Kris Van Den Abeele

FACULTY OF ENGINEERING

Department of Mechanical Engineering

Development of high-order

accurate schemes for

unstructured grids

Thesis submitted in fulfillment of the requirements for the

award of the degree of Doctor in de Ingenieurswetenschappen

(Doctor in Engineering) by

Kris Van den Abeele

May 2009

Advisor: Prof. Dr. Ir. Chris Lacor

Abstract

The past decade, there has been a surge of research activities on high-

order methods for unstructured grids in the computational fluid dynamics

(CFD) community. The driving motivation for this surge is the expectation

that these methods have the potential of delivering the required accuracy

for flow problems with complex physics and geometry more efficiently, i.e.

with less CPU-time, than traditional first- and second-order accurate fi-

nite volume (FV) methods. Typical examples of such problems are flows in

which turbulent phenomena play an important role, for instance for tur-

bulent combustion or for the generation of aeroacoustic noise. High-order

methods are also more suited than lower-order ones for the simulation of

the propagation of acoustic waves from a sound source to the observer of

the sound. These acoustic waves typically propagate over a large number

of acoustic wave lengths and possibly undergo various refraction, inter-

ference and scattering effects, which make them quite difficult to resolve

accurately.

These high-order methods for unstructured grids are currently not yet

mature enough to be used for actual industrial applications. They lack the

robustness and ease of use displayed by traditional lower-order CFDmeth-

ods. Furthermore, there are a number of high-order methods under devel-

opment and it is far from clear which method will eventually prove to be

the optimal one. The discontinuous Galerkin (DG) method is arguably the

most popular method. Other high-order methods are the residual distribu-

tion or fluctuation splitting method, the continuous finite element method

and the high-order FV method. The subject of the present PhD research

consists of two relatively new methods, namely the spectral volume (SV)

and the spectral difference (SD) method. The contents of each of the eleven

chapters of this thesis is briefly summarized below.

Chapter 1 gives a brief introduction to the research field of high-order

i

accurate methods. The need for high-order methods specialized for un-

structured grids is illustrated. A summary of their merits and remaining

challenges is given. The issue of efficient algebraic solvers for high-order

methods is also briefly touched.

A literature survey is included in Chapter 2. For completeness, an overview

of the most important literature on the DG method, to which the SV and

SD methods are strongly related, is given. The survey then proceeds with

the available literature for the SV and SD methods themselves. The most

important contributions to algebraic solver algorithms for high-ordermeth-

ods are also mentioned.

The governing equations describing the problems that are solved in the

present thesis are discussed in Chapter 3. The linear advection equation

and Burgers equation are simple model equations that are used to assess

the accuracy of the high-order methods. More practical flow problems are

governed by the Euler equations, the Navier-Stokes (N-S) equations and

the linearized Euler equations.

In Chapter 4, a short summary of the classical FV method is given. Sev-

eral important general concepts, like structured and unstructured grids

and approximate Riemann solvers, are also introduced in this chapter.

An extensive discussion of the SV methodology for the discretization of

convective, diffusive and source terms and for the imposition of boundary

conditions is included in Chapter 5. The quadrature-free formulation of

the SV method is also described. Finally, some criteria for the appropriate

partitioning of a cell into sub-cells or control volumes (CVs), as required

for the SV method, are given.

Chapter 6 contains an analogous discussion for the SD methodology, in-

cluding two new approaches for the discretization of diffusive terms. An

important result of the present PhD research is the solution point indepen-

dence property of the SD method, which is proven and illustrated in this

chapter. The flux point distributions that are used by the SD method are

also discussed.

Another significant result is presented in Chapter 7, where the connec-

tion between the SV and the SD method is investigated. It is shown that

for one-dimensional problems, the SV and the SD method are completely

equivalent if the CV faces of the SV method coincide with the flux points

ii

of the SD method.

The main results of this research are discussed in Chapter 8, where the

conclusions of analyses of the stability and accuracy of the SV and SDmeth-

ods are presented. Several weak instabilities in previously used 1D, 2D

and 3D SV and SD schemes are identified. Where possible, new schemes

that are stable and accurate are designed. For third-order SD schemes on

triangular grids, the stability analysis indicates that there is no flux point

distribution that results in a stable scheme. A similar result was found for

the third-order SV schemes on tetrahedral grids, for which there exists no

stable partitioning into CVs. The results of the analyses are confirmed by

numerical tests.

The issue of efficient solution algorithms for the nonlinear algebraic sys-

tems that arise from any high-order spatial discretization is addressed in

Chapter 9. The Newton-GMRES algorithm and the nonlinear LU-SGS al-

gorithm are discussed, along with their strengths and weaknesses.

The SV and SD methods have been implemented in a C++ code, named

COOLFluiD and developed at the von Karman Institute for Fluid Dynam-

ics. Solutions for flow problems governed by the Euler, N-S and linearized

Euler equations, obtained with the SV and the SD implementations in

COOLFluiD, are presented and discussed in Chapter 10. These results

clearly illustrate the capabilities of these high-order methods.

The final chapter of this thesis, Chapter 11, summarizes the conclusions

of the present PhD research and discusses future challenges for the SV

and SD methods, and for high-order methods in general.

iii

Acknowledgments

The first person whom I would like to gratefully acknowledge is my pro-

moter, professor Chris Lacor, who gave me the opportunity to do a PhD

under his guidance. I thank him for his valuable suggestions during my

PhD research, as well as for granting me sufficient freedom to pursue my

own ideas.

Secondly, I am greatly indebted to professor Z.J. Wang. He invited me

for a stay at Iowa State University in the beginning of 2007, and the two

months I spent with his research group were without a single doubt the

most productive of the past four years. It was a pleasure and a privilege to

work with him.

My gratitude also goes to professor Herman Deconinck, who introduced

me to the COOLFluiD framework, which proved to be an extremely valu-

able asset for my research. On the same note, I would like to thank Tiago

Quintino, Andrea Lani, Thomas Wuilbaut, Nade`ge Villedieu and the other

members of the COOLteam for their support during the implementation

of the spectral volume and spectral difference modules inside COOLFluiD.

The IT support of our system administrator Alain Wery is invaluable for

the research at our department. I greatly appreciate him for his good mood

and everlasting patience through the perpetual stream of requests and

computer problems coming towards him. Thank you very much, Alain!

Our secretary Jenny Dhaes also deserves a very big thank you. The ad-

ministrative support she gives is what allows the PhD students to focus on

their research. And the pleasant conversation she offers when there is a

need to take the mind of the research is much appreciated.

I am pleased to acknowledge my colleagues, or rather, friends. I have been

v

working with Matteo Parsani for more than two years now. Together, we

have faced and won many battles against programming bugs and unco-

operative algorithms. Moreover, he spent a lot of his time to proofread

my thesis. I wish him the best of luck with his further research. The

many discussions on mathematics and physics I have had with Ghader

Ghorbaniasl were always fruitful. His input to this thesis is very much

appreciated. I also think back fondly of the mathematical, but also the

philosophical discussions I have had with Mahdi Zakyani. Santhosh Ja-

yaraju gave me the template that was used for this thesis, and in doing

so, saved a lot of much needed time for me. I also enjoyed working with

him on the teaching assignments we carried out together. In this regard, I

should mention Patryk Widera as well. I guided the wind turbine projects

of the bachelor students with him. He made it fun to do so, even though I

was under the pressure of writing my thesis at that time. My most recent

colleague, Willem Deconinck, made the process of writing my thesis more

bearable with his pleasant mood and sense of humour. In the three years

during which we were office mates, I have shared many laughs and a lot of

joy with Mark Brouns. I can honestly say that the department was never

the same again after he left... I would like to acknowledge Jan Ramboer,

who guided me through my master thesis and initially introduced me to

the world of computational aeroacoustics and high-order methods. I also

thank my colleagues Dean Vucinic, Khairy Elsayed and Sergey Smirnov,

and former colleague Tim Broeckhoven.

Last, but certainly not least, I would like to thank my parents, my grand-

parents, my sisters and brother, and my girlfriend, for the support they

have given me throughout my education and PhD research.

vi

Jury members

President Prof. Gert DESMET

Vrije Universiteit Brussel

Vice-president Prof. Rik PINTELON


Secretary Prof. Patrick KOOL


Internal members Prof. Stefaan CAENEPEEL


External members Prof. Gerard DEGREZ

Universite Libre de Bruxelles

Prof. Z. J. WANG

Iowa State University

Promoter Prof. Chris LACOR


vii

Contents

1 Introduction 1

2 Literature survey 7

2.1 Discontinuous Galerkin method . . . . . . . . . . . . . . . . . 7

2.2 Spectral volume and spectral difference methods . . . . . . . 8

2.3 Time marching and algebraic solvers . . . . . . . . . . . . . . 10

3 Governing equations 13

3.1 Linear advection equation . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Exact solution . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Dimensionless numbers . . . . . . . . . . . . . . . . . 17

3.3.3 Characteristic wave solutions . . . . . . . . . . . . . . 17

3.3.4 Interaction with solid walls . . . . . . . . . . . . . . . 18

3.3.5 Far field boundary conditions . . . . . . . . . . . . . . 18

3.3.6 Exact solution . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 21

3.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.2 Dimensionless numbers . . . . . . . . . . . . . . . . . 23

3.4.3 Interaction with solid walls . . . . . . . . . . . . . . . 24

3.4.4 Exact solution . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Linearized Euler equations . . . . . . . . . . . . . . . . . . . 25

3.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5.2 Far field boundary conditions . . . . . . . . . . . . . . 26

3.5.3 Exact solution . . . . . . . . . . . . . . . . . . . . . . . 27

ix

4 Classical finite volume method 29

4.1 Discretization of the domain . . . . . . . . . . . . . . . . . . . 29

4.2 Finite volume method . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 First-order accurate scheme . . . . . . . . . . . . . . . 32

4.2.2 Higher-order accuracy . . . . . . . . . . . . . . . . . . 34

4.3 Discretization of diffusive terms . . . . . . . . . . . . . . . . . 36

5 Spectral volume method 39

5.1 Discretization of convective and source terms . . . . . . . . . 39

5.2 SV basis polynomials . . . . . . . . . . . . . . . . . . . . . . . 44


5.3.1 Local SV approach . . . . . . . . . . . . . . . . . . . . 47

5.3.2 Second approach of Bassi and Rebay . . . . . . . . . . 48

5.3.3 Interior penalty approach . . . . . . . . . . . . . . . . 49

5.4 Imposition of boundary conditions . . . . . . . . . . . . . . . 49

5.4.1 Dirichlet boundary condition . . . . . . . . . . . . . . 50

5.4.2 Neumann boundary condition . . . . . . . . . . . . . . 51

5.4.3 Mirror boundary condition . . . . . . . . . . . . . . . . 51

5.4.4 Simplified curved slip-wall boundary treatment . . . 52

5.5 Quadrature-free implementation . . . . . . . . . . . . . . . . 54

5.6 Criteria for the selection of partitions . . . . . . . . . . . . . 56

5.7 Partition definitions . . . . . . . . . . . . . . . . . . . . . . . . 58

5.7.1 Partitions for 1D cells . . . . . . . . . . . . . . . . . . . 59

5.7.2 Partitions for triangles . . . . . . . . . . . . . . . . . . 59

5.7.3 Partitions for tetrahedra . . . . . . . . . . . . . . . . . 59

6 Spectral difference method 63

6.1 Discretization of convective and source terms . . . . . . . . . 63

6.2 SD basis polynomials . . . . . . . . . . . . . . . . . . . . . . . 69


6.3.1 Local SD approach . . . . . . . . . . . . . . . . . . . . 71

6.3.2 Second approach of Bassi and Rebay . . . . . . . . . . 71

6.3.3 Interior penalty approach . . . . . . . . . . . . . . . . 71

6.4 Imposition of boundary conditions . . . . . . . . . . . . . . . 72

6.5 Quadrilateral and hexahedral cells . . . . . . . . . . . . . . . 72

6.6 Solution point independence property . . . . . . . . . . . . . 74

6.6.1 Simplex cells . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6.2 Quadrilateral and hexahedral cells . . . . . . . . . . . 77

6.7 Flux point distribution definitions . . . . . . . . . . . . . . . 83

6.7.1 Flux point distributions for 1D . . . . . . . . . . . . . 83

6.7.2 Flux point distributions for triangles . . . . . . . . . . 84

x

6.7.3 Flux point distributions for quadrilaterals . . . . . . . 84

6.7.4 Flux point distributions for hexahedrons . . . . . . . 85

7 Connection between SV and SD methods 87

7.1 SV-SD equivalence in 1D . . . . . . . . . . . . . . . . . . . . . 87

7.2 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8 Stability and accuracy analysis of spatial discretizations 93

8.1 Wave propagation analysis of 1D schemes . . . . . . . . . . . 93

8.1.1 Second-order SV and SD schemes . . . . . . . . . . . . 94

8.1.2 Third-order SV and SD schemes . . . . . . . . . . . . 97

8.1.3 Fourth-order SV and SD schemes . . . . . . . . . . . . 101

8.1.4 Higher-order SV and SD schemes . . . . . . . . . . . . 104

8.1.5 Comparison with DG schemes . . . . . . . . . . . . . . 106

8.1.6 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.2 Wave propagation analysis of 2D schemes . . . . . . . . . . . 112

8.2.1 SV schemes for triangular cells . . . . . . . . . . . . . 113

8.2.2 SD schemes for triangular cells . . . . . . . . . . . . . 126

8.2.3 SD schemes for quadrilateral cells . . . . . . . . . . . 131

8.3 Stability analysis of 3D SV schemes for tetrahedral cells . . 137

8.3.1 Second-order SV schemes . . . . . . . . . . . . . . . . 138

8.3.2 Third-order SV schemes . . . . . . . . . . . . . . . . . 138

8.3.3 Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 144

9 Time marching and algebraic solvers 145

9.1 Newton-GMRES solver . . . . . . . . . . . . . . . . . . . . . . 148

9.2 Nonlinear LU-SGS solver . . . . . . . . . . . . . . . . . . . . 150

9.3 Global and local time stepping . . . . . . . . . . . . . . . . . . 152

10 Applications 155

10.1 Euler test cases . . . . . . . . . . . . . . . . . . . . . . . . . . 156

10.1.1 Acoustic wave propagation . . . . . . . . . . . . . . . . 156

10.1.2 Inviscid cylinder flow . . . . . . . . . . . . . . . . . . . 166

10.2 Navier-Stokes test cases . . . . . . . . . . . . . . . . . . . . . 180

10.2.1 Laminar cylinder flow . . . . . . . . . . . . . . . . . . 180

10.2.2 Laminar pipe bend flow . . . . . . . . . . . . . . . . . 197

10.3 Linearized Euler test cases . . . . . . . . . . . . . . . . . . . 206

10.3.1 Multipolar sound sources in a stagnant fluid . . . . . 206

10.3.2 Multipolar sound sources in a shear flow . . . . . . . 211

xi

11 Conclusions and perspectives 215

11.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

11.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

11.2.1 Spectral volume method . . . . . . . . . . . . . . . . . 218

11.2.2 Spectral difference method . . . . . . . . . . . . . . . . 218

11.2.3 Developments for general high-order methods . . . . 219

A Discontinuous Galerkin method 221

A.1 Discretization of convective and source terms . . . . . . . . . 221

A.2 DG basis functions . . . . . . . . . . . . . . . . . . . . . . . . 223

A.3 Discretization of diffusive terms . . . . . . . . . . . . . . . . . 224

B Methods for stability analysis 225

B.1 Analysis of wave propagation properties . . . . . . . . . . . . 225

B.1.1 1D wave propagation properties . . . . . . . . . . . . . 226

B.1.2 2D wave propagation properties . . . . . . . . . . . . . 229

B.2 Matrix method . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

B.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . 233

B.3 Stability of the time discretization . . . . . . . . . . . . . . . 234

B.3.1 Forward Euler time marching scheme . . . . . . . . . 234

B.3.2 General time marching schemes . . . . . . . . . . . . 237

C p-Multigrid 243C.1 Full approximation scheme . . . . . . . . . . . . . . . . . . . 243

C.2 V-cycles and full multigrid . . . . . . . . . . . . . . . . . . . . 244

C.3 Transfer operators . . . . . . . . . . . . . . . . . . . . . . . . 244

xii

Nomenclature

BDF2 Second-order backward differencing

BR2 Second approach of Bassi and Rebay

CAA Computational aeroacoustics

CEM Computational electromagnetics

CFD Computational fluid dynamics

CFL Courant-Friedrichs-Lewy

CV Control volume

DG Discontinuous Galerkin

DNS Direct numerical simulation

DOF Degree of freedom

FD Finite difference

FDS Flux difference splitting

FV Finite volume

GMRES Generalized minimum residual

IP Interior penalty

LDG Local discontinuous Galerkin

LEE Linearized Euler equations

LES Large eddy simulation

LSD Local spectral difference

LSV Local spectral volume

LU-SGS Lower-upper symmetric Gauss-Seidel

N-S Navier-Stokes

ODE Ordinary differential equation

R-K Runge-Kutta

RAM Random access memory

RANS Reynolds averaged Navier-Stokes

SD Spectral difference

SGS Symmetric Gauss-Seidel

SSP Strong stability preserving

SV Spectral volume

xiii

TVB Total variation bounded

TVD Total variation diminishing

Subscripts

0 Mean value Value in the far fieldac Acoustic pressureac Acoustic pressurebnd Boundary valueC Convective fluxc Characteristic valuecc Current cellD Diffusive fluxgho Ghost valueI Imaginary part of a complex numberi Cell or generating pattern indexi Generating pattern indexin Value at an inletint Internal valuej Solution variable or generating pattern indexj Generating pattern indexL Left celll Flux point indexm Face or eigenmode indexn Component normal to a wall or a facenb Neighbouring cellsR Real part of a complex numberR Right cellt Tangential componentwall Value associated to a wall

Symbols

Logical and Cross section~1d Unit vector defining a direction~1k Unit vector in direction of the wave vector~1n Unit vector normal to a wall or a face~1n,~

Unit normal to a face in a cell-mapped coordinate system

Damping factor in averaging operator for diff. term treatment3 DOF of a 3rd-order SV partition/SD flux point distribution4 DOF of a 4th-order SV partition/SD flux point distributionmax Maximum absolute eigenvalue of the flux Jacobian matrix

xiv

Bias in the averaging operator for the diffusive term treatment3 DOF of a 3rd-order SV partition/SD flux point distribution4 DOF of a 4rd-order SV partition/SD flux point distributionB Length of ~B1, m

~QL(R) Polynomial used in the BR2 lifting operator definition for SD

S Surface of a face, m2

t Time step, sVi Volume of cell with index i, m

3

V~,j Volume of CV j in the mapped coordinate system~

x 1D cell size, m4 DOF of a 4rd-order SV partition/SD flux point distributionjm Kronecker delta functions Entropy error Specific heat capacity ratio, about 1.4 for air3 DOF of a 3rd-order SV partition/SD flux point distribution4 DOF of a 4rd-order SV partition/SD flux point distribution|| || Lebesgue constant of SV partition/basis polynomial set Thermal conductivity, JK1m1s1

Wave length, m~ Lifting operator used with the BR2 diff. term treatmentM0 Spatial discretization matrix for linear advection equation

M+1

Spatial discretization matrix for linear advection equation

M1 Spatial discretization matrix for linear advection equation Dynamic viscosity coefficient, kgm1s1

v Bulk viscosity coefficient, kgm1s1

~ Divergence operator, m1~ Gradient operator, m1~~ Divergence operator in mapped coordinate system Kinematic viscosity coefficient,m2s1

Dimensionless exact angular frequency Angular frequency, s1

Dimensionless numerical or modified angular frequency Numerical or modified angular frequency, s1

Riemann flux upwinding parameter 2D advection speed vector orientation angle Mass density, kg~ Averaged gradient approximations on a face Courant Friedrichs-Lewy (CFL) number, dimensionless~ Polynomial approximation of the conserved variable gradients~ Polynomial approximation of a single conserved variable gradi-

xv

ent

xx, ... Viscous stress tensor elements, Pa Dimensionless exact eigenvalue from wave propagation analysis 2D wave vector orientation angle Exact eigenvalue from wave propagation analysis, s1

m Eigenvalue of dimensionless matrixM~ Vector of mapped coordinates [, , ]

, , Mapped coordinates~sj Mapped coordinates of solution point j

~a Advection speed vector, ms1

a (Amplitude of) advection speed, ms1~B1 Vector defining a 2D generating pattern, m~B1 Dimensionless vector defining a 2D generating pattern~B2 Vector defining a 2D generating pattern, m~B2 Dimensionless vector defining a 2D generating patternc Speed of sound, ms1

CD Drag coefficient, dimensionlessCL Lift coefficient, dimensionlessCP Pressure coefficient, dimensionlesscP Specific heat capacity at constant static pressure, Jkg

1K1

cv Specific heat capacity at constant specific volume, Jkg1K1

d Dimensionalityt

Partial derivative with respect to time, s1x

Partial derivative with respect to spatial x-coordinate,m1

E Specific total energy, Jkg1

e Specific internal energy, Jkg1

FR 1D Riemann flux

FD Drag force, NFi Polynomial approximation of fi~Fi Mapped flux polynomial in cell i

~Fi,l Mapped flux vector at flux point l in cell iFL Lift force, Nfs Vortex shedding frequency, s

1

fC,i -components of convective flux vectors in mapped coordinatesystem

~fC,i Convective flux vector projected in mapped coordinate systemfC x-components of convective flux vectors~fC Convective flux vectorfD x-components of diffusive flux vectors~fD Diffusive flux vector

xvi

~FR ~1n Riemann flux through a face with unit normal ~1nG Amplification factor of a full discretization in space and timeGi Polynomial approximation of gigC,i -components of convective flux vectors in mapped coordinate

system

gC y-components of convective flux vectorsgD y-components of diffusive flux vectorsH Specific total enthalpy, Jkg1

h Specific internal enthalpy, Jkg1

hf Local length scale associated to a face, m

Hi Polynomial approximation of hihC,i -components of convective flux vectors in mapped coordinate

system

hC z-components of convective flux vectorshD z-components of diffusive flux vectorsI Unity matrix

I Imaginary unit number, square root of 1~k Wave vector, mK Dimensionless wave numberk Wave number, m1

Lc Characteristic length scale, mLsj Lagrangian basis polynomial associated to solution point j

Lfl Lagrangian basis polynomial associated to flux point lLi,j SV basis polynomial with index j in cell iM Linear advection spatial discretization matrix

M Mach number, dimensionless~M intj,l Coefficients for quadrature-free computation of internal face resid-

ual contributions

N Number of cells in a gridNf Number of flux points in a cellNs Number of solution variables in a cellNs,GP Number of solution variables in a generating patternNs,tot Total number of solution variables on a meshNfaci Number of faces enclosing cell with index iNGP Number of generating patterns defining a gridP Static pressure, Pap Solution polynomial degreePt Total pressure, PaPr Prandtl number, dimensionlessQ Averaged conserved variables, used for diffusive term treatment

q Spatial-temporal Fourier wave exact complex amplitude

xvii

Q Coefficients of expansion of Q in terms of eigenvectors

Q Spatial Fourier wave numerical complex amplitude as a function

of time

q Spatial Fourier wave exact complex amplitude as a function oftime

q Set of conserved variables

Q Polynomial approximation of a single conserved variableq A single (scalar) conserved variableq0 Initial values of conserved variables q

q0 Initial value of a single conserved variable qqH Heat flux through a surfaceqi Conserved variables in mapped coordinate system

QGPi Solution variables in generating pattern with index i

Qm Eigenmode solution of semi-discretized linear advection equa-

tion

Qm Eigenvector of dimensionless matrixM

Q0m Coefficients of initial solution expansion in terms of eigenvectorsQi,j SD solution variable, solution at solution point j in cell i

Qi,j SD solution variable, mapped coordinate system conserved vari-

ables at solution point j in cell iQi FV solution variable, averaged solution in cell i

Qi Polynomial of degree p+ 1, for gradient computation with SD

Qi

(~)

Solution polynomial in cell i

Qi,j SV solution variable, averaged solution in CV j in cell i

QGPi,j Solution variables in generating pattern with indices i, j

R Spatial residual

r Specific gas constant, about 287Jkg1K1 for airRe Reynolds number, dimensionlessS Discretized source terms

s Source terms

S Strouhal number, dimensionlesss Entropy, Jkg1K1

Sm Coefficients of source term expansion in terms of eigenvectorsSproj Projected surface of an object, m

2

T Temperature, Kt Time, st Dimensionless timeTt Total temperature,K~u Velocity vector [ux uy uz]

T, ms1

ux x-component of velocity, ms1

xviii

uy y-component of velocity,ms1

uz z-component of velocity, ms1

V Computational domainv Specific volume,m3kg1

~x Position vector [x y z]T , mx, y, z Spatial coordinates, mz A complex number

Superscripts Perturbation value Latest available, l if not updated, l + 1 if updated0 Initial quantityl Solver iteration indexn Time iteration indexT Transpose

xix

Chapter 1

Introduction

Computational fluid dynamics (CFD) has known an impressive growth

over the past decades, thanks to the progress in the fields of numerical

solution techniques and computer sciences. Nowadays, commercial flow

solver software packages, which are typically based on second-order ac-

curate finite volume (FV) discretizations of the Reynolds averaged Navier-

Stokes (RANS) equations, can solve large turbulent flow problems for com-

plicated geometries up to engineering accuracy within a few hours on par-

allel computing systems. This has enabled a significant reduction in the

required cost and time for the design process of flow devices such as pumps,

compressors, turbines, or even complete aircraft, by eliminating the need

for the building and testing of prototypes in the early stages of the design

process. These early prototypes have been replaced by cheaper and faster

flow simulations. Consequently, prototypes are now only used in the final

stages, when they are necessary for the validation of the flow simulations

and for the fine tuning of the design.

However, classical second-order accurate algorithms are insufficient to ac-

curately predict the flow in modern applications with complicated geome-

tries, multidisciplinary aspects and complex physics, such as computa-

tional aeroacoustics (CAA) or turbulent combustion problems. These appli-

cations require a more accurate prediction of turbulent phenomena than

what is attainable with second-order RANS simulations. With CAA prob-

lems, this is necessary for an accurate prediction of the sound that is pro-

duced by turbulence. With turbulent combustion, the small scales of tur-

bulence have a significant influence on the mixing of fuel and air, and con-

sequently on the efficiency of the combustion. One then has to resort to

1

CHAPTER 1. INTRODUCTION

a direct numerical simulation (DNS) of the turbulence, or to a large eddy

simulation (LES), since a DNS is often too expensive. These techniques

are based on a direct simulation of the propagation of respectively all or

only the larger turbulent vortices. Second-order algorithms are mostly too

dissipative to resolve these vortices accurately. Higher-order methods are

more suited for such applications, since they have much better wave prop-

agation properties.

In the field of CAA, apart from the production of sound, the propaga-

tion of sound also has to be predicted accurately. The distance between

a sound source and an observer of the sound is typically a large number of

acoustic wave lengths. Thus, the propagation of acoustic waves over large

distances should be simulated. Moreover, these acoustic waves generally

have a small amplitude compared to the mean flow values. Whether one

computes this acoustic wave propagation in the same DNS or LES simu-

lation as the sound production, or in a separate simulation based on the

linearized Euler equations, these small acoustic waves would be entirely

damped out and/or incorrectly dispersed before they reach the observer

position with second-order methods, unless a restrictively large amount of

cells or degrees of freedom (DOFs) is used. Higher-order schemes, because

of their better wave propagation properties, need less DOFs and less com-

putational time to predict the propagation of acoustic waves with sufficient

accuracy.

On structured grids, higher-order accuracy can be achieved relatively eas-

ily with the FV method, by extending the stencil that is used for the re-

construction of the solution variables at the cell faces. This can readily be

done, since information about neighboring cells is immediately available

on structured grids. However, for the complicated geometries of typical in-

dustrial flow problems, the generation of structured grids is very difficult,

requires lots of experience, and often takes days, weeks or even months.

On the other hand, the generation of unstructured grids is much easier,

can be automated, and often takes no longer than a few hours at most.

The price for this easier grid generation is that the immediate accessi-

bility of neighboring cell data is lost. Consequently, increasing the order

of accuracy of a FV method by using larger reconstruction stencil sizes

is much more difficult. Moreover, such reconstruction procedures on un-

structured grids often lead to ill-conditioned or even singular linear alge-

braic systems, which causes a serious degradation of numerical accuracy.

Additional difficulties with this approach arise for the parallelization on

different CPUs. The number of cells of which the solution must be commu-

2

nicated to other CPUs increases with the order of accuracy, and eventually

grows very large. Furthermore, these cells are increasingly further away

from the boundary of the grid partition on a certain CPU, and thus more

and more difficult to access.

Other high-order methods, which are more suited for unstructured grids,

are thus required. Since high-order accuracy always requires the construc-

tion of a high-degree polynomial, which is only possible if enough infor-

mation is available, the solution lies with methods that have a sufficient

amount of DOFs locally in each cell. Such methods thus approximate the

solution by a polynomial of a certain degree on each cell. They are called

compact methods, since only data local to the cell, and possibly data of its

immediate neighbors, is required for the evaluation of the fluxes. It is ob-

vious that this eliminates the need for access to cells that are further away

and that such methods are consequently much more easily parallelizable.

One then distinguishes between methods with which the solution approx-

imation is continuous between two neighboring cells and methods with

which this is not the case. Examples of the former are the continuous finite

element method and the residual distribution method. The discontinuous

Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods

belong to the latter class of methods.

The class of methods with solution representations by cellwise continuous

polynomials is the subject of the present thesis, and more specifically the

SV and SD methods. These two methods were proposed a few years ago

as alternatives to the popular DG method. The DG method has been un-

der development since the 1980s, and consequently has reached a certain

level of maturity. It enjoys a firm mathematical basis and many interest-

ing properties, such as general nonlinear stability for arbitrary cell shapes

and superconvergence properties of certain functionals of its numerical so-

lution. However, its formulation is rather complicated, making it difficult

to interpret physically, and also quite expensive, due to the numerical eval-

uations of surface and volume integrals that are required. The formulation

of the SV method is based on the total sum of fluxes through the enclosing

surface of a control volume (CV), like the FV method. Consequently, it has

a clear physical interpretation and requires only the evaluation of surface

integrals. The SD method directly computes the divergence of the flux vec-

tors in certain solution points, like the finite difference method. Thus, the

SD method is also easily physically interpretable and requires no numer-

ical evaluation of any integrals. The main disadvantages of the SV and

the SD methods are that they do not have as firm a mathematical basis as

3


the DG method yet and that they are not uniquely defined. For the SV

method, partitions of the cells into CVs have to be chosen, while for the SD

method, solution and flux point distributions have to be selected. These

CV partitions and point distributions have a certain number of identifying

parameters, depending on the order of accuracy, which must be specified to

define the SV or SD schemes. The stability and accuracy properties of both

methods depend strongly on these parameters and consequently, a suitable

choice for them is of paramount importance. The proper definition of CV

partitions for the SV method and of solution and flux point distributions

for the SD method is the main focus of the present thesis.

An issue that requires careful attention with high-order spatial methods

is the design of efficient time marching techniques. High-order spatial dis-

cretization operators are usually much stiffer than their low-order counter-

parts. Classical explicit time marching algorithms, such as explicit Runge-

Kutta schemes, have an upper limit for the time step that can be taken for

stability reasons, which is prescribed by the Courant-Friedrichs-Lewy con-

dition or CFL-condition. Such classical algorithms can be very inefficient

in combination with high-order spatial schemes, with which the maximum

time step tends to be very small. Such restrictively small time steps can

be avoided by the use of implicit time marching algorithms, some of which

are stable with arbitrarily large time steps, e.g. the second-order backward

differencing scheme. However, with such algorithms, a nonlinear algebraic

system must be solved at each time step. To reap the benefits of the large

time steps corresponding to implicit schemes, an efficient solver for these

systems is critical. Two algebraic solvers for such systems, namely the

Newton-GMRES and the nonlinear LU-SGS algorithms, are considered in

this thesis.

The outline of the remainder of this thesis is as follows. A survey of the

available literature on the DG, SV and SD methods, and efficient alge-

braic solvers for these high-order methods, is included in Chapter 2. The

governing equations that describe the physical problems which are consid-

ered and solved in the present thesis, namely the linear advection equa-

tion, Burgers 1D inviscid equation, the Euler equations, the Navier-Stokes

equations and the linearized Euler equations, are discussed in Chapter 3.

A short overview of the classical FV method, which introduces some con-

cepts and techniques that are also used by the SV and SD methods, is

given in Chapter 4. The formulations and properties of the SV and SD

methods are then discussed in respectively Chapter 5 and 6. Chapter 7

deals with the relation between the SV and the SD methods, which are

4

shown to be equivalent in the one-dimensional case. An extensive study

of the stability and accuracy properties of the SV and SD methods, in-

cluding illustrations for linear advection problems, is presented in Chap-

ter 8. The issue of efficient time marching with high-order accurate spa-

tial methods is addressed in Chapter 9, where the Newton-GMRES and

nonlinear LU-SGS solution algorithms for nonlinear algebraic systems as-

sociated to implicit time marching schemes and the local time stepping

technique are discussed. The capabilities of the SV and SD methods are

illustrated in Chapter 10, with results for problems governed by the Eu-

ler, Navier-Stokes and linearized Euler equations. Finally, conclusions are

drawn and future perspectives are considered in Chapter 11.

5


6

Chapter 2

Literature survey

A survey of the available literature on high-order methods for the solution

of partial differential equations on unstructured grids is given in this chap-

ter. More specifically, this survey includes an overview of the literature on

the discontinuous Galerkin (DG) method, to which the methods that form

the subject of the present thesis, namely the spectral volume (SV) and the

spectral difference (SD) methods, are strongly related. Subsequently, the

state of the art of the SV and SD methods themselves is discussed. Finally,

a short overview of the literature on time marching and efficient algebraic

solvers for high-order methods has been included.

A nice overview on high-order methods for the solution of the Euler and

the Navier-Stokes (N-S) equations on unstructured grids is given in a re-

view article by Wang [114]. The interested reader is refered to this article

for a review of other high-order methods which have not been considered

here.

2.1 Discontinuous Galerkin method

The DG method is probably the most popular and most developed high-

order accurate method for unstructured grids. It was introduced in 1973

by Reed and Hill [83] for a steady conservation law, namely the neutron

transport equation. It was first used for unsteady advection laws by Van

Leer [108] in 1978.

Important contributions to the development of the DG method for hyper-

bolic conservation laws were made by Cockburn, Shu et al. [23, 2527, 29],

7

CHAPTER 2. LITERATURE SURVEY

with the development of the Runge-Kutta DG (RKDG) methods. A compre-

hensive overview of these RKDG methods can be found in a review article

by Cockburn and Shu [30].

High-order accurate DG results for the compressible Euler and N-S equa-

tions were demonstrated by Bassi and Rebay [9, 10, 12]. Many other ap-

proaches for the discretization of the diffusion equation and the diffusive

terms of the N-S equations were considered. These include interior penalty

(IP) approaches, see e.g. Douglas and Dupont [57], the approach by Bau-

mann and Oden [14, 15] and the local DG approach by Cockburn and Shu

[28]. An interesting overview and study within a unifying framework of

all these approaches can be found in Arnold et al. [2], where their consis-

tency, stability and order of accuracy are discussed. The order of accuracy

of all these approaches for the diffusion equation is limited to p + 1, withp the degree of the solution polynomials. The recovery methods and thesuboptimal or poor mans recovery methods, developed by Van Leer et al.

[110112], based on a better understanding of the physical nature of the

diffusion equation, are capable of achieving higher orders of accuracy, up

to 2p+2. These recovery methods do not fit inside the unifying frameworkproposed by Arnold et al. [2].

Many other researchers made significant contributions to the DG method.

A quadrature-free DG formulation was developed by Atkins and Shu [3].

Hu et al. [53] performed an analysis of the wave propagation properties of

the DG method. A simplified treatment of curved wall boundaries for the

Euler equations with the DG method was proposed by Krivodonova and

Berger [65]. Space-time implicit DG methods for hyperbolic conservation

laws were presented by Lowrie et al. [69], Van der Vegt and Van der Ven

[107], and Klaij et al. [62]. General overviews of the DG method can be

found in lecture notes by Cockburn et al. [24] and by Hartmann [45]. A

short summary of the DG methodology has been included in Appendix A.

2.2 Spectral volume and spectral difference

methods

The basic methodology of the SV method was first presented byWang [113]

in 2002, along with its application to one-dimensional scalar hyperbolic

conservation laws. The extension of the SV method to two-dimensional

scalar equations and a study of different limiting strategies to capture dis-

continuities in the solution, was reported byWang and Liu [116]. The same

8

2.2. SPECTRAL VOLUME AND SPECTRAL DIFFERENCE METHODS

authors presented the extension of the SV method to one-dimensional sys-

tems of conservation laws, along with an optimization study of the SV

partitions, in [117]. Subsequently, a first application to two-dimensional

inviscid flow problems, governed by the 2D Euler equations, was reported

by Wang et al. [121]. A very important issue with all high-order meth-

ods, namely the appropriate treatment of curved wall boundaries, was ad-

dressed for the 2D SVmethod byWang and Liu [118], by using a high-order

geometric mapping of the SV cells near such boundaries. The extension to

three-dimensional systems of conservation laws was then carried out by

Liu et al. [68], who applied the SV method to 3D computational electro-

magnetics (CEM) problems. A first formulation of the SV method for the

N-S equations was developed and presented by Sun et al. [93]. Haga et al.

solved 3D Euler and N-S problems with the SV method, using a supercom-

puter, namely Japans Earth Simulator Computer. A further contribution

to the SV method for diffusive problems was made by Kannan et al. [61],

who investigated different formulations for the discretization of diffusive

terms with the SV method.

Comparisons of the SV method with the DG method were made by Sun

and Wang [91] and Zhang and Shu [125]. The 3D SV method is poten-

tially very expensive if a standard formulation based on Gauss quadrature

rules is used. In analogy with the quadrature-free formulation of the DG

method, a quadrature-free formulation of the SV method, which is more ef-

ficient than the standard formulation in terms of computational time, was

developed by Harris et al. [43].

An effort that was specifically oriented towards the appropriate defini-

tion of high-order accurate SV partitions of simplex cells, based on the

Lebesgue constant criterion formulated by Wang and Liu [116], was made

by Chen [20, 21]. The author of the present thesis also contributed to this

goal, by using stability analysis techniques to assess the accuracy and sta-

bility properties of schemes corresponding to 1D, 2D and 3D SV partitions

of simplex cells. These analyses and their results have been reported in

Van den Abeele et al. [100102] and will be discussed in Chapter 8 of

the present thesis. In recent work by Harris and Wang [42], this analysis

technique was coupled to an optimization algorithm, with the goal of iden-

tifying optimal SV partitions.

The first work on the method which is now known as the SD method dates

from 1996 and is due to Kopriva and Kolias [64] and Kopriva [63], who

called the method conservative staggered-grid Chebyshev multidomain

9


method. Their formulation was for quadrilateral cells and they solved two-

dimensional compressible flow problems based on the Euler equations. A

general formulation of the method, including simplex cells, was given in

2006 by Liu et al. [67], who first called it SD method, and applied it to

two-dimensional scalar conservation laws and CEM problems. The SD

method for simplex cells was then successfully extended to the 2D Euler

equations by Wang et al. [119] and to the 2D N-S equations by May and

Jameson [73] and Wang et al. [120]. An implementation of the SD method

on hexahedral cells for the 3D N-S equations was reported by Sun et al.

[94]. Different approaches for the discretization of the diffusive terms in

the N-S equations with the SD method, based on similar approaches that

were developed for the DG method, were investigated by the present au-

thor, Van den Abeele et al. [105]. Huang et al. [54] reported an implicit

space-time implementation of the SD method.

In collaboration withWang, the present author proved an interesting prop-

erty of the SD method, namely that it is independent of the positions of its

solution points. This property is discussed in Chapter 6. They also per-

formed an extensive study of the stability and accuracy properties of the

SD method, the results of which are discussed in Chapter 8. The solution

point independence property and the stability and accuracy analysis were

published in Van den Abeele et al. [104].

The present author, again in collaboration with Wang, discovered that the

1D SV and SD methods are equivalent. This equivalence was reported in

Van den Abeele et al. [103] and is discussed in Chapter 7. Huynh [55] pro-

posed a set of 1D SV and SD schemes based on Legendre-Gauss quadrature

points and proved that these are stable for arbitrary orders of accuracy.

2.3 Time marching and algebraic solvers

High-order accurate methods are in general much stiffer than low-order

methods. High-order explicit time marching algorithms, such as the popu-

lar four-stage fourth-order accurate Runge-Kutta (R-K) scheme, which has

been very popular since the famous article by Jameson et al. [59], suffer

from restrictively small time steps due to the Courant-Friedrichs-Lewy

(CFL) stability limit, when combined with high-order spatial schemes.

This is especially true for viscous problems, where cells with high aspect

ratios are needed to resolve boundary layers. Strong stability preserving

(SSP) or total variation diminishing (TVD) R-K schemes, originally devel-

10

2.3. TIME MARCHING AND ALGEBRAIC SOLVERS

oped by Shu [89] and Shu and Osher [76], have also been used extensively

in combination with high-order spatial methods, but also suffer from a low

upper limit for the time step.

These restrictive CFL limits can be overcome by using implicit time march-

ing schemes. However, such schemes require the solution of a nonlinear

algebraic system at each time step and consequently, efficient algebraic

solvers are a necessity. Several algorithms have been used in literature.

Element Jacobi methods were used by e.g. Helenbrook and Atkins [46] for

the DG method. Newton-GMRES solvers with preconditioners were used

in combination with DG schemes by Bassi and Rebay [8], and in combi-

nation with SV and SD schemes by Van den Abeele et al. [105, 106]. A

matrix-free Krylov method was applied to DG schemes by Rasetarinera

and Hussaini [82] and to SD schemes by May et al. [72]. Nonlinear lower-

upper symmetric Gauss-Seidel (LU-SGS) solvers, see Chen and Wang [22],

were used with SD schemes by Sun et al. [95, 96] and Van den Abeele et al.

[105], and with SV schemes by Kannan et al. [61], Parsani et al. [77] and

Haga et al. [40]. Line-implicit solvers were developed by Mavriplis [71],

who applied his algorithm to finite volume schemes, and by Fidkowski et

al. [36], who combined theirs with a DG method.

A class of very important convergence acceleration techniques is formed by

the geometric, or h-, and p-multigridmethods, which exploit the strength ofalgebraic solvers, namely removing high-frequency errors, on subsequent

levels of coarser solution approximations. Geometricmultigrid, with which

coarser solution approximations are obtained by switching to coarser grids,

has been widely used in CFD since Jamesons paper [58] in 1983. The p-multigrid algorithm, with which the coarser solution approximations are

obtained by switching to lower-degree polynomials, was introduced by Ron-

quist and Patera [85] in 1987, and further developments were reported by

Maday and Munoz [70]. Since then, it has been applied to the DG method

by Bassi and Rebay [11] and by many other researchers, e.g. Helenbrook

et al. [46, 47] and Fidkowski et al. [36]. It was also used with the SV

method by Van den Abeele et al. [100], Parsani et al. [77] and Kannan et

al. [61]. Applications to the SD method were reported by Premasuthan et

al. [79] and May et al. [72].

11


12

Chapter 3

Governing equations

All physical problems considered in this thesis are governed by a system

of conservation laws, valid on a domain V , and mathematically describedby a set of convection-diffusion equations with source terms, of the form

q

t+ ~ ~fC (q) = ~ ~fD

(q, ~q

)+ s (q) . (3.1)

In this expression, q is the set of conserved variables,~fC (q) the set of con-

vective flux vectors, ~fD

(q, ~q

)the set of diffusive flux vectors and s (q)

the source terms. To complete the definition of a physical problem, appro-

priate initial and boundary conditions must also be specified in addition to

(3.1).

The interpretations of the different terms are as follows. The partial de-

rivative to time of the conserved variables qt

represents the local change

in time of the conserved variables. The divergence of the convective flux

vectors ~ ~fC (q) models the transport (convection) of the conserved quan-tities. The mechanisms that dissipate the conserved quantities (diffusion)

are described by the divergence of the diffusive flux vectors ~~fD(q, ~q

).

Finally, the source terms s (q) model sources or sinks for the conservedvariables.

The definitions of the aforementioned quantities, corresponding to the dif-

ferent physical models under consideration in the present thesis, are given

in the following sections.

13

CHAPTER 3. GOVERNING EQUATIONS

3.1 Linear advection equation

This conservation law models the advection of a scalar conserved variable

q with constant advection speed ~a. Simple though it may be, the linearadvection equation does describe a very important physical phenomenon,

namely the propagation of waves. This phenomenon occurs with all of the

physical models described in the following sections.

3.1.1 Definition

The definitions of q and~fC are

q = (q) and ~fC = (~aq) . (3.2)

The diffusive fluxes~fD and the source terms s are zero.

3.1.2 Exact solution

The exact solution of the linear advection equation is readily available.

Given an initial profile q0 (~x) = q (0, ~x), the exact solution at time t is theinitial profile, translated by the vector ~at:

q (t, ~x) = q0 (~x ~at) . (3.3)

Because of its simplicity and the availability of an exact solution, the linear

advection equation is ideal for testing the accuracy of numerical schemes

for the solution of systems of conservation laws like (3.1).

3.2 Burgers equation

In the present thesis, only the 1D inviscid version of Burgers equation is

considered. It models the nonlinear advection of a single conserved vari-

able q, where the initial profile of q deforms as it is advected, and disconti-nuities are formed after a certain amount of time.

3.2.1 Definition

The 1D inviscid version of Burgers equation is defined by

q = (q) and ~fC =

([q2

2

]), (3.4)

14

3.3. EULER EQUATIONS

and ~fD and s are again zero. With these definitions, (3.1) can be rewrittenin the so-called convection form of Burgers equation:

q

t+ q

q

x= 0, (3.5)

which shows that Burgers equation in 1D does indeed correspond to the

nonlinear advection of a conserved scalar variable q, with the local advec-tion speed equal to the value of q itself. Consequently, parts of the profileof q with higher values are advected faster than parts with lower values.In regions with q

x> 0, the profile of q is then expanded in the x-direction,

while it is compressed in regions with qx

< 0. This explains why, aftera certain amount of time, a discontinuity or shock is formed in the latter

regions.


Given an initial profile q0 (x) = q (0, ~x), the solution at time t obeys

q0 (x) = q(t, x+ tq0 (x)

). (3.6)

From this expression, the exact profile of q can be computed, which makesBurgers equation an interesting model problem to test the performance of

numerical schemes for nonlinear conservation laws like (3.1).

3.3 Euler equations

The Euler equations are used to model the flow of a compressible inviscid

fluid, or the flow of a compressible viscous fluid in flow regions where the

effects of viscosity and heat conduction are negligible. A brief summary of

the Euler equations and its properties is given below. More information

can be found in Callen [19] and Hirsch [50].

3.3.1 Definition

The Euler equations are obtained by expressing three physical conserva-

tion laws. The first is conservation of mass, also known as the continuity

equation. The second law expresses conservation of momentum in the x-, y-

15


and z-direction. The third and last law is the law of conservation of energy.

Thus, the definitions of q and~fC = [fC gC hC ]Tare

q =

uxuyuzE

, fC =

uxu2x + Puxuyuxuz

ux (E + P )

, (3.7)

gC =

uyuxuyu2y + Puyuz

uy (E + P )

and hC =

uzuxuzuyuzu2z + P

uz (E + P )

. (3.8)

There are again no diffusive fluxes ~fD and no source terms s. In theseexpressions, is mass density, P is static pressure, ux, uy and uz are theCartesian velocity components, and E is specific total energy. The velocityvector ~u is [ux uy uz]

Tand its magnitude is denoted by |~u|. Specific total

energy E is related to the specific internal energy e by

E = e+u2x + u

2y + u

2z

2. (3.9)

Another usefull state variable is the specific internal enthalpy h. It is de-fined as

h = e+P

= e+ Pv, (3.10)

where the specific volume v is equal to 1/. Specific internal energy andenthalpy are related to temperature T by the specific heat capacities, re-spectively at constant specific volume, denoted cv, and at constant staticpressure, denoted cP :

e = cvT and h = cPT. (3.11)

The ratio cP /cv is called specific heat capacity ratio and is about 1.4 forair. For an ideal gas, the following relation (ideal gas law) between P , and T holds

p = rT, (3.12)

16


where r is the specific gas constant, which is about 287Jkg1K1 for air.It can then easily be shown that, given values for and r, cv and cP areconstants given by

cv =r

1 and cP =r

1 . (3.13)

The relation between , P , ux, uy, uz, and E in (3.7) and (3.8) is then

E =1

1P

+u2x + u

2y + u

2z

2, (3.14)

resulting in a closed system of five nonlinear partial differential equations

with five unknowns.

3.3.2 Dimensionless numbers

From dimensional analysis one can derive that one dimensionless number

is needed to fully characterize the flow. The dimensionless number that is

commonly used is theMach numberM , defined by

M =|~u|c, (3.15)

where the speed of sound c is given by

c =

P

. (3.16)

The interpretation of the Mach number is straightforward: if it is less

than one, then the local flow is subsonic, and if it is greater than one, then

the local flow is supersonic. It is also an indication of the importance of

compressibility effects in a flow. Typically, a flow can be treated as incom-

pressible if M < 0.3. For higher Mach numbers, compressibility must betaken into account. Given a certain geometry and a specific heat capacity

ratio , it suffices to give a characteristic Mach numberMc, like for exam-ple the Mach number in the far fieldM or at an inletMin, to fully definethe flow regime for an inviscid fluid.

3.3.3 Characteristic wave solutions

The compressible Euler equations support five characteristic wave solu-

tions, the so-called Riemann invariants. Three of these waves propagate

17


with the same wave speed, namely the flow velocity ~u. These characteristicwave solutions represent the advection of entropy and vortexes by the flow.

The other two characteristic wave solutions correspond to acoustic waves.

They propagate in every direction, with a wave speed relative to the mov-

ing fluid equal to the speed of sound c, defined by (3.16). Thus, the absoluteacoustic wave speed is ~u+ c~1d, where the unit vector ~1d can point in any di-rection. Acoustic waves are not only the sound waves that can be heard by

the human ear, they are also the mechanism by which information prop-

agates through a compressible fluid, and no information can travel faster

than the acoustic wave speed. If an obstacle is placed in a fluid flowing

at subsonic speed, then knowledge about this is transported upstream by

acoustic waves, and the flow will adapt itself before the obstacle and go

around it. If the flow is supersonic (|~u| > c) though, then no informationcan travel upstream. Consequently, a supersonic flow upstream of a dis-

turbance cannot adapt itself to this disturbance. In that case, a bow shock,

behind which the flow is subsonic, will form before the obstacle. The sub-

sonic flow between the shock and the obstacle can and will adapt itself and

go around the obstacle.

3.3.4 Interaction with solid walls

Naturally, a fluid cannot penetrate a solid wall. The mathematical formu-

lation of this fact is simply that the fluid velocity component normal to the

wall must be the same as that of the wall itself:

un uwall,n = (~u ~uwall) ~1n = 0, (3.17)

where ~1n is the unit vector normal to the wall. This expression serves as aboundary condition to the Euler equations (3.1), (3.7) and (3.8).

3.3.5 Far field boundary conditions

With external flows, in reality the domain extends to infinity or at least

far enough to justify this assumption. When the Euler equations are solved

approximately using a computer, it is not possible to represent an infinite

domain, and a far field boundary at a finite distance should be introduced.

At this boundary, a suitable far field boundary condition that does not pol-

lute the solution in the internal domain by for instance spurious wave re-

flections, must be introduced. At present, there is no method available that

is capable of completely avoiding such spurious reflections.

18


The simplest and crudest approach is to simply impose the free flow solu-

tion at the far field boundary, and is in fact a Dirichlet boundary condition.

This does not take into account the flow physics, but it generally works

well if the far field boundary is sufficiently far away.

The most commonly used approach, called the characteristics boundary

condition, does take the flow physics into account partially, through a local

1D approximation for the direction normal to the boundary. A number of

physical variables equal to the number of outgoing Riemann invariants is

extrapolated from the internal domain. The remaining physical variables

at the boundary are computed using the expressions that state that the

incoming Riemann invariants corresponding to the 1D approximation are

zero. This approach causes less reflections than a simple Dirichlet bound-

ary condition, but the boundary should still be far enough away. Moreover,

the approach does fail and reflections are generated if outgoing waves con-

taining a tangential velocity component reach the boundary. This is the

case for acoustic waves that do not propagate normal to the boundary, and

for vorticity waves, which might possibly even reverse the flow direction

locally.

A similar difficulty arises for internal flows, where suitable boundary con-

ditions must be used at in- and outflow boundaries.


The exact solution of the Euler equations is in general not available. How-

ever, the following properties can be used to assess the quality of an ap-

proximate solution.

Because the diffusive flux vectors are uniquely zero, there is no dissipa-

tion mechanism that produces heat in regions of smooth flow (away from

shocks). If there is no injection of external heat into the flow, then the

flow is called adiabatic and it follows from the first law of thermodynamics

(conservation of energy) that entropy s, given by

s = cv ln

(P

), (3.18)

is constant throughout the flow field if no shocks are present. Therefore,

19


the entropy error s, defined as

s =P

Pc

c 1, (3.19)

where the subscript c again denotes characteristic values, is a good mea-sure of the accuracy of a numerical solution obtained with a method to

approximately solve the Euler equations.

Following the law of conservation of energy, specific total enthalpy

H = E +P

= h+

u2x + u2y + u

2z

2(3.20)

is preserved at steady state. The associated total temperature, or stagna-

tion temperature, Tt = H/cP is also constant. This is the temperature at astagnation point of the flow, where ~u is zero. Since an Euler flow withoutshocks is isentropic, P/ is constant. From the ideal gas law (3.12), it

then follows that pT

1 is also constant. The total pressure or stagnation

pressure Pt associated to Tt is given by

Pt = P

(TtT

) 1

= P

(1 +

12

M2)

1(3.21)

and is also preserved. The ratio Pt/Pt,c, called total pressure loss, shouldthus be equal to one everywhere.

Other dimensionless quantities which are often used to assess the accu-

racy of a numerical method to solve the Euler equations are the pressure

coefficient CP , the lift coefficient CL and the drag coefficient CD. The pres-sure coefficient is defined as

CP =P Pc12c |~uc|2

. (3.22)

The lift coefficient CL is the dimensionless version of the lift force FL,which is the force exerted by the flow on an object, perpendicular to the

direction of the flow. Its definition is

CL =FL

12c |~uc|2 Sproj

, (3.23)

20

3.4. NAVIER-STOKES EQUATIONS

where Sproj is the projected surface of the object, on a plane perpendicularto the flow direction. The drag coefficient CD is the dimensionless ver-sion of the drag force FD, which is exerted on an object by the flow, in thedirection of the flow. It is given by

CD =FD

12c |~uc|

2Sproj

. (3.24)

The numerically obtained pressure, lift or drag coefficients are compared

to experimental data or to data obtained from a very accurate simulation

for validation.

3.4 Navier-Stokes equations

The Navier-Stokes (N-S) equations model the behaviour of a compressible

viscous fluid. They are the same as the Euler equations that were dis-

cussed in the previous section, with additional diffusive terms to model

the effect of viscosity and heat conduction. Consequently, the N-S equa-

tions inherit a number of the properties of the Euler equations, namely

the definitions of the thermodynamic variables like pressure, internal en-

ergy, entropy, etc., the Mach number, the characteristic wave solutions and

the treatment of far field boundary conditions.

3.4.1 Definition

The conserved variables q and the convective fluxes ~fC are defined in the

same way as for the Euler equations. The diffusive flux vectors ~fD =[fD gD hD]

Tare

fD =

0xxyxzx

xxux + yxuy + zxuz + Tx

, (3.25)

gD =

0xyyyzy

xyux + yyuy + zyuz + Ty

(3.26)

21


and

hD =

0xzyzzz

xzux + yzuy + zzuz + Tz

, (3.27)

with the thermal conductivity of the fluid. The shear stresses are definedby

xx = 2uxx

+ v

(uxx

+uyy

+uzz

),

yy = 2uyy

+ v

(uxx

+uyy

+uzz

), (3.28)

zz = 2uzz

+ v

(uxx

+uyy

+uzz

),

and

xy = yx =

(uyx

+uxy

),

xz = zx =

(uzx

+uxz

), (3.29)

zy = yz =

(uyz

+uzy

),

where is the dynamic viscosity coefficient and v is the bulk viscositycoefficient. The latter is related to a viscous stress caused by a volume

change. Usually, it is assumed that there is no such stress, which leads

to v = 2/3 and a vanishing trace of the shear stress tensor. One alsodefines the kinematic viscosity coefficient as

=

. (3.30)

The gradients of the velocity components and temperature, required for

the evaluation of the diffusive flux vectors, can be computed from the gra-

22

3.4. NAVIER-STOKES EQUATIONS

dients of the conserved variables as follows:

~ux = 1~ (ux) ux

~,

~uy = 1~ (uy) uy

~,

~uz = 1~ (uz) uz

~, (3.31)

~T = 1cv

~ (E) 1cv

[E (u2x + u2y + u2z)] ~

1cv

[ux~ (ux) + uy ~ (uy) + uz ~ (uz)

].

These formulas are usefull if the velocity and temperature gradients can-

not be computed directly.

3.4.2 Dimensionless numbers

Dimensional analysis shows that a compressible N-S flow for a certain ge-

ometry is completely characterized by three dimensionless numbers. The

first dimensionless number is the Mach number, which was discussed in

the previous section and defined by (3.15).

The second dimensionless number that is used is the Reynolds number

Re, given by

Re =c |~uc|Lc

c=|~uc|Lcc

, (3.32)

with Lc a characteristic length scale of the flow problem. The Reynoldsnumber can be interpreted as a ratio of typical inertial stresses and typical

viscous stresses, as is obvious when its definition is rewritten as

Re =c |~uc|2c

|~uc|Lc

inertial forcesviscous forces

. (3.33)

If the Reynolds number is low, then the flow is dominated by the vis-

cous stresses, which results in a laminar flow with smooth streamlines.

If the Reynolds number is high, then the inertial stresses dominate and

the flow will be turbulent, characterized by random vortexes and stochas-

tically changing streamlines.

23


The third dimensionless number is the Prandtl number Pr and is definedas

Pr =ccP,cc

. (3.34)

It is a measure of the ratio between momentum diffusivity and thermal

diffusivity, or the ratio of the rate by which momentum is transferred by

viscosity and the rate by which heat is transferred by conduction. The

latter interpretation can be understood from the following relation:

Pr =cc

ccP,c

momentum diffusivitythermal diffusivity

. (3.35)

There is no length scale involved in the definition of the Prandtl number

and thus it is more a property of a fluid than a measure of a flow state. For

air, the Prandtl number is about 0.72.

3.4.3 Interaction with solid walls

Like an inviscid fluid, a viscous fluid cannot penetrate a solid wall. More-

over, because of viscosity, the velocity component tangential to the wall

should be the same as that of the wall itself. This translates into the ex-

pression

~u ~uwall = 0. (3.36)

Figure 3.1: A typical velocity profile in a

laminar boundary layer. [1]

Near the wall, a thin layer in

the flow will form, where the flow

velocity rapidly changes from the

one imposed by the solid wall

to that of the (mostly inviscid)

flow away from the wall. This

thin layer is called a (momen-

tum) boundary layer and is il-

lustrated in Figure 3.1. The

thickness of a boundary layer de-

creases with increasing Reynolds

number. Its existence must be

taken into account by methods that

solve viscous flow problems, be-

cause a fine resolution is required

to resolve the steep velocity gra-

dients that occur in the boundary

layer.

24

3.5. LINEARIZED EULER EQUATIONS

For a viscous flow, either the wall temperature

T Twall = 0 (3.37)

or the heat flux qH through the wall into the fluid

(~1n ~T

) qH,wall = 0 (3.38)

should also be specified. In the former case, a thermal boundary layer

will form, which is a thin layer near the wall where temperature rapidly

changes from the value imposed by the wall to the one of the flow away

from the wall. The Prandtl number is a measure of the relative thickness

of the momentum and thermal boundary layers, where Pr > 1 correspondsto a thicker momentum boundary layer and Pr < 1 to a thicker thermalboundary layer.

Expression (3.36), combined with (3.37) or (3.38), serves as a boundary

condition to the N-S equations (3.1), (3.7),(3.8) and (3.25) to (3.27).


In a very limited number of test cases, the exact solution of the N-S equa-

tions is known. Mostly, approximate solutions are validated by compar-

ison with results obtained from experiments or from very accurate but

expensive simulations. The dimensionless coefficients, like the pressure

coefficient CP , the lift coefficient CL and the drag coefficient CD, all ofwhich were defined in Section 3.3, can of course also be used with N-S

simulations.

3.5 Linearized Euler equations

The linearized Euler equations (LEEs) model the propagation of small per-

turbations on a known mean flow field. They are used to compute the

propagation of acoustic waves in the far field, where there are no acoustic

sources. More information on the LEEs can be found in Tam [97].

3.5.1 Definition

The LEEs are obtained from the Euler equations (3.1), (3.7) and (3.8), by

decomposing the flow variables , ~u and P into a mean flow value and a

25


perturbation to this mean flow:

= 0 + ,

~u = ~u0 + ~u,

P = P0 + P.

(3.39)

Substituting these variables in the Euler equations, subtracting the mean

flow terms and neglecting products of perturbations, the following sets of

conserved variables q and convective fluxes~fC are obtained

q =

0ux

0uy

0uz

P

, fC =

0ux + ux,0

0ux,0ux + P

0ux,0uy

0ux,0uz

ux,0P + P0ux

, (3.40)

gC =

0u

y + uy,0

0uy,0ux

0uy,0uy + P

0uy,0uz

uy,0P + P0uy

and hC =

0uz + uz,0

0uz,0ux

0uz,0uy

0uz,0uz + P

uz,0P + P0uz

. (3.41)This procedure also leads to a source term

s =

0

0ux + ux,0

ux,0x

+0u

y + uy,0

ux,0y

+0u

z + uz,0

ux,0z

0ux + ux,0

uy,0x

+0u

y + uy,0

uy,0y

+0u

z + uz,0

uy,0z

0ux + ux,0

uz,0x

+0u

y + uy,0

uz,0y

+0u

z + uz,0

uz,0z

( 1)P

u0,xx

+u0,yy

+u0,zz

ux

P0x

uyP0y

uzP0z

, (3.42)which models part of the refraction effects and is zero in the case of a uni-

form mean flow. This term is also responsible for the excitation of linear

instability waves in cases where the mean flow is a shear flow. Possible

solutions to this problem include neglecting this mean flow source term,

Bogey et al. [16], or using a modified formulation called acoustic pertur-

bation equations based on source term filtering, Ewert and Schroder [34].

Additional source terms can be added to introduce sound sources, Bailly

and Juve [4]. The diffusive fluxes~fD are zero.

3.5.2 Far field boundary conditions

The formulation of nonreflective boundary conditions for far field bound-

aries may be evenmore critical for the LEEs than for the Euler and the N-S

26

3.5. LINEARIZED EULER EQUATIONS

equations, since the whole purpose of the LEEs is to simulate the propaga-

tion of acoustic perturbations. Any spuriously reflected wave immediately

pollutes the sound field in the interior domain significantly.

A simple approach is to use a characteristics boundary condition based

on a local 1D approximation, as for the Euler equations. This approach

causes a significant amount of reflections when vorticity waves and oblique

acoustic waves pass through the boundary, and in practice, it should be

combined with a buffer layer, or sponge layer. Such a buffer layer is a

zone between the domain of interest and the actual far field boundary,

where outgoing and spuriously reflected waves are damped. This damping

can be encouraged by progressively increasing the size of the cells towards

the outflow, which increases the numerical damping introduced by the so-

lution method, or by introducing additional damping terms in the buffer

layer. The perfectly matched layer approach, see for instance Hu [52], is

an example of a buffer layer.

Another approach is to use the radiation and outflow boundary conditions

of Tam andWeb [51], which are based on asymptotic solutions of the LEEs.

The outflow boundary conditions are imposed where the mean flow exits

the domain, since entropy, vorticity and acoustic waves pass through this

boundary. At other boundaries, only acoustic waves exit, and there, the

simpler radiation boundary conditions can be used. These boundary con-

ditions can also be combined with a buffer layer.


The analytical solution of the LEEs is known for many problems with

a uniform mean flow, including the advection of entropy, vorticity and

acoustic waves, and acoustic fields due to monopole, dipole and quadrupole

sound sources. A number of benchmark test cases for numerical methods

to solve the LEEs and for the far field boundary conditions can be found in

the proceedings of the NASA workshops on benchmark problems for CAA,

[31, 32, 41], along with analytical or reference solutions.

27


28

Chapter 4

Classical finite volume

method

As discussed in the previous chapter, most physical problems are described

by systems of convection-diffusion equations that are valid on a certain do-

main V . Analytical solutions to these systems are available in only a verylimited number of cases and in general, the domain and the governing

equations have to be discretized to obtain an approximate solution. In the

present chapter, first the discretization of the domain is discussed. Then,

a short overview of the most widely used method for the spatial discretiza-

tion of the governing equations, namely the finite volume (FV) method, is

given.

4.1 Discretization of the domain

Withmost methods, the discretization of the domain is obtained by a space-

filling subdivision into small volumes, called cells, which are polygons, like

triangles and quadrilaterals, in 2D, and polyhedra, like tetrahedra, pyra-

mids, prisms and hexahedra, in 3D. Such a subdivision is called a grid or

mesh. Other geometrical entities that characterize a grid, apart from cells,

are nodes or vertices (the corners of cells), edges (the lines defining cells)

and, in 3D, faces (the polygons enclosing a cell). For 2D grids, as is com-

monly done, the edges will be refered to as faces in the present thesis. A

distinction is made between structured and unstructured grids.

With a structured grid, the cells are always quadrilaterals or hexahedra,

29

CHAPTER 4. CLASSICAL FINITE VOLUME METHOD

XYZ

(a) Structured grid.

XYZ

(b) Unstructured grid.

Figure 4.1: A structured grid (left) and an unstructured grid (right) on a rectangu-

lar domain. Grids generated with Gmsh [37].

and the subdivision is done such that each cell and each node is uniquely

identified by two indices in 2D and by three indices in 3D (one index per

coordinate direction). Each cell index ranges from 1 to a certain numberNl and each node index from 1 to Nl + 1, with l = 1, 2(, 3). A grid con-sequently contains N1 N2[N3] cells and (N1 + 1) (N2 + 1) [ (N3 + 1)]nodes. An example of a structured grid for a rectangle is shown in Figure

4.1(a). The advantage of such a grid is that for a certain cell, information

about neighboring cells is easily available, without needing an extensive

data structure. Only the node coordinates should be stored for each set of

node indices. This allows for very efficient and fast algorithms to discretize

the governing equations. The flexibility of structured grid methods can be

improved by using multiple connected blocks, each of which has its own

set of cells, defined by two or three indices. This also makes it possible

to apply these methods on parallel computing systems, by assigning each

block to a different processor, and in practice, every sizable code that uses

a structured grid can handle multiple blocks of cells. However, the gener-

ation of such multi-block structured grids for the complex geometries that

are common in industrial applications, is very cumbersome, possibly takes

weeks or even months, and is consequently often practically impossible.

The other class of grids, on which the present thesis is focused, consists

of the unstructured grids. With such grids, the cells can have any shape

and are not ordered in any way. Each cell is identified by its own unique

index, denoted i, which ranges from 1 to N , whereN is the number of cells.Similarly, each node is identified by a unique index, which ranges from 1to the total number of nodes. There is no fixed relation between the num-

ber of cells and the number of nodes in an unstructured grid. An example

of an unstructured grid is shown in Figure 4.1(b). Such grids are much

30

4.2. FINITE VOLUME METHOD

more flexible than their structured counterparts and their generation can

be automated relatively easily for complex geometries. Moreover, the ap-

plication to parallel computing systems is fairly straightforward, since any

unstructured grid can be partitioned into an arbitrary number of smaller

unstructured grids, each of which can be assigned to a different processor.

The price of this higher flexibility is that the ready availability of neigh-

boring cell data associated to a structured grid is lost, and that a more ex-

tensive data structure is required. The minimum data sets that should be

stored to define an unstructured grid are, apart from the node coordinates

for each node index, the indices of the connected nodes for each cell index,

and, for each boundary condition, the indices of the connected nodes for

each boundary face index. However, with most unstructured grid meth-

ods, the efficiency would be very bad if only these data sets were avail-

able, and consequently, additional connectivities are usually stored. For

instance, with methods that use a solution representation that is allowed

to be discontinuous at faces, like the finite volume and the discontinuous

Galerkin (DG) method, the indices of the connected cells for each face in-

dex are commonly stored, and sometimes also the indices of the connected

faces for each cell index. Because of this greater difficulty to gain access

to information about neighbouring cells, unstructured grid methods are

generally slower than structured grid methods for a given problem, once

the grid has been generated. Furthermore, the design of high-order accu-

rate spatial methods for unstructured grids is much more difficult than for

structured grids.

4.2 Finite volume method

The topic of the present PhD research consists of two high-order methods

for the spatial discretization of the governing equations on unstructured

grids, namely the spectral volume (SV) and the spectral difference (SD)

method. Before proceeding with a discussion about these two methods in

the following chapters, a brief overview of the classical cell-centered FV

method is given. The aim is to illustrate its capabilities and limitations

for unstructured grids, and the need for new methods that are specifically

designed for high-order accuracy on unstructured grids. Furthermore, a

number of FV techniques that are introduced in the present chapter are

also used by the SV and SD method.

The FV method is the most widely used method for spatial discretization.

This method has been under development since the early 1960s, and con-

31

CHAPTER 4. CLASSICAL FINITE VOLUME METHOD

sequently has reached a considerable level of maturity. It is sufficiently

flexible to solve physical problems with complex geometries with second-

order accuracy on unstructured grids, which is adequate for the error level

required for most engineering problems. Furthermore, it possesses enough

robustness to result in a solution for almost every problem. For these rea-

sons, the majority of commercial software packages for the simulation of

flows in existence today is based on the FV method.

4.2.1 First-order accurate scheme

Consider the general form of a conservation law with only convective fluxes

and source terms:q

t+ ~ ~fC (q) = s (q) . (4.1)

Integrating this expression over each cell, also called control volume in the

case of the FV method, with volume Vi, boundary Vi and the index iranging from 1 to N , and applying Gausss theorem results in

dQidt

= 1Vi

Vi

~fC ~1n dS + 1Vi

Vi

s dV = Ri, (4.2)

where the cell-averaged conserved variables

Qi =1

Vi

Vi

q dV (4.3)

are the solution variables of the FV method and the Ri are the correspond-

ing residuals, which go

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