equation(Menter, 1994).

∂(ρω)∂t

+∂(ρωUj)∂xj

− ∂

∂xj

[(µ+ µtαΩ)

∂ω

∂xj

]=

C1Pk

νt− ρC2ω

2 +2ραε(1− F1)

ω

∂k

∂xj

∂ω

∂xj, (4.106)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+ µtαK)

∂k

∂xj

]= Pk − β∗ρωk . (4.107)

First the blending function F1 should be implemented, so we need some auxiliaryfunctions such as:

CDkω = max[2αεω

−1 ∂k

∂xj

∂ω

∂xj, 10−20

], (4.108)

arg1 = min

max

[ √k

0.5ωy,500νωy2

],

4αεk

CDkωy2

, (4.109)

to nally write:

F1 = tanh(arg4

1

). (4.110)

The four constants are hence blended:

αΩ = F1αω + (1− F1)αε , (4.111)αK = F1αkω + (1− F1)αkε , (4.112)C1 = F1Cω1 + (1− F1)Cε1 , (4.113)C2 = F1Cω2 + (1− F1)Cε2 . (4.114)

Values for the introduced constants can be found in Tab. 4.10.At the rst near wall node ω is specied to be:

ωw = 10.06.0µ

ρCω2y2. (4.115)

71


β∗ Cε1 Cε2 Cω1 Cω2 αkε αkω αε αω

0.09 0.4404 0.0828 0.5976 0.075 1 0.5 0.856 0.5

Table 4.10: Standard values for k − ω BSL coecients.

4.9.3 Shear Stress Transport modelOne of the major dierences between eddy-viscosity and full Reynolds-stress

models is that the latter accounts for the important eect of the transport of theprincipal turbulent shear stress τ12 = −ρu′v′. The base for such an equation isBradshaw's assumption that the shear stress in boundary layer is proportionalto the turbulent kinetic energy:

τ12 = ρa1k . (4.116)

For conventional two-equation models the principal shear stress can be com-puted as:

τ12 = µtΩ = ρ

√Productionk

Dissipationk

a1k , (4.117)

where Ω is the vorticity and a1 is a constant.In adverse pressure gradient ows the ratio of production to dissipation could

be much larger than one, meaning that Bradshaw's hypothesis is substantiallyviolated. To satisfy Eq. 4.116 within the framework of eddy-viscosity models µt

should be bounded in such a way:

µt =a1k

max[a1ω,ΩF2]. (4.118)

Of course original formulation of the eddy-viscosity must be maintained for freeshear layers so the same blending function approach as for the baseline modelis adopted.

First come the equation of conservation for turbulent properties:

∂(ρω)∂t

+∂(ρωUj)∂xj

− ∂

∂xj

[(µ+ µtαΩ)

∂ω

∂xj

]=

C1Pk

νt− ρC2ω

2 +2ραε(1− F1)

ω

∂k

∂xj

∂ω

∂xj, (4.119)

∂(ρk)∂t

+∂(ρkUj)∂xj

− ∂

∂xj

[(µ+ µtαK)

∂k

∂xj

]= Pk − β∗ρωk . (4.120)

Second the bounding function F2 is constructed:

arg2 = max

[2

√k

0.09ωy,500νωy2

], (4.121)

F2 = tanh(arg2

2

). (4.122)

72


Then the blending function is built:

CDkω = max[2ραεω

−1 ∂k

∂xj

∂ω

∂xj, 10−20

], (4.123)

arg1 = min

max

[ √k

0.09ωy,500νωy2

],

4αεk

CDkωy2

, (4.124)

F1 = tanh(arg4

1

). (4.125)

Finally constants are blended like in BSL model:

αΩ = F1αω + (1− F1)αε , (4.126)αK = F1αkω + (1− F1)αkε , (4.127)C1 = F1Cω1 + (1− F1)Cε1 , (4.128)C2 = F1Cω2 + (1− F1)Cε2 . (4.129)

β∗ Cε1 Cε2 Cω1 Cω2 αkε αkω αε αω a1

0.09 0.44035 0.0828 0.59761 0.075 1 0.5 0.856 0.5 0.31

Table 4.11: Standard values for k − ω SST coecients.

Wall boundary condition for ω is specied in such a way:

ωw = 10.06.0µ

ρCω2y2. (4.130)

Another variant of the k − ω SST, later in this thesis also called simply SST,proposed to extend applicability beyond aerodynamic applications has beenimplemented following the instructions given in CFX guide CFX (2004).

Modications regards eddy viscosity:

µt =a1k

max[a1ω,√

2SF2], (4.131)

the limiter CDkω:

CDkω = max[2ραεω

−1 ∂k

∂xj

∂ω

∂xj, 10−10

], (4.132)

a bound imposed on production term:

Pk = min[G− 2

3ρk∂Uj

∂xj, 10ε

], (4.133)

and the value of Cω1 = 0.5532.

4.9.4 Automatic Near Wall TreatmentAll previously cited models are based on a Low Reynolds formulation and

therefore they require a quite ne grid at walls (y+ ≈ 1) but for cases of a cer-tain complexity this results to be a quite strict constraint. Unfortunately one

73


of the most common issues in the preparation of accurate calculation meshesfor heat transfer analysis of complex geometries is the assurance of a proper y+

distribution, particularly when the ow eld is distinguished by relevant vari-ation of ow Mach number. Therefore, in order to increase grid independencea mixed approach between wall-function and Low Reynolds was added to thek − ω SST model. The idea, see Menter (1993), is to blend the two approachesvia a blending function Γ calculated algebraically from the non dimensional walldistance. Both turbulent production and turbulent specic dissipation are im-posed on the rst node mixing the eects of the Low Reynoldsand the HighReynolds contributions:

Pawt,p = PLR,pe−Γ + PHR,pe

− 1Γ , (4.134)

ωawt,p = ωLR,pe−Γ + ωHR,pe

− 1Γ , (4.135)

where Γ is calculated with an algebraic expression for y+ dened with uτ =max(uτ,LR, uτ,HR) . The same blending is applied to thermal quantities follow-ing Kader universal law B. (1982).

74

Chapter 5

Combustion and FlameModelling

This chapter introduces and discusses the main aspects of ames type froma general overview. The three main characteristics of the ame burning can besummarized in diusion, premixed and partially premixed type. In the nextsection we analyze in detail these three ames. At the same time we focus ourattention in the continuum mechanic description the range of validity of eachame type and related main turbulent combustion models.

Contents5.1 Reynolds Averaged Navier-Stokes equations of mass

fraction . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Perfect gases law and thermodynamical properties 775.3 Chemical Kinetics . . . . . . . . . . . . . . . . . . . 78

5.3.1 Arrhenius model . . . . . . . . . . . . . . . . . . . . 785.3.2 Reaction mechanisms . . . . . . . . . . . . . . . . . 795.3.3 Chemical kinetics in turbulent ow . . . . . . . . . . 79

5.4 Flame Stretch . . . . . . . . . . . . . . . . . . . . . 805.5 Diusion Flames . . . . . . . . . . . . . . . . . . . . 81

5.5.1 Soot . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.5.2 Laminar diusion ames modelling . . . . . . . . . . 845.5.3 Mixture fraction . . . . . . . . . . . . . . . . . . . . 845.5.4 Diusive amelets . . . . . . . . . . . . . . . . . . . 865.5.5 Turbulent combustion diagram . . . . . . . . . . . . 875.5.6 Turbulent diusion ames modelling . . . . . . . . . 895.5.7 Presumed pdf amelets models . . . . . . . . . . . . 90

5.6 Premixed Flames . . . . . . . . . . . . . . . . . . . 935.6.1 Combustion regimes and types of ame . . . . . . . 945.6.2 Turbulence and ame scales . . . . . . . . . . . . . . 945.6.3 Spectral Diagram . . . . . . . . . . . . . . . . . . . . 965.6.4 Premixed turbulent combustion models . . . . . . . 975.6.5 Bray-Moss-Libby model . . . . . . . . . . . . . . . . 98

75

Chapter 5. Combustion and Flame Modelling

5.6.6 Level-Set Model: G-Equation . . . . . . . . . . . . . 1015.7 Partially Premixed Flames . . . . . . . . . . . . . . 102

5.7.1 Partially Premixed Flames Modelling . . . . . . . . 102

76

Combustion and Flame Modelling Chapter 5.

5.1 Reynolds Averaged Navier-Stokes equationsof mass fraction

The equation of conservation of mass fraction of a generic k− th specie withρk the k − th specie density and ρ the mixture average density takes the form:

∂ρYk

∂t+∂

(ρuiYk

)

∂xi=

∂

∂xi

[(µ

Sck+

µT

ScT

)∂Yk

∂xi

]+ ρ˜ωk (5.1)

where Sck and ScT are respectively the laminar and turbulent Schimdt numbersdened as:

Sck =µ

ρDkScT =

µT

ρDT(5.2)

where Dk and DT are respectively the laminar and turbulent diusions of k−thspecie that in this work, following a common procedure, are assumed to be equalto corresponding Prandtl's numbers dened in Eq. 5.4.

Finally the last term of RHS ρ˜ωk is the average reaction rate for specie kthat requires a supplementary model.

Energy conservation is ensured by the thermochemical enthalpy equation:

∂ρh

∂t+∂

(ρuih

)

∂xi=∂p

∂t+∂ (pui)∂xi

(5.3)

+∂

∂xi

[(µ

Pr+µT

Prt

)∂h

∂xi

]− ∂qR

∂xi+ T V

with viscous heating T V neglectable if Ma2 ¿ 1.0 and ∂qR

∂xirepresenting

the heat sink due to radiative exchange. Pr and PrT are laminar andturbulent Prandt's numbers dened as:

Pr =µCp

λPrT =

µTCp

λ(5.4)

where Cp(T ) =∑

k YkCpk(T ) is the mixture mean specic heat at constant

pressure. The thermochemical enthalpy is written as:

h(T ) =∫ T

Tref

Cp(T )dT +∑

Yk∆h0k(Tref ) (5.5)

having dened ∆h0k(Tref ) as the formation enthalpy of k − th specie.

The above reported RANS equations set, coupled with a turbulence modelrepresents a powerful instrument to model reacting turbulent ow.

5.2 Perfect gases law and thermodynamical prop-erties

In Eq. 5.5 has been introduced the denition of thermochemical enthalpy,that makes it possible to take in account of energy amounts produced and ab-sorbed by reaction. In this denition has been used the mixture mean specic

77


heat at constant pressure as the weighted average of species specic heats, with-out dropping their dependence on temperature. Indeed while considering uxeswith high temperature gradients, such as those involving combustion, it is neces-sary to model each k − th species specic heat at constant pressure as functionof temperature, while the eect of pressure is neglected by using the perfectgases model.

hk =R

Wk

(N∑

n=1

ankTn

n+ aN+1,k

)and Cpk =

∂hk

∂T

∣∣∣∣p

(5.6)

Specic heats can be evaluated starting from molecular properties of each specie,unfortunately this approach involves highly complicated functions, then thedependence on temperature is modelled with polynomial expressions. In presentwork, for each k − th specie, has been used the expression reported in Eq. 5.6where R = 8.31445 [kJ/(kg · K)] is the universal perfect gases constant andWk is the molar mass of specie k. Again from Eq. 5.6 we can notice thatthermochemical enthalpy is formed by two part: the rst one, dependant ontemperature, is the sensible enthalpy while the other, temperature independent,is the formation enthalpy referred to standard conditions (usually T = 298.15Kand p = 101325Pa) The model of perfect gases gives also a relation betweentemperature, pressure and density, the perfect gases equation of state:

pk =ρkRT

Wk(5.7)

where Wk is the molar weight of k − th species, and pk is its partial pressure.Using the Dalton's law for ideal gases mixtures (p =

∑k pk) and dening the

mixture averaged molar mass as in Eq. 5.8, we can dene the equation of statefor the mixture as reported in Eq. 5.9.

W =1∑

kYk

Wk

(5.8)

p =ρRT

W(5.9)

5.3 Chemical KineticsIn species conservation equations (Eq. 5.1) has been introduced the reaction

rate that, in its unaveraged form, is: ρωk. This term requires an additionalmodel that is a chemical kinetic model (for further details see (Peters, 2000)).

5.3.1 Arrhenius modelArrhenius model is commonly accepted in the eld classical inorganic chem-

istry. Consider the general reaction:

ν′1χ1 + ν′2χ2 + ν′3χ3 + . . .+ ν′kχk ↔ ν′′1χ1 + ν′′2χ2 + ν′′3χ3 + . . .+ ν′′kχk

78


The speed of reaction is dened as the variation, in the unit of time, of eachspecie concentration weighted to its own stoichiometrical coecient:

q = − 1ν′1

dc1dt

= − 1ν′2

dc2dt

= . . . =1ν′′1

dc1dt

=1ν′′2

dc2dt

= . . .1ν′′k

dckdt

where ck indicates the molar concentration of a generic specie. The net reactionspeed is dened as:

q = qf − qr

where the subscript f and r stand respectively for forward and rearward reac-tions. In equilibrium condition the net speed is zero so qf = qr. Rearward andforward reactions speeds are usually modelled as a function of species concen-trations:

qf = kf

K∏

k=1

cλk

k qr = kr

K∏

k=1

cλk

k

Where λk is are the reaction order with respect to various components and theirsum gives the global reaction order. kf and kr are the kinetics constants of thereaction and are dependant on temperature. These constants are evaluated bythe Arrhenius' equation:

k = AT βexp

(Ea

RT

)(5.10)

where A is the frequency factor while Ea is the activation energy. A, Ea and βare characteristics of each reaction.

5.3.2 Reaction mechanismsWith a reaction mechanism is intended that set of reactions justifying the

presence, the endurance and the concentration of all experimentally detectedspecies. Those chemical mechanisms obtained by this approach are called de-tailed. For each reaction presents in those mechanism are experimentally evalu-ated the characteristic constants, A, Ea and β appearing in Arrhenius equation.An accurate expression of chemical equation together with the knowledge of ki-netics constants, enables to evaluate each ωk. Despite of their accuracy, detailedmechanisms are computationally too expensive, hence while analyzing reactingow by CFD it is usually preferred to use simplied models.

5.3.3 Chemical kinetics in turbulent owThe RANS form of species conservation equations (Eq. 5.1) needs the av-

erage reaction rate: ρ˜ωk. Unfortunately, due to its strong non linearity, theuse of Arrhenius expression with averaged values of temperature and speciesconcentrations, would provide a wrong estimation of the average reaction rates.

A rst solution is obtained by approximating the reaction rate expressionby a Taylor's series, having introduced instantaneous variables decomposition

79


into their averaged and uctuating components. This is a direct approach forρ˜ωk closure called moment closure that is very rarely used. Representation byTaylor's series introduces a new series of correlations among uctuating terms(such as Y ′′k T

′′) that cannot be neglected a priori, hence a specical algebraicallike modelling procedure is required, that is very hardly generalizable. Then it isusually preferred to close the expression of ρ˜ωk by models that keep in account,in a more or less empirical way, the problem physics: these are the so-calledturbulent combustion models that are described in following sections.

5.4 Flame StretchFlame stretch is that set of tridimensional eects of distortion perturbing the

ame front. The ensemble of these eects submits the ame front to a stretchingthat inuences the combustion processes there happening ((Andreini, 2004) and(Poinsot and Veynante, 2001)). Flame stretch is dened as the speed of amearea (A) variation that for a laminar ame it can be expressed as:

K =1A

dA

dt(5.11)

Considering Fig. 5.1, we can dene ~n as the normal to ame front, ~u the

Figure 5.1: Premixed ame propagation speed

reactants stream speed, and SL as the laminar ame front speed (zero for diu-sion ames, subsection 5.5), relative to reactants stream so, ~w, the ame frontabsolute speed, can be calculated as:

~w = ~u+ SL~n

This denition allows to rewritten the ame stretch as:

K = ~∇t · ~u+ SL~∇t · ~n (5.12)

where ~∇t is the gradient operator component tangential to ame front.

• The rst term of Eq. 5.12 represents the distortion eect of ame surfacedue to reactants speed gradients. It is a term owed to the lack of ow

80


uniformity and it is also present in steady ames: it is commonly calledstrain.

Most common examples of strain aected laminar ames are the diusionsteady strained ame (Fig 5.2(a)) and the premixed symmetrical counterow ames (Fig. 5.2(b)). The rst one having a strain rate equal toa = ∂ux

∂x , is steady due to strain itself. The second one having a strainrate equal to a = ∂uy

∂y is a twin ame stabilized by two opposite reactantjets, where the ame fronts runs parallel to the stagnation plane.

• The second term of Eq. 5.12 is due to curvature eect of ame surface.And it is equal to zero in diusion ames, because of the absence of aame speed.

(a) (b)

Figure 5.2: Examples of strained ames: a) steady strained one-dimensional diusion ameand b) counterow premixed ames

Stretching induces modication of chemical processes then it cause modicationsof laminar premixed ames speed.

If reactants ow is turbulent, the local ame stretch dened as in Eq. 5.12represents the instantaneous stretching (for premixed ame and, by droppingthe term dependant on ame speed, it can be representative of diusion ametoo). Theoretically it should be possible to dene a Favre averaged form ofstretching (K), but practically the averaging procedure would produce a largenumber of Reynolds stresses terms. Hence it is, usually, preferred to modelthe ame stretch with the Favre averaged strain rate, expressed as function ofturbulence eld characteristic variables ((Andreini, 2004)).

5.5 Diusion FlamesIn many combustion applications fuel and oxidizer enter separately into the

combustion chamber where they mix and burn during continuous interdiusion.

81


Figure 5.3: Diusion ame structure

These ame are called non-premixed ames or diusion ames since diusionof reacting species is the rate-controlling processes ((Peters, 2000)).

Diusion combustion has its main applications in Diesel engines, furnacesand airplanes engines. In many applications non-premixed ames not onlyinvolves separated uxes but fuel and oxidizer may be solids or liquid, thenchemical-physical changes, such as evaporation, are needed. In Fig. 5.3 isshown a prototype of the diusion ame congurations, where from the rightside comes the oxidizer (which may be diluted in other gases) while form theleft side comes the fuel (diluted or not). Fuel and oxidizer diuse towards thereaction zone where they burn and generate heat. Temperature reaches its max-imum in this zone and diuses away from the ame front towards the fuel andoxidizer streams.

Again keeping the attention on Fig. 5.3 we can make some important con-siderations ((Poinsot and Veynante, 2001, chap. 3)):

• Far away on each ame side, the gas is either too rich or too lean toburn. Chemical reactions take place only in those limited regions, wherefuel and oxidizer are mixed adequately. The most favorable conditionsoccur where fuel and oxidizer are in stoichiometric proportions: a diusioname usually lies along the points where mixing produces a stoichiometricmixture.

• Flames as the one reported in Fig. 5.3 are steady only when strain isapplied to the ame, i.e. when the fuel and oxidizer streams are pushedagainst each other at given speeds.

• Diusion ames do not exhibit a reference speed, contrarily to premixedones (chapt. 5.6): ames are unable to propagate towards fuel because ofthe lack of oxidizer and the lack of fuel in oxidizer stream makes impossiblethe propagation even in this side. The lack of propagation implies thatthe reaction zone does not move signicantly relatively to the ow eld.

82


• A diusion ame does not have a reference thickness: an unstretchedame grows up indenitely with time while the thickness of a stretchedame depends essentially on stretch and can assume a very wide range ofvalue.

This chapter rstly introduces the methods suitable to treat laminar diu-sion ames, then an extension of these model is performed to treat turbulentcombustion.

5.5.1 SootA peculiar product of diusion combustion is soot. In ideal conditions hy-

drocarbons combustion processes lead to the formation of carbon dioxide (CO2)and water vapor (H2O). Instead, while occurring real conditions, in fuel reachzones, combustion and pyrolysis of hydrocarbons produce intermediated speciesand radicals that may lead to soot formation. Soot formation analysis shouldbe consider since ((Da Soghe, 2006)):

• soot is a product of incomplete combustion processes so the more it isproduced the worse is the combustion process productivity;

• soot is one the most dangerous pollutant for human health;

• soot is an highly radiatively active specie (cfr. subsubsection 6.5.2.2) thatexplain why it is considered in present work, and moreover this feature isexploited in steam generators and furnaces;

• soot is used in the production of inks and pneumatics.

Soot is mainly made of carbon and, in smaller proportions, of hydrogen,oxygen and nitrogen. Soot particles density is smaller that carbon ones, it mayvary between 1700 − 1900 kg/m3 and depends on particles porosity. Usualdimensions of soot lay interval between 5− 80 nm.

A detailed description of soot formation processes and relative models canbe found in (Da Soghe, 2006), (Pasquini, 2006) and (Lautenberger, 2002) andthe processes involving soot can be summarized as:

Particles inception owed to reactions, in gaseous phase, between low weightmolecules

Surface growth due to the merger of soot molecules to already formed parti-cles

Particles coagulation and fragmentation provided by collisions between par-ticles that may merge or fragment

Particles oxidation is the only process that may reduce soot mass and isperformed on particles surface involving reactions with oxygen and OH

that produce CO and CO2.

83


These processes are usually occurring at the same time and are stronglydependant on combustion system features. Usually, in common combusted, theproduced soot (rising over 99 %) is almost completely oxidized before exitingthe combustion system.

5.5.2 Laminar diusion ames modellingIn this section will be demonstrated how the treatment of non-premixed

ame can be split in two separated tasks:

• A mixing problem that provides the mixture fraction Z, introduced insubsection 5.5.3, as a function of space and time coordinates

• A ame structure problem where the links between ame variables andthe mixture fraction are used to construct all other ame variables, i.e.the ame structure in Z-space

5.5.3 Mixture fractionAs has already been hinted the the mixing of reacting species is studied by

the mixture fraction (Z) then here is reported its deriving procedure.Following the approach proposed in (Peters, 2000), transport of each chem-

ical element is considered (mass of elements is conserved while the mass ofchemical species may change due to chemical reactions).

Denoting akj as the number of atoms of element j in a molecule of speciek, and Wj as the atomic weight of j, the mass of all atoms of j present in thecombustion system is

mj =n∑

k=1

akjWj

Wkmk (5.13)

The mass fraction of element j is then

Zj =mj

m=

n∑

k=1

akjWj

WkYk (5.14)

The laminar equation of conservation for species k, that has its RANS formin Eq. 5.1, using the Fick's law for diusion term, is

∂ρYk

∂t+∂ (ρuiYk)∂xi

=∂

∂xi

(µ

Sck

∂Yk

∂xi

)+ ρωk (5.15)

Adding species conservation equations Yk in the same way as in Eq. 5.14 thej − th element conservation equation is

ρ∂Zj

∂t+∂ρuiZj

∂xi=

∂

∂xi

[n∑

k=1

akjWj

Wk

(ρDk

∂Yk

∂xi

)](5.16)

84


where the chemical source terms ρωks do not appear since, for the mass conser-vation of each element j, present in any reaction l in k species, having stoichio-metric coecient νkl is true that

n∑

k=1

akjWkνkl = 0 (5.17)

Again relying on the mass conservation of each element, if all diusivities areconsidered equal (Dk = D), the balance equation for the element mass fractionbecomes

ρ∂Zj

∂t+∂ρuiZj

∂xi=

∂

∂xi

(ρD

∂Zj

∂xi

)(5.18)

Considering the global reaction equation (omitting intermediates reactions) forthe complete combustion of an hydrocarbon fuel CmHn

ν′FCmHn + ν′O2O2 → ν′′CO2

CO2 + ν′′H2OH2O (5.19)

we can writeZC

mWC=

ZH

nWH=YF,u

WFand ZO = YO2,u (5.20)

Using the original approach of Burke and Schumann of 1928, we can dene theconserved scalar β can be dened as

β =ZC

ν′FmWC+

ZH

ν′FnWH− 2

ZO

ν′O2W ′

O2

(5.21)

this scalar becomes equal to 0 in stoichiometric condition. Normalizing beta weobtain the denition of mixture fraction:

Z =β − β2

β1 − β2(5.22)

that is equal to 0 in the oxidizer stream (denoted by the subscript 2) and equalto 1 in the fuel stream (denoted by subscript 1). After having dened themixture fraction we can now write its transport equation remembering that wehave considered all species diusivities equal to D.

ρ∂Z

∂t+∂ρuiZ

∂xi=

∂

∂xi

(ρD

∂Z

∂xi

)(5.23)

Normalization let state that every specie conservation can be achived with Eq.5.23 with the boundary conditions:

- Z=1 on fuel side

- Z=0 on oxidizer side

- other boundaries, such as walls and outlets, are all characterized by zeroux of each species, elements and mixture fraction. So the normal compo-nent of mixture fraction gradient is equal to zero on each of these bound-aries

85


This last statements on boundary conditions together with the nature of Zequation, that is the same for all species, implies that by solving only the Zequation we have the concentration of all species Yk ,and consequently (section5.7) the mixture mean molar weight, specic heat and formation enthalpy:

Yk = f(Z, t) for k = 1, N (5.24)

Mixture fraction equation is also representative of enthalpy equation if ther-

Figure 5.4: Example of ame structure in Z-space, methane-air equilibrium ame

mal diusivity is assumed to be equal to the species one and if boundaries areadiabatic ((Poinsot and Veynante, 2001)).

Moreover the diusive ame front is identied by the isosurface:

Z (~x, t) = Zst

where Zst is the stoichiometric mixture fraction value.

5.5.4 Diusive ameletsIn previous subsection has been shown how the ame structure can be rep-

resented by just one variable: the mixture fraction. In multi-dimensional ows,this representation requires that the ame is thin compared to other ows andwrinkling scales. Each element of the ame front can be viewed as a small lam-inar ame also called amelet : this amelet view is the basis of many modelsfor turbulent combustion. Keeping in mind this hypothesis, the function 5.24allows to rewrite the mass fraction balance equation (Eq. 5.15) as:

ρ∂Yk

∂t+ Yk

[∂ρ

∂t+∂ρui

∂xi

]

︸︷︷︸continuity

+∂Yk

∂Z

[ρ∂Z

∂t+ ρui

∂Z

∂xi− ∂

∂xi

(ρD

∂Z

∂xi

)]

︸︷︷︸Z−equation

(5.25)

−ρD(∂Z

∂xi

∂Z

∂xi

)∂2Yk

∂Z2= ρωk

86


The two terms within brackets in the LHS vanish because of continuity andmixture fraction equation (Eq. 5.23) so we obtain the amelet equation as:

ρ∂Yk

∂t= ρD

(∂Z

∂xi

∂Z

∂xi

)∂2Yk

∂Z2+ ρωk =

12ρχ∂2Yk

∂Z2+ ρωk (5.26)

Where has been introduced the scalar dissipation rate χ (Eq. 5.27) that hasthe dimension of an inverse of time (like strain) and can be interpreted as theinverse of a characteristic diusion time ((Peters, 2000)).

χ = 2D(∂Z

∂xi

∂Z

∂xi

)(5.27)

It measures the Z-gradient and the molecular uxes of species towards theame. Standing to (Poinsot and Veynante, 2001) it is directly inuenced bystrain, growing aside with this; moreover it gives an estimate of mixing layerthickness by

√D/χ.

In the assumption of steady amelet, always true in present work, the speciesreactions rates can be written as:

ωk = −12

χ︸︷︷︸mixing

∂2Yk

∂Z2︸︷︷︸reaction

(5.28)

that shows how diusive ame reaction rate only depends on Z. At this point,stated that the mixture fraction appears to be the basic variable for most the-ories of laminar diusive ames, the modelling of this kind of ames can bedecoupled into two problems.

5.5.5 Turbulent combustion diagram

Figure 5.5: Regimes in nonpremixed turbulent combustion

In diusion combustion, in contrast to premixed combustion, there is nocharacteristic velocity scale such as the laminar burning velocity, and therefore

87


there is no ame thickness dening a characteristic length scale. There existsthe strain rate a, locally imposed by the ow eld which is the inverse of acharacteristic time. This can be used together with the diusion coecient Dst

to dene a diusion thickness:

ld =(Dst

a

)1/2

There also exists the gradient of the mixture fraction eld |∇Z| which relatesthe thickness ld, dened in physical space, to a diusion thickness (∆Z)F inmixture fraction space:

(∆Z)F = |∇Z| ld

that, using the denition of the scalar dissipation rate (Eq. 5.27), becomes:

(∆Z)F =(χst

2a

)1/2

In addition to the (∆Z)F it is possible to dene a reaction zone thickness (Fig.5.3) as the sum of the thicknesses of fuel consumption layer and oxidation layerboth dened in Z-space, even if the last one is the much thicker than the rst.

Figure 5.5 (log-log diagram) reports the Peters' combustion regimes dia-gram for diusion ames where (∆Z)F and (∆Z)R represent the just denedthicknesses of diusion zone and reaction zones.

The mixture fraction uctuation Z ′ (Z ′st stand for its value at stoichiometricconditions) is dened as the r.m.s. of mixture fraction variance: Z ′ =

(Z ′′2

)1/2

.With χq and χst are indicated scalar dissipation rates (Eq. 5.27, (Peters, 2000))at quenching and stoichiometric conditions (Favre averaged).

For large mixture fraction uctuations, where Z ′ > (∆Z)F , uctuations inmixture fraction space extend to suciently lean and rich mixture such thatthe diusion layers surrounding the reaction zone are separated, this is theseparated amelets zone. For small mixture fraction uctuations, where Z ′ <(∆Z)F , which may be due to intense mixing or to partial premixing of the fuelstream, the reaction zones are connected, then we are in connected ameletszone. The line of equation Z ′st/ (∆Z)R = 1 represents those uctuations thatare equal to the thickness of the reaction zone. Moreover for χq < χst the ameis quenched.

Also in gure is reported the shape of a ame that shows how local condi-tions, along the contour of mean stoichiometric mixture in a jet diusion ame,would t into dierent regimes.

This diagram has to be used with great care ((Poinsot and Veynante, 2001))since the local ame thickness and speed depend on the local ow conditionssuch as local strain rates and may be aected by unsteady eects.

In a given combustor, the ame structure may strongly depend on the spatiallocation. Moreover some strong approximation have been introduced in itsderivation.

88


5.5.6 Turbulent diusion ames modellingWhile introducing the modelling of laminar diusion ames it has been re-

ported than this kind of ame does not propagate. This feature implies thatdiusion ames cannot impose their own dynamics but they are completelycontrolled by the ow eld ((Poinsot and Veynante, 2001)), then they are verysensitive to stretching and easily quenched by turbulent uctuations; moreoverthe amelets assumption is sometimes not justied.

The splitting of diusion ames modelling into mixing and ame structureissues, remains correct when analyzing turbulent ames, but it has to be revisedto take account of turbulence eects:

• The mixing problem has to provide the Favre's average of mixture fractionZ (xi, t) (Eq. 5.33)

• A ame structure problem where species mass fractions Yk or reactionrates ρωk are expressed as function of mixture fraction. These functionsare formulated as conditional expressions (Yk|z∗, ρωk|z∗), because of theinuence of multiple parameters in a turbulent ow.

To determine the average values of Yk (by which can be obtained all othervariables to solve Navier-Stokes Equations, subsec. 5.2) or of ρωk, the meanvalue of Z is no longer sucient: higher Z moments are need and, if possible,a probability density function (pdf ) of Z. When the pdf of Z, p(Z) is known,average species mass fraction (Yk) or average reactions rate (ρωk = ρ˜ωk) aregiven by:

ρYk =∫ 1

0

(ρYk|Z∗

)p (Z∗) dZ∗ (5.29)

ρ˜ωk =∫ 1

0

(ρωk|Z∗

)p (Z∗) dZ∗ (5.30)

where(Q|Z∗

)denotes the conditional average of quantity Q for a given value

of the mixture fraction Z = Z∗, depending on Z∗ and various quantities suchas the scalar dissipation rate. p (Z∗) is the Z-pdf that complies with:

∫ 1

0

p (Z∗) dZ∗ = 1

In RANS of diusion ames, the solution of species conservation can beobtained by two families of method:

Primitive variable methods based on Eq. 5.29. Assumptions are made onthe ame structure to provide conditional quantities

(ρYk|Z∗

), for ex-

ample from amelet libraries or through balance equations (CMC mod-elling). Species mass fraction conservations equations (Eq. 5.1) are nolonger required and mean reaction rates ρωk are not modelled: RANScodes solve only for ow variables (ρ, ui, h . . .) and mixture fraction vari-ables (Z, Z ′′2, . . .) to estimate, directly or indirectly, the probability den-sity function.

89


Reaction rate approach implies the solution of species mass fraction conser-vations equations (Eq. 5.1) that requires the reaction rates ρωk. Theseare provided by libraries built with Eq. 5.30 or by the Eddy DiusivityConcept model that assumes reaction rates to be function of species massfraction and of turbulent mixing ((Magnussen and Mjertag, 1976)).

5.5.7 Presumed pdf amelets modelsThis model is a primitive variables method where the pdf of Eq. 5.29 (p (Z∗))

is assumed a priori.The basic idea of amelet modelling is to assume that a small instantaneous

ame element embedded in a turbulent ow has got the structure of a laminarame (Peters (2000)). This assumption requires all reaction zones to be thincompared to turbulent length. Some parameters must be conserved in orderto represent correctly turbulent amelets using laminar diusion ames. Theidentication of these parameters is hard to be performed so simplications arerequired and only stretching will be considered.

In present section we will show how it is possible to describe turbulent amestructure by conditional averaging of those functions dening the structure ofthe laminar ame (Eq. 5.24), and considering the eect of strain. This approachallows to express the averaged mass fractions:

ρYk =∫ ∞

0

∫ 1

0

ρYk (Z,K) p (Z,K) dZdK (5.31)

where p (Z,K) is the pdf of mixture fraction and strain rate that are usuallyconsidered statistically independent:

p (Z,K) = p (Z) p (K)

The shapes of these pdf s are usually assumed a priori (from that comes thename of the model). For mixture fraction is used a β-function (Fig.5.6), basedon Favre averaged mixture fraction Z and its variance Z ′′2:

p (Z) =Zα−1 (1− Z)β−1

Γ (α) Γ (β)Γ (α+ β)

With:

Γ (x) =∫ ∞

0

e−ttx−1dt α = Zγ β =(1− Z

)γ

and

γ =Z

(1− Z

)

Z ′′2− 1 ≥ 0

while the mixture fraction variance is dened as:

Z ′′2 =ρ(Z − Z

)2

ρ(5.32)

90


Figure 5.6: Shapes of the β-function for dierent value of Z and γ

The pdf for the stretching is usually the Dirac-delta function, assuming thatthe strain rate does not change along the ame front:

p (K) = δ(K − K

)

Once that pdf s have been dened, solutions of Eq. 5.31 can be computed andby those hypothesis presented in section 5.2, we can build amelet librariescontaining the variables

(Q = Q

(Z, Z ′′2, K

))needed for the solution of the

RANS and consequently of the mixing problem.In order to dene the ame structure in space is required the evaluation

of mixture fraction Favre's average, mixture fraction variance and strain rateelds. Strain rate can be calculated as function of turbulence variables, whilefor mixture fraction Favre's average and variance are needed proper transportequations that are solved together with RANS equations:

Favre averaged mixture fraction equation is obtained by Favre averagingof Eq. 5.23 and can be expressed as:

∂

∂t

(ρZ

)+

∂

∂xi

(ρuiZ

)=

∂

∂xi

(ρD

∂Z

∂xi− ρu′′i Z ′′

)(5.33)

where the turbulent transport term −ρu′′i Z ′′ is modelled by the sameapproach used for RANS (section 5.1):

− ρu′′i Z ′′ =µT

ScZ

∂Z

∂xi(5.34)

ScZis the Schimdt turbulent number for Z and µT is the turbulent vis-

cosity. By this assumption the closed Favre averaged mixture fractionequation form is:

∂

∂t

(ρZ

)+

∂

∂xi

(ρuiZ

)=

∂

∂xi

[(ρD +

µT

ScZ

)∂Z

∂xi

](5.35)

91


Mixture fraction variance equation is obtained by its denition (Eq. 5.32)applied on instantaneous and Favre's average Z-equations (Eq. 5.23 andEq. 5.33) and after some rearrangements can be written as:

∂

∂t

(ρZ ′′2

)+

∂

∂xi

(ρuiZ ′′2

)= − ∂

∂xi

(ρu′′i Z ′′2

)

︸︷︷︸turbulent transport

(5.36)

+∂

∂xi

(ρD

∂Z ′′2

∂xi

)+ 2Z ′′

∂

∂xi

(ρD

∂Z

∂xi

)

︸︷︷︸molecular diffusion

− 2ρu′′i Z ′′∂Z

∂xi︸︷︷︸production

− 2ρD∂Z ′′

∂xi

∂Z ′′

∂xi︸︷︷︸dissipation

In this equation appear many unclosed term that can be modelled as follow((Poinsot and Veynante, 2001)):

• Turbulent transport term modelled in the same fashion of RANS :

− ρu′′i Z ′′2 =µT

ScZ′′2

∂Z ′′2

∂xi

where ScZ′′2 is the turbulent Schmidt's number for mixture fraction

variance.

• Molecular diusion terms can be usually neglected against turbulenttransport.

• The unknown contribution in the production term −ρu′′i Z ′′ is mod-elled using Eq. 5.34:

− ρu′′i Z ′′∂Z

∂xi=

µT

ScZ

∂Z

∂xi

∂Z

∂xi

• The dissipation term requires more assumptions: it can be meantto be the scalar dissipation rate of mixture fraction uctuations Z ′′,and it can be assumed to play the same role as the dissipation ofturbulent kinetic energy ε with the velocity eld. It can be modelledas:

2ρD∂Z ′′

∂xi

∂Z ′′

∂xi= cρ

ε

kZ ′′2

where c is a model constant.

Having modelled all these terms the mixture fraction variance equationcan be rewritten as:

∂

∂t

(ρZ ′′2

)+

∂

∂xi

(ρuiZ ′′2

)= (5.37)

+µT

ScZ′′2

∂Z ′′2

∂xi+

µT

ScZ

∂Z

∂xi

∂Z

∂xi− cρ

ε

kZ ′′2

92


5.6 Premixed Flames

Pratical applications of premixed combustion are spark-ignition engines andDry Low NOx gas turbines combustors based lean − burn technology in or-der to comply with the strict legal requirements on NOx emission achievedthanks to the low temperature at which such combustors work. Indeed tur-bines combustors design represents the main task of CFD analysis on premixedcombustion systems since the need to keep temperature lower and lower, tocontinuously reduce the nitrogen oxides emissions, makes combustion processesbecome strongly unstable leading to ame blow-o and local quenching whichmay induce the formation of carbon monoxide that also is highly polluting.

Premixed combustion requires that fuel and oxidizer be completely mixedbefore reaching the zone where combustion reactions takes place.

Fuel and oxidizer premixing is only possible at suciently low tempera-tures where the chain-branching mechanism, that drives the reaction chain inhydrocarbon oxidation, is unable to complete. Under such low temperatureconditions, combustion reactions are said to be frozen. The frozen state ismetastable, because a suciently strong heat source, a spark for example, canraise the temperature at values that allow the combustion process to start.

Once fuel and oxidizer have homogeneously been mixed and if their mixingratio lies between the ammable limits, i.e. between blow-o and ash-backow conditions, as an heat source is supplied, it becomes possible for a amefront to propagate through the mixture. Owing to the temperature sensitivity

Figure 5.7: Schematic representation of a premixed ame

of the reaction rate, the gas behind the ame front (Fig. 5.7) rapidly approachesthe burnt state close to chemical equilibrium, while the mixture in front of theame typically remains in the unburnt state. Therefore the combustion systemon the whole contains two stable states, the unburnt and the burnt gases state.In premixed combustion both states exist in the system at the same time; theyare spatially separated by the ame front where the transition from one to theother takes place.

In present chapter are presented the most common turbulent premixed com-bustion models, with particular attention to the used Bray Moss Libby one.Anyway each model is suitable to be used in particular conditions so a combus-tion regimes classication is needed.

93


5.6.1 Combustion regimes and types of ameTurbulence and combustion processes can interact in very dierent ways,

hence it is very dicult to build models (based on some, restrictive assumptionsby their nature) able to model premixed turbulent combustion at all regimes.It then becomes essential to perform a classication of turbulence combustionregimes. This classication is achieved by combustion regimes diagrams relatingturbulence and combustion length scales.

5.6.2 Turbulence and ame scalesAs already stated combustion regimes classication requires turbulence and

combustion scales, here are introduced those variables.If r is the dimension of a generic eddy belonging to the considered turbulent

eld (assuming values in the range between Kolmogorov length scale, Lk andthe integral length scale, LT , (Mangani, 2003)), the dissipation rate of turbulentkinetic energy, associated to those eddies of dimension r is expressed as:

ε =u′3(r)r

being u′ =√

2/3 k representative of eddies rotational speed. Similarly a turbu-lent mixing characteristic time of eddies of dimension r can be dened as:

τm =r

u′(r)=r2/3

ε1/3

This time scale can be compared to the one characteristic of a ame (of thicknessδL and with speed S0

L) propagating in the considered (τL = δL/S0L), by the

denition of the Damköhler nondimensional number:

Da(r) =τm(r)τ

L

The correct denition of the Damköhler's number, Da (Eq. 5.38), is basedr = LT , at which correspond τm(LT ) = τT = k/ε (being k = RMS(u′)).

Da =τT

τL

(5.38)

The Da, considering the denitions of τTand τm can be rewritten as:

Da =LT

δL

S0L

u′(5.39)

Damköhler's number allows to determine two limit conditions.If Da >> 1 chemical reactions have a much higher speed than turbulent

mixing, giving a just lightly distorted ame front but still laminar in its innerstructure. This is the so called amelet regime, that allows to treat describe thecombustion with single step innitely fast chemistry.

94


In the opposite case, with Da << 1 reactions are slower than turbulentmixing. In this situation the combustion system can be described with the Per-fectly Stirred Reactor hypothesis ((Andreini, 2004)), that requires the Arrheniuschemical model.

In order to describe the interaction between ame front and eddies havingdimension comparable to Kolmogorov length scale Lk (small eddies), is intro-duced the Karlovitz number:

Ka =1

Da(Lk)=τ

L

τk

(5.40)

that can be rewritten as:

Ka =(LT

δL

)−1/2 (S0

L

u′

)−3/2

(5.41)

that implies:

ReT = Ka2Da2

and

Ka =(δLLk

)2

(5.42)

The Karlovitz's number is the proportional to the ratio between ame front

Figure 5.8: Schematic representation of both amelet regimes. If Ka < 1 amelets arecorrugated (a) while for 1 < Ka < 100 amelets are thickened (b)

thickness and small eddies characteristic length. The condition Ka = 1 is theKlimov-Williams criterion delimiting the amelet combustion regime. ForKa <1 also smaller eddies are greater than ame front thickness that completely keepsits laminar structure. Finally, for Ka > 1 Lk is comparable with ame thicknessand then small eddies start to interact with laminar ame inner structure. Fora clearer description of this phenomena is convenient, as proposed by (Peters,2000), to introduce a Karlovitz's number in function of the ame inner layerthickness (δIN ):

KaIN =(δIN

Lk

)2

(5.43)

95


Figure 5.9: Peters' combustion regimes diagram

It usually assumed that δIN ≈ 0.1δL ((Andreini, 2004)) so:

KaIN ≈ Ka

100(5.44)

Until Ka < 100 (or KaIN < 1), the inner layer is not aected by eddiesaction, keeping its laminar structure. Then the introduction of the inner layerKarlovitz's number and the consequent more detailed analysis, allow to extendthe amelet regimes to be within 1 < Ka < 100 where small eddies havea smaller thickness than ame diusion thickness δL, having then the abilityto penetrate in the preheat zone (Fig. 5.8) but having a smaller dimensionthan reactions zone (inner layer and oxidation layer) that practically keeps astructure similar to a laminar ame one. The combustion regime in the range1 < Ka < 100 is called Thin Reaction Zone and we are dealing with Thickened-Wrinkled Flamelets ((Peters, 2000)). These two regimes are reported in Fig.5.8.

5.6.3 Spectral DiagramThe introduced combustion regimes can be represented with suitable spec-

tral diagrams. These diagrams were rstly introduced by Borghi, and in morerecent times experiments and Direct Numerical Simulations have lead to a newerdiagram proposed by (Peters, 2000), that is here described (Fig. 5.9). This di-agram is built with the LT /δL as abscissa and the ratio u′/S0

L as ordinate, andusing a log-log representation.

In Peters' diagram can be identied these regimes:

Ka<1: Flamelets regime. Usually split in:

• u′/S0L < 1: wrinkled amelets. Bigger turbulence structure associ-

ated to a rotational speed u′, are not able to corrugate the ame

96


front that would implies an interaction (between ame and turbu-lence). Flame front practically propagates in laminar way.

• u′/S0L > 1: corrugated amelets. Aside the rising of turbulent uctu-

ation u′, turbulent motion structures are able to corrugate the amefront. Interactions between eddies and ame front are now present:can be spotted pocket of burnt and fresh gases

1 < Ka < 100: thin reaction zones (thickened wrinkled amelets) as al-ready hinted, smaller eddies dimensions (Lk) allow uctuations to pene-trate the laminar ame preheat zone, anyway without modifying the reac-tion zone that keeps its well dened shape and small thickness, as well asin laminar regime. The overall eect is that the ame front is thickenedand still subject to the wrinkling action of large eddies (of dimension LT )

Ka>100: Distributed reactions (perfectly stirred reactor). Kolmogorovlength scale is small enough that the whole ame laminar structure is sub-ject to turbulent uctuations actions: eddies are even smaller than reac-tion zone and then no longer exists a laminar structure. Flame is near thePerfectly Stirred Reactor state where the mixing process can be consid-ered innitely fast, then the process is completely controlled by chemicalreactions. Practical realizations of combustion systems operating in theDistributed Reaction Zone are prevented by quenching phenomenon, dis-tinguishing this regime. There exist many systems working near the bor-der between Thin Reaction Zone and Distributed Reaction Zone such asgas turbine combustors, indeed getting nearer to the Distributed ReactionZone allows to limit the reaction zone and the of the whole apparatus.

5.6.4 Premixed turbulent combustion modelsHere are described the premixed ames modelling by RANS approach, with

particular care to the Bray-Moss-Libby model.Models for premixed combustion can be grouped in three families:

Local source models corresponding to reaction rate approach for diusionames (5.5.7). In this models a direct modelling of the reaction rates toallows the solution of species mass conservation equations (Eq. 5.1). Theclassical example is the EBU described in (Andreini, 2004). This modelscan be used also to provide the source term for the progress variableequation, described below.

Flamelet models corresponding to primitive variable model for diusion ames(5.5.7). In this models the combustion analysis is split in a mixing prob-lem and in a ame structure problem. Among these models can be foundthe G-equation model ((Mangani, 2003)), and the Bray Moss Libby modelthat is based on the progress variable.

97


5.6.5 Bray-Moss-Libby modelThe Bray-Moss-Libby's (since now on referred to as BML), proposed in 1977,

is the classical premixed turbulent combustion amelet model. It is based ona presumed pdf for a progress variable ((D.Veynante and L.Vervisch, 2002))dened as:

c =T − Tu

Tb − Tuc =

YF − YFu

YFb− YFu

(5.45)

where the subscript u indicates the unburnt gas state, and b the burnt one,clearly the progress variable c is equal to 1 in the burnt gases ow, and to 0 inthe fresh gases ow. If species diusivities are assumed equal to correspondingthermal diusivities, and the specic heat (Cp) is constant, these two denitionscan be demonstrated to be analogous.

In BML model is assumed an innitely fast chemistry with a single step(F +O→P ) mechanism, implicitly assuming that DaÀ 1. Innitely fast chem-istry assumption implies that the amelet has got an innitesimal thickness andthat its internal structure induces the instantaneous crossing from unburnt toburnt gas states. With these hypothesis the probability of having intermediatestates is virtually null, so the pdf is made of two Dirac-delta (δ), one in c = 0and the other in c = 1:

p (c, ~x, t) = α (~x, t) δ (c) + β (~x, t) δ (1− c) (5.46)

with

α (~x, t) + β (~x, t) = 1

where α (~x, t) and β (~x, t) are respectively the probability of having c = 1 andc = 0, in point ~x at time instant t. The BML model also assumes a constantpressure for the system (nearly true in gas turbine combustors) so the perfectgases equation of state (Eq. 5.7) can be expressed as follow:

ρuTu = ρbTb

Then is useful the denition of the heat release factor or expansion factor, τ as:

τ =ρu

ρb− 1 =

Tb

Tu− 1 (5.47)

Therefore Reynolds average density is expressed as:

ρ = αρu + βρb = (1− β) ρu + βρb

Stated that

ρc = ρc =∫ 1

0

ρ c p(c) dc = ρbβ = ρbc (5.48)

average density may be rewritten, using the denition of τ , as follow:

ρ =(

(1− c)ρu

+c

ρb

)−1

= ρb1 + τ

1 + τ c=

ρu

1 + τ c

98


By this last expression follows that:

α =1− c

1 + τ cβ =

(1 + τ) c1 + τ c

Using the found relation between Favre's and Reynolds average of c (Eq. 5.48),the Favre's average of generic variable Q is obtained by:

Q = αρu

ρQu + β

ρb

ρQb = (1− c)Qu + c Qb (5.49)

where the unburnt and burnt values are calculated by conditional averagingrespect to corresponding progress variable values, being p (Q|c = c∗) the pdf ofQ if c = c∗:

Qu =∫ +∞

−∞Qp (Q|c = 0) dQ Qb =

∫ +∞

−∞Qp (Q|c = 1) dQ

Since scalar variables such as k− th mass fraction are associated to correspond-ing value of laminar amelet that, for considered hypothesis only allow burntand unburnt state, conditional averages of mass fraction is coincident with cor-responding value in fresh gases and burnt gases stream so we can write:

Yk = (1− c)Yku + cYkb (5.50)

that allows, by those hypothesis presented in section 5.2, to calculate the vari-ables needed for the solution of the RANS. All these denition allows to statethat BML model completely describes the turbulent ame by the solution of theFavre's averaged transport equation for c:

∂

∂t(ρc) +

∂

∂xi(ρcui) = − ∂

∂xi

(ρc′′u′′i

)+ Ωc (5.51)

that is performed aside RANS equations, once that models are provided for itstwo unclosed terms, turbulent transport −ρc′′u′′i and source Ωc.

• Usually the turbulent transport is modelled by the common Boussinesq'shypothesis:

− ρc′′u′′i =µT

ScT

∂c

∂xi(5.52)

This formulation can be a source of errors, indeed may occur that tur-bulent uctuation leads to diusion of c in opposite sense respect to thesign of progress variable gradient (Gradient Diusion, GD): this situationis called Counter Gradient Diusion (CGD) (a detailed explanation ofthis phenomenon can be found in (D.Veynante and L.Vervisch, 2002) and(Peters, 2000)).

• For the source terms exist a number of models:

Flame Surface Density is probably the most used model where Ωc isexpressed as the product between the Flame Surface Density Σ (i.e.

99


ame surface area for each volume unit) and the ame front reactionspeed ωL:

Ωc = ρuωLΣ (5.53)

An evaluation of ωL is obtained by the laminar ame speed of thecorresponding amelet (S0

L), keeping in consideration the stretching.The simplest expression was proposed by Bray:

ωL = I0S0L

Where I0 is the laminar ame stretch factor needing an adjunctmodel. An expression for Flame Surface Density is the Bray one:

Σ =g

σy

c (1− c)Ly

where g and σy are model constant and Ly is the wrinkling lengthand is usually related to the turbulence integral scale LT . A moreaccurate model is the Coherent Flame Model that implies the solutionof a transport equation for Σ ((Andreini, 2004) and (Poinsot andVeynante, 2001))

Turbulent Flame Closure is based on the closure of Ωc by the use ofan algebraic expression, empirically dened, for the turbulent amefront propagation speed ST ((Zimont et al., 1998)):

Ωc = ρuSTFk|~∇c| (5.54)

where FK is a coecient that takes on account laminar ame stretch-ing (0 < Fk < 1). The proposed ST expression is:

ST = Au′3/4S01/2

L D−1/4T L

1/4T (5.55)

where A is a constant (usually assumed equal to 0.5), u′ is the velocityturbulent uctuation, S0

L is the unstretched laminar ame speed, DT

is the turbulent diusivity and LT is the integral turbulence scale.Kolmogorov Petroski Piskunov closure considers the progress vari-

able source term as a function of c itself. It derivation is based onthe asymptotic analysis of the progress variable equation (Eq. 5.51,for details on derivation procedure (Andreini, 2004)). The nal formis:

Ωc = ρuCKPPS2

T

DTc (1− c) (5.56)

where the turbulent ame speed is dened as function of laminarspeed and Damköhler number:

ST = S0L + u′ · (1 +Da−2

)1/4 (5.57)

where u′ =√

2/3 k.

100


5.6.6 Level-Set Model: G-Equation

To overcome the restrictions of the progress variable formulations, a modelbased on a pure kinematic equilibrium was considered. The model is centeredon the so-called G-Equation which is solved in this work with an innovativerst order ame front tracking technique. The Level-Set model for premixedames was originally developed by Peters Peters (2000): it is based on thehypothesis that instantaneous turbulent premixed ames may be described byan ensemble of zones where species molecular diusion and chemical reactionscoexist in thin reactive-diusive layers (amelets). The Level-Set approach isvalid in the entire amelet combustion regime and, as demonstrated by Peters,also in the thin reaction regime Peters (2000). To track the position of the meanturbulent ame front, Eqn. (5.58) is used to describe, from a kinematic point ofview, the balance between the ame front propagation (with burning velocityST ), and the chemically inert turbulent ow eld.

∂ρG

∂t+∇ ·

(ρUG

)=

(ρS0

T

) |∇G| − ρDT κ|∇G| (5.58)

The transport equation for passive scalar G is a conditional equation, as it isdened only on the ame front (G = G0): the additional condition of unicity ofthe gradient normally to the ame front has to be assured, so that G providesthe Cartesian distance from the mean ame front at any point of the domain.

|∇G| = 1, G 6= G0 (5.59)

According to Mangani (2003), an experimentally veried Gaussian distributionhas been adopted to model the Probability Density Function ofG. An additionaltransport equation for the mean variance G′′2 it's therefore required to closePDF function (Eqn. (5.60)).

∂ρG′′2

∂t+∇ ·

(ρUG′′2

)= ∇‖ ·

(ρDT∇‖G′′2

)+

2ρDT

(∇G

)2

− csρε

kG′′2

(5.60)

To ensure the condition of unicity for the PDF, G′′2 has to maintain constanttowards the ame front normal direction:

∇G′′2 · ∇G = 0 (5.61)

Turbulent ame speed ST is evaluated from the solution of an additional trans-port equation for the turbulent term σT of the ame surface area ratio σ:

ST

SL=AT

A= σ → σ ∼= |∇G|+ σT = 1 + σT

→ ST = SL(1 + σT )(5.62)

101


Transport equation for σT was adapted from Peters Peters (2000) and requiresa re-distribution step identical to that described for G′′2.

∂ρσT

∂t+∇ · (ρUσT ) = ∇‖ ·

(ρDT∇‖σT

)+

c0ρ(−u′′u′′) : ∇U

k· σT + c1ρ

DT (∇G)2

G′′2· σT

2+

−c2ρ S0LσT

2

(G′′2)1/2− c3ρ

DσT3

G′′2

(5.63)

5.7 Partially Premixed FlamesIn previous sections have been treated the two mixing extremes involving

combustion. In many technical applications the mixing lies somewhere betweenthese two extremes being sometimes a desidered or otherwise an unavoidablesituation. Indeed a perfect premixing is usually just an ideal situation especiallywhen the fuel is a liquid, as in jet engines that work with prevaporized kerosene.In other situations, as in the case of some power turbines combustors, that areusually fed by a premixed methan-air mixture, the ame holding is provided bya diusion jet ame pilot, leading to local temperature rising and consequentlyto NOx formation making this holding method less and less used. In this casethe pilot ame is a lift-ame ((Peters, 2000)) and has a structure similar to theone reported in g. 5.10.

A key element of partially premixed ames is the triple ame: while plot-ting the laminar burning velocity of a partially premixed ame, as function ofmixture fraction, can be spotted the presence of a maximum burning velocitythat lies close to the stoichiometric mixture. If a range of the mixture fractionsZmin < Z < Zmax exists in a partially premixed eld, and the stoichiometricmixture fraction lies between these limits, ame propagation generates a amestructure called a triple ame. The leading edge of the ame, called the triplepoint, propagates along a surface that is in the vicinity of stoichiometric mix-ture. On the lean side of that surface there is a lean premixed ame branch andon the rich side there is a rich premixed ame branch, both propagating witha lower burning velocity. Behind the triple ame point, a diusion ame devel-ops into which unburnt intermediates such as H2 and CO diuse from the richpremixed ame branch and the left over oxygen diuses from the lean premixedame branch (for further details see (Echekki and J. H. Chen, 1998)).

5.7.1 Partially Premixed Flames ModellingThe analysis of partially premixed ames can be performed by an EDC

model or by coupling diusion and premixed ames models. (Peters, 2000)shows how it can be performed considering the mixture fraction and the G (xi, t)scalar elds (G-equation model (Mangani, 2003)): the mixture fraction deter-mines the local equivalence ratio and thereby the ame speed used in the equa-tion of scalar G that determines the location of the premixed ame front.

102


Figure 5.10: Schematic representation of a lifted jet diusion ame

In present work has been chosen to treat the partial premixed ame by cou-pling the presumed shape pdf amelet model, described in subsubsection 5.5.7,with the Bray-Moss-Libby model and the G-Equation models.

The followed approach is provided by (D.Veynante and L.Vervisch, 2002):the idea is that burnt (indicated by the subscript b) and unburnt (indicatedby the subscript u) states variables are no longer constant but are aected bymixing process.

The pdf for BML model (Eq. 5.46) has to be rewritten as, also, functionof mixture fraction Z. In the hypothesis of validity of amelet approach itbecomes:

p (c, Z, ~x, t) = α (~x, t) δ (c) pu (Z, ~x, t) + β (~x, t) δ (1− c) pb (Z, ~x, t) (5.64)

where pu (Z, ~x, t), pb (Z, ~x, t) are respectively the unburnt and burnt pdf fordiusion ames (Eq. 5.31).

The Favre's average for the generic variable Q (Eq. 5.49) has to be rewrittenin terms of Qu(Z,K) and Qb(Z,K), stated that now:

Qu =∫ 1

0

Qp (Z|c = 0) dZ Qb =∫ 1

0

Qp (Z|c = 1) dZ (5.65)

Under these hypothesis Favre's average of Q is calculated with:

Q = (1− c)Qu (Z) + c Qb (Z) (5.66)

for the G-Equation model the probability c is readily determined by integratingthe normal distribution of scalar G, P (G;x), across ame front:

cu =∫ G0

−∞P (G, x)dG (5.67)

In this way c has the same meaning of a progress variable and it assumes thevalue 0 in burned gas and 1 in unburned gas.

103


Only these modications are carried out, and all other denition and as-sumption of BML G-Equation model are still valid. Finally for both models,the dependence on mixture fraction has to be considered also in source termscalculations that requires the unburnt state density and the laminar ame speedthat now are both functions of the mixture fraction.

104

Chapter 6

Radiative Heat Transfer

Present chapter is contains a description of common radiation modellingprocedures with a more detailed description of the used method to modelparticipating media properties and to solve the Radiative Transfer Equation.

Contents6.1 Radiation physic . . . . . . . . . . . . . . . . . . . . 1076.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 Radiative Intensity . . . . . . . . . . . . . . . . . . . 1076.2.2 Irradiance . . . . . . . . . . . . . . . . . . . . . . . . 1086.2.3 Radiant energy ux vector . . . . . . . . . . . . . . . 1086.2.4 Black Body Emissive Power . . . . . . . . . . . . . . 108

6.3 Radiative Properties of Surfaces . . . . . . . . . . 1096.3.1 Emittance . . . . . . . . . . . . . . . . . . . . . . . . 1096.3.2 Absorbance . . . . . . . . . . . . . . . . . . . . . . . 1106.3.3 Reectance . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Radiative Transfer Equation . . . . . . . . . . . . . 1106.4.1 Interaction with Participating Media . . . . . . . . . 110

6.4.1.1 Absorption . . . . . . . . . . . . . . . . . . 1116.4.1.2 Emission . . . . . . . . . . . . . . . . . . . 1116.4.1.3 Scattering . . . . . . . . . . . . . . . . . . . 111

6.4.2 Derivation of RTE . . . . . . . . . . . . . . . . . . . 1126.4.3 Boundary conditions for RTE . . . . . . . . . . . . . 112

6.5 Solution Methods For RTE . . . . . . . . . . . . . 1136.5.1 Optical Thickness . . . . . . . . . . . . . . . . . . . 1136.5.2 Spectral Nature Modelling . . . . . . . . . . . . . . . 113

6.5.2.1 Modelling of Radiative Properties GaseousCombustion Product . . . . . . . . . . . . 113

6.5.2.2 Radiative properties of soot . . . . . . . . . 1156.5.3 Directional Nature Modelling . . . . . . . . . . . . . 1166.5.4 Spherical Armonics Method . . . . . . . . . . . . . . 1176.5.5 Finite Volume Method . . . . . . . . . . . . . . . . . 119

6.6 Radiative Heat Transfer Eect On Overall SystemEnergy Balance . . . . . . . . . . . . . . . . . . . . . 121

105

Chapter 6. Radiative Heat Transfer

106

Radiative Heat Transfer Chapter 6.

6.1 Radiation physicAll materials continuously emit and absorb electromagnetic waves, or pho-

ton, by lowering or raising their atomic energy levels. The strength and wave-length of emission depend on the temperature of the emitting material and onmaterial itself. For heat transfer application the most important wavelengthsare in the range 10−7 ÷ 10−3m that embraces ultraviolet, visible and infraredregions. As will be shown in present chapter, the strength of radiation dependson the fourth (at least) power of temperature, while conduction and convec-tion depend linearly on temperature, hence it is clear that in high temperaturesystems radiation is the most important way of energy transfer.

Among these high temperature systems, combustion ones are of the mostinuenced by radiative exchange, being also interested by the presence of highlyactive species such as soot, CO2, H2O, NOx and SOx.

The importance of radiative exchange implies its modelling, that, anyway,is computationally expensive.

In order to keep a link with combustion modelling, it is important to statethat those quantities related to combustion and uid ows, such as temperature,pressure, species concentration are Favre averaged if turbulence is involved.

6.2 DenitionsIn this section will be dened those quantities that describe radiative heat

transfer, further details can be found in (Viskanta, 2005) and (Modest, 2003).

6.2.1 Radiative Intensity

Figure 6.1: Symbols for the denition of radiative intensity

The fundamental quantity that governs the radiation eld is the spectralintensity of radiation or spectral radiance.

Considering eλ+∆λ(~r, t+ ∆t)− eλ(~r, t) as the amount of energy in the wavelength interval λ and λ + ∆λ transported across an arbitrary oriented imagi-

107


nary element of area A at a point P (of coordinates (~r, Fig. 6.1) during thetime interval (t + ∆t) − t and conned to an element of solid angle Ω aboutthe direction ~s. The vertex of the elementary cone is on the surface element Aand the outward normal ~n the surface makes an angle θ with the direction ~s.The limit in Eq. 6.1 tend to a nite value, as it has been demonstrated experi-mentally, provided that the point P and the direction ~s are kept constant. Thevalue obtained by this limit is the spectral radiative intensity. The integrationof spectral radiance on the whole wave length spectrum (Eq. 6.2) leads to thedenition of total radiative intensity I(~r,~s, t).

Iλ(~r,~s, t) = limA,Ω,∆λ,∆t→0

∣∣∣eλ+∆λ(~r, t+ ∆t)− eλ(~r, t)cosθ A Ω ∆λ ∆t

∣∣∣ (6.1)

I(~r,~s, t) =∫ ∞

0

Iλ(~r,~s, t) dλ (6.2)

6.2.2 IrradianceIrradiance is dened as the rate of radiant energy incident on a volume

element per unit area, so it is obtained by integrating the spectral intensity onthe whole solid angle as in the equation:

Gλ(~r, t) =∫

4π

Iλ(~r,~s, t) dΩ (6.3)

To calculate the total irradiance an integration of the irradiance on the wholewave length spectrum has to be performed as:

G(~r, t) =∫ ∞

0

Gλ(~r, t) dλ =∫

4π

I (~r,~s, t) dΩ (6.4)

where has been used the denition of total radiative intensity (Eq. 6.2).

6.2.3 Radiant energy ux vectorRadiant energy heat ux vector (in direction ~s) can be dened as shown in

Eq. 6.5 in its spectral form and in its total form reported in Eq. 6.6 using Eq.6.2.

~qR,λ

=∫

4π

Iλ(~r,~s, t)~s dΩ (6.5)

~qR =∫ ∞

0

~qR,λ

dλ =∫

4π

I(~r,~s, t)~s dΩ (6.6)

6.2.4 Black Body Emissive PowerIn 1901 Max Planck published his work on quantum statistics assuming that

a molecule can emit photons only at distinct energy levels, being photon energyrelated to wavelength by the Planck's law:

E = ~c

λ

108


where ~ is the Planck's constant. In his work Planck found that the spectralemissivity power for a black surface (of normal ~n) bounded by a transparentmedia with refractive index m is:

ebλ(T, t) =∫

2π

Ibλ(T, t)~s · ~n dΩ =2πhc20

m2λ5[ehc0kT/mλ − 1](6.7)

where k = 1.3807 · 10−23 J/K is the Boltzmann's constant and c0 is light speedin vacuum. The total black body emissivity power is obtained by the integrationon the whole spectrum of the spectral emissivity power as follow:

eb(T, t) =∫ ∞

0

ebλ(T, t) dλ = n2σT 4 (6.8)

where σ = 2π5k4

15h3c20

= 5.670 · 10−8 W/m2K4 is the Stefan-Boltzmann constant.Dened the spectral black body emissivity power, the denition of the blackbody spectral irradiance is easily obtained as (by the rst two terms of Eq.6.7): Ibλ(T, t) = ebλ(T, t)/π, in which has to be pointed out that black bodyirradiance is isotropic so the dependence on direction is neglected. Calculationof total black body radiative intensity can also be performed with Eq. 6.2 thatbecomes:

Ib(T, t) =∫ ∞

0

Ibλ(T, t) dλ =∫ ∞

0

ebλ(T, t)π

dλ =n2σT 4

π(6.9)

6.3 Radiative Properties of SurfacesIn this section are reported the radiative properties of surface, being these

representative of the solid boundaries of combustion systems analyzed in thepresent work.

6.3.1 EmittanceThe rate of radiant energy emission per unit area from an isothermal body at

temperature T into the solid angle ∆Ω about the direction ~s in the wavelengthinterval λ and λ+ dλ is given by

deλ(~s, T ) = Ieλ(~s, T )cosθ dΩ (6.10)

where Ieλ is the spectral intensity of emitted radiation. It is possible (and usu-ally done) to express the energy truly emitted by a surface as proportional tothe maximum possible value that any surface would emit, if it were in thermo-dynamic equilibrium at temperature T, that is the black body emission (Eq.6.7):

deλ(~s, T ) = ελ(~s, T )Ibλ(~s, T )cosθ dΩ (6.11)

where the proportionality coecient

ελ(~s, T ) =deλ(~s, T )debλ(~s, T )

=Ieλ(~s, T )Ibλ(~s, T )

(6.12)

109


is the spectral directional emittance.Similarly can be dened the spectral hemispherical emittance as ελ(T ) =

Iλ(T )Ibλ(T ) and the total hemispherical emittance as ε(T ) = I(T )

Ib(T ) . If the spectralemittance is the same for all wavelengths the surface is dened as a grey bodyand if the emittance is the same for all directions the surface is a grey diusiveone that is the kind of surface used in this work.

6.3.2 AbsorbanceSimilar considerations to those used in previous section can be used for

the denition of absorbance that represents the proportionality factor betweenthe energy amount (coming from direction ~s) absorbed from a surface and theamount that would be absorbed by a black surface, that absorbs the radiationcompletely. Thermodynamical arguments has lead to the knowledge that ab-sorptivity is equal, in value, to the emittance, so in most general case can bewritten that

αλ(~s, T ) = ελ(~s, T ) (6.13)

If a surface is grey diusely emitting one it is also grey diusely absorbingand then:

α(T ) = ε(T ) (6.14)

6.3.3 ReectanceThe reectance of a surface is the rate of incoming radiation that leave the

surface without being emitted or transmitted. By its denition it depends ontwo direction: the incoming radiation one and the outgoing one, and the lawof reciprocity says that the directions can be exchanged without varying thevalue of reectance ρ. Energy balance for an opaque (only emitting, absorbingand reecting) diusely grey absorbing-emitting surface implies that: ρ(T ) =1− α(T ) = 1− ε(T ).

6.4 Radiative Transfer EquationWhile section 6.3 has presented the behavior of radiative transfer phenomenon

between surfaces in absence of participating media, now we are going to developthe mathematical model for the radiative exchange through a participating me-dia. Firstly are presented the radiative properties of participating media thatwill be used for the following development of the Radiative Transfer Equation,since this point on simply RTE.

6.4.1 Interaction with Participating MediaSimilarly to a surface, a participating media can emit and absorb the radi-

ation, while the transmitted part of radiation is represented by that share of

110


radiation crossing a unit of volume without being absorbed, emitted or reected(scattered). The phenomenon described below can be justied by the solutionof the Maxwell's equations set.

6.4.1.1 Absorption

Experimental experiences evidence that the amount of radiation, expressedas radiative intensity, absorbed while crossing a participating media is propor-tional to radiative intensity itself and to the distance the radiation beam travelsthrough the media. This statement allows to write that

(dIλ)abs = −kλIλds (6.15)

where the proportionality constant kλ is the absorption coecient and negativesign is used because absorption induces a radiative intensity decrease.

6.4.1.2 Emission

As already reported in introduction a participating media can also emitradiation. Considerations similar to those used for surfaces, dealing with blackbody emission and simply emission, has lead to dene the rate of emission asproportional to the emission of a black body, at the same temperature of themedia and, as for absorption, proportional to the length of beam path.

(dIλ)em = kλIbλds (6.16)

where the proportionality constant is again equal to the absorption coecientdue to, as for surfaces, thermodynamical guesses.

6.4.1.3 Scattering

Figure 6.2: Scattering due to a small particles

Scattering is a phenomenon that involves direction changes (see gure 6.2)of radiation beams then it represents an attenuation (similar to absorption)for a direction and at the same time it represents an augmentation for someother directions. Since its modelling is complex and in present work can beneglected because either gases molecules or soot particles are much smaller thanwavelengths of interest (Rayleigh's theory) here are not reported further detailthat can be found in (Modest, 2003) and (Viskanta, 2005).

111


6.4.2 Derivation of RTEThe just introduced relation for Absorbtion and Emission of the participating

media can be used to built a radiative energy balance for a radiation beamtravelling in the direction ~s as:

dIλds

= kλIbλ − kλIλ (6.17)

This equation is Lagrangian in nature since it is written as we were followinga ray travelling at speed of light c, hence the length of the path can be expressedas ds = c dt, and transformed in an Eulerian coordinates the RTE is:

1c

dIλdt

=1c

∂Iλ∂t

+ ~s · ~∇Iλ = −kλIλ + kλIbλ (6.18)

The unsteady term ∂Iλ

∂t can be usually dropped stated that the speed oflight is much greater than those typical of uid ows so that the radiativephenomenon can be considered instantaneous.

Even if an explicit relationship is not reported, each quantity may be depend-ing on location in space, time and wave length while the intensity of radiationis also function of direction.

Notwithstanding its complexity this relation is built on strong simplications((Modest, 2003)) since the media is considered:

• at rest (comparing its speed the light one)

• non polarizing

• in local thermodynamical equilibrium

• with constant index of reection

• non-scattering

6.4.3 Boundary conditions for RTEIf the participating media is surrounded by diusive reective and emitting

opaque surfaces, RTE is subject to the boundary condition reported in Eq. 6.19,where ~rw is the coordinates vector of an emitting point on the surface normal to~n and ~s is the outgoing direction. The second term of LHS represent the reectedshare of total incoming radiation, obtained by the integration of incident rayradiation intensities on the solid angle including all possible incoming directions(~s′ with ~s′ ·~n < 0). This kind of surfaces, as already stated in section 6.3, emitsand reects radiation with the same intensity in all directions.

Iλ(~rw, ~s) = ε(~rw)Ibλ(~rw) +ρ(~rw)π

∫

~n·~s′<0

Iλ(~rw, ~s′)∣∣~n · ~s′

∣∣dΩ′ (6.19)

This kind of boundary condition is, unfortunately, dependant on domainvalue of radiative intensities, except for black surfaces that emit independentlyfrom direction and incoming radiation while it is dependant only on tempera-ture.

112


6.5 Solution Methods For RTEAs for many other of problems of engineering interest, also RTE has not

got an easy analytical solution for geometries and participating media involvedin combustion systems, thus numerical solutions have to be found ((Viskanta,2005)). In order to study radiative transfer in engineering problems two kindsof model have to be used, one to describe the directional nature of radiationand another to describe its spectral nature.

6.5.1 Optical ThicknessBefore getting involved in the description of RTE solution methods it is

useful to introduce a variable that allows to describe the behavior of radiativesystem: the optical thickness. It takes in account the attenuation of beam dueto absorption and is dened as:

τλ =∫ s

0

kλds

where s is the beam path length, normally it is more useful to write it in termof a system reference length (L):

τλ =∫ L

0

kλds (6.20)

6.5.2 Spectral Nature ModellingModelling of spectral nature mainly involves the calculation of participating

media absorption coecient, an accurate prediction is necessary in order toobtain a correct estimate of radiative exchange. Anyway a strong dependenceon wavelength is present but at the same time, modelling this phenomenon ishighly computationally expensive. In combustion systems are present manyradiative active species gaseous, solid and liquid. The most active species areCO2, H2O, CO, NO2 and CH4, while solids and liquids are soot, liquid droplets,pulverized coal, char, ash dust etc.

In this section will be described the properties of gases and of soot, beingthe species present in systems analyzed in this work.

6.5.2.1 Modelling of Radiative Properties Gaseous Combustion Prod-uct

Among gaseous combustion products CO2 and H2O represent the most im-portant for radiative exchange, also NOx and SO2 have high absorption co-ecient, but due to their little concentrations, their contribution is negligible.Scattering from gaseous molecules, as already stated, is unimportant due totheir dimensions that are much smaller that wavelengths of interest.

Quantum physics, governing radiation phenomenon, allows to state thatspecies emit and absorb radiation of proper wave lengths, that are those inducing

113


excitement of electrons. So a proper estimation of absorption coecient is hardto achieve since it will be highly uneven in the spectrum, and interaction withsurrounding molecules can induce modications on energy levels, introducinganother source of uncertainty. Then if a surface can be easily and correctlyrepresented by a grey body, a gas cannot be.

Even with great diculties and approximation many models have been re-alized for spectral treatment of gases, and can be divided in four groups:

Line-by-line calculations rely on the knowledge (today and in near futurenot completely availed) of absorption coecient at every spectral line,then the RTE has to be solved for every wavelength (in order of millions).Then even if the directional treatment were omitted, this approach is infact computationally too expensive for combustion system analysis.

Narrow band calculations based on the observation that absorption coe-cient has rapid change across the spectrum, especially when compared tothose of black body emission. Then can be used a proper average valueof absorption coecient on small wave length intervals where black bodyemission is constant. Theoretically an average that exactly represents theline-by-line absorption coecient can be realized but practically, especiallywhen gases mixture are involved, it cannot be done. Anyway modern im-provements, and simplications, of this model are obtained by noting thatsame values of absorption coecient are repeated in the spectrum, tak-ing in account this repetitive behaviour has been realized the correlatedk−distribution model ((Zhanga and Shib, 2005)).

Wide band calculations represent an extension of the hypothesis used fornarrow band models, indeed looking at the infrared spectral region (overallthe most important) the absorption coecient is practically constant forevery rotational-vibrational bands, then can be used and averaged valuefor each level.

Global calculations represent a wide group of method since it has had manydevelopment being the most suitable family for analysis of combustionsystems. In these systems the radiative heat ux or its divergence are themost interesting variable, and indeed global calculations are aimed to givea good prediction of them. Most important models of this category are:

- Planck mean absorption (further details are reported below becausethis is the model used in present work)

- Weighted Sum of Grey Gases ((Trivic, 2004)) that replace the ab-sorption coecient of the non grey gas with the weighted sum ofgrey gases absorption coecients

- Spectral line based weighted sum of grey and Absorption DistributionFunction both based on previous model and on the use of spectralline databases to calculate grey gases absorption coecients.

114


- Full spectrum k−distribution based uses the narrow band k− distri-bution on the whole spectrum by the use of fractional Planck function

As already stated, in present work has been used the Planck mean absorptionmethod that denes the absorption coecient as:

k =

∫∞ kλIbλ dλ∫∞ Ibλ dλ

=π

σT 4

∫

∞kλIbλ dλ (6.21)

To calculate the global absorption coecient has been used a correlationby International Workshop on Measurement and computation of Nonpremixedames ((Da Soghe, 2006, pages 101-102)). In this correlation the absorptioncoecient is dependant on CO2 and H2O partial pressures, and is dependanton the reciprocal of temperature by a polynomial series. Further details onspectral treatment of gases can be found in (Viskanta, 2005) and in (Modest,2003).

6.5.2.2 Radiative properties of soot

While describing Diusion ames (subsection 5.5.1) has been pointed outthat may occur the formation of soot particles that experiments have evidencedto have, generally, a bigger emissivity power than gases (a detailed descrip-tion of soot radiative properties is availed in (Viskanta, 2005) and in (Modest,2003)) then its eect has to be considered and here are reported the modellingprocedure used in present work.

Soot emits in visible spectral region many hydrocarbon ames is visible tothe human eye being soot particles small, they are in thermal equilibrium withthe ux they are transported by, then they emit large amounts of energy inthe infrared spectral region. Experiments have evidenced that soot, generally,emits much more than gases do. Because of the big width of emission spectrumand of variable dimensions and shapes, soot radiative properties are hard to bestudied and modelled.

Usually soot particles dimensions are much smaller that those of most im-portant wavelength for radiative thermal transfer. This last statement has twoimportant implication: the scattering owed to soot is negligible, the absorp-tion coecient is directly proportional to soot volume fraction and inversely towavelength.

This assumptions lead to write the spectral soot absorption coecient as:

kλ,soot = C0fv

λ(6.22)

where fv is the soot volume fraction (Eq. 10.3), and C0 is a constant dependanton complex index of refraction. The total absorption coecient is dened as:

ksoot = 3.83fvC0T

C2(6.23)

where C2 = 1.4388 cm ·K is the second Planck function constant.

115


Since soot is usually present together with gaseous media, contributions toabsorption coecient from both have to be considered. In present work, themixture absorption coecient has been as the sum of gases absorption coecient(see previous section) and the soot one:

kmixture = ksoot + kgases (6.24)

6.5.3 Directional Nature ModellingIn this section will be reported a review on methods suitable to model the

directional nature of RTE. In gure 6.3 is reported a classication of thesemethods ((Viskanta, 2005)), in particular can be found out three major groups:

Figure 6.3: Solution methods for the dierential radiative equation

Optically Thin Approximation based on the observation that in the opti-cally thin limit (τλ → 0), the media absorbs radiant energy leaving theboundaries as well as emits itself but is not optically dense enough toreceive any from other elements of volume, and the system is said to beemission dominated. Then irradiance, appearing in the heat sink due toradiative transfer (Eq. 6.46), is calculated by Eq. 6.3 considering onlyradiative intensity leaving boundaries.

Optically Thick Medium represents the opposite of previous situation (τλ À1), in this case the system is absorption dominated. Being radiation ex-tremely rapidly absorbed, radiative intensity depends only on local con-ditions. The heat ux vector can be expressed as function of radiativeintensity gradient and then, its divergence is function of radiative inten-sity laplacian.

Self-Absorbing Medium in this case neither absorption nor emission can beneglected, so the whole RTE has to be solved. This group is the widestone, and being representative of most accurate method, it will be detailedbelow.

116


Keeping in consideration the classication from Viskanta, Self-AbsorbingMedium methods can be divided in four groups:

1. Directional Averaging

2. Dierential Approximation

3. Energy Balance

4. Hybrid

All directional approximation methods involve some type of averaging of theradiative intensity eld with directions.

The multiux methods are probably the simplest to apply and still are beingused, to model radiative transfer in combustion systems, even if they suer frompoor accuracy. Also DOM belongs to this group and are one the most used andaccurate method used together with CFD.

The rst order Dierential Approximations of the RTE yield predictions ofreasonable accuracy and can be adopted for spectral and band calculations.

Energy Balance Methods, such as the Zone Method, the statistical MonteCarlo Method and stochastic approaches are incompatible with CFD, due totheir heaviness coming from their high accuracy. Anyway the Finite VolumeMethod is a good compromise between accuracy and suitability to be used withCFD, and being one of the used method, in present work, it is discussed insubsection 6.5.5.

Hybrid Methods have advantages and disadvantages of the method fromwhich they are derived, to this group belongs the P1-approximation that is theother method used in this work and further detail are in subsection 6.5.4.

6.5.4 Spherical Armonics MethodSpherical armonics method (Pn) represents an approximated strategy to

solve RTE. It is convenient because it neglects the treatment of directionalbehaviour of radiative transfer, reducing the number of variables to be taken inaccount. It was rst proposed by (Jeans, 1917) in his work on radiative transferof stars.The rst step is to rewrite RTE in nondimensional form (6.26) by dening:

kλ =kλ

krefε =

1krefxref

~∇ε = xref~∇ (6.25)

where kref , xref and ε are respectively the reference absorption coecient, thereference leght and the reciprocal of optical thickness.

ε~s · ~∇εIλ = −kλ(Iλ − Ibλ) (6.26)

After some rearrangements nondimensional form of RTE (Eq. 6.26) be-comes:

(1 +

ε

kλ

~s · ~∇ε

)Iλ = Ibλ (6.27)

117


(1− a)−j =∞∑

j=1

aj (6.28)

Then the radiative intensity can be calculated by inverting Eq. 6.27 usingthe Newmann's series (Eq. 6.28):

Iλ =(

1 +ε

kλ

~s · ~∇ε

)−1

Ibλ (6.29)

≈(

1− ε

kλ

~s · ~∇ε +ε2

k2λ

(~s · ~∇ε

)2

+ · · · (−1)i εi

kiλ

(~s · ~∇ε

)i)Ibλ

Integrating Eq. 6.29 on the whole solid angle and stated that∫

4π

(~s · ~∇ε

)i

dΩ =[1 + (−1)i

] 2πi+ 1

∣∣∣~∇iε

∣∣∣ (6.30)

and using the denition of irradiance (Eq. 6.3) we obtain

Gλ = 4π

(1 +

ε2

3k2λ

∣∣∣~∇ε

∣∣∣2

+ε4

5k4λ

∣∣∣~∇ε

∣∣∣4

+ε6

7k6λ

∣∣∣~∇ε

∣∣∣6

+ · · · (6.31)

+[1 + (−1)i

] 2π4(i+ 1)

∣∣∣~∇ε

∣∣∣i)Ibλ

Inverting Eq. 6.31 and using again the Newmann's series (Eq. 6.28) weobtain the approximated Simplied Spherical Armonics Equation(SPn):

4πIbλ =

(1− ε2

3k2λ

∣∣∣~∇ε

∣∣∣2

+4ε2

45k4λ

∣∣∣~∇ε

∣∣∣4

− 44ε6

945k6λ

∣∣∣~∇ε

∣∣∣6)Gλ +O(ε8) (6.32)

If the series on RHS of Eq. 6.32 were truncated at rst order we can writethe SP1 equation in dimensional form as:

13kλ

∆Gλ = kλ (Gλ − 4πIbλ) (6.33)

the SP1, is an approximated form of RTE with and error comparable to O(ε4),then it is easy to understand that this method is suitable for optically thickmedia (ε→ 0).

Also the RTE boundary condition has to be modied to be used with SP1, thiswas done by Marshak ((Modest, 2003, pages 475-476)) for a diusive emitting-reecting surface. The boundary condition for SP1 is:

− 2(

2ελ− 1

)(1

3kλ

)~n · ~∇Gλ +Gλ = 4πIbλ(Tw). (6.34)

As already stated, this equation is valid for diusive surfaces and it couldnot be in a dierent way since the SP1 neglect directional behaviour of radiativetransfer.

118


In present work has been derived the SP1 form of RTE that was proposedby (Gelbard, 1961) for nuclear reactor modelling, and that is equal to sphericalarmonics P1 approximation and in eect it is better known with this last name.In this section has been described the SP1 asymptotic derivation, reported in(Larsen et al., 2002), due to its simple form, at least compared to the Pn deriva-tion, involving Legendre's series ((Modest, 2003).

A result of SP1 approximation, that instead represents an hypothesis for thederivation of P1 approximation, is that heat ux vector can be calculated asthe gradient of irradiance (Eq. 6.35). This can be demonstrated comparing theRHS s of divergence of radiative heat ux vector equation (6.46) and of SP1equation (6.33) it can be found out that they are equal, and it is, obviously,correct also for LHS s.

~qR,λ

= − 13kλ

~∇Gλ. (6.35)

6.5.5 Finite Volume MethodThe nite volume method ((Mishra and Roy, 2007)), represents a strong

enhancement of DOM by which inherits the spatial discretization of RTE. Itwas rst proposed by (Briggs et al., 1975) in the eld of neutron transport. Itsuse was extended to radiative heat transfer by (Raithby and Chui, 1990).

The nite volume method numerically solves the exact RTE. IntegratingLHS of Eq. 6.18 for direction ~s in a control volume V (bounded by Γ) by theuse of Gauss-Green theorem we obtain:

∫

V

~s · ~∇I dV =∫

V

~∇ · (~sI) dV −∫

V

I(~∇ · ~s

)dV

︸︷︷︸=0

=∫

Γ

I ~s · ~n dΓ (6.36)

where the second term of RHS is dropped since the direction versor ~s is constantin space ((Modest, 2003)), and ~n represents the normal to Γ. Then integratingRTE over the control volume and solid angle Ωi leads to:

∫

Ωi

∫

Γ

I~s · ~n dΓ dΩ =∫

Ωi

∫

V

kIb dV dΩ−∫

Ωi

∫

V

kI dV dΩ (6.37)

In the simplest implementation of the nite volume method it is assumed,for the LHS of Eq. 6.37, that the intensity is constant across each face of thediscrete control volume (leading to problems reported in chap. 9.2) as wellas over the solid angle. While RHS components are considered constant in thewhole control volume and equal to value assumed in the control volume centroid.By this hypothesis, equal to those of nite volume method applied to CFD, Eq.6.37 becomes:

∑

k

Iki

(~Di · ~nk

)Ak = kp (Ibpi − Ipi)V Ωi (6.38)

119


where, Iki is the radiative intensity through face k, of area Ak, in the i − th

direction, while Ibpi, Ipi and kp represent respectively the centroid black bodyradiative intensity, the radiative intensity in i − th direction and the centroidabsorption coecient. ~Di is a versor pointing into an average direction withinsolid angle element Ωi and is dened as:

~Di =∫

Ωi

~s dΩ (6.39)

Until this point the method is the same of DOM, and the discriminationbetween these two methods lies on the treatment of term 6.39: in DOM it isdiscretized with approximated quadrature (not ensuring energy conservation)while in FVM it is discretized with exact relations. In present work has beenfollowed the method suggested by (Mishra and Roy, 2007).

Figure 6.4: Coordinate system for FVM angular discretization

Taking a look at gure 6.4 we can write:

~s = (sinθ cosφ)~i+ (sinθ sinφ)~j + (cosθ)~k (6.40)

And performing the change of integration variable: dΩ = sinθ dθ dφ

While integration domain becomes:

Ωi =[φi − ∆φi

2, φi +

∆φi

2

]×

[θi − ∆θi

2, θi +

∆θi

2

](6.41)

After this modications, the average direction versor for nite volume methodcan be rewritten as:

~Di =∫ φi+

∆φi2

φi−∆φi2

∫ θi+∆θi2

θi−∆θi2

~s sinθ dθ dφ (6.42)

Once that a numerical form for RTE has been obtained it is now needed anumerical formulation for boundary conditions. For a grey diusive emitting-absorbing-reecting opaque surface this means discretizing Eq. 6.19. It can be

120


do by multiplying Eq. 6.19 by ~s · ~n and integrating over all outgoing directionsand considered that

∫~n·~s<0

~n · ~s dΩ = π we obtain:∫

~n·~s′>0

I ~n · ~s dΩ = εwIbw

∫

~n·~s′>0

~n · ~s dΩ + ρw

∫

~n·~s′<0

I|~n · ~s|dΩ (6.43)

Then by those approximations applied to discretize the RTE Eq. 6.43 forface, q, becomes:

Iq0∑

i,out

( ~Di · ~nq)Aq = εqIbq

∑

i,in

( ~Di · ~nq)Aq (6.44)

+(1− εq)∑

i,out

Iqi

∣∣ ~Di · ~nq

∣∣Aq

And after some rearrangements the value of radiative intensity for boundaryface q becomes:

Iq0 = εqIbq + (1− εq)

∑i,out Iqi

∣∣ ~Di · ~nq

∣∣∑

i,in( ~Di · ~nq)(6.45)

6.6 Radiative Heat Transfer Eect On OverallSystem Energy Balance

The eect of radiative heat exchange on the system energy balance is takenin account by the enthalpy equation (Eq. 5.3), where appears the divergence ofradiative heat ux vector, as well as for the conductive heat ux.

The calculation of the divergence of radiative heat ux vector has beenomitted in section 6.2, while dening other radiation variables, because it canbe written by the RTE. The derivation of divergence of heat ux comes by thedenition of the radiative heat ux vector (Eq. 6.5) and for a for a steadysystem, by the use of RTE (Eq.6.18), it can be written:

~∇ · ~qR,λ

= ~∇ ·∫

4π

Iλ(~r,~s)~s dΩ =∫

4π

~s · ~∇IλdΩ = (6.46)

−kλ

∫

4π

IλdΩ + kλ

∫

4π

IbλdΩ

By some calculation, the use of irradiance denition (Eq. 6.3) and consid-ering that a black body emits in the same way in all direction (subsec. 6.2.4),the divergence of the radiative heat ux vector becomes:

~∇ · ~qR,λ

= kλ

(4πIbλ −

∫

4π

IλdΩ)

= kλ (4πIbλ −Gλ) (6.47)

The total form of this equation, that is the one to be used in Eq. 5.3, can beobtained by integrating over the all wave length spectrum:

~∇ · ~qR

=∫ ∞

0

~∇ · ~qR,λdλ = k

(4πIb −

∫

4π

IdΩ)

= k (4πIb −G) (6.48)

121

Chapter 7

Results: Heat Transfer

In this chapter the runs for validation of the proposed Low Reynoldsmodels will be presented. First a general test consisting of a steadyincompressible turbulent ow over a at plate with the aim of selecting thebest performing Low Reynolds k − ε models. After this rst selection,attention was addressed to cases of interests for turbomachinery coolingsystems and to models of better known performance. Simulations have beenconducted rst against standard test, available from literature as referencecases, and secondly against experiments performed at the Energy EngineeringDepartment of the University of Florence.

Contents7.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . 1247.2 Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . 1257.3 Impingement Cooling . . . . . . . . . . . . . . . . . 127

7.3.1 ERCOFTAC C25 Axial-symmetric Impingement . . 1277.3.2 Single hole case . . . . . . . . . . . . . . . . . . . . . 1317.3.3 Multirow Impingement Cooling . . . . . . . . . . . . 134

7.4 Film Cooling . . . . . . . . . . . . . . . . . . . . . . 1377.4.1 Sinha Test . . . . . . . . . . . . . . . . . . . . . . . . 1387.4.2 Multirow Eusion Cooling . . . . . . . . . . . . . . . 1417.4.3 Fan-Shaped Cooling Holes . . . . . . . . . . . . . . . 143

7.4.3.1 SAA Geometry . . . . . . . . . . . . . . . . 1457.4.3.2 PAA Geometry . . . . . . . . . . . . . . . . 150

7.5 Blade internal cooling - Trailing Edge . . . . . . 1537.5.1 Trailing Edge Cooling System: AITEB2 . . . . . . . 153

7.5.1.1 Geometry G1.0 . . . . . . . . . . . . . . . . 1587.5.1.2 Geometry G1.2A and geometry comparison 159

123

Chapter 7. Results: Heat Transfer

7.1 GeneralitiesBefore starting the analysis of the obtained results, not to weigh down the

treatment of the subject, some features common to all simulations will be dis-cussed in this section.

Due to the huge number of implemented turbulence models, a shortcut hasbeen used to name most of them: the acronyms before dened and summarizedin Tab. 7.1 are widely used in substitution of authors' full name.

Table 7.1: Acronyms for the various turbulence models.

k − ε Low Reynolds by Abe et al. (1994) AKNk − ε Low Reynolds by Chien (1982) CH

k − ε Low Reynolds by Chen et al. (1996) CLLk − ε Low Reynolds by Hwang and Lin (1998) HW

k − ε Low Reynolds by Lam and Bremhorst (1981) LBk − ε Low Reynolds by Lien and Leschziner (1993) LW

k − ε Low Reynolds by Lien (1992) LNRRealizability constraint postscript Real

Two Layer TLk − ω SST SST

k − ω SST Automatic Wall Treatment SSTawt

All cases are run with the second order advection scheme SFCD describedin Sec. 1.4 with a deferred approach mixing implicit and explicit derivatives:

ΦDef,ne = ΦUDS,n

e + γ(ΦSFCD,n−1

e − ΦUDS,n−1e

), (7.1)

to obtain blended second order accuracy at convergence, ΦUDS,ne = ΦUDS,n−1

e ,but convergence properties typical of upwind schemes.

Even if case dependent, it is preferred to give a general indication for thevalues of suggested underrelaxation factors of the various quantities in Tab. 7.2.Convergence criteria for the inner cycle are imposed as a relative tolerance, ratio

Table 7.2: Typical values for the underrelaxation factors used.

Velocity 0.5− 0.7

Enthalpy 0.5− 0.7

Turbulent kinetic energy 0.4− 0.6

Turbulent dissipation 0.4− 0.6

Turbulent specic dissipation 0.6− 0.8

Turbulent viscosity 0.8− 1.0

Pressure corrector 0.005− 0.01

Density 0.1

αmix 0.9− 1.0

of nal residual on initial residual, of [10−3−10−5] for all quantities but pressurecorrector that can be solved with a lower accuracy. For the outer iteration a goodconvergence criterion was considered initial residual for all quantities equal to10−6. However not always has been possible to obtain such levels of convergencefor all quantities.

124

Results: Heat Transfer Chapter 7.

7.2 Flat PlateA simple test of a ow over a at plate has been considered the best choice to

start the validation and selection process of the Low Reynolds k− ε turbulencemodels. Because of the large amount of accessible data, the ease and velocityof the test, corrections and tuning could be carried out quickly and accurately.

The ow eld being modeled is tested against experimental data reportedby Wieghardt and Tillman (1951) and later included in the 1968 AFOSR-IFPStanford Conference (Coles and Hirst, 1969).

Of particular interest is the possibility of testing correct near wall behaviorof dicult to measure quantities such as turbulent kinetic energy and turbulentdissipation. Due to experimental limitation in fact, physical proles of k and εcan be found almost only for at plate simulations. Results are so reported interms of non dimensional k and ε plotted versus non dimensional wall distanceat an axial location where ow is fully developed, see Fig. 7.1(a) and Fig. 7.1(b).

The geometry of reference is a at plate with test section in the fully de-veloped zone at a distance X = 4.687m from the inlet, posed directly at thebeginning of the solid wall. Test conditions result in a weakly compressibleow, inlet Ma = 0.2, fully turbulent ow, inlet turbulent intensity Tu = 5.76%and length scale lt = 34.1mm. These inlet conditions were in practice trans-formed in conditions for velocity, density, turbulent kinetic energy and turbulentdissipation:

U = Ma√γRT , (7.2)

ρ =p

RT, (7.3)

k =32‖~U‖2Tu2 , (7.4)

ε =k

32

lt. (7.5)

The plate has been considered adiabatic and the inlet temperature was Tin = 294.4 K.Details about applied boundary conditions are listed in Tab. 7.3.

Table 7.3: Flat plate - Computational boundary conditions.

Inlet temperature 294.4 KInlet velocity 68.9 ms−1

Outlet pressure 101400 PaInlet turbulent kinetic energy - k 23.6 m2s−2

Inlet turbulent dissipation - ε 3365 m2s−3

Near wall behavior of both turbulent kinetic energy and turbulent kineticenergy dissipation has been properly checked, through the boundary layer upto a very little minimum y+. Due to the relative simplicity of the case, runscould be performed on a very ne grid: rst node y+ ≈ 0.1. The total numberof cells for the mesh was 17641. To drop dimensions from the turbulent kinetic

125


energy and dissipation, the following non-dimensionalizations has been used:

k+ =k

u2τ

, (7.6)

ε+ =νε

u4τ

, (7.7)

where:

uτ =√τwρ

(7.8)

and τw is the proper wall shear stress. It is better underline the world proper topoint out the dierences between Low Reynolds and wall functions treatment.Shear stress is in fact directly computed with velocity wall normal gradient:

τw = ‖~τw‖ = ‖T ~n− [(T ~n) · ~n]~n‖ , (7.9)

where T is the Reynolds stress tensor dened in Eq. 4.20, calculated at thewall. The abscissa the non-dimensional wall distance:

y+ =uτy

ν. (7.10)

Both the non dimensional k and ε were checked in their behavior against the

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

k+

y+

exp. AKN CH CLL HW LB LW LNR

(a) k+ prole

0 20 40 60 80 1000.00

0.05

0.10

0.15

0.20

0.25

0.30

+

y+

1/(xy)+

exp Patel AKN CH CLL HW LB LW LNR

(b) ε+ prole

Figure 7.1: Turbulence quantities proles.

non dimensional wall distance at an axial location where the ow can be con-sidered fully developed, Fig. 7.1. The level of agreement with the experimentalis fairly good, especially for k+, with free stream values matching for almostall models. However the discrepancy between experimental and numerical sim-ulation follows exactly the proles obtained by other researchers some of whichare directly models' authors Patel et al. (1985); Chien (1982); Hwang and Lin(1998). Both for a higher guarantee of stability and for the best reliability onmatching experimental results, a further test was conducted only on CLL andAKN.

For the same geometry another case involving heat transfer was set up tovalidate mesh independence of the k−ω SST with the automatic wall treatment.

126


0 1 2 3 4 5 60.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

Crawford y+ = 10-6

y+ = 10-4

y+ = 5x10-6

Sta

nton

num

ber [-]

X [m]

Figure 7.2: Stanton number prole.

Three dierent grids with increasing rst node distance from the wall were tested(y+

p = 0.1; y+p = 9; y+

p = 35). Results are reported in terms of Stanton numberand tested against Incropera Incropera and DeWitt (2001), see Fig.7.2. Even ifthe three proles are not exactly coincident the mesh dependence is decreasing asthe ow develops and moreover the agreement is greatly improved from standardwall function approach.

7.3 Impingement CoolingAs already known in literature, impinging jets ows are a specic class of

ows for which the modeling of turbulence results determinant in obtainingacceptable agreement with experimental data. In fact, the ow conditions inthe area around the stagnation point result pretty dependent on the turbulencemodel used. The numerical estimation of the heat transfer coecient, for ex-ample, can lead to errors of about 100% in the evaluation of the peak, choosingeither one or another model. This class of ows is so suitable for deeply testingnew turbulence models.

Two dierent impinging geometries were tested: a 2-D normal impingingjet of air has been performed following a test case by ERCOFTAC and aftera ve row array of holes reproducing experimental results obtained during theEuropean project LOPOCOTEP.

7.3.1 ERCOFTAC C25 Axial-symmetric ImpingementAs above reported, this case has been modeled to reproduce experimental

data collected in an extensive set of measurements of a turbulent jet impingingorthogonally onto a large plane surface. In the experimental campaign tests havebeen done against dierent changeable parameters such as Reynolds numberRe = 2.3 · 104 and 7 · 104, diameter of the pipe D, the eight of the jet abovethe plate Z = 2 − 10 D, see Fig. 7.3 to clarify notation. Further reference

127


for the experimental campaign can be found in (Cooper et al., 1993) as wellas in the ERCOFTAC online database. Geometry has been simplied in an

Figure 7.3: ERCOFTAC - Schematic view of experimental set up.

axial-symmetric jet. Correct velocity and turbulence conditions at the end ofthe pipe were obtained from the development of a 50D upstream extruded inlethole, see Fig. 7.4 for details on the grid. The total number of cells for this casewas 70000. The plate, heated at constant heat ux q = 200 W m−2, has been

(a) Total (b) Mesh Detail

Figure 7.4: ERCOFTAC - Geometry detail of the grid around the stagnation point.

computed using a xed gradient temperature wall (Neumann type boundarycondition), while all other walls has been considered adiabatic, zero heat ux.Under these conditions the ow results almost incompressible. Only one testcondition was reproduced for validation purposes, with ow conditions reportedin Tab. 7.4. Results for this test are reported in terms of Nusselt number as afunction of radial distance from the center. Nusselt number is calculated fromthe heat transfer coecient. Heat ux is calculated again using enthalpy wall

128


Table 7.4: ERCOFTAC - Flow conditions.

Re number 23000

D 101.6 mmZ/D 2

Temperature wall normal gradient 7057.16 Km−1

Inlet temperature 293 KInlet velocity 4.53 ms−1

Outlet pressure 100000 PaInlet turbulent kinetic energy - k 0.07695 m2s−2

Inlet turbulent dissipation - ε 0.3531 m2s−3

gradient. Briey, equations to calculate Nu from the known ow eld are:

q = α∂h

∂n, (7.11)

HTC =q

T − Tad, (7.12)

Nu =HTC · L

k, (7.13)

where HTC is the heat transfer coecient, Tad is the adiabatic wall temperature,for this low speed simulation approximately equal to inlet static temperature,L is the characteristic length: the diameter of the pipe, and k = Cp α is thethermal conductivity.

The turbulence models tested for this case are, after the selection over theat plate: CLL, CLL realizable, AKN, Two Layer, k−ω SST, k−ω SSTawt. The

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

50

100

150

200

250

300

350

400

450

Nu D

r/D

Exp CLL AKN CLLReal SST TL SSTawt

Figure 7.5: ERCOFTAC - Nusselt number distribution along radius.

predictions of all two-equation models used in this validation case are in goodagreement with the experimental data far from the stagnation point. As knownin literature (Durbin and M. Benhia, 1998), in the area around the stagnationpoint, k − ε Low Reynolds models without the realizability constraint fail and

129


dramatically overpredict the peak in the heat transfer coecient (error of almost200%).

At the same time Two Layer, SST and realizable models show about thesame peak value. The local maximum at r/D ≈ 2 is not seen by the Two Layerand is slightly predicted by the SST. On the contrary, the realizable CLL iswell predicting such peak only shifting a bit towards higher values of the radius.Furthermore it is possible to compare the behavior of the SSTawt model withthe other models. When the same Low Reynolds mesh is used its predictionsare obviously collapsing on the standard SST model and agreement with thebest performing models is good.

(a) ERCOFTAC - AKN (b) ERCOFTAC - CLL

(c) ERCOFTAC - CLLReal (d) ERCOFTAC - SST

(e) ERCOFTAC - TL

Figure 7.6: Turbulent kinetic energy distribution on symmetry plane [m2/s2].

For turbulent kinetic energy we can see in Figs. from 7.6(a) to 7.6(e) asturbulence models dier to each other: rst of all it may be noticed how twoof the models, the realizable CLL and the k − ω SST, have an almost laminar

130


core in the jet that is going up to the wall. So that both these two models limitproduction term in k equation when the rate of strain is high in this case nearthe stagnation point. It is shown nally how the k − ω SST implicitly satisfythe realizability constraint proposed in Sec. 4.6.7 thanks to the limit on theeddy viscosity, see Eq. 4.118.

7.3.2 Single hole caseFor further validation of the three best performing models from the previous

simulation: Two Layer, k−ω SST, realizable CLL model, a test against in-houseexperimental data seemed the best solution. This case is therefore modeled toreproduce the following situation: coolant is injected from a plenum through aperforated plate and impacts over a at plate at uniform heat ux. The holes onthe plenum compose an array of 10− 11 spanwise rows per 9 streamwise holes.This test is simulating the behavior of the rst row only, while in the following3 − 2 rows (the array of holes is staggered) for a total of 5 jets are impinging.For further details refer to (Facchini and Surace, 2006; Surace, 2004).

As can be seen from the grid, see Fig. 7.7, jet spreading is bounded by awall from one side; the other two vertical boundaries are symmetry planes: acutting hole plane and a hole-to-hole symmetry. Main geometrical parametersare the diameter of the pipe D = 4mm and the height of the jet over the plateZ = 3.75D. The grid on which simulations were run was globally composed by386732 cells.

Figure 7.7: Impingement single hole - Grid.

Main ow parameters can be seen in Tab. 7.5, these conditions have beenapplied at the computational domain as the boundary conditions presented inTab. 7.6.

Simulations have been validated in terms of heat transfer coecient calcu-lated with respect to inlet static temperature circa coincident to inlet total tem-perature. Adiabatic simulations have been done too, in order to check whetherthis approximation could be done or not. Heat transfer coecient is plotted

131


Table 7.5: Impingement single hole - Flow conditions.

Inlet Total Temperature 308.2 K

Inlet Total Pressure 86341 Pa

Outlet Pressure 85101 Pa

Inlet Turbulence intensity - Tu unknown →≤ 0.5%Turbulence length scale - lt unknown → 0.10 D

Inlet mass ux - m 0.452 kg m−1

Wall Heat ux 3000 W m−2

Table 7.6: Impingement single hole - Computational boundary conditions.

Inlet Temperature 308.2 K

Outlet Pressure 85101 Pa

Inlet turbulent kinetic energy 3.145 · 10−6 m2s−2

Inlet turbulent dissipation 2.292 · 10−6 m2 s−3

Inlet Velocity 0.28956 m s−1

Temperature wall normal gradient 111649 K m−1

along the cutting hole symmetry line versus non-dimensional distance from thestagnation point.

-3 -2 -1 0 1 2 3 4 5

0

100

200

300

400

500

600

700

HTC

[Wm

-2K

-1]

X / D

Exp TL SST CLLReal

Figure 7.8: Impingement single hole - Heat transfer coecient on impinged wall alongsymmetry line.

First thing to notice from Fig. 7.8 is that, contrarily to ERCOFTAC test,realizable CLL model fails in well predicting heat transfer coecient around thestagnation point. Moreover, due to the potential core that is not extinguishedat the wall, two unphysical spurious peaks are predicted at X/D ≈ 1.

Two Layer and SST result in being almost equivalent both for the peak leveland the far from the stagnation point values, with the Two Layer predictionsslightly lower everywhere on the impinged surface.

From a designer point of view however, what is more important is a param-

132


eter that well represent some mean value of the heat transfer coecient. Aver-aging has not been done for this case so, to give an idea of the bi-dimensionaldistribution of HTC over the plate, map of wall heat transfer coecient arecontrasted in Fig. 7.9. Scale of reference is the same used for the experimen-tal results so it is obviously scaled on its maximum 375 and minimum value50 [W m−2K−1].

(a) Realizable Chen Lien Leschziner (b) Two Layer

(c) k-ω SST

Figure 7.9: Impingement single hole - Heat transfer coecient distribution on impinged wall[W/(m2K)].

It can be noticed how the shapes of these isolines dier: Two Layer presentsalmost circular contour lines while realizable CLL tends to have higher valuesdownstream on the symmetry line at the same distance from the stagnationpoint. The SST model lies in between this two bounds. This tendency ishowever smoothed downstream. What is interesting is the bubble of higherHTC right on the hole-to-hole symmetry plane. This is due by the interactionof the jet with the wall conning the ow in addition at the jet to jet interaction.The eect of non constrained single jet in fact decreases with the distance but inthis case practically the mass owed towards the right side, see Fig. 7.9, cannotbe drained on that side because of the solid wall. Streamlines are thus curved bythe wall and, not powerful enough to penetrate the jets, goes in between them.Crossow rounds the reference jet and increases the velocity near the symmetryplane. This results in an higher than expected HTC. In order to check which ofthe three distributions is more physical, probably comparisons are better doneevaluating the ow eld on the symmetry plane: velocity and turbulent kineticenergy are reported in Fig. 7.10 and Fig. 7.11.

Both velocity magnitude and k do not perfectly agree in terms of qualitativedistributions, but if for velocity contour levels pretty much coincide, for tur-bulent kinetic energy this is not true, and realizable CLL presents a maximumvalue more than two times lower with respect to the other two models. It hasalready been underlined in Sec. 7.3.1, that also k − ω SST uses a sort of real-izability constraint, in this case though, the low k zone is much less developed

133


(a) Realizable CLL (b) Two Layer

(c) k-ω SST

Figure 7.10: Impingement single hole - Velocity magnitude on cutting hole symmetry plane[m/s].

and values stand in the middle between the Two Layer and the realizable CLL.It is interesting to see how the really low values of viscosity inside the potentialcore of the k − ε Low Reynolds model, increase velocity and the higher heattransfer coecient is easily explained.

Very important in validation process for new CFD packages is the testingagainst equivalent models released in commercial codes. A very wide investiga-tion of the same geometry has been done with StarCD by Di Carmine (2004).Comparisons between the results are in perfect agreement one with the other interms of a qualitative analysis. Quantitatively speaking the dierences betweenthe two results are however more than acceptable with maximum errors for theheat transfer coecient, on peak values, of the order of 5%.

7.3.3 Multirow Impingement CoolingThis case refer to the same set of experiments of the previous test. As

stated the number of jets is increased to 5: the grid is doubled and basicallytwo consecutive rows of holes are cut by the symmetry lines. Being a staggeredpattern of holes, the holes result three on the rst symmetry line and two on thesecond one. For this multi-hole simulation the plenum is bigger and the imposedinlet mass ow is approximately ve fold than single hole case. Computationalboundary conditions follow exactly the previous single hole test. In any case amuch heavier grid, 1704740 cells, was necessary, see Fig. 7.12. For this geometry,

134


(a) Realizable CLL (b) Two Layer

(c) k-ω SST

Figure 7.11: Impingement single hole - Turbulent kinetic energy on cutting hole symmetryplane [m2/s2].

Figure 7.12: Impingement ve holes - Grid.

only the Two Layer and k-ω SST models have been tested against experimentalresults, again in terms of heat transfer coecient on the impinged wall.

First the heat transfer coecient on the symmetry line is plotted as a func-tion of the non-dimensional distance from the rst hole stagnation point. How-ever in this case, there are two cutting holes symmetry lines so both experimentaland numerical data are sampled onto the two dierent symmetry lines (path1and path2) and then merged together in the zone where a relative minimumis located, see Fig. 7.13. Even if obtained results are in good agreement withexperimental data far from the stagnation point, it should be noticed that pre-dictions for the peak values are quite dierent from measured data. Higherdiscrepancies on the even peaks are probably due to errors in the experimentalmeasurements (Facchini and Surace, 2006). Comparing the two models, Two

135


0 5 10 15 20 25 30 35 40 45 500

100

200

300

400

500

600

700

HTC

[Wm

-2K

-1]

X / D

exp. TL path1 TL path2 SST path1 SST path2

Figure 7.13: Impingement ve holes - Heat transfer coecient along center lines.

Layer predicts peak values a 10% better of the k-ω SST giving basically identicalresults outside the stagnation points area. In any case, it should be consideredthat temperature gradients are quite small: a better agreement is expected forhigher values of wall heat ux. Symmetry line behavior of HTC is not a re-

(a) Two Layer (b) k-ω SST

Figure 7.14: Impingement ve holes - Velocity magnitude on symmetry plane [m/s].

ally good parameter to establish whether or not the cooling device is eective.Even though it quite well shows maximum and minimum values, no informationon the mean value, felt by the wall, are obtained. A wall HTC map is hencepresented in Fig. 7.16 to both give an idea of an averaged value and permitan evaluation of ow structures. For this last purpose is of particular help thepicture of the ow eld reported in Fig. 7.14 and Fig. 7.15. It could be noticedeither in the velocity or in the temperature plot, the two models are perfectly inagreement in the prediction of the structure of motion. The eects due to crossow over the plate magnify from left to right obviously following the increasein mass ow. Furthermore two columns of holes are not completely decoupled,the eect of such coupling is seen in the smearing on the symmetry plane atthe height of the corresponding jet as well as in the increase of heat transfercoecient, see Fig. 7.13, on the opposite path.

As already pointed out for a single hole conguration, the decrease of HTC

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(a) Two Layer (b) k-ω SST

Figure 7.15: Impingement ve holes - Temperature distribution on symmetry plane [K].

downstream the jet is faster for the Two Layer than the other model, resulting inmore roundish contour plots. The bubbles of higher heat transfer coecient dueto passage of more streamlines deviated by the jet results now multiplied becauseof the interaction of the same phenomenon on dierent adjacent holes. This canbe noticed also in Fig. 7.13, looking at the local maximum in correspondencewith the hole on the opposite path.

(a) Experimental (b) Two Layer

(c) k-ω SST

Figure 7.16: Impingement ve holes - Heat transfer coecient distribution on impinged wall[W/(m2K)].

7.4 Film CoolingAmong the dierent techniques used to cool the hot parts of combustion,

one of the most used is the so called lm cooling. It basically consists of aninjection of cooling air from the wall to cool, into the hot gas ow with the aimof producing a thin lm of cold uid that isolates the walls from the main ow.

Because of the complex interaction between air and hot gases during mixing,many dierent injection hole shapes and distributions have been studied and agreat amount of research work is still on going.

First step done has been again the validation of some of the models proposedagainst a case well-known in literature. For this family of situations, classiedas lm cooling simulations, the main reference is the experimental investigation

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done by Sinha et al.. It simply is a study of a single jet mixing in cross ow overa at plate. Single row conguration was mainly considered in order to have areference geometry to compare results for dierent turbulence models.

In fact, single jet simulation over a at plate are not of particular interest forturbomachinery design. Multi jet congurations are the standard and shouldtherefore be investigated.

In particular, most recent developments in drilling capabilities allow themanufacturing of wide arrays of micro-holes (diameters less than 1mm), cur-rently referred to as eusion cooling. Even if this technique does not producea lm wall protection as in standard lm cooling, its most ecient cooling fea-ture is the heat removed by the passage of coolant inside the holes (heat sinkeect): the high number of holes and their high length/diameter ratio (withangles below 30Â) allows to heavily increase the overall cooling eectiveness(Gustafsson, 2001). Eusion cooling represents the base in the thermal designof modern aero-engine combustors and its use in the cooling of turbine endwallsis also investigated. Even if lm wall protection may not represent the maincooling eect in eusion technique, the prediction of mixing between coolantand cross ow and the corresponding assessment of adiabatic eectiveness, stillrepresent one of the most dicult task in CFD analysis Andreini et al. (2005).

Second case for the lm cooling family was so chosen to represent experi-mental investigations done on an eusion cooling system done at the EnergyEngineering Department of the University of Florence Arcangeli et al. (2006).These experiments represent a real full coverage turbine end-wall cooling device.

7.4.1 Sinha TestAs above introduced, experimental data and geometries for this case are

based on tests made by Sinha et al. (1991). The geometry is a at plate with asingle row of circular holes obliquely connecting the main uid domain with aplenum at higher pressure. A jet of coolant is as a consequence entering fromthe hole into the hot gas domain, see Fig. 7.17. It is important to report some

Figure 7.17: Sinha - Calculation domain.

fundamental geometrical dimensions such as hole diameter D = 12.7mm andcoolant inlet area Aci = 1.210mm2.

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Computational domain, as usual, reproduce the minimum periodic geometrymeaning that two symmetry planes are imposed, one passing through hole axisand the other half spanwise pitch. Of particular importance for this case is thecomputational grid in the zone around the hole, where the jet entrains in themean ow. Magnications of the used mesh are reported in Fig. 7.18, globallythe mesh is composed by 177468 cells. Experimental ow conditions are listed

Figure 7.18: Sinha - Near wall mesh details.

in Tab. 7.7, and are traduced in computational boundary conditions listed inTab. 7.8. Results were checked in terms of adiabatic eectiveness, a parameter

Table 7.7: Sinha - Flow conditions.

Cross ow temperature 300 K

Coolant temperature 153 K

Pressure 101325 Pa

Density ratio - DR 2.0

Blowing-rate - M 0.5

Momentum ratio - I 0.125

Turbulence level - Tu ≤ 0.2 %Cross ow velocity 20 m s−1

Rec 15700

Wall heat ux - q 0 W m2

that describe the inuence by hot gas temperature or by the coolant on the walltemperature:

η =Tw − Tg

Tc − Tg, (7.14)

where Tw is wall local temperature, Tg stands for hot gas temperature and Tc

for coolant temperature.The performances of standard turbulence models (STD model in Figs.7.19(a)

and 7.19(b) were analyzed in comparison with the anisotropic model ATL interms of laterally-averaged and local span-wise and centerline eectiveness.

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Table 7.8: Sinha - Computational boundary conditions.

Freestream inlet temperature 300 K

Coolant inlet temperature 153 K

Outlet pressure 101325 Pa

Freestream inlet velocity 20 ms−1

Coolant inlet velocity 0.2618 ms−1

Freestream inlet turbulent kinetic energy 0.002385 m2s−2

Coolant inlet turbulent kinetic energy 4.19 · 10−7 m2s−2

Freestream inlet turbulent dissipation 7.021 m2s−3

Coolant inlet turbulent dissipation 0.09305 m2s−3

Temperature wall normal gradient 0 K m−1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

exp. TL ATL

X/D

(a) 10D

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

exp. TL ATL

X/D

(b) 15D

Figure 7.19: Spanwise distribution of eectiveness.

0 2 4 6 8 10 12 14 16 18 20 22 240.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

exp. TL ATL

<>

X/D

(a) Laterally averaged

0 2 4 6 8 10 12 14 16 18 20 22 240.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

X/D

exp. TL ATL

(b) Center line

Figure 7.20: Laterally averaged and local center line eectiveness.

Local lateral eectiveness 10 and 15 diameters downstream is shown in Fig. 7.19,local centerline and laterally averaged eectiveness is also presented in Fig. 7.20.Finally a map of wall eectiveness is reported in Fig. 7.21. The improvementsobtained using the anisotropic model are remarkable. The experimental lat-eral proles of eectiveness are perfectly reproduced both 10 and 15 diametersdownstream while all the other isotropic models where failing such predictionsdramatically. The centerline values also tend to be much closer to experiments

140


Figure 7.21: Eectiveness distribution over the wall.

especially when the distance from the hole is increasing. It is clear how theeddy viscosity model predicts a too coherent jet Fig. 7.21, thus severally un-derestimating lateral cooling performances. However this error is smoothed inthe laterally-averaged parameters and as expected the anisotropic factor justdistribute in dierent way the coolant over the plate not inuencing that muchthe averaged eectiveness. This consideration allows the use of isotropic modelswhen the averaged parameters are considered but as expected conrms how thelocal values of wall temperature cannot be considered reliable.

7.4.2 Multirow Eusion CoolingEven if this case was set up to reproduce an eusion cooling system, whose

main cooling eect is related with the heat-sink behavior of the coolant owinginto the channels, it must be underlined that numerical simulations were donein adiabatic conditions. This kind of simulations help designers in predictingow development and temperature distributions also in cases where the wallheat ux is not null. The geometry for this case is a six staggered holes atplate interposed in between a plenum and a channel at lower pressure. To en-hance numerical stability, the plenum has been meshed as six dierent smallerplena each one with the same inlet mass ow imposed to respect total exper-imental cooling air mass ow. The computational grid shown in Fig. 7.22 istwo million elements. A summary of experimental ow conditions can be foundin Tab. 7.9. This set of physical conditions was translated into computationalboundary conditions in the manner shown in Tab. 7.10. Results are reported

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(a)

(b)

Figure 7.22: Mesh details

Table 7.9: Eusion - Flow conditions.

Cross ow temperature 323 K

Coolant temperature 298 K

Wall heat ux - q 0 W m−2

Pressure 6.0 · 104 Pa

Density ratio - DR 1.103

Blowing-rate - M 0.2

Table 7.10: Eusion - Computational boundary conditions.

Freestream inlet temperature 323 K

Coolant inlet temperature 298 K

Wall temperature normal gradient 0 K m−1

Outlet pressure 6.0 · 104 Pa

Freestream inlet velocity 42.0 m s−1

Coolant inlet velocity 0.1505 m s−1

Freestream inlet turbulent kinetic energy 6.615 m2 s−2

Coolant inlet turbulent kinetic energy 3.398 · 10−6 m2 s−2

Freestream inlet turbulent dissipation 873.6 m2 s−3

Coolant inlet turbulent dissipation 2.573 · 10−6 m2 s−3

in terms of spanwise averaged adiabatic eectiveness, see Fig. 7.23. Togetherwith experimental data, correlative approach predictions using L'Ecuyer andSoechting correlation with Sellers superposition criterion have been reportedArcangeli et al. (2006). It is in fact quite dicult to reproduce exactly suchexperimental proles that are better used as a reference mean value for η. Inagreement to what has been already noticed in the previous case, the laterallyaveraged eectiveness is not that much inuenced by the anisotropic factor, see

142


Fig.7.23. All models in fact qualitatively well reproduce the correlative decayof the spanwise averaged eectiveness downstream the holes. By looking at thetwo-dimensional eectiveness map in Fig. 7.24 however it's evident how the lat-eral diusion of coolant is more correct in the anisotropic case than in the othertwo with a more qualitative agreement with experimental data.

0 10 20 30 40 50 600.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

X / D

exp. TL ATL SST L'Ecuyer-Soechting

Figure 7.23: Spanwise averaged adiabatic eectiveness.

(a) Two LayerAnis (b) Two Layer

(c) SST (d) experimental

Figure 7.24: Comparison between laterally averaged and local center line lm cooling eec-tiveness.

7.4.3 Fan-Shaped Cooling HolesThe application of the CFD analysis is aimed to study the performances

in terms of adiabatic eectiveness of dierent manufacturing solutions for fan-shaped cooling holes. In particular the eect of introducing a corner radiusinside the hole shaping for a given geometry and the inuence of a small com-pound angle has been investigated (Carmine et al., 2007).

The test case data are available from experiments performed at the "Di-partimento di Energetica" of the University of Florence and are reported in(Facchini et al. (2007)), where a complete description of experimental facility,

143


instrumentation and data is available. Comparisons with numerical simulationshave been performed on two sets of experimental results, respectively BR ∼ 0.74and BR ∼ 1.30 − 1.35 for both geometries. Since gas and coolant density arevery close, respectively 0.82 and 0.89 kg/m3, BR and VR are also close. Com-putational results are compared with experimental data in terms of adiabaticeectiveness. The maximum absolute error in measuring the experimental adi-abatic eectiveness is about 0.05 (see Facchini et al. (2007)). Both SAA shapedcongurations and PAA smooth conguration with compound angle deviationwere simulated (see Table 7.11), (Carmine et al., 2007). In agreement with

Table 7.11: Blowing Ratio and Reynolds number for hot tests

BR Re

SAA Smooth 0.73 1.31 10795 18837Sharp 0.74 1.29 11129 19097

PAA Smooth 0.75 1.36 11783 20800

turbulence model requirements, grid were clustered near walls so that y+ < 1.This feature was obtained using at least 15-20 cells inside the boundary layerthickness. The degree of resolution of the ne mesh was obtained followingthe results of previous similar works (Mangani et al. (2007); Bacci and Fac-chini (2007); Andreini et al. (2006)). For SAA sharp and smooth congurations

(a) (b)

Figure 7.25: SAA Computational grids: coarser 7.25(a) and ner 7.25(b)

the calculation domain for computational analysis has then been chosen basedon a symmetry plane boundary condition (Acharya (2007); Bacci and Facchini(2007); Andreini et al. (2006)) and only one half of the whole test section wasreproduced, resulting in a reduction of calculation time. The choice of a symme-try plane also avoided convergence problems due to numerical jet shedding andoscillating phenomena that prevent a proper convergence of the RANS analysis.In order to reproduce the 2 degrees compound angle eect of real geometry onlythe PAA smooth conguration was chosen. The entire shaped hole with cyclicpatches was simulated instead of mid-hole with symmetry patches congura-

144


Figure 7.26: PAA Computational domain

Figure 7.27: Dierences between numerical and experimental domain

tion and a correspondent velocity component was set on inlet boundary (Figure7.26). This strategy was chosen to have a single geometry model to performfurther analysis with dierent compound angles without mesh modications.Clearly the use of the cyclic boundary conditions instead of walls inuences theow eld near the lateral wall but reproduce correctly the relative interactionbetween the main ow and the jet cooling (Figure 7.27). Therefore the localinteraction between the jets is conserved in the simulation.

7.4.3.1 SAA Geometry

Eectiveness results have been compared with experimental data in termsof spanwise distributions and laterally averaged values. Spanwise distributionsof numerical η for three locations downstream injection hole in comparison withexperimental data are shown in Figures 7.28, 7.29, 7.30, 7.31 for all turbu-lence models, for the two blowing ratio considered and for the two congurationtested. The well known under prediction of lateral spreading and over pre-diction of peak values shown by numerical RANS approach is evident in allnumerical results. However data obtained applying the anisotropic model seemto have the better agreement with experimental data. On the contrary pro-les obtained with SST and TL turbulence models show a structure that isnot visible in experimental data. Near the symmetry plane, in fact, eec-tiveness shows a relative maximum associated to the shaping. The evidence ofthis maximum is greater for sharp edge hole and tends to atten proceedingdownstream injection so that, at the lower blowing ratio, for X/D = 25 is dis-appeared for smooth geometry and strongly reduced for the sharp one. At thehigher blowing ratio, this maximum does not disappear in the smooth geometry

145


-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpSSTTLTL Anis

X/D = 5

Y/D

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

X/D = 15

Y/D

ExpSSTTLTL Anis

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpSSTTLTL Anis

Y/D

X/D = 25

(c)

Figure 7.28: Local lateral η for three location downstream for SAA smooth congurationBR=0.73

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpSSTTLTL Anis

X/D = 5

Y/D

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpSSTTLTL Anis

X/D = 15

Y/D

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0ExpSSTTLTL Anis

X/D = 25

Y/D

(c)

Figure 7.29: Local lateral η for three location downstream for SAA smooth congurationBR=1.31

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

X/D = 5

Y/D

ExpSSTTLTL Anis

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

X/D = 15 ExpSSTTLTL Anis

Y/D

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(c)

Figure 7.30: Local lateral η for three location downstream for SAA sharp congurationBR=0.74

with SST model. Should be to noticed that the unexpected asymmetric trendof the experimental data for y/D < −3 of the sharp conguration is a spuriouseect due to an uncorrect TLC color response in the area closest to the channelwall Facchini et al. (2007). Anisotropic correction to turbulence model allowsto greatly improve accuracy of numerical prediction. In particular the mostinteresting result is the disappearing of the double peak structure predicted bystandard models. The increase in lateral diusion of jet permits a more physicalredistribution of coolant between centerline and peripherals regions resulting in

146


-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpSSTTLTL Anis

X/D = 5

Y/D

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(c)

Figure 7.31: Local lateral η for three location downstream for SAA sharp congurationBR=1.29

Figure 7.32: Numerical adiabatic eectiveness map for SAA smooth conguration

Figure 7.33: Experimental adiabatic eectiveness map for SAA smooth conguration

147


Figure 7.34: Numerical adiabatic eectiveness map for SAA sharp conguration

Figure 7.35: Experimental adiabatic eectiveness map for SAA sharp conguration

a more smoothed prole. Anisotropic model shows the best agreement withexperimental data for X/D = 5 for lower blowing ratio and for X/D = 25 forhigher blowing ratio. Figures 7.32, 7.33, 7.34 7.35 report numerical and exper-imental data in terms of 2D contour maps which allow a more comprehensivecomparison among dierent data sets. The maps comparison shows that fornumerical and experimental results there is no uid interaction between the jetcooling and the lateral walls. That validates the use of a symmetry boundarycondition instead of a no-slip type boundary for the lateral domain. It can beobserved how computed jet's structure is much more coherent than in exper-iments: experimental eectiveness iso-lines tend to smooth quickly, especiallyfor the higher blowing ratio, showing for X/D > 15 a fairly uniform prole. It

148


also appears clearly the greater lateral spreading of anisotropic model and itsrelated smoothing of double peak structure. It's interesting to point out a lower

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80 Exp SST TL TL Anis

BR=0.73

X/D

(a)

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80BR=1.31

X/D

Exp SST TL TL Anis

(b)

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80 Exp SST TL TL Anis

BR=0.74

X/D

(c)

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80BR=1.29 Exp

SST TL TL Anis

X/D

(d)

Figure 7.36: Spanwise averaged eectiveness < η > for SAA geometry for smooth (7.36(a),7.36(b)) and sharp (7.36(c), 7.36(d)) congurations

maximum value reported by experiments in terms of eectiveness in the regionjust downstream the hole (0 < X/D < 5) when blowing ratio increases, while allCFD results predict a region with η ∼ 0.8. The discrepancy may be due to heatconduction within plate material during experiments which is not accounted incalculations. In fact as blowing ratio increases convective heat transfer growsand tend to smooth η peaks and probably to modify jet decay. In order to havea more proper comparison we should perform conjugate computations takinginto account solid conduction. Finally gure 7.36 reports the comparison withexperimental data in terms of averaged spanwise eectiveness. The most in-teresting observation is the monotonic progressive decay of η predicted by allcomputations against a more complex distribution shown by experiments, whichis almost constant over wide zones. Analyzing single results we can point outthat k−ωSST better predicts the spanwise averaged data. It probably dependson a deeper penetration of jet within freestream boundary layer. Taking intoaccount its lack of lateral diusion it could be interesting to verify the behaviorof the model with the anisotropic correction: such modication will be the sub-ject of future investigation. Contrary to the experiments, minor dierences areobserved between sharp and smooth congurations for the spanwise averagedeectiveness, since the η decay is similar for both congurations.

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7.4.3.2 PAA Geometry

In order to reproduce the 2 degrees compound angle eect only the PAAsmooth conguration was chosen. Numerical simulations with this congura-tion have been performed only with the Two-Layer turbulence model (bothstandard and anisotropic). Eectiveness results have been compared with ex-perimental data in terms of spanwise distributions (see Figures 7.37, 7.38) andlaterally averaged values for two dierent blowing ratio. The maps comparison

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

ExpTL TL Anis

X/D = 5

Y/D

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

X/D = 15 ExpTL TL Anis

Y/D

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(c)

Figure 7.37: Local lateral η for three location downstream for PAA smooth congurationBR=0.75

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(a)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(b)

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.2

0.4

0.6

0.8

1.0


Y/D

(c)

Figure 7.38: Local lateral η for three location downstream for PAA smooth congurationBR=1.36

shows that for numerical and experimental results there is no uid interactionbetween the jet cooling and the lateral walls. That validates the use of a cyclicboundary condition in order to simulate the PAA compound angle congura-tion. Most of the discussions reported for SAA geometry can be repeated inthis case. The comparison with experimental data shows a general overestima-tion of central peak eectiveness and an underestimation of lateral spreadingof the jet, resulting in an underestimation of eectiveness far from centerline(Y/D > 2). The most important observation is a great underestimation of theeect of compound angle predicted by computations (see Figures 7.39, 7.40).Especially at the exit of the hole experimental maps show a strong inuence ofcompound angle on eectiveness values, while numerical eectiveness distribu-tion seems to be inuenced by crossow inlet angle mainly far downstream the

150


Figure 7.39: Numerical adiabatic eectiveness map for PAA smooth conguration

Figure 7.40: Experimental adiabatic eectiveness map for PAA smooth conguration

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

BR=0.75

X/D

Exp TL TL Anis

(a)

-5 0 5 10 15 20 250.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80 Exp TL TL Anis

BR=1.36

X/D

(b)

Figure 7.41: Spanwise averaged eectiveness < η > for PAA geometry and smooth cong-uration

151


hole. The double peak structure is very reduced using the anisotropic turbu-lence model, but is always present unlike the SAA simulation. Various authorsexplain the missing of the bimodal eectiveness pattern by the inuence of thehole length on the velocity prole, involving the presence or less of recirculationbubbles downstream the diuser wall (Saimweber and Schulz (2007)). Also anon-uniform ow eld upstream the plenum of the coolant jet can lead to thesuppression of the double peak in the adiabatic wall eectiveness (Gritsch et al.(2001)). Figure 7.41 shows results in terms of laterally averaged eectiveness.A quite good agreement is observed at least for lower blowing ratio. With highBR conductive eects already noticed for the SAA geometry become relevant.

152


7.5 Blade internal cooling - Trailing EdgeEven though only hinted in previous sections, it should be clear how cooling

of hot parts of turbomachinery is reached using simultaneously many dierenttechniques, see also Fig. 7.42. The most ecient and eective cooling techniques,lm and impingement cooling, have already been presented but there is anotherimportant cooling method to be tested.

Even if not very well suited for the high temperature typical of leadingedge of rst stage turbine blades, internal cooling with ducts is still used inless extreme situations because of its low manufacturing cost and the ease ofdesign. Moreover in zones where geometrical constraints are particularly strict

Figure 7.42: Typical compound cooling system of a blade.

and interaction with external ow risky in term of aerodynamic performances,other techniques results very complicated to be inserted. That is why attentionwas posed on the axial cooling duct at the trailing edge of a turbine blade.The channel is ribbed and interaction with pedestals is also simulated. Ribsare usually inserted to increase the overall eciency and eectiveness of thecooling device, to reduce the air mass ow and more uniformly distribute theheat transfer coecient on the wall. At the same time some drawbacks, such ashigher manufacturing costs and higher pressure loss, should be considered.

7.5.1 Trailing Edge Cooling System: AITEB2This case refers to a typical cooling system of the trailing edge of a blade,

chosen as a case for the European project AITEB2 and deals with the investi-gation of the usage of enlarged pedestals, inserted in a wedge shaped duct, inconjunction with rib turbulators. Three ribs were arranged, orthogonally with

153


respect to the ow direction, inside the channel dened by a couple of pedestals.Two congurations with the same pedestal arrangement but with dierent ribshapes were considered. Square and circular ribs were compared with a smoothnot ribbed surface Carmine et al. (2008).

Numerical data were compared with in-house collected experimental dataand geometries G1.0, G1.1A and G1.2A were studied (see Tab.7.12). Geometryparameters are summarized in Facchini and Tarchi (2008). Figure 7.43 is a

Table 7.12: Geometries

Geometry CharacteristicsG1.0 not ribbedG1.1A square ribsG1.2A circular ribs

sketch of the computational domain together with boundary condition typeused. The choice of a symmetry plane also avoided convergence problems due

Figure 7.43: Computationl domain and boundaries

to numerical jet shedding and oscillating phenomena that prevent a properconvergence of the RANS analysis. A summary of the boundary conditions isreported in Tab.7.13 Constant uid properties were considered. For geometryG1.1A unstructured grid was generated while for geometries G1.0 and G1.2Aonly the structured multi-block grid one. Table 7.14 reports grid characteristicsin terms of mesh type and number of elements. Grids were clustered near thewalls so that y+ << 1: gure 7.44 shows details of the computational grids.

Comparisons with experimental simulations were performed for Re = 18000and Ma = 0.3 in the throat section. Computational results are compared withexperimental data in terms of HTC distribution and streamwise pressure drop.The maximum absolute error in measuring the experimental data is 12.2% forthe HTC and 0.2% for the static pressure.

154


Table 7.13: Boundary conditions

Location Variable Units ValueInlet m kg s−1 0.00231528Inlet T K 326.24Inlet Tu − 0.01− 0.05Inlet l m 0.001Wall Tw K 273− 293− 313Outlet p Pa 43000

Table 7.14: Grid characteristics

Name Geometry Type ElementsA G1.1A unstructured 1800000D G1.0 structured 1100000E G1.2A structured 1500000

(a) (b) (c) (d)

Figure 7.44: Computational grids: pedestal and rib details for unstructured and structuredmeshes

For geometry G1.1A rst a turbulence model sensitivity analysis was per-formed with mesh A and Tw = 313K. Figure 7.45 shows the heat transfercoecient maps over the endwall while Fig. 7.46 reports the HTC distributionon the centerline. All turbulence models predict some spurious spots in theheat transfer coecient due to the eect of tetrahedral mesh. We can see anover prediction of the heat transfer coecient in the leading edge zone of therst enlarged pedestals and upstream the rst rib just near the lateral pedestalwall Bacci et al. (2005). These over predictions can be attributed to the excessof turbulence production in the stagnation regions typical of the two equationseddy viscosity models Durbin (1996). Downstream, this highly turbulent owcollides on the frontal surface of the rst rib. The ow in this region is sensiblyaccelerated because of the curvature of the leading edge of the enlarged pedestaland because of the reduction of the channel passage section due to the blockage

155


EXP

CFX

SST

FOAM

SST

FOAM

RLR

FOAM

TL

HTC

[W/m2K]

Figure 7.45: Geometry G1.1A: heat transfer coecient map

HTC centerline down

0

50

100

150

200

250

300

350

400

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

x/D [-]

HTC [W/m

2K]

CFX-SST

FOAM-SST

FOAM-RLR

FOAM-TL

EXP1

EXP2

Figure 7.46: Geometry G1.1A: heat transfer coecient distribution

eect of the pedestal. When this accelerated and turbulent ow impinges onthe rst rib we have the formation of a new recirculating ow zone and again alot of turbulence production. This leads to a sensibly higher value of the HTCwith respect to that resulting from the experimentations. Observing the HTCbehavior near the second enlarged pedestal, we can see typical peak values dueto impinging ow and vorticity produced by the horse-shoe vortex Ireland andJones (1986). The exceeding turbulence produced in the stagnation region infront of the pedestal is then convected along the pedestal surface and lead to theoverestimation of the HTC in the whole leading edge region. This disagreementwith experimental results is made larger by the experimental setup. In fact, it

156


EXP

FOAM

RLR

FOAM

TL

FOAM

LR

HTC

[W/m2K]

Figure 7.47: Geometry G1.0: heat transfer coecient map

must be pointed out that the presence of aluminum pedestals leads to an under-estimation of the heat transfer coecient around them, where an appreciable airtemperature drop occurs because the pedestal temperature is, on the average,lower than the endwall one. Such temperature decrease is not considered in theexperimental data reduction (see Facchini and Tarchi (2008)). The comparisonbetween numerical and experimental data shows that the SST turbulence modelof both codes doesn't capture the rate of growth of the HTC peaks after therst two ribs and underestimates it in the sloping at plate after the rib region.Furthermore the CFX SST overestimates the heat transfer after every rib whileFOAM SST has a better accuracy. The discrepancy between the two codes canbe explained by the universal temperature law used in CFX code Menter et al.(2003) while FOAM uses the standard low Reynolds formulation.

The FOAM RLR doesn't predicts the rate of growth of the HTC peaks afterthe rst two ribs and underestimates the heat transfer downstream the third riband in the sloping at plate after the rib region. The FOAM RLR is the onlyturbulence model that correctly predicts the heat transfer on the at plate inthe inlet. The FOAM TL shows the better agreement with experimental dataeven if over predicts the heat transfer coecient in the rst part of the channelupstream the ribs. In fact HTC is well predicted compared to experimentaldata in the rib region as we can see from HTC map (see Fig.7.45) and HTCcenterline prole (see Fig.7.46). Furthermore the FOAM TL is the only modelthat correctly evaluates the rate of growth in the ribbed channel.

157


HTC centerline down

0

50

100

150

200

250

300

350

400

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18x/D [-]

HTC [W/m

2K]

FOAM-TL

FOAM-LR

FOAM-RLR

EXP1

EXP2Section changeSlope change

Figure 7.48: Geometry G1.0: heat transfer coecient distribution

7.5.1.1 Geometry G1.0

The geometry G1.0 was simulated in order to better understand the HTCover prediction of the FOAM TL upstream the ribbed channel. Figure 7.47shows the heat transfer coecient maps over the endwall while Fig. 7.48 re-ports the HTC distribution on the centerline. The FOAM TL doesn't correctlypredict the heat transfer distribution and shows a great overestimation in com-parison with experimental results. The reason for that is the cross sectionalarea reduction of the channel that leads to a mean ow acceleration. Insteadthe FOAM LR presents a better agreement with experimental data. In fact theturbulent boundary layer thickness is reduced because of the main ow accel-eration from about 40 to 100 m/s through the 40 mm long wedge shaped zone:acceleration parameter K ranges from 5e-5 to 6e-6. Therefore K values for thepresent ow are within the range where decreasing in turbulent heat transferassociated with acceleration would be expected Metzger et al. (1986); Andreiniet al. (2004): LR is the only model among the used ones that can account for itand this could explain both such good agreement with experimental data andthe over estimation when using TL. FOAM LR presents a good agreement withexperimental data, even if the heat transfer coecient in the stagnation regionsis over predicted. This over prediction is limited using the realizability con-straint (FOAM RLR). As we can see in Fig.7.48, some dierences are presentalso where there are the slope change and the section change in the geometry.High HTC peak values close to the pedestals are due to the well known presenceof horse-shoe vortex secondary ow Ireland and Jones (1986). In particular, theconsequent local impinging ow conguration is the principal responsible of theenhanced heat transfer. Furthermore, accordingly to correspondent secondaryow intensity reduction, HTC enhancement close to the pedestal reduces faraway from its leading edge. The presence of horse-shoe vortex is conrmedfrom contour plot of x vorticity component on six sections normal to x axis7.49. Positive and negative values of Wx are related to clockwise and counter-clockwise rotation of the ow (with respect to a stream wise axis). Furthermore

158


Wx

[1/s]

Figure 7.49: Geometry G1.0: contour of x vorticity component on six sections normal to xaxis

EXP

FOAM

TL

HTC

[W/m2K]

Figure 7.50: Geometry G1.2a: heat transfer coecient map

we can appreciate secondary ow intensity reduction from pedestal leading edgeto its trailing edge: this causes HTC streamwise reduction.

7.5.1.2 Geometry G1.2A and geometry comparison

For geometryG1.2A only the FOAM-TL was used. Numerical results slightlyover predict experimental results (see Fig.7.50). Figure 7.51 reports the nalcomparison between the three geometries in term of heat transfer coecientdistribution. While in the experimental results geometry G1.2A shows lower

HTC centerline down

0

50

100

150

200

250

300

350

400

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

x/d [-]

HTC [W/m

2K]

G1.0-RLR

G1.1A-TL

G1.2A-TL

G1.0-EXP

G1.1A-EXP

G1.2A-EXP

Figure 7.51: Geometry comparison: heat transfer coecient distribution

159


heat transfer coecient values then geometry G1.1A, in the numerical resultsthis geometry shows higher values. This behavior can be explained looking atFig.7.52, that shows a comparison between geometry G1.1A and G1.2A in termof velocity vector plot on the symmetry plane. The conventional ow pattern

Reattachment

pointVelocity peak

v

[m/s]

Figure 7.52: Geometry comparison: velocity vector plot

is established between two ribs. A primary recirculation zone is formed justdownstream the rst turbulator with much more little secondary recirculationin the corner created by the side of the rib and the endwall. Downstream wehave the ow reattachment and a new little recirculation zone arise in front ofthe following rib. Geometry G1.1A presents bigger vorticity zones downstreamand upstream the ribs. Instead geometry G1.2A presents a higher velocitypeak near the wall, where there is the heat transfer coecient higher value. Sothe ow eld is coherent with the heat transfer distribution. We also noticethat little recirculation zones are present on the top of both rib type. Figure7.53 shows the static pressure distribution on the centerline of the upper wall.Clearly geometry G1.1A presents more pressure losses then geometry G1.2A and

Static Pressure centerline up

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

x/d [-]

p/pin [-]

G1.0-RLR

G1.1A-TL

G1.2A-TL

G1.0-EXP

G1.1A-EXP

G1.2A-EXP

Figure 7.53: Geometry comparison: static pressure distribution

geometry G1.2A presents more pressure losses then geometry G1.0. There isalso a quite good agreement with experimental data. All the turbulence modelsused show the same trend.

160

Chapter 8

Results: TurbulentCombustion Modeling

In this chapter are described those procedures used to implement thepresumed pdf amelets models for diusion and premixed and partiallypremixed ames. First of all are described the amelet libraries generationprocedures, followed by the description of the coupling between combustionmodelling and uid ow modelling. After the model implementation has beendescribed, a validation testcase and application will be reported.

Contents8.1 Results: Diusion Flames . . . . . . . . . . . . . . 162

8.1.1 Numerical Modeling . . . . . . . . . . . . . . . . . . 1628.1.1.1 Libraries generation . . . . . . . . . . . . . 1628.1.1.2 Coupling with CFD code . . . . . . . . . . 163

8.1.2 Model Validation: Blu Body SANDIA Burner . . . 1648.2 Results: Premixed Flames . . . . . . . . . . . . . . 176

8.2.1 Numerical Modeling . . . . . . . . . . . . . . . . . . 1768.2.2 Model Validation: Lean Premixed Test Rig . . . . . 177

8.3 Results: Partially Premixed Flames Model . . . 1808.3.1 Numerical Modeling . . . . . . . . . . . . . . . . . . 1808.3.2 Model Validation: GE10 Burner . . . . . . . . . . . 181

161

Chapter 8. Results: Turbulent Combustion Modeling

8.1 Results: Diusion Flames8.1.1 Numerical Modeling

The presumed pdf amelets models for diusion ames has been describedin subsection 5.5.7. This model requires the solution of the RANS equationsset and the solution of the ame structure problem (section 5.5.6).

The solution of RANS gives the elds of mixture fraction Favre mean valueand variance and of thermochemical enthalpy, other than speed, pressure andturbulence elds.

The solution of the ame structure problem gives the eld of ow densityeld, needed by the RANS code, and the desired species elds. Flame struc-ture is solved before getting involved in the complete ame simulation and thesolution is stored in libraries to those we will refer as amelet libraries.

8.1.1.1 Libraries generation

The rst step in amelet libraries generation is the solution of quasi-1D lam-inar ames, obtained by simplifying the tridimensional Navier-Stokes equationsto particular geometrical congurations. In present model implementation issolved the counter ow opposed ame conguration (Fig. 5.2(b)). The solu-tion of quasi-1D laminar ames is performed by the Chem-1D code by de Goey((Andreini, 2004), (Mangani, 2003)) that uses the CHEMKIN thermochemicaldatabase. The Chem-1D output is:

Yk = Yk(Z,K) T = T (Z,K) (8.1)

where Yk is the generic mass fraction of the k species, T the temperature and Zthe mixture fraction Anyway in the present work the strain rate K is negligibleand hence is not considered.

The quasi-1D laminar ames solution output (Eq. 8.1) are then integratedby:

Q =∫ 1

0

Q(Z)p(Z)dZ (8.2)

where p(Z) is the Z-pdf.The integration of Eq. 8.2, performed during preprocessing by the in house

FORTRAN code prePdf, gives libraries of the generic variable Q as:

Q = Q(Z, Z ′′2

)

Equations 8.1 tighter with equations 5.7, 5.8 and 5.9 allow to calculate the owdensity (that becomes function of only Z and Z ′′2) needed to solve the RANSequations, anyway if the ame is not adiabatic it is necessary to consider theheat exchange eects.

To make this, the conservation of thermochemical enthalpy is enforced (Eq.8.3) through a proper transport equation (5.3) solved by the CFD code and is

162

Results: Turbulent Combustion Modeling Chapter 8.

explicitly considered the dependence of mean specic heat on temperature (Eq.8.4).

h =K∑

k=1

hk = Cp(T )(T − Tref ) + ψ (8.3)

ψ =K∑

k=1

Yk∆hok

Cp(T ) =1

T − Tref

∫ T

Tref

Cp(T )dT (8.4)

Cp(T ) =K∑

k=1

YkCpk(T )

The libraries are:

Mean specic heat with constant pressure with Tref = 0 K

Cp = Cp

(Z, Z ′′2, T

)(8.5)

Formation enthalpy

ψ = ψ(Z, Z ′′2

)(8.6)

Mean molecular weight

W = W(Z, Z ′′2

)(8.7)

Species mass fraction

Yk = Yk

(Z, Z ′′2

)(8.8)

8.1.1.2 Coupling with CFD code

The CFD code solves RANS equations, together with mixture fraction Favreaverage (Eq. 5.33) and variance (Eq. 5.36). In gure 8.1 is reported theowchart of the overall model.

As already been hinted in previous section the RANS solutions needs theow density (see Fig. 8.1) that is calculated by the perfect gas law :

ρ =pflameletW

RT

where the pressure pflamelet is an input (which is the same used to solve thelaminar ames, being usually nearly constant in combustion systems) whileboth molecular weight W and temperature T are given after interrogating thelibraries:

163


Figure 8.1: Diusion ame model owchart

1. Z and Z ′′2 obtained by the RANS solution are used to read the librariesto obtain the values of W , Yk and ψ and the vector Cp

(T

), and usually

bilinear interpolations are required;

2. a guessed temperature is used to calculate a Cp (again interpolations areusually needed) that together with the formation enthalpy is used to invertEq. 8.3 to obtain again the temperature value;

3. guessed and calculated temperature are compared and if tolerance require-ment is met, follow point 4 otherwise steps back to point 2;

4. the calculated temperature and the mean molecular weight are used inperfect gases law.

8.1.2 Model Validation: Blu Body SANDIA BurnerThe validation testcase is the Sidney BBody burner from the International

Workshop on Measurement and Computation of Turbulent Nonpremixed Flames(TNF). It is an open workshop (started in 1996) aimed to share experimental andnumerical experiences of international researchers, in the eld of diusion andpartially premixed ames. One purpose is to built a database of experimentalmeasures, on laboratory ames, suitable to be used as validation testcase fornew numerical models available on the web site (http://www.aeromech.usyd.edu.au/thermofluids/main_frame.htm).

The BBody burner (identication code HM1E in TNF database) is one of thebest among examples of atmospheric laboratory ame: it has a simple geometry,it is axial symmetrical and it reproduces all features of a real combustor. Theburner consists of a cylindrical blu body, surrounded by an air stream (coow)and is characterized by an orice in its centre that inject the fuel (see Fig. 8.2where are reported burner dimensions (Dally et al., 1998)).

Downstream the blu-body there is a complex ow eld, made of a widerecirculation zone that ensures, in the upstream reaction zones, the presence ofhot gases enough to stabilize the ame (see gures 8.2 and 8.3).

164

http://www.aeromech.usyd.edu.au/thermofluids/main_frame.htm

http://www.aeromech.usyd.edu.au/thermofluids/main_frame.htm


Figure 8.2: BBody burner: geometry and ame images

The burned is fed with a mixture of CH4/H2 (1:1 in volume), having Zst =0.04997.

This ame represents a suciently complex testcase. Just the prediction ofthe recirculation zone extension is hardly achievable with two equation turbu-lence models, but it is key of the estimation of ame stabilization. Low value ofstoichiometric mixture fraction (Zst) may cause strong discrepancies on temper-ature eld even if small errors on mixture fraction occurred ((Andreini, 2004)).In (Hossain et al., 2001), it has been shown that the radiative transfer doesnot induce sensible eects on the temperature eld (∆T < 50 K) and simula-tions performed with radiative exchange model (for implementation details seeChapt. 10), whose results are not reported here, conrm it. The simulated ame

Figure 8.3: BBody burner: schematic representation of the ame zones with some samplingsections

is divided into three dierent zones (Fig. 8.3):

1. The recirculation zone where are overriding the turbulent mixing eects

2. An extinction zone (neck zone) just downstream the recirculation zone,where the ux regains its uniform jet-like structure: the chemical reactivityis in this case the limiting phenomenon.

165


3. Farther from injection orice the ame is stabilized (somehow it is reignited)due to the weakening of turbulent mixing allowing a real jet like propaga-tion.

Figure 8.4: BBody mesh

The mesh used in simulations is shown in gure 8.4. It is made of 24948hexaedrons, and one in circumferential direction due to the axial-symmetry. Theinlet boundary conditions are the averaged experimental values of temperature,velocity and turbulent kinetic energy (table 8.1) while the dissipation has beenchosen to comply with:

k3/2

ε≈ Ldissipation = 0.07 ·Rjet = 0.13 [mm]

while all wall are adiabatic and no-slip and at the outlet has been imposedthe atmospheric pressure. The laminar amelets are built with the detailed

Inlet v[m/s] k[m2/s2] ε[m2/s3] T [K]

Fuel 118 208.88 1.6380 · 105 300Air 40 1.8950 224.72 300

Table 8.1: BBody INLETS boundary conditions

mechanism of Gas Research Institute. It contemplates 32 chemical species and177 reactions. It is capable to manage steady combustion with high temperatureof methane and hydrogen but it neglects nitrogen oxides formation mechanisms(anyway not interesting since these have small inuences on energy balances).

Simulations have been performed with the standard high-Reynolds turbu-lence model ((OpenCFD, 2006b)) where the coecient Cε1 has been changedfrom the usual value of 1.44 to 1.6 in order to better estimate jet penetrationas suggested in (Masri, 1998). The obtained results, reported in the plots ingures 8.9-8.17, are compared to results from experiments, and from a ameletmodel implemented in STAR − CDTM (by eective members of the group inwhich present work has been carried out).

The model implemented in STAR− CDTM used the same chemical mech-anism (GRI) and the same combustion model of the present work but neglectsthe dependence on mixture fraction variance of mean specic heat (Eq. 8.5) i.e.the Z−pdf for Cp is a Dirac-delta. Calculation results for mixture fraction Favreaverage (g. 8.9) match experimental results in each section, except for an over-estimation close the burner axis in the recirculation zone (x < 65 mm), this may

166


Figure 8.5: BBody burner: mixture fraction (left) and temperature contours (right)

be explained by an excessive penetration of the fuel stream. STAR− CDTM

correctly evaluates the axis value of Z together with a global underestimationthat grows with the distance from the burner contrarily to FOAM that, on thecontrary, improves moving away from the burner. The implemented model cor-rectly spots the positions of stoichiometric condition even if a very small errormay cause much larger error on temperature and species.

The shape of mixture fraction variance (g. 8.10) is well reproduced forthe rst two sections (x = 13 [mm] and x = 30 [mm]), with an higher peakvalue. Moving downstream, results become worse for the last two sections inthe recirculation zone (x = 45 [mm] and x = 65 [mm]) while they improve innext extinction and jet zones (x = 90 [mm] and x = 120 [mm]). A similar, butstronger, behaviour is shown by STAR− CDTM .

The good prediction of mixture fraction moments for the rst two sectionsimplies the matching of temperature eld (g. 8.11) even if the reactivity isunderestimated, how can be spotted by the CO and CO2 plots (g. 8.12, 8.13),where CO has higher value than experimental results (at second section g. 8.13)while CO2 has a lower mass fraction than experiments, except where a local highreactivity is present. This localized higher reactivity maybe due to the neglec-tion of strain in a zone where the ow is highly curved is also spotted by thehigh OH concentration, that is much lower elsewhere. Hydroxide is producedwhere the CO is oxidized to CO2 where the ame is far from extinction; thiscan also be noticed in gure 8.6, that shows higher OH concentration where ispresent a gradient of CO mass fraction on the air side. STAR− CDTM showsa similar behaviour for the rst section (g. 8.12) and the temperature peakis lower aside with less steep shapes of CO2 and OH plots, that anyway havehigher concentrations resulting in an overall higher temperature. In the secondsection (g. 8.13) the reactivity is largely overestimated by STAR − CDTM .

167


Figure 8.6: BBody burner: OH and CO mass fraction contours

The H2O mass fraction matches experimental and STAR − CDTM results atboth sections.

The section in the recirculation zone (clearly visible in g. 8.7) at x =45 [mm] (g. 8.14) shows lower temperature close to the axis, this is providedby either STAR − CDTM or present work models, so it probably cannot beattributed to the local overestimation of Z since STAR well predicts it on theaxis, instead it maybe due the local overestimation of the variance that giveslocal higher concentration of CO and the lower one of CO2. Moving away fromthe axis temperature eld is well predicted until the zone near to the stoichio-metric point that is better localized by STAR − CDTM that indeed a moreprecise estimation of the peak value. FOAM much better predicted the COwhile STAR−CDTM shows a large decrease, for radial distances greater than10 mm, of its concentration sided by the matching of CO2 values that anywayseems to be met by a larger reactivity demonstrated by an higher concentra-tion of OH that is better predicted by the present work model that also giveslower mass fraction of carbon dioxide. The lack of a very good prediction ofCO may seem in contrast with the good prediction of temperature, especiallyby STAR − CDTM , but it has to be considered that this is a section stronglyaected by recirculation and then by the mixing with hot burnt species fromother zones. For the right downstream section (x = 65 [mm], g. 8.15) can bedone similar considerations although STAR−CDTM works much better on COand for zone close to the stoichiometric point it matches experimental results,and now it is FOAM that overestimate the reactivity.

For both sections, both codes can predict H2O concentration with a highprecision.

In the extinction and jet-like propagation zones (x = 90 mm, g. 8.16 andx = 120 mm, g. 8.17) STAR − CDTM matches the temperature eld while

168


Figure 8.7: BBody burner: details of velocity eld contour in the recirculation zone withstreamlines

Figure 8.8: BBody burner: mixture fraction variance contour near blu body.

FOAM strongly underestimates it closer to the axis and overestimates it nearthe stoichiometric zone. These statements strongly disagree with the predictionof mixture fraction moments so that the amelet model may be inadequatein these zones indeed either (Hossain et al., 2001) or (Masri, 1998) fails here,while (Andreini, 2004) matches experimental results with an EDC model withdetailed chemistry. Moreover, aside with a good prediction of mixture fractionmoments, and with much dierent values than STAR − CDTM ones, CO isstrongly overestimated by both codes (calculated concentrations are nearly twofolded than the experimental ones). CO2 concentration is well predicted bySTAR−CDTM while FOAM strongly overestimates it near the stoichiometricvalue that is better localized by STAR−CDTM that also work better on OH.H2O mass fraction is matched by both codes.

The analysis of results shows the good behaviour of the implemented model.Previous simulations, whose results are not reported here, where conducted withthe same model implemented in STAR−CDTM and the obtained results werecompared the above reported ones and the improvement in temperature eldcalculation has allowed to demonstrate that the specic heat dependence onmixture fraction variance cannot be neglected.

169


0 5 10 15 20 25 300,0

0,2

0,4

0,6

0,8

1,0

Mix

ture

frac

tion

[-]

radial distance [mm]

exp HTC STAR-CD HTC FOAM Zst

Y=13 [mm]

0 5 10 15 20 25 300,0

0,2

0,4

0,6

0,8

1,0

Y=30 [mm]


Mix

ture

frac

tion

[-]


0 5 10 15 20 25 300,0

0,2

0,4

0,6

0,8


Y=45 [mm]

Mix

ture

frac

tion

[-]


0 5 10 15 20 25 300,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7


Y=65 [mm]

Mix

ture

frac

tion

[-]


0 5 10 15 20 25 300,0

0,1

0,2

0,3

0,4

0,5

Y=90 [mm]

Mix

ture

frac

tion

[-]



0 5 10 15 20 25 300,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

Y=120 [mm]

Mix

ture

frac

tion

[-]



Figure 8.9: BBody burner: mixture fraction average proles.

170


0 5 10 15 20 25 300,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

0,016

0,018

0,020

Y=13 [mm]

mix

ture

frac

tion

varia

nce


exp HTC STAR-CD HTC FOAM

0 5 10 15 20 25 300,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

0,016

Y=30 [mm]

mix

ture

frac

tion

varia

nce



0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

Y=45 [mm]mix

ture

frac

tion

varia

nce



0 5 10 15 20 25 300,000

0,002

0,004

0,006

0,008

0,010

Y=65 [mm]mix

ture

frac

tion

varia

nce



0 5 10 15 20 25 300,000

0,002

0,004

0,006

0,008

0,010

0,012

0,014

Y=90 [mm]mix

ture

frac

tion

varia

nce



0 5 10 15 20 25 300,000

0,002

0,004

0,006

0,008

0,010

0,012

Y=120 [mm]mix

ture

frac

tion

varia

nce



Figure 8.10: BBody burner: mixture fraction variance proles.

171


0 5 10 15 20 25 30200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30200

400

600

800

1000

1200

1400

1600

1800

2000

0 5 10 15 20 25 30

400

600

800

1000

1200

1400

1600

1800

2000

EXP HTC STAR-CD HTC FOAM

Tem

pera

ture

[K]


Y=13 mm Y=30 mm


Tem

pera

ture

[K]


radial distance [mm]radial distance [mm]


Tem

pera

ture

[K]

Tem

pera

ture

[K]

Y=45 mm


Tem

pera

ture

[K]

Tem

pera

ture

[K]

radial distance [mm]0 5 10 15 20 25 30

400

600

800

1000

1200

1400

1600

1800

Y=65 mm


0 5 10 15 20 25 30

400

600

800

1000

1200

1400

1600

Y=90 mm


0 5 10 15 20 25 30200

400

600

800

1000

1200

1400

1600

Y=120 mm


Figure 8.11: BBody burner: temperature proles.

172


0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08


CO

[mas

s fra

ctio

n]



0 5 10 15 20 25 30-0,02

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

exp HTC STAR-CD HTC FOAMH

2O [m

ass

fract

ion]

0 5 10 15 20 25 30-0,01

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08



CO

2 [m

ass

fract

ion]

0 5 10 15 20 25 30

-0,00010,00000,00010,00020,00030,00040,00050,00060,00070,00080,00090,00100,00110,00120,00130,00140,0015



OH

[mas

s fra

ctio

n]

Figure 8.12: BBody burner: species proles at Y=13 mm.

0 5 10 15 20 25 300,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

H2O

[mas

s fra

ctio

n]



0 5 10 15 20 25 30-0,01

0,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

radial distance (mm)radial distance (mm)


CO

2 [mas

s fra

ctio

n]


0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

CO

[mas

s fra

ctio

n]


0 5 10 15 20 25 300,0000

0,0002

0,0004

0,0006

0,0008

0,0010

0,0012

0,0014

0,0016

0,0018

0,0020

OH

[mas

s fra

ctio

n]

EXP HTC STAR-CD HTC-FOAM


173


0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

exp HTC STAR-CD HTC FOAMC

O [m

ass

fract

ion]

radial distance [mm]0 5 10 15 20 25 30

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

exp HTC STAR-CD HTC FOAMH

2O [m

ass

fract

ion]


0 5 10 15 20 25 300,00

0,02

0,04

0,06

0,08



CO

2 [m

ass

fract

ion]

0 5 10 15 20 25 30

0,0

5,0x10-4

1,0x10-3

1,5x10-3

2,0x10-3



OH

[mas

s fra

ctio

n]


0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

0 5 10 15 20 25 300,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

CO

2 [mas

s fra

ctio

n]


CO

[mas

s fra

ctio

n]




0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

OH

[mas

s fra

ctio

n]H

2O [m

ass

fract

ion]



0 5 10 15 20 25 30-2,0x10-4

0,0

2,0x10-4

4,0x10-4

6,0x10-4

8,0x10-4

1,0x10-3

1,2x10-3

1,4x10-3

1,6x10-3

1,8x10-3

2,0x10-3




174


0 5 10 15 20 250,00

0,01

0,02

0,03

0,04

0,05

0,06

CO

[mas

s fra

ctio

n]



0 5 10 15 20 25 30

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14


H2O

[mas

s fra

ctio

n]

0 5 10 15 20 25 30

0,00

0,02

0,04

0,06


radial distance [mm]radial distance [mm]

CO

2 [m

ass

fract

ion]

0 5 10 15 20 25 30-0,00010,00000,00010,00020,00030,00040,00050,00060,00070,00080,00090,00100,00110,0012



OH

[mas

s fra

ctio

n]


0 5 10 15 20 25 300,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

H2O

[mas

s fra

ctio

n]



0 20

0,00

0,01

0,02

0,03

0,04

0,05

0,06

CO

2 [mas

s fra

ctio

n]



0 5 10 15 20 25 300,00

0,01

0,02

0,03

0,04

0,05

0,06

CO

[mas

s fra

ctio

n]



0 200,0000

0,0002

0,0004

0,0006

0,0008

0,0010


OH

[mas

s fra

ctio

n]



175


8.2 Results: Premixed Flames8.2.1 Numerical Modeling

In this case the solution of RANS gives the elds in terms of progress variableFavre mean value and variance and of thermochemical enthalpy, other thanspeed, pressure and turbulence elds.

The solution of the ame structure problem gives the eld of ow densityeld, needed by the RANS code, and the desired species elds. Flame struc-ture is solved before getting involved in the complete ame simulation and thesolution is stored in libraries to those we will refer as premixed amelet libraries.

The numerical implementation of Eqn. (5.58), as well as for G′′2 and σT donot represent a dicult task; on the contrary, ensuring gradient of G condition,this from Eqn. (5.61) and a similar for σT needs particular care, leading to are-initialization and a re-distribution of the scalar values all over the domain.

The procedure of redistribution of scalars may be summarized as follows:

1. Localization of the mean ame front position (G = G0 = 0)

2. Evaluation of G′′2 and σT ame front values

3. Evaluation of minimum distance dmin from the mean ame front for allcell centers

4. Re-initialization of G eld (G = dmin, G 6= G0)

5. Re-distribution of mean ame front G′′2 and σT scalar values in the amefront normal direction

Figure 8.18: Localization of mean ame front by linear interpolation.

Referring to Fig. 8.18, in which A and B denote two adjacent volume cells, suchas to satisfy Eqn. (8.9), mean ame front Cartesian coordinates X = (x1, x2, x3)may be found by rst order interpolation algorithm of Eqn. (8.10)

G(A) · G(B) < 1 (8.9)

176


X eG=0 = XA +G(A)

G(A)− G(B)(XB −XA). (8.10)

A G weighted average is performed to evaluate mean ame front values for G′′2and σT (Eqn. (8.11)).

Φ eG=0 =G(A)

G(A)− G(B)Φ(A) +

(1− G(A)

G(A)− G(B)

)Φ(B),

Φ = G′′2, σT

(8.11)

As mentioned before, in this Level-set method the characterization of ther-mochemical properties of reacting mixture is determined following a laminaramelet approach. In particular laminar back-to-back amelets are solved exter-nally from CFD computation. This type of amelets allows to set aerodynamicstretch as a function of strain K, which only depends on inlet conditions.

Laminar amelet data are integrated before the CFD computation and thenstored into external look-up tables. The generic mixture averaged scalar is givenby Eqns. (8.12) and (8.13).

Q(G, G′′2, K) =∫ +∞

−∞Q(G−G0, K) · P (G)dG, (8.12)

P (G) =1

(2πG′′2)0.5exp

(− (G− G)2

2G′′2

). (8.13)

To limit the size of look-up tables, the minimal set of data is stored, which isconstituted by laminar ame speed, mean molecular weight, enthalpy of forma-tion and averaged mixture specic heat like in the previous section. Becauseof PDF integration, table data set is stored as a function of G, its varianceG′′2 and the turbulent parameter stretch K, dened in chapter 5.4: this last,during the CFD calculation is approximated by integral turbulent frequencyε/k and during calculations the value of each scalar is evaluated by a tri-linearinterpolation.

8.2.2 Model Validation: Lean Premixed Test RigThe validation test case proposed for premixed combustion is an experimen-

tal burner from Vanderbilt University Nandula (2003): the geometry consistsof a blu-body placed as recirculation promoter to stabilize a natural gas leanpremixed ame conned in a quartz duct (see Fig. 8.19 for geometry and testboundary conditions). The very high inlet turbulence level provides a fully tur-bulent ow eld, determining a Damkhöler number lower than unity and placingthe developed ame in the thin reaction regime Borghi (1990). The adoptedstabilization mechanism and ame regime allow to properly realize the typicalcombustion characteristics of a modern DryLow NOx gas turbine combustor.

177


Figure 8.19: Vanderbilt combustor geometry and boundary conditions.

Calculations were performed with the two selected models Level-Set 5.6.5(GEq) and Turbulent Flame Closure (TFC) 5.54: laminar amelets for Level-Set model have been prepared using the reaction mechanisms of Gas ResearchInstitute (GRIMech 2.11). For TFC model the unstretched laminar ame speedwas assumed equal to 0.108[m/s]. Computational mesh was prepared assumingthe axis symmetry of the ame: about ten thousands hexahedral elements hadbeen proved to satisfy mesh independence. Fig. 8.20 reports a direct comparison

Figure 8.20: Contour plots of temperature and progress variable source for tfc model (left)and of temperature and iso-g for level-set model (right).

178


between the two models. A qualitative comparison of the temperature distribu-tions points out a very similar behavior of the two models, with a comparableame length and brush thickness. Interesting to observe the contour plot of theprogress variable source term for the TFC model: the maximum values are lo-cated in the shear layer region where local increasing of turbulent kinetic energyimplies high values of turbulent ame speed (see Eqn. 5.63). Iso-lines plot ofscalar G allows to properly point out the location of averaged ame front aspredicted by Geq model.

Fig. 8.21 reports a comparison among temperature radial proles predictedby the two models and measured on test rig: three dierent axial locationswere considered, being D the diameter of blu-body. An overall agreement isobserved even if, locally, some discrepancies arise. Both models underpredicthot gas temperature in the central region for the rst axial section. A possi-ble explanation is a local heat loss in the actual ame due to the cooling ofblu-body plate which was not modeled in the computations. A wrong mod-elization of aerodynamic stretch may also locally inuence the accuracy: this isconrmed by recalling that implemented TFC model do not account for amestretch giving an explanation to the underestimated ame brush thickness andto the higher values of ame temperature predicted by this model. Movingdownstream the blu-body, the Geq model seems to slightly overpredict amebrush thickness: also in this case it is possible to explain the issue with an over-prediction of ame stretch which it is probably due to an inaccurate predictionof turbulent eld. Nevertheless, Geq model appears to more properly predictsthe real length of ame with respect to measured data. The real value added

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

EXP GEq TFC

x/D = 0.1

T [K

]

r/D [-]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

x/D = 0.8

T [K

]

r/D [-]

EXP GEq TFC

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

200300400500600700800900

10001100120013001400150016001700

EXP GEq TFC

x/D = 1.0

T [K

]

r/D [-]

Figure 8.21: Averaged temperature proles at three dierent locations x/d = 0.1, x/d = 0.8

and x/d = 1.0.

179


0.0 0.1 0.2 0.3 0.4 0.5 0.6100

1000

10000

0.0 0.1 0.2 0.3 0.4 0.5 0.6100

1000

10000

x/D = 0.3

x/D = 1.0

r/D [-]

r/D [-]

CO

[ppm

]CO

[ppm

]

EXP OF-GEq

EXP OF-GEq

Figure 8.22: CO Mole fraction.

by the use of Level-Set approach is the opportunity to predict non-equilibriumeects on main chemical species due to the interaction with turbulent ow eld.Fig. 8.22 reports a comparison between measured and computed CO mole frac-tion. A noticeable overprediction is observed even when the model is able tocatch the location of local maxima in radial proles. The inaccurate quantitativeprediction of CO distribution is partly due to the cited issues in the computa-tion of stretch eld, but it aso depend on the use of scalar G as abscissa inlaminar amelet without the suggested normalization proposed by HerrmannHermann (2006). A detailed investigation on this topic is beyond the scope ofthis work. In fact it is important to recall that actual combustors are usuallyoperated with partially premixed ame that will be modeled with non-premixedlaminar amelets based on mixture fraction as primitive scalar.

8.3 Results: Partially Premixed Flames Model8.3.1 Numerical Modeling

The model has been built starting from the diusion ame solver (chapt.5.5) it consists of:

Progress variable equation steady formulation of Eq. 5.51, with the turbu-lent transport term modelled with Eq. 5.52, is solved aside RANS.

Library interpolation performed on burnt state amelets libraries and onunburnt state amelets libraries.

Flamelet libraries for the generic variable Q in burnt state are built inthe same way of the diusion ames ones (subsection 8.1.1.1): Qb =Qb

(Z, Z ′′2

)

180


Figure 8.23: Flowchart of BML for partial premixed ame.

Flamelet libraries for unburnt state are built in the same way of the diu-sion ames ones but without the dependence on mixture fraction variance(i.e. the pdf is a Dirac's Delta) since its eect is negligible: Qu = Qu

(Z

)

Interpolation performed on burnt state amelets libraries gives: Cpb, ψb,Wb.Interpolation performed on unburnt state amelets libraries gives: Cpu,ψu, Wu.Unburnt state values are used to invert equation 8.3 to obtain Tu thattogether with Wu, is used to calculate, by perfect gases law (Eq. 5.9), theunburnt gases density ρu needed by c equation (5.51) source term.By burnt and unburnt values, through Eq. 5.66, the ow values of Cp, ψ,W are obtained. The resulting specic heat is used to calculate the owtemperature by inverting Eq. 8.3, and then, by perfect gases law (Eq.5.9), the ow density needed by the the RANS code.

c−equation source term both Eq. 5.54 and 5.56 models, presented in chap-ter 5.6.5, have been implemented.The laminar ame speed (SL) is obtained by interpolating on librariesfunction of mixture fraction Favre's average, built with and experimentalcorrelation from (Elia et al., 2001).The velocity uctuation, needed by equations 5.55 and 5.57 is dened asu′ =

√32k, diusivity is calculated by DT = Cµ

k2

ε where Cµ is a modelconstant usually setted to 0.09.The turbulent length scale is dened as: LT = Cµ

k2

ε .The Damköhler number is calculated through equation 5.39.

8.3.2 Model Validation: GE10 BurnerThe partially premixed model has been validated with the analysis of the

K-ONE DLN combustor of the gas turbine GE10 produced by GE Oil & Gas

181


Figure 8.24: GE 10: schematic representation

Nuovo Pignone (GE-NP).It is a single tubular combustor, settled with its axis orthogonal to the tur-

bine one (g. 8.24) and burns an air-methane premixed ame, that is stabilizedby a diusion pilot jet.

The device then belongs to the Dry-Low NOx combustors category based onthe lean premixed technology.

The combustor has been analysed by a simplied geometrical representationallowing to treat it as axial symmetrical.

The used geometrical model, even when simplied, can analyse every mostimportant combustor subsystems (g. 8.25), keeping in consideration the reac-tants mixing processes and considering the air injection needed for wall coolingand for combustion products dilution, performed to reduce the temperature tovalues allowed in turbine rst stages. The combustor is made of a cylindrical

Figure 8.25: GE 10: schematic representation of the burner, property of GE-NP

182


enclosure having, on its top, a conical cover (g. 8.25).The main stream reaches the reaction zone through an annular duct, that

connect the so-called burner cap to the cylindrical enclosure.The pilot ame is realized by the injection of methane through 32 nozzles

disposed on the cylindrical cover around the already described duct.Burner cap is made of an annular duct where reactant premixing occurs

while the stream has got a centripetal motion (g. 8.25). At the duct inlet aredisposed the fuel injection devices, which are 24 small pipes, with their axis inradial direction, equally distributed around the burner axis. Each of these pipeshas got 7 holes through which the fuel is injected.

In this premixing zone are also present some vanes, with variable staggerangle making them capable of creating a swirl of the reacting mixture. Indesign condition, at which simulations have been performed, the vane are settedin other to remove the swirl, induced by the non uniform incoming air velocity:the reacting ow get into the main reaction zone in, nearby, axial direction.

The cylindrical liner is cooled, mainly, by lm cooling even if, in the coldair side, turbulence promoters are present in order to enhance the heat ex-change. The exceeding air dilution is performed thanks to 4 holes that injectit in correspondence of the combustor outlet. The mesh on which runs have

TCD [K] PCD [bar] mair [kg/s] mfuel [kg/s]686 15.7 37.0 0.728

Table 8.2: GE10: inlet air condition, TCD and PCD stand for Temperature and Pressureat Compressor Discharge

been performed has been made available by GE-NP for (Andreini, 2004). Themodel has a single layer of hexaedrical cells representing a 5 angular portion,for a total of 80000 elements (g.8.26).

In order to represent the whole combustor geometry, by a bidimensionalmesh, each cooling and dilution air holes and the 32 fuel injector have beenrepresented by slots having equivalent eective areas. Fuel injection nozzles,on the 24 pipes, has been represented as injection points equally distributedin circumferential direction, keeping in consideration their real radial position:the fuel injection has been simulated by local mass source on single cells (seeg. 8.26).

The combustor treats an air mass ow of about 37 kg/s shared at 70% bythe main inlet (combustion air), at 20% by the cooling holes and the remaining10% by the dilution holes.

The fuel mass ow is about 0.728 [kg/s]: 5% for the pilots and 95% for themain ame. Fuel temperature has been assumed equal to 293K. All simulationshave been carried out with the standard k− ε high-Reynolds turbulence model(see g. 8.27): for the main inlet has been considered a turbulence intensityof 15% and 10% for all other inlets, while for all inlet has been considered adissipation length of 7% of the hydraulic radius.

183


Figure 8.26: GE 10: used mesh, property of GE-NP

Figure 8.27: GE 10: schematic representation of geometrical model and ame

184


Simulations results have been compared to those by (Andreini, 2004) comingfrom an implementation of the KPP-BML model in STAR − CDTM , with aslightly dierent approach: the model is adiabatic and temperature and density,in both burnt and unburnt states, are adiabatic equilibrium values obtained asfunction of only mixture fraction, for further details see (Andreini, 2004).

In gure 8.27 is shown the schematic representation of expected K-ONEame. Due to the lack of reactant stream swirl, it assumes a conical like shape,extending in axial direction up to far sections from the inlet. The ame is heldin correspondence of the diusion pilot ame.

It has also been shown that the holding may occur in correspondence of thenose (see g. 8.26).

(Andreini, 2004) places the combustion regime that interests the K-ONE, inthe thin reaction zone with huge Karlovitz's numbers that place the ame veryclose to the distributed reaction zone.

The BML model assumes high value of Damköhler's number being formallycorrect in the corrugated ame zone. Then it has to be expected an underesti-mation of the overall reactivity.In reported plots KPP stands for Kolmogorov-Petrovski-Piskunov closure, TFC stands for Turbulent Flame Closure and STAR− CDTM

stands for result from (Andreini, 2004).Considering the mixture fraction plots (Fig. 8.28) at the end of the premixing

duct, results from implemented models are identical and nearly match the re-sults by STAR−CDTM . FOAM predict a slightly more uniform mixing whilethe manufacturer declares that the mixing is uniform. The velocity elds by

0,00 0,02 0,04 0,06 0,08 0,10

0,018

0,020

0,022

0,024

0,026

0,028

0,030

Mix

ture

Fra

ctio

n [-

]

HTC-STAR-CDTM

HTC-FOAM

Radial distance

Figure 8.28: GE 10: plot of mixture fraction near combustor inlet

the implemented models are identical and nearly equal to the STAR− CDTM

one (g 8.29): the main stream from inlet and the lower speed beneath the noseand in correspondence of the recirculation zones are clearly visible. Comparingg. 8.30 and g. 8.31, reporting the vector plot of velocity, is noticeable theagreement in the prediction of the recirculation zone that (Andreini, 2004) hasdemonstrated to be unable to induce a ame holding, instead assured by thepilot ame. Figure 8.32 reporting the progress variable contours and gure 8.33

185


Figure 8.29: GE 10: velocity contours, left STAR− CDTM right FOAM

Figure 8.30: GE 10: vector plot of velocity eld near combustor cap by (Andreini, 2004)

Figure 8.31: GE 10: vector plot of velocity eld near combustor cap by the implementmodel

reporting temperature contours show how the ame structures predicted by theimplemented model are very similar to (Andreini, 2004) results. In particular,TFC closure gives a longer ame respect to STAR− CDTM results while theKPP gives a shorter ame and is the only one that achieve to complete the reac-tion before the end of the combustion dedicated zone localized in correspondenceof dilution air inlets. STAR− CDTM gives a maximum pilot ame temperatureof 2476K, the KPP model 2320K and the TFC model 2307K these dierenceswill be discussed later. Discrepancies between the elds of progress variable,and consequently of temperature, provided by the implemented models have tobe attributed mainly to the progress variable source terms, whose contours arereported in g. 8.37 where is clearly noticeable the links between the sourceterm and the progress variable itself: as the source term lm is longer the c willhave a longer zone with values greater than 0. Indeed, the implemented modelsdier just by the source term denition (Eq. 5.54-5.56) while dierences withSTAR− CDTM are also due to discrepancies between turbulent kinetic energy

186


Figure 8.32: GE 10: progress variablecontours from right to left: KPP closure,TFC closure and STAR− CDTM KPPclosure

Figure 8.33: GE 10: temperature con-tours from right to left: KPP closure, TFCclosure and STAR− CDTM KPP closure

Figure 8.34: GE 10: ame lift, STAR−CDTM

Figure 8.35: GE 10: ame lift, FOAM-KPP

Figure 8.36: GE 10: ame lift, FOAM-TFC

and dissipation elds.Progress variable source terms by implemented models also dier because

of the dierent turbulent ame speeds by which they are calculated (Eq. 5.55-5.57). Contours of these two ame speeds are reported in g. 8.38 that showsthe agreement, in shapes, among the runs results, anyway are also noticeabledierences in magnitude, with the higher provide by the KPP model and thelower by STAR− CDTM , again the turbulence elds play a key role.

Moreover is noticeable that there are no reaction beneath the nose, wherethe presence of a shear layer would justify the reacting mixture ignition that isprevented by the there occurring strong dilution. The manufacturer conrmesthese considerations also informing about the necessity of injecting a small airow in correspondence of the nose produced recirculation zone (purge air) inorder to avoid any reaction. The interaction between the cooling air stream, the

187


Figure 8.37: GE 10: progress variablesource term from right to left: KPP clo-sure, TFC closure and STAR− CDTM

KPP closure

Figure 8.38: GE 10: progress variablesource term from right to left: KPP clo-sure, TFC closure and STAR− CDTM

KPP closure

main reactant stream and the pilot jet produces a lifted ame (see chapter 5.7)that can be spotted in g 8.34-8.35-8.36 that show how the implemented modelsgives result very close to those coming from STAR− CDTM .

The here described implementation of the partially premixed ames modelis part of a larger work not yet completed.

The analysis performed on the GE-10 combustor has shown the good capa-bility of the implemented models moreover the non-adiabiticity of the model,aside with the complete analysis of the mixing process performed by a ameletapproach (considering also mixture fraction variance) allows the prediction ofmore plausible pilot ame temperature that indeed where surely overestimatedby (Andreini, 2004).

188

Chapter 9

Results: Radiative HeatTransfer

This chapter deals with the implementation following by validations for theFinite Volume Method and P1 for RTE. Firstly are described theimplementation procedures, then is described a problem aecting this method,the false scattering, followed by the two RADIARE testcases and theisothermal sphere test case performed to evaluate the eect of false scattering.As well as for FVM the implemented form of RTE is suitable to be used withglobal models and every radiative exchange related variables have to beintendend to be expressed in their total form.

Contents9.1 Results: P1 Model . . . . . . . . . . . . . . . . . . 190

9.1.1 RTE Numerical Modeling . . . . . . . . . . . . . . . 1909.1.2 Model Validation: Concentrical Cylinders Testcase . 191

9.2 Results: Finite Volume . . . . . . . . . . . . . . . 1949.2.1 RTE Numerical Modeling . . . . . . . . . . . . . . . 1949.2.2 False Scattering . . . . . . . . . . . . . . . . . . . . . 1969.2.3 Model Validation: RADIARE . . . . . . . . . . . . . 197

9.2.3.1 Case 1 . . . . . . . . . . . . . . . . . . . . . 1979.2.3.2 Case 2 . . . . . . . . . . . . . . . . . . . . . 199

9.2.4 Model Validation: Spherical Isothermal Domain . . 202

189

Chapter 9. Results: Radiative Heat Transfer

9.1 Results: P1 Model9.1.1 RTE Numerical Modeling

The implemented form of RTE is the approximated one reported in Eq.6.33. The FVM, the present model does not require particular implementingprocedures since it neglects the directional treatment of RTE.

Laplacian term in Eq. 6.33 is represented in normal FOAM formalism, thesame has been done for the RHS. The only modication is the multiplicationof the whole equation for k since its presence at the denominator of LHS mayproduce singularities, especially in combustion systems where for the fresh gasesusually occurs that k → 0, because there the concentrations of main radiativelyactive species CO2, H2O and soot, are practically null.

The real challenging part of the P1 implementation is represented by theMarshak 's boundary condition (Eq. 6.34), that is imposed only when:

• ε > 0

• k > 0

because otherwise we would have singularities, and moreover there would be noneed to solve Eq. 6.34 since it would be simply expressed as:

~n · ~∇G = 0,

where ~n is the normal to the wall with a domain incoming direction.Even if sometimes Marshak 's boundary condition can be simplied, when

it has to be solved it gives many problems. Being an implicit equation in G

it requires an iterative procedure, in the present work it has been chosen torearrange it in order to calculate the Gi value as a function of the gradient ofits previous iteration value Gi−1 and of wall black body emission:

Gi = 4πIb(Tw) + 2(

2ε− 1

)(13k

)~n · ~∇Gi−1.

Unfortunately the term ~∇Gi−1 shows an highly instable behaviour, so a strongexplicit under relaxation is needed to obtain its convergence. The solutionprocedure is reported in Fig. 9.1 where it can be easily spotted the presence oftwo iterative loops, one to solve the Marshak 's equation with G eld from P1last solution, and the other to solve the RTE with the last calculated boundaryconditions.

Marshak 's condition is suitable for diusive surfaces, but P1 equation can besolved together other kind of boundary conditions such as inlet, outlet, symmetryand cyclic that can be used in their normal form.

The wall heat ux, that exists only when emissivity is higher than zero, isnot directly calculated as the wall normal component of Eq. 6.35 again due tothe singularities occurring as k → 0. To avoid singularities Eq. 6.35 has been

190

Results: Radiative Heat Transfer Chapter 9.

Figure 9.1: Flowchart of P1 solving algorithm

rearranged by the use of Marshak 's boundary condition (Eq. 6.34), that hasthe RHS written in term of ~n · ~∇G as well as the wall heat ux that becomes:

qw = ~n · ~qwR= − 1

3k~n · ~∇G|w =

ε

2 (2− ε)(4πIb(Tw)−G)

9.1.2 Model Validation: Concentrical Cylinders TestcaseP1 is a simplied model that behaves well under restrictive hypothesis hence

it may be impossible and even misleading to validate it with experimental test-cases. This is the reason why in the present work has been chosen to validate itsimplementation by comparing the simulation results to the solution obtained byan analytical solution of P1 equation. This may seem a simple task but consid-ering that two iterative loops are required and that implementation ofMarshak 'sboundary condition is troublesome this is not so easy.

The performed testcase is provided by (Modest, 2003, page 477) and repre-sents a grey media at radiative equilibrium placed between two black concentric,innitely long, cylinders of radius R1 and R2 that are isothermal at temperatureT1 and T2 (T1 > T2). The absorption coecient k is constant and uniform. The

191


temperature eld is obtained by:

Φ =T 4 − T 4

2

T 41 − T 4

2

=1 + 3

2τ2lnτ2τ

1 + τ2τ1

+ 32τ2ln

τ2τ1

as function of radial coordinate r (τ = kr) once with the values τ1 = kR1 andτ2 = kR2.

The testcase presented by (Modest, 2003) gives results in adimensional form.The present validation procedure has been performed with cylinders having

R1 = 0.25 m, R2 = 0.5 m and a height of 3 m. The mesh is a multiblockstructured one with 10 equally spaced elements in radial direction, 36 equallyspaced elements in circumferential direction and 29 equally spaced elements inaxial direction.

Four sets of simulations has been carried out with dierent cylinders temper-atures (table 9.1), and the same range of optical thicknesses. The two cylinders

case T1 [K] T2 [K]a) 1000 800b) 1000 500c) 1500 500d) 2500 500

Table 9.1: P1 cylinders: temperatures of inner T1 and outer T2 cylinders.

have been represented by two black diusely emitting-absorbing surfaces thenMarshak 's boundary condition is there imposed, while top and bottom are rep-resented by symmetry-planes in order to simulate innite cylinders.

The temperature elds have been dened by a properly realized utility.The results, represented by the nondimensional radiative heat ux at inner

cylinder

Ψsimulation =qw(r = R1)σ (T 4

1 − T 42 )

are compared to those obtained through the analytical solution of P1 :

Ψanalytical =2

1 + τ2τ1

+ 32τ2ln

τ2τ1

(τ2τ

)

where τ = τ1 = kR1.The percentage errors versus the optical thickness τ = k(R1 − R2) = 0.25k

are reported in table 9.2 . The error is expressed as:

%Err =∣∣∣∣Ψsimulation −Ψanalytical

Ψanalytical

∣∣∣∣ · 100

Simulation results are characterized by small errors (less than 3%) that increasewith optical thickness while the dierence between the temperatures of the twocylinders compensates the error.

192


case τ 0.25 1 2.5 4.0 5.0a) Err% 0.81 0.80 1.84 2.40 2.98b) Err% 0.81 0.88 1.64 2.17 2.87c) Err% 0.83 0.81 1.53 2.16 2.62d) Err% 0.09 0.80 1.53 2.20 2.57

Table 9.2: P1 cylinders: %−Error on nondimensional heat ux Psi versus optical thickness

The matching of analytical solutions implies the good behaviour of the nu-merical model but it is important to state that this does not imply the goodprediction of an eventual analytical solution of the exact form of the RTE.

193


9.2 Results: Finite Volume9.2.1 RTE Numerical Modeling

Here the implementation of the Finite Volume Method for RTEwill be showed.The implemented form of RTE is the one reported in Eq. 6.38, where both

absorption coecient k, the temperatures needed to calculate the black bodyemissions Ibpi and the walls emissivities are given as solver inputs.

While describing the Finite Volume Method has been shown how, the RTEthat is a Lagrangian equation, is treated in its Eulerian form. Some math hadbeen performed on LHS since the gradient operator has not got an implicitformulation while the divergence operator has.

The RHS is integrated in volume by FOAM itself, while the integrationon solid angle is performed by computing Eq. 6.41, where both ∆φi and ∆θi

are calculated considering the number of divisions setted in RTEModelSettingsdictionary, in particular:

∆φi =π

2 · number of divisions for φ∆θi =

π

number of division for θ

Then being φ ∈ [0; 2π] and θ ∈ [0;π] the total number of direction is:

4 · n φ · n θ

The LHS requires more attention: the divergence operator integration is per-formed by FOAM, but again the directional treatment has to be explicitly done.Directional treatment requires the computing of Eq. 6.42 on every faces of everymesh cells, that is done by considering again the division setted by the user.

Boundary conditions for diusely absorbing-emitting opaque surfaces areimposed through Eq. 6.45 that also requires the computation of Eq. 6.42 andfurthermore its component normal to boundary surface ~Di ·~nq. Every directioncontribution to the second term of RHS is added to the stored previouses di-rections value, as well as is done for heat ux (Eq. 6.6) and enthalpy heat sink(Eq. 6.48).

The condition for an opaque surface is not completely exhaustive: aim ofthe present work is to develop radiative heat transfer modelling coupled withcombustion modelling; hence conditions, such as inlet, outlet, symmetry andcyclic, may be needed.

Literature usually deals with only diusive emitting-absorbing surfaces, andit is common use to represent inlets and outlets as diusive surfaces with zeroemissivity since neither absorption nor emission occur on these patches, exceptfor those of participating media that is considered by RTE iteself. Insteadsymmetry and cyclic conditions, in their rigorous formulations, are hardly im-plemented due to the directional nature of radiative exchange:

- symmetry plane is the same of a mirror so every beam is specular reectedthen, on symmetry planes, directions result coupled. And worse a beam

194


Figure 9.2: Flowchart of FMV-RTE solution

impinging on a patch might not have a specular outgoing direction amongthose considered, then interpolations among directional radiative inten-sities are needed: the algorithm is memory consuming and convergencecould be hardly achieved.

- a cyclic condition can be represented by a transmissive surface and samedrawbacks of symmetry have to be faced: directions are coupled, interpo-lations are needed.

These considerations, at least for symmetry conditions, give the necessity andthe possibility to simplify. Indeed, if a symmetry is a reecting surface it couldbe represented by a diusive surface having null emissivity, obviously the ne-glection of directionality would give some error but we have demonstrated thatthis approximation does not give poor results (see subsection 9.2.3.2). If com-bustion processes would be taken into account, other sources of errors would beintroduced such as absorption coecients calculations and neglection of tem-perature oscillation due to RANS approach, each of them of big inuence, sothe error due to symmetry treatment as diusive surface might be one of theless important.

In present work cyclic condition treatment has not been handled.

195


9.2.2 False ScatteringIf the FVM eliminated the problems rising from approximated angular dis-

cretization performed in DOM, the spatial discretization error, due to nitevolume approximation, is still present.

The nite volume method formulation of RTE, as well as other CFD trans-port equations, contains radiative intensity values of cells centroids and faces(Eq. 6.38), then an interpolation scheme is needed. Standing to cited lit-erature ((Modest, 2003), (Viskanta, 2005)), commonly, some CFD schemesare used (step-upwind, CLAM and MUSCL) but also specic schemes havebeen developed (exponential scheme, diamond mean ux scheme and hybridscheme). Again from cited literature can be stated that the rst order stepscheme, counter part of upwind scheme of CFD, as well as the second orderCLAM, ensures a boundedness of solution, while other second order schemesproduce oscillations. Anyway neither the most sophisticated schemes are ableto eliminate the false scattering ((Chai et al., 1993), (Coelho, 2002) and (Josephet al., 2005)) except for the appropriately realized hybrid scheme ((Hong-Shung,2004)).

In this section how nite volume spatial discretization produces errors will bedescribed. Considering a bidimensional enclosure (9.3), lled with an isothermalgrey media, h represents height and κ the absorption coecient, between twoisothermal black walls of surface area S. The system is divided into two controlvolumes of centroids C1 and C2 sharing the face AB non orthogonal to beamdirection. Consider the calculation of the total radiant energy If,total impingingon the upper surface in a direction ~si orthogonal to the two parallel surfaces.The analytical solution is:

Figure 9.3: Bidimensional control volume for RTE

If,total =∫

S

IfdS = IfS (9.1)

with If = Ib,we−κh + κ

∫ h

0

Ibe−κsds = Ib,we

−τ + Ib(1− e−τ) (9.2)

196


where Ib,w is the lower surface black body emission, Ib is the media absorptioncoecient and τ is the media optical thickness (Eq. 6.20). The nite volumediscretization assumes that the radiative intensity is constant across the wholeAB surface but it can be easily demonstrated that radiative intensity at pointA is equal to Ib,w while at point B is equal to the value calculated with Eq. 9.2,and the dierence between them grows aside the optical thickness of the media(Eq. 6.20). This error is surely increased by the step (upwind) scheme, sinceit simply puts on the AB face value the C1 one, however (Joseph et al., 2005)this error was demonstrated to be present even when using the exponentialscheme that exactly solves the RTE for each cell, while hybrid interpolationscheme able to solve the problem (Hong-Shung, 2004) has been developed. Thissimple example allows to understand that the solution to false scattering comesfrom a mesh renement and after by the use of a more sophisticated exponentialscheme, but this is not always feasible, due to oscillations and the usual heavinessof schemes such as the exponential one. While dealing with non isothermalmedia characterized by a variable absorption coecient, that is the usual caseof combustion systems, the radiative exchange take place at shorter distancesthan those between surrounding surfaces so the eective optical thickness isreduced and false scattering phenomenon is mitigated. This, together with theadvanced schemes, induces the use of only the step-upwind scheme in presentwork.

9.2.3 Model Validation: RADIARE9.2.3.1 Case 1

The rst testcase is the Case 1 of RADIARE programme kindly made avail-able by Prof. Bialecki of Silesian University of Technology (Poland). RADIARE((K.P. Helbig et al., 2001)) is an European cooperative research programmeaimed to compare various radiative transfer methods for a set of four bench-mark test cases in order to asseses the model predictions. The calculation wereperformed by the following institutions:

• ENEL Produzione - Research (ENEL), Pisa, Italy; Discrete Transfer Method(DTM) and Discrete Ordinates Method (DOM)

• Lehrsthul fuer Stroemungsmechanik (LSTM), Erlangen, Germany; Monte-Carlo Method (MC)

• Instituto Superior Tecnico (IST), Lisbon, Portugal; Discrete Transfer Method(DTM) and Discrete Ordinates Method (DOM)

• International Flame Research Foundation (IFRF), IJmuiden, Netherland;Boundary Element Method (BEM).

The present testcase represents a well-stirred combustion chamber. Its geometryis quite simple (Fig. 9.4): it is a rectangular combustion chamber with size 1mx 1m x 3m. Participating media temperature is uniform with a value of 1773

197


Figure 9.4: RADIARE case 1: well-stirred combustion chamber

K while its absorption coecient is equal to 0.1 m−1, wall temperature is 1273K everywhere and wall emissivities are equal to 0.5 and 0.8 for two dierentsimulations.

The burner opening and furnace outlet shown in Fig. 9.4 are ignored in thecalculations. Simulations have been performed on a structured mesh made of30 x 30 x 90 uniformly distributed cubic control volumes.

Both runs were performed with 16 directions (nφ = 2 and nθ = 2) andupwind-STEP was the interpolation scheme. The sampled results are the wall

Figure 9.5: RADIARE perfectly stirred combustor wall radiative heat ux contourt. Emis-sivity=0.5

radiative uxes on one of the side-walls (all four having the same values) cen-terline parallel to combustor axis (Fig. 9.4), with a pitch of 0.25 m. Wallradiative ux represents the wall-normal component of radiative heat ux vec-tor (Eq. 6.6). Obtained results are compared to those by DOM and DTMmodels, both implemented by either ENEL or Instituto Superior Tecnico (IST).DOM model has been chosen because is very similar to the used FVM (seesubsec. 6.5.5), while DTM is very accurate and is often used to validate othersolution methods. In Fig. 9.6 and Fig. 9.7 are plotted the results of the imple-mented model (HTC-FVM) and of benchmark simulations. For sections closerto the combustor inlet and outlet results from every solution method are moreor less identical with both emissivities. At intermediate distances the wall heatux is underestimated in magnitude for both emissivities, this is an eect ofthe false scattering and of the diusivity of upwind, the ENEL-DOM shows asimilar but stronger trend enhanced by the higher emissivity, anyway the lackof details on simulation settings used for benchmark models makes impossible

198


0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75

-35

-34

-33

-32

-31

-30

Wal

l rad

iativ

e flu

x (k

W)

Distance from burner (m)

ENEL DOM ENEL DTM IST DOM IST DTM HTC-FVM

Figure 9.6: RADIARE perfectly stirred combustor wall radiative heat ux vs z coordinate.Emissivity=0.5

0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75-39

-38

-37

-36

-35

-34

-33

-32

Wal

l rad

iativ

e flu

x (k

W) ENEL DOM

ENEL DTM IST DOM IST DTM HTC-FVM

Distance from burner (m)

Figure 9.7: RADIARE perfectly stirred combustor wall radiative heat ux vs z coordinate.Emissivity=0.8

deeper analyses.

9.2.3.2 Case 2

The second testcase also belongs to RADIARE programme and concernsa cylindrical combustion chamber. The dimensions of the furnace are 1 m indiameter and 3 m in length (Figure 9.8). The furnace is axially red and sym-metrically cooled. The constant wall emissivity is equal to 0.7 and absorptioncoecient of the medium is equal 0.1 m−1.

The temperature distribution of the wall and the medium temperatures isknown and dened (T (x, y, z)), by keeping in consideration gure 9.8, by the

199


Figure 9.8: RADIARE case 2: cylindrical combustion chamber.

following equations:

T (x, y, z) = To(x, y, z) + (Ta − To(x, y, z))g(x, y, z)

To(p) =T1

2

(1 +

12R

√x2 + y2

)

g(x, y, z) =L

zexp

(−m0 − 1

2

((ln(z/L) +m0

s

)2

+x2 + y2

(σR)2

))

where R is the radius of the chamber, L is the Lent of the furnace, m0 and sare the parameters determining the the position of the maximum temperaturealong the axis Z and σ is a parameter determining the jet width and the spreadof the jet. All parameters assumes the values reported in table 9.3, it should benoticed that two ame lengths are considered by parameter m0. In the above

T1 1200CTa 1800Cm0 0.75 (short ame) 0.25 (long ame)s 1σ 0.375

Table 9.3: Constants value for RADIARE case 2 temperature distribution

Figure 9.9: RADIARE case 2 temperature distribution for short-ame (upper) and long-ame (lower)

equations temperatures are expressed in degrees Celsius then, in order to beused in RTE code, they have to be translated in the Kelvin scale.

200


Four simulations were performed: we have considered both ame lengths ontwo meshes, both structured. One mesh is representative of the whole com-bustion chamber and is made of 9 control volumes in radial direction, 72 oncircumferential direction and 49 in axial direction. The second one is repre-sentative of a quarter of the domain and obtained by splitting the combustionchamber, and the mesh, with two planes intersecting each other on the chamberaxis (Fig. 9.10). These two meshes have been used in order to evaluate theeect of approximations made for symmetry planes (see sec. 9.2.1).

Figure 9.10: RADIARE case 2 quarter mesh

For all simulations walls emissivity was uniform and equal to 0.7.Simulations were performed on 64 directions (nφ = 4 and nθ = 4) and with

upwind-STEP as interpolation scheme.The ame structure, i.e. the temperature eld, has been dened by using a

properly realized utility.In this case are considered the irradiances (Eq. 6.4) on a generating line of

the cylinder with a pitch on 0.25 m on both sides.Obtained results are compared to DOM and DTM from RADIARE even

if the ENEL-DOM is not available. Both Fig. 9.11 and Fig. 9.12 show thegood agreement with benchmark models of the simulations performed on thewhole mesh. Results are better than those obtained through the RADIAREtestcase 1, this can surely be partially attributed to the false scattering indeed,since the temperature eld is not uniform, radiative exchange also takes placebetween zones with dierent gases temperature and not only between gases andsurfaces as happened in the isothermal domain, so the eective optical thicknessis reduced implying a false scattering softening. This is conrmed by the betterperformances while analysing the longer ame where also the distances betweename and surfaces are reduces, while for the shorter ame the error increasesfarther from the ame.

Greater discrepancies between the two meshes were found in those zone closerto the ame where walls reect more being aected by a higher irradiance,this explain the larger dierences for the longer ame. Anyway the adoptedapproximation for symmetry-plane (see sec. 9.2.1) does not give unacceptableerrors.

201


0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75110

115

120

125

130

135

140

145

150

Wal

l Irr

adia

nce

(kW

)

Distance from burner z (m)

IST DOM IST DTM ENEL DTM HTC FVM quarter mesh HTC FVM whole mesh

Figure 9.11: RADIARE case 2 wall irradiance by short-ame

As well as for case 1 the lack of further details on benchmark models simula-tions does not allow to go deeper in the analysis, so it cannot be established howdiscrepancies among models are aected by the denition of the temperatureelds, that may change with mesh renement level.

After this analysis it can be stated that both RADIARE testcases haveshown the good behaviour of the implemented model while used to analyzeoptically thin systems, clearly showing the eect of false scattering moreover therepresentation of symmetry-planes with a diusive absorbing reecting emittingwall has been shown not to compromise the results.

9.2.4 Model Validation: Spherical Isothermal DomainAs already hinted this testcase has been performed to evaluate the eect of

false scattering and indeed it is reported in (Joseph et al., 2005) where is ana-lyzed the behaviour of dierent interpolation schemes for DOM runs performedon unstructured meshes.

The domain is spherical, with a radius equal to R = 1 m, and is surroundedby an isothermal black wall (at Tw = 300 K and Iw(r) = σTw

4/π) and lled ofan isothermal media (at Tm = 1200 K and Ib(r) = σTm

4/π).For such a system exists an analytical solution for the heat sink (Eq. 6.48)

in the sphere centre (r = 0) that is provided by:

~∇ · ~q(r = 0) = 4πk

[Ib(r = 0)−

(Ib,we

−kR + k

∫ R

0

Ib(r)e−krdr

)]= (9.3)

4ke−kRσ(Tm

4 − Tw4)

Furthermore in order to study the eect of false scattering this testcase isintented to show the behaviour of the implemented RTE code while used on

202


0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75125

130

135

140

145

150

155

160

Distance from burner z (m)

Wal

l Irr

adia

nce

(kW

) IST DOM IST DTM ENEL DTM HTC-FVM quarter mesh HTC-FVM whole mesh

Figure 9.12: RADIARE case 2 wall irradiance by long-ame

Figure 9.13: Spherical domain unstructured mesh

unstructured meshes like the one on which simulations have been carried outthat is made of 130088 tetrahedral.

The used interpolation scheme is upwind-STEP. Either φ or θ are dividedin 20 parts for a total of 1600 directions, that large number of direction wererequired to obtain as uniform as possible elds value on the wall.

Runs were performed with a wide range of absorption coecient values tosimulate dierent optical thickness (Eq. 6.20). In this testcase the radiative en-thalpy sinks obtained by simulations (Eq. 6.48) are compared to those obtainedby analytical solutions (Eq. 9.3).The comparison is performed by calculating the error as:

%Err =

∣∣∣∣∣~∇ · ~q(r = 0)simulation − ~∇ · ~q(r = 0)exact

~∇ · ~q(r = 0)exact

∣∣∣∣∣ · 100

whose results are plotted in Fig. 9.14. The error exponentially grows asideoptical thickness, evidencing a strong eect of false scattering. For kabs < 3,that is representative of case involving combustion, error is below 10%.

203


-1 0 1 2 3 4 5 6 7 8 9 10 110

10

20

30

40

50

60

70

80

90

100

HTC-FOAM [Joseph et al., 2005]

k [1/m]

% E

rror

Figure 9.14: Spherical domain %-error on enthalpy sink vs absorption coecient

Same trends, or worse, were obtained by (Joseph et al., 2005). Moreover thistestcase evidenced the need of large number of discretization directions withunstructured meshes. This testcase beside having shown the behaviour of theimplemented model while analysing optically thick systems, has demonstratedthe capability to use unstructured meshes, anyway sided by the requirement ofa great number of directions.

204

Chapter 10

Results: Combustion & RTEModels Coupling

This chapter deals with the radiative transfer modelling coupled with thecombustion modelling, in present section are described those procedures usedto couple the implemented RTE models (see chapt. 9.2 and 9.1) with theimplemented diusion ames model (see chap. 5.5).

The obtained model is able of simulating diusion ames that producessoot.

Contents10.1 Numerical Modeling . . . . . . . . . . . . . . . . . . 206

10.1.1 Participating Media Absorption Coecient Model:Gaseous Media . . . . . . . . . . . . . . . . . . . . . 206

10.1.2 Soot absorption coecient . . . . . . . . . . . . . . . 20610.1.3 Soot modelling . . . . . . . . . . . . . . . . . . . . . 20610.1.4 Coupling with CFD code . . . . . . . . . . . . . . . 207

10.2 Model Validation: Craneld Jet-k Burner . . . . 208

205

Chapter 10. Results: Combustion & RTE Models Coupling

10.1 Numerical Modeling10.1.1 Participating Media Absorption Coecient Model:

Gaseous MediaThe global participating media absorption coecient is obtained by adding

the contributions of gaseous media and soot (Eq. 6.24).The absorption coecient of gaseous media is evaluated with a global model

(Eq. 6.21) by the International Workshop on Measurement and Computation ofTurbulent Nonpremixed Flames (TNF). The global gases absorption coecientis given by the sum of the contributions (aj) of CO2 and H2O weighted withrespective partial pressures (pj = Yj

WWk

p):

k =J∑

j=1

pjaj (10.1)

where:

aj =I∑

i=0

Cji

(1000T

)i

The correlation is reliable in the temperatures range 300− 2500 K.

specie Cj0 Cj1 Cj2 Cj3 Cj4 Cj5

CO2 18.741 −121.310 273.5 −194.050 56.310 −5.8169H2O −0.23093 −1.1239 9.4153 −2.9988 0.51382 −1.8684 · 10−5

Table 10.1: Jet-K: Sandia absorption coecient constants

10.1.2 Soot absorption coecientSoot absorption coecient is evaluated through equation 6.23 with C0 =

5.102652532 and the soot volume fraction fv (Eq. 10.3) provided by a propermodel described in following subsection.

10.1.3 Soot modellingIn order to evaluate the soot concentration (needed to calculate soot volume

fraction Eq. 10.3) a dedicated model is needed, a detailed description of availablemodels can be found in (Pasquini, 2006) and (Lautenberger, 2002) and amongthese only the Mauss' one ((F. Mauss et. al., 2002)) has been implemented inpresent work.

Mauss' model is based on a detailed description of inception, coagulation andfragmentation, surface growth and oxidation processes, that is performed by anutility (written in FORTRAN ) already used in (Da Soghe, 2006) and kindly

206

Results: Combustion & RTE Models Coupling Chapter 10.

made available by Mauss. Here is reported the description of the performedimplementation procedures.

It has to be stated that soot is not considered in amelet libraries so itseect on combustion processes is taken in account only through its enhancingeect on radiative heat exchange.

The soot mass fraction (Ys) is given by a proper conservation equation:

∂ρYs

∂t+∂

(ρuiYs

)

∂xi=

∂

∂xi

(µT

Scs

∂Ys

∂xi

)+ ρs

dfv

dt(10.2)

that has got similar features of other species conservation equations (Eq. 5.1)with the term dffv

dt representing the reaction rate. This equation is not correctfrom a theoretical point of view, as conservation equations are derived fromthe Eulerian analysis of continuum systems, and if this approach is practicallycorrect for uids, its use to represent a particulate matter cannot be completelyjustied.

The reaction rate is expressed as:

dfv

dt=dfv,sg

dt− dfv,fr

dt− dfv,ox

dt+dfv,pi

dt

that contemplates surface growth (sg), fragmentation and coagulation (fr) andoxidation (ox) processes that are linearly dependant on soot volume fraction(Eq. 10.3) and the particle inception (pi).

fv =ρYs

ρswith soot density ρs = 1700− 1900 [kg/m3] (10.3)

In Mauss utility the soot reaction rate terms are given by a pdf of mixturefraction and scalar dissipation rate (Eq. 5.27), as is done for diusion ames(section 5.5.6).

10.1.4 Coupling with CFD codeThe core is the diusion ame code described in chapter 5.5. In gure 10.1

is reported the layout of solver with the added parts.The mixture fraction moments (Z and Z ′′2) and scalar dissipation rate (Eq.

5.27) are given, as inputs, to the Mauss' utility that gives back the soot reactionrate needed, by the CFD code, to solve the Ys equation (10.2). It is interestingto point out that Mauss' utility is written in FORTRAN language while, as wellknown, the present work is developed in FOAM-C++ language hence a linkingbetween dierent programming languages has been performed.

The Ys is used in Eq. 10.3 to calculate the soot volume fraction needed, to-gether with temperature, to calculate the soot absorption coecient (Eq. 6.23).

Mixture fraction moments are also used to calculate CO2 and H2O massfraction that, together with temperature, are used to calculate the gases ab-sorption coecient (Eq. 10.1) that is added to the soot one to obtain the globalparticipating media absorption coecient (Eq. 6.24).

207


Figure 10.1: Diusion ame combined with soot and radiative heat exchange modellingowchart

The global absorption coecient and temperature are given to the RTEsolver (see chap. 9.2 and 9.1) that gives back the radiative enthalpy sink (Eq.6.48) that is used, by the CFD solver, in the enthalpy equation (5.3).

10.2 Model Validation: Craneld Jet-k BurnerIn order to test the implemented model has been chosen the Craneld Uni-

versity (UK) Jet-k testcase ((Z. Wena et al., 2003) and (Da Soghe, 2006)). Ithas got a very simple geometry representing a free-propagating kerosene ame.Fuel characteristics are described in (Da Soghe, 2006), in which simulations havebeen carried out by the same diusion amelet model used for BBody testcase(chapter 5.5) developed in STAR − CDTM whose results (HTC-STAR) arecompared to those obtained in present work. In the work of (Da Soghe, 2006)were used an EDC model too that is the one used on BBody by (Andreini, 2004),but are not reported here. The STAR−CDTM runs had been performed withDOM as RTE model, while for the absorption coecient had been used thesame models of present work. In g 10.2 is reported a schematic representationof the ame experimentally analyzed by (J.B Moss et al., 1994). The ame,burning with C/H moles ratio of 12.5/24.4 and 20% of aromatic hydrocarbonsin volume, is fed through a 1.5 mm orice and conned by a boron silicate pipeof 155 mm.

The fuel, that is liquid at standard conditions, is pre-evaporized in a chamberjust upstream the injection orice. This chamber is kept at 800 K and the fuelgets into the combustion chamber at, approximately 600 K.

The orice is surrounded by a ring shaped slot, with a thickness of 0.25 mm,that injects an homogeneous mixture of oxygen and acetylene intended to stabi-lize the ame. The acetylene pilot mass ow corresponds to the 2% of main fuelmass ow and is not considered in simulation as well as is done by (Da Soghe,2006) that justies this neglecting since the used combustion model are not able

208


of predicting quenching that makes the pilot necessary in the experimental ap-paratus. As well as has been done by (Da Soghe, 2006) the fuel is represented

Figure 10.2: Jet-k: schematically representation

by the civil aeronautical Jet-A, the chemical mechanism is the 16 steps by theNASA and the Mauss' utility represents it as the normal-Decane (C10H22).

The mesh is the same used by (Da Soghe, 2006) and is made of a 10 portion ofa cylinder (bounded by two symmetry planes) and is made of 29000 hexaedrons.

All wall are considered adiabatic from the convection point of view. Whilethe pipe is a black wall for RTE solution. The outlet pressure is atmospheric.

In table 10.2 are reported the boundary conditions applied considering that(Z. Wena et al., 2003) has shown that proles of inlets velocities can be consid-ered uniform. The second coecient of turbulent energy dissipation equation

patch U [m/s] k [m2/s2] ε [m2/s3] T [K] ε

Fuel inlet 22.28 0.6701 4.507 598 0Air inlet 0.234 7.3920 · 10−5 5.2220 · 10−6 288 0

Table 10.2: Jet-K: boundary conditions.

Cε2 (relative to the dissipation term) is changed from 1.92 to 1.84 in order totake in account the jet spreading. While comparing with STAR − CDTM re-sults it is important to consider that (Da Soghe, 2006) did not perform thismodication since that work was aimed to evaluate the dierent behaviour ofcombustion models, chemical mechanism and radiative exchange model ratherthan matching experimental results.

Two runs where performed, one with the Finite Volume Method (with nΦ =2 and nΘ = 6 for a total of 48 directions) and the other with the P1 method.

Experimental values (J.B Moss et al., 1994), are available for:

• Mixture fraction

• Temperature

209


• Soot volume fraction

Measurements had been carried out on the ame axis and on two radius locatedat 100 and 300 mm from orice. Also CO2 and H2O mass fractions, Z ′′2and soot reaction rates are reported (Fig. 10.3-10.4-10.5) to be compared tothose obtained by (Da Soghe, 2006). In order to compare the behaviour theRTE are reported the relative enthalpy heat sinks (Fig. 10.3-10.4-10.5) and theabsorption coecients (Fig. 10.6).

If not explicitly denied results from the two RTE models have to be con-sidered equal or very close. Plots relative to the ame axis (g. 10.3) clearlyshow how every eld is shifted away from the jet orice respect to experimentsand to STAR − CDTM , this is an eect of the turbulence model coecientmodication and of the code itself, being excessive jet penetration also noticedin BBody testcase (chap 5.5). Jet enhanced penetration results in a mixturefraction overestimation, on the axis, respect to either experiments or (Da Soghe,2006) and no dierences are present among the two RTE models.

Turbulence model modication also acts on mixture fraction variance that isunderpredicted respect to STAR−CDTM being its production and dissipationaected by turbulent kinetic energy and its dissipation (see equation 5.37). Axialtemperature is underestimated for distances from burner below 150 mm due to Zoverestimation since signicantly dierent heat sinks by the two models togetherwith same temperatures allows to states that in this zone the eect of radiativeexchange is negligible.

For farther sections axial temperature is more heavily overestimated due alsoto the neglecting of enthalpy sink while solving amelets equations (enthalpy de-fect, (Hossain et al., 2001)), indeed simulations conducted with the EDC modelby (Da Soghe, 2006), that considers the eect of radiative heat exchange onchemistry, show good results in those sections. P1 model gives axial lower val-ues of heat sink, with negative values near to the injection orice, and highertemperatures than FVM does. These eects due to the RTE P1 model itself,are enhanced by the overestimation of temperature that leads to much lowerabsorption coecients (g. 10.6) indeed, where it occurs, soot contribution isoverwhelmed by gases ones.

Axial soot volume fraction is overpredicted due to mixture fraction overes-timation, indeed soot reaction rate is higher than STAR − CDTM one and isshifted downstream. Also the reaction rate in oxidation zone (negative reac-tion rate) is higher in magnitude coherently with higher value of soot volumefraction.

Considering values on the radius at 100 mm from burner (g. 10.4) can benoticed that mixture fraction is very well predicted as well as temperature, thatin this section is mainly aected by the mixture fraction being radiative eectneglectable again as can be stated by the dierences between the heat sinks bythe two models that do not gives dierent temperatures. This good behaviour isan eect of the reduced jet spreading assured by the Cε2 that is clearly visible inall plot especially versus STAR−CDTM . P1 gives overall lower values of heat

210


0 100 200 300 400-0,10,00,10,20,30,40,50,60,70,80,91,01,1

Mix

ture

frac

tion

Z [-]

Distance from burner [mm]

EXP STAR-CD DOM OpenFOAM FVM OpenFOAM P1

0 50 100 150 200 250 300 350 400 450 500 550 6000,000

0,002

0,004

0,006

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0,010

0,012

0,014

0,016

0,018

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ture

Fra

ctio

n V

aria

nce

[-]


HTC FOAM FVM HTC FOAM P1 HTC STAR-CD

0 50 100 150 200 250 300 350 400500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

Tem

pera

ture

[K]



0 50 100 150 200 250 300 350 400 450 500 550 6000,000,010,020,030,040,050,060,070,080,090,100,110,120,130,140,15

CO

2 [mas

s fra

ctio

n]

Distance from burner


0 50 100 150 200 250 300 350 400 450 500 550 6000,00

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

H2O

[mas

s fra

ctio

n]


HTC FOAM-FVM HTC FOAM-P1 HTC STAR-CD

0 100 200 300 400 500 6000,0

2,0x10-6

4,0x10-6

6,0x10-6

8,0x10-6

1,0x10-5

1,2x10-5

soot

vol

ume

fract

ion

f v [-]

axial distance [mm]


0 100 200 300 400 500 600-4,00x105

-2,00x105

0,002,00x105

4,00x105

6,00x105

8,00x105

1,00x106

1,20x106

1,40x106

1,60x106

1,80x106

Rad

iativ

e en

thal

py s

ink

[W]

Distance from bruner [mm]

HTC FOAM-FVM HTC FOAM-P1

0 100 200 300 400 500 600-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5


Soo

t rea

ctio

n ra

te [s

-1]

Radial distance [mm]

Figure 10.3: Jet-K burner : axial plots

211


0 5 10 15 20 250,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40M

ixtu

re fr

actio

n Z

[-]



x=100 [mm]

0 5 10 15 20 25 30-0,001

0,000

0,001

0,002

0,003

0,004

0,005

0,006

0,007

HTC FOAM FVM HTC FOAM P1

Mix

ture

Fra

ctio

n V

aria

nce

[-]


x=100 [mm]

0 5 10 15 20 25 30

400

600

800

1000

1200

1400

1600

1800

Tem

pera

ture

[K]



x=100 [mm]

0 5 10 15 20 25 30

0,00

0,02

0,04

0,06

0,08

0,10

x=100 [mm]C

O2 [m

ass

fract

ion]



0 5 10 15 20 25 30

0,00

0,01

0,02

0,03

0,04

0,05

0,06

x=100 [mm]

H2O

[mas

s fra

ctio

n]



0 5 10 15 20 250,0

5,0x10-7

1,0x10-6

1,5x10-6

2,0x10-6

2,5x10-6

Soo

t vol

ume

fract

ion

f v [-]



x=100 [mm]

0 5 10 15 20 25 30-2,0x105-1,0x105

0,01,0x1052,0x1053,0x1054,0x1055,0x1056,0x1057,0x1058,0x1059,0x1051,0x1061,1x1061,2x1061,3x106

Rad

iativ

e en

thal

py s

ink

[W]



x=100 [mm]

0 5 10 15 20 25 300,00

0,05

0,10

0,15

0,20

0,25

0,30

x=100 [mm]


Soo

t rea

ctio

n ra

te [s

-1]


Figure 10.4: Jet-K burner : radial plots at 100 mm from burner

sink that become negative on ame-air border, and being very close either the

212


0 10 20 30 40 50 60

0,00

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0,04

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0,12

0,14

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ture

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x=300 [mm]

0 10 20 30 40 50 600,0

1,0x10-4

2,0x10-4

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4,0x10-4

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9,0x10-4

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Varianc

e [-]



x=300 [mm]

0 10 20 30 40 50 60

200

400

600

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1200

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[K]



x=300 [mm]

0 10 20 30 40 50 60 70 80

-0,010,000,010,020,030,040,050,060,070,080,090,100,110,120,130,14

x=300 [mm]C

O2 [m

ass

fract

ion]



0 10 20 30 40 50 60 70 800,00

0,01

0,02

0,03

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0,05

0,06

0,07

x=300 [mm]

H2O

[mas

s fra

ctio

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0 5 10 15 20 250,0

1,0x10-6

2,0x10-6

3,0x10-6

4,0x10-6

5,0x10-6

6,0x10-6

7,0x10-6

8,0x10-6

9,0x10-6

1,0x10-5

Soo

t vol

ume

fract

ion

f v [-]



x=300 [mm]

0 5 10 15 20 25 30 35 40 45 50 55 60-2,0x105-1,0x105

0,01,0x1052,0x1053,0x1054,0x1055,0x1056,0x1057,0x1058,0x1059,0x1051,0x1061,1x1061,2x1061,3x1061,4x1061,5x1061,6x106


Rad

iativ

e en

thal

py s

ink

[W]


x=300 [mm]

0 10 20 30 40 50 60

-0,10-0,050,000,050,100,150,200,250,300,350,400,450,500,55



Soo

t rea

ctio

n ra

te [s

-1]

Figure 10.5: Jet-K burner : radial plots at 300 mm from burner

213


0 5 10 15 20 25 300,0

0,2

0,4

0,6

0,8

1,0

1,2


Abs

orpt

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coef

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nt [m

-1]


x=100 [mm]

0 10 20 30 40 50 60 700,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4



Abs

orpt

ion

coef

ficie

nt [m

-1]

x=300 [mm]

0 100 200 300 400 500 6000,0

0,5

1,0

1,5

2,0


Abs

orpt

ion

coef

ficie

nt [m

-1]


axial trend

Figure 10.6: Jet-K burner : absorption coecient plots

temperatures or the absorption coecients, predicted by the two RTE models,these dierences have to be attributed to RTE models only. It can also bespotted a slight shifting toward the axis of mixture fraction variance, carbondioxide and water vapour of results by FVM that induces a slight shifting of theabsorption coecient (g. 10.6). P1 model gives lower values of soot reactionrate that has been found to be due to an higher value of scalar dissipation rate.

Bigger dierences can be found at radius 300 mm far from burner (g 10.5),here mixture fraction has greater values than experiments and again this is aclear eect of enhanced jet penetration. The shapes of obtained results, butnot the values, are closer to experiments than STAR−CDTM ones. The samething has to be said for temperature that is higher with P1. Mixture fractionvariance is higher than in STAR−CDTM and also has got steeper shapes. CO2

and H2O have got lower concentrations than in (Da Soghe, 2006) that also hadobtained more diusive shapes. Soot volume fraction is closer to experimentalresults than STAR−CDTM value, soot is oxidised and still produced while forSTAR−CDTM it is only oxidised, this is due to the still high value of mixturefraction.

Again P1 gives lower value of radiative enthalpy sink, enhanced by a smallerabsorption coecient (g. 10.6), due to higher temperatures in those radialdistances where soot volume fraction is negligible (r > 30 mm).

By an overall analysis of results it can be stated that the spreading of theame is lower than predicted by STAR − CDTM this can be noticed by theoverprediction of mixture fraction on the axis and with the tighter shapes ofmixture fraction variance, species mass fractions and temperature plots in radialdistances, and has several times stated, this is an eect of the turbulence modelmodication. Figure 10.2 clearly shows that P1 model gives lower enthalpy sinkand consequently larger temperature, in gure 10.9 are reported the contours

214


Figure 10.7: Jet-K burner : temperaturecontours left P1, right FVM

Figure 10.8: Jet-K burner : enthalpy ra-diative sink contours left P1, right FVM

Figure 10.9: Jet-K. From left to right contours of: mixture fraction Favre's average, hydrox-ide mass fraction, soot source term [s−1] and soot mass fraction

of mixture fraction Favre average, hydroxide and soot mass fraction and sootreaction rate that clearly shows formation and oxidation zones.

The results show that the implemented models behave well better than canbe stated by simply looking at the reported graphs. Indeed many levels ofapproximation are present, and it could not be otherwise because of the madesimplications.

The fuel is a mixture so a complete modelling would be unaordable, hencethe combustion mechanism is a semi-detailed one, the lack of species experi-mental concentrations impedes to directly evaluate its eects.

The soot modelling is simplied since a detailed modelling would be heavy,and soot formation has not been completely understood yet.

215


The RTE models are both aected by approximation by their nature: the P1is suitable to be used with optically thick systems and Jet-K is not one of thesewhile the FVM is solved with an approximated form of symmetry conditions,the gases absorption coecient is calculated with a global model.

The jet-like nature of the ame cannot be completely reproduced by the twoequations turbulence models although some coecients modications improvesresults.

216

Conclusions

A RANS CFD multipurpose code was developed for heat transfer and com-bustion modeling. Object oriented continuum mechanics libraries based onOpenFOAM has been improved to predict heat transfer and turbulent com-bustion phenomena in gas turbine, many dierent turbulence and combustionmodels has been developed and calculations were performed on representativetests.

Numerical results have been compared mainly with experimental data ob-taining in some cases good agreement. Disagreements might be explain to a pooradherence of the model to the physics, typical of RANS simulations. An a prioriknowledge of the models behavior in the dierent situations is thus required todecouple these eects. That is why models have always been tested againstcases, representing the state of art for turbomachinery cooling and combustionmodeling, known in literature.

The real aim of this investigation was to render OpenFOAM as reliable ascommercial codes in the numerical predictions for heat transfer simulations. Soof primary importance was the adherence to experimental data in the caseswhere commercial solvers are in good agreement too. In fact for the in-housecases, a quite wide database of previous simulations, done with commercialsolvers (CFX (2004); Sta (2004)), was available. Results of commercial andOpenFOAM solvers are in substantial agreement, with major disagreementsconcentrated in zones where trusting commercial solvers is questionable, mean-ing that more evident failures can be attributed to the lack of modeling morethan incapacity in solving the model equations.

In research and industrial environment, it is fundamental to correctly predictthe eorts necessary to obtain a goal, in order to establish which are the mostconvenient developments to invest on. From this point of view, OpenFOAMsurely oers great advantages to the programmers in comparison with othercodes. With the acquired knowledge of language structure, programming isbecoming faster and major objectives can be met.

In implementing such models the adopted strategy has been mainly to sepa-rate the activity of programming, namely developing new models, and running,that is solving cases of interest.

After this long testing OpenFOAM is proven very exible for the implemen-tation of new, solver algorithms, boundary condition types and physical models.

217

Chapter 10. Conclusions

Furthermore it shows a quite good stability and precision both for single proces-sor and multi processor parallel calculations. Nowadays however OpenFOAMstandard release is not ready to be treated as a ready to run code, its selectionof models is too small and only rough estimates can be done. For complex casessuch the ones presented in this thesis, specic model implementation is neces-sary. Theoretically there are no limitations on changes to be done in the code,all les are opened and built in a way to facilitate the necessary additions andmodications. The problem stands in the complexity of the code itself: even ifrough, it has a wide variety of dierent tools, and in the programming languageoften quite dicult to interpret.

The used object oriented language results to be very exible for implementingnew turbulence and combustion models, solver algorithms, boundary conditiontypes and physical combustion models. The modied version of OpenFOAMwas an ecient solution framework for CFD calculation and it take place as areal alternative to the CFD commercial software.

218

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