COMPUTATIONAL AEROELASTICITY USING A PRESSURE-BASED SOLVER

By

RAMJI KAMAKOTI

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2004

Copyright 2004

by

Ramji Kamakoti

This dissertation is dedicated to my parents and sister.

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Professor Wei Shyy for his

constant support and guidance throughout this work. Equally, I would like to thank Dr.

Bhavani Sankar, Dr. Andrew Kurdila, Dr. Renwei Mei, Dr. Nagaraj Arakere, and Dr.

Michael Frank for serving on my committee and providing their support in completing

this work. I would like to extend my sincere gratitude to Dr. Siddarth Thakur and other

members of the computational thermo-fluids laboratory for making the work environment

very lively and enjoyable to work in. Lastly, I would like to acknowledge the support

given by my family throughout my career.

v

TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................................................................................. iv

LIST OF TABLES........................................................................................................... viii

LIST OF FIGURES ........................................................................................................... ix

ABSTRACT..................................................................................................................... xiii

CHAPTER

1 INTRODUCTION ...........................................................................................................1

Aeroelasticity and the Fluid-Structure Interaction Problem.........................................1 Problem Statement........................................................................................................4

2 LITERATURE REVIEW ..............................................................................................10

Aerodynamic Models..................................................................................................10 Physical Models...................................................................................................10 Reduced-Order Models .......................................................................................12

Review of Coupled Computational Aeroelasticity (CAE) Models ............................13 Fully coupled Analysis ........................................................................................14 Loosely and Closely Coupled Analysis...............................................................16

Loosely coupled analysis .............................................................................16 Closely coupled analysis ..............................................................................17

Review of Moving Boundary Models ........................................................................22 Review of Geometric Conservation Law ...................................................................24 Review of Interfacing Techniques..............................................................................26

3 GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM.........................32

Governing Equations ..................................................................................................32 Flow Module .......................................................................................................32

Navier-Stokes equations...............................................................................32 Transformation to curvilinear coordinates ...................................................33

Geometric Conservation Law..............................................................................37 Turbulence Modeling ..........................................................................................38

The k-ε transport equations ..........................................................................40

vi

Filter-based turbulence model for unsteady Reynolds-Averaged Navier-Stokes (RANS) computations .................................................................42

Boundary conditions ....................................................................................43 Wall treatment ..............................................................................................43

Structural Dynamics Model.................................................................................44 Moving Grid Module...........................................................................................48

Overview of Algorithm...............................................................................................50 Discretized Form of Equations............................................................................50 Evaluation of Contravariant velocities on Non-staggered Grid ..........................52 Pressure-Based Flow Solver (Semi-Implicit Method for Pressure-Linked

Equations, SIMPLE) ........................................................................................55 Pressure-Implicit Splitting of Operators (PISO) Algorithm for unsteady

computations ....................................................................................................58 Updating Jacobian values for moving boundary treatment .................................60

First order Implicit Scheme:.........................................................................61 First-order time-averaged scheme:...............................................................62 Second order implicit scheme ......................................................................62 Second order time-averaged evaluation of Jacobian....................................63

Newmark Integration Method for Structure Solver.............................................64 4 COMPUTATIONAL PROCEDURE AND CODE VALIDATION.............................66

Computational Procedure ...........................................................................................66 Geometry definition and Computational Grids ..........................................................67

Geometry Definition............................................................................................67 Computational Grids ...........................................................................................68

Computational fluid dynamic (CFD) grid ....................................................68 Computational structural dynamic (CSD) grid ............................................69

Coupling and Interfacing Procedure ...........................................................................70 Code Validation ..........................................................................................................74

Steady-state CFD Computations .........................................................................75 Unsteady Computations using PISO Algorithm..................................................77

Effect of number of stages on accuracy and stability of PISO algorithm ....80 Momentum Interpolation Techniques for Computing Contravariant Velocities 84 Geometric Conservation Law..............................................................................88

Two-dimensional channel flow: First order backward Euler.......................89 Two-dimensional channel flow: PISO algorithm.........................................94 Three-dimensional elastic wing: AGARD 445.6 .........................................95

Moving Boundary Module ................................................................................100 Structure Solver .................................................................................................102

5 RESULTS AND DISCUSSION..................................................................................105

Coupled Simulation for Incompressible Flow Conditions .......................................105 Comparison of PISO and SIMPLE Algorithms........................................................111 Coupled Simulation for Compressible Flow Conditions..........................................112

vii

Time Scales and Choice of Time Step Size for the Coupled Problem..............113 Flutter Boundary Prediction for AGARD Wing at a Transonic Mach Number116 Flutter Computations Using a Filter-Based Turbulence Model (M=0.96)........124 Summary of Flutter Boundary Prediction for AGARD Wing...........................128

6 CONCLUSIONS AND FUTURE WORK ..................................................................132

Conclusions...............................................................................................................132 Future Directions ......................................................................................................137

LIST OF REFERENCES.................................................................................................138

BIOGRAPHICAL SKETCH ...........................................................................................144

viii

LIST OF TABLES

Table page 2-1. Description and key results of a few fully-coupled analysis methods .......................15

2-2. Description of CAE simulations using CAP-TSD, ENS3DAE and CFL3DAE ........19

2-3. Summary of work with a moving mesh algorithm.....................................................21

2-4. Summary of work related to ALE formulation ..........................................................22

2-5. Comparison of moving mesh algorithms....................................................................24

2-6. Summary of representative interface techniques........................................................28

2-7. Summary of boundary element methods ....................................................................30

4-1. Error Norm versus grid velocity for the four GCL schemes for 3-D wing case using Backward Euler method .................................................................................95

4-2. Error Norm versus grid velocity for the four GCL schemes for 3-D wing case ........98

4-3. Tip deflection at two different time instants for different GCL schemes for 3-D wing case ..................................................................................................................99

4-4. Comparison of wing mode shapes between 10 element beam model (present study) and 120 element plate model.......................................................................102

5-1. Comparison of critical flutter speed and dynamic pressure with experiment and other numerical results ...........................................................................................130

ix

LIST OF FIGURES

Figure page 1-1. Aeroelastic diagram of forces and associated phenomena ...........................................2

1-2. Flutter speed index prediction for AGARD 445.6 wing using several methods..........7

2-1. Sample MDICE environment for aeroelastic simulation ...........................................17

2-2. Coupled fluid-structure flow diagram ........................................................................27

2-3. Varying levels of complexity in modeling for fluids and structures ..........................27

3-1. Displacements Measured with respect to the Elastic Axis ........................................46

3-2. Location of variables u, v and p on a 2-D non-staggered grid for the pressure–based algorithm. .......................................................................................................50

3-3. Overview of the SIMPLE algorithm ..........................................................................58

4-1. Schematic of the AGARD 445.6 wing used in the wind tunnel.................................67

4-2. Overview of the Multi-block CFD grid......................................................................69

4-3. CFD surface grid along with grid distributions at the leading and trailing edges ......69

4-4. Schematic of the FEM grid on the AGARD wing......................................................70

4-5. Schematic to demonstrate interpolation technique.....................................................71

4-6. Schematic of a super element: Portion of the entire structure....................................72

4-7. Sample CFD mesh superimposed on the discretized beam structure.........................73

4-8. Schematic to demonstrate the extrapolation procedure..............................................74

4-9. Top view of the CFD domain showing the type of boundary conditions specified at different surfaces .................................................................................................75

4-10. Steady state surface pressure contours on the AGARD wing ..................................76

x

4-11. Steady state pressure coefficient distribution at different spanwise locations on the top surface ..........................................................................................................76

4-12. Computational domain for flow past square cylinder along with imposed boundary conditions .................................................................................................77

4-13. Periodic oscillation of the cross-stream (v) component of velocity using PISO algorithm for square cylinder case at Re=215..........................................................79

4-14. Vordex shedding past a square cylinder using PISO algorithm for Re=215. A) ∆t=0.001, B) ∆t=0.0005 ...........................................................................................79

4-15. Periodic oscillation of the cross-stream (v) component of velocity using SIMPLE algorithm for square cylinder case at Re=215. .........................................80

4-16. Pressure residual history for unsteady flow over a square cylinder (Re=215).........81

4-17. Periodic oscillation of Cross-stream velocity (v) using different number of stages for PISO algorithm ........................................................................................82

4-18. Computational domain and boundary conditions imposed for flow over a circular cylinder........................................................................................................83

4-19. Pressure residual history for unsteady flow over a circular cylinder (Re=100) .......83

4-20. Periodic oscillation of cross-stream velocity (v) for different number of corrector stages.........................................................................................................84

4-21. Schematic of Cavity flow grid along with boundary conditions ..............................85

4-22. Velocity and pressure contours for cavity flow at Re=100 using different momentum interpolation schemes for various time step sizes at y=0.5 location in the cavity. .............................................................................................................86

4-23. Schematic of computational domain surrounding a cylinder ...................................87

4-24. Velocity and pressure plot for flow around a cylinder at Re=40 using different momentum interpolation schemes for various time step sizes at the symmetry line downstream of the cylinder ...............................................................................88

4-25. Computational grids for channel flow at different time instants ..............................90

4-26. Velocity profile for channel flow with Re=100 at different time instants for coarse grid (151×11) using Backward Euler method...............................................91

4-27. Error norm versus grid velocity using various schemes for channel flow for 151×11 grid using Backward Euler method ............................................................91

xi

4-28. Velocity profile for channel flow with Re=100 at different time instants for fine grid (301×21) using Backward Euler method..........................................................93

4-29. Error norm versus grid velocity using various schemes for channel flow for 301×21 grid using Backward Euler method ............................................................93

4-30. Velocity profile for channel flow at different time instants for 151x11 grid using PISO method ............................................................................................................95

4-31. Plot depicting the arbitrary movement of the wing in the spanwise direction ........97

4-32. Error norm versus grid velocity for the various schemes for AGARD wing using Backward Euler method .................................................................................97

4-33. Spanwise deflection of AGARD wing at four different time instants......................99

4-34. Schematic of multi-block grid used to validate moving mesh module ..................100

4-35. Effect of the 2 parameters, FACMIN and β, on the re-meshing.............................101

4-36. Tip deflection of AGARD wing versus number of time steps for ∆t=0.0001........104

4-37. Tip deflection of AGARD wing versus number of time steps for ∆t=0.001..........104

5-1. Spanwise wing shapes at different time instants (Grid configuration I) ..................106

5-2. Time varying displacement of wing at different spanwise locations (Grid configuration I).......................................................................................................107

5-3. Time history of lift coefficient for AGARD 445.6 wing subject to 1-degree angle of attack for both grid configurations.....................................................................108

5-4. Time history of lift/drag ratio for AGARD 445.6 wing subject to 1-degree angle of attack for both grid configurations. ....................................................................108

5-5. Pressure contour on the surface of the wing at steady state .....................................109

5-6. Comparison of lift coefficient time history for AGARD wing subject to different angles of attack for grid configuration I.................................................................110

5-7. Comparison of lift coefficient time history for AGARD wing subject to different angles of attack for grid configuration I.................................................................110

5-8. Spanwise displacements at three different time instants to compare PISO and SIMPLE methods using incompressible flow around an AGARD wing example at a Re=3.64x105 based on unit root chord. ...........................................................112

5-9. Diffusive and convective time scales near wing tip region for different time step sizes and grids ........................................................................................................114

xii

5-10. Diffusive and convective nondimensional paramter at wing tip spanwise location for different grids and time step sizes ......................................................116

5-11. Generalized displacement versus time for three different dynamic pressures for ∆t=5x10-5................................................................................................................118

5-12. Generalized displacement versus time for three different dynamic pressures for ∆t=1x10-5................................................................................................................119

5-13. Effect of grid resolution on generalized displacements using similar CFL numbers ..................................................................................................................120

5-14. Flutter points for different choice of time step sizes (1x10-5 and 5x10-5 for grid I (350 K points) and grid configurations (350 K grid and 800 K grid) ....................121

5-15. Tip deflection of wing versus time for grid I (350 K points) using a time step size of 5x10-5 ..........................................................................................................122

5-16. Wing shapes at maximum and minimum tip deflection points ..............................123

5-17. Mach number contours representing supersonic region in the flow domain at mid-span plane. ......................................................................................................123

5-18. Surface pressure contours indicating supercritical region or region of supersonic flow on the surface of the wing. .............................................................................124

5-19. Blending function plot at mid-span plane...............................................................126

5-20. Comparison of filter-based model and standard k-e model for q/qe=0.98 with ∆t=5x10-5 and filter size, ∆=0.15. ..........................................................................127

5-21. Flutter boundary comparison between filter-based turbulence model and standard k-ε model using grid I and ∆t=5x10-5 ......................................................127

5-22. Spanwise wing shape at maximum and minimum tip deflection for (left) M=0.678 and (right) M=1.072 ...............................................................................129

5-23. Mach number contours at maximum and minimum tip deflection points..............129

5-24. Summary of flutter speed index prediction for AGARD 445.6 wing ....................131

xiii

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

COMPUTATIONAL AEROELASTICITY USING A PRESSURE-BASED SOLVER

By

Ramji Kamakoti

August 2004

Chair: Wei Shyy Major Department: Mechanical and Aerospace Engineering

A computational methodology for performing fluid-structure interaction

computations for three-dimensional elastic wing geometries is presented. The flow solver

used is based on an unsteady Reynolds-Averaged Navier-Stokes (RANS) model. A well-

validated k-ε turbulence model with wall function treatment for near wall region was

used to perform turbulent flow calculations. Relative merits of alternative flow solvers

were investigated. The predictor-corrector-based Pressure Implicit Splitting of Operators

(PISO) algorithm was found to be computationally economic for unsteady flow

computations. Wing structure was modeled using Bernoulli-Euler beam theory. A fully

implicit time-marching scheme (using the Newmark integration method) was used to

integrate the equations of motion for structure. Bilinear interpolation and linear

extrapolation techniques were used to transfer necessary information between fluid and

structure solvers. Geometry deformation was accounted for by using a moving boundary

module. The moving grid capability was based on a master/slave concept and transfinite

xiv

interpolation techniques. Since computations were performed on a moving mesh system,

the geometric conservation law must be preserved. This is achieved by appropriately

evaluating the Jacobian values associated with each cell. Accurate computation of

contravariant velocities for unsteady flows using the momentum interpolation method on

collocated, curvilinear grids was also addressed. Flutter computations were performed for

the AGARD 445.6 wing at subsonic, transonic and supersonic Mach numbers. Unsteady

computations were performed at various dynamic pressures to predict the flutter

boundary. Results showed favorable agreement of experiment and previous numerical

results. The computational methodology exhibited capabilities to predict both qualitative

and quantitative features of aeroelasticity.

1

CHAPTER 1 INTRODUCTION

The term computational aeroelasticity (CAE) generally refers to coupling high-

level computational fluid dynamic (CFD) methods with structural dynamic tools to

perform aeroelastic analysis. Recently, CAE has gained interest as considerable progress

has been made in CFD, computational structural dynamics (CSD), and in computer

technologies. Extensive research on CFD and CSD has already been done. The aim of our

study was to develop a closely coupled CAE model (comprising a detailed CFD model

with a simplified CSD model) to perform fluid-structure interaction computations on

three-dimensional wing bodies.

Aeroelasticity and the Fluid-Structure Interaction Problem

Aeroelasticity can be defined as the phenomena associated with the interaction of

aerodynamic forces and inertial forces within elastic structural systems. There are also

aeroelastic phenomena associated with interaction between aerodynamic and elastic

forces alone (Bisplingoff et al. 1955). Aeroelastic problems mainly arise from the flexible

nature of the structure. In other words, rigid structures do not experience aeroelasticity of

any sort. It is well known that external forces acting on a flexible structural system (such

as a wing) lead to a deformation in the wing geometry, and this structural deformation

thereby leads to additional aerodynamic loads.

2

Figure 1-1. Aeroelastic diagram of forces and associated phenomena

Generally, aeroelasticity has two major classes: dynamic and static. Dynamic

aeroelasticity usually involves interactions among inertial, aerodynamic and elastic

forces; whereas static aeroelasticity involves interaction between aerodynamic and elastic

forces. Figure 1 shows different kinds of aeroelastic phenomena depending on how the

different forces interact.

Dynamic aeroelasticity: includes phenomena such as flutter, buffeting, and

dynamic response. Flutter is an oscillatory dynamic instability (primarily caused by the

elasticity of the structure) that occurs in an aircraft in flight at high speeds. The speed at

which this occurs is called flutter speed or critical speed. Buffeting is a phenomenon that

occurs because of transient vibrations of aircraft structural components (due to

aerodynamic impulses such as wake behind wings). Dynamic response includes transient

response associated with aircraft components; and is caused by rapidly applied loads

(such as gusts, moving shock waves, or other dynamic loads). All of the above-mentioned

phenomena are all transient phenomena; hence the term dynamic aeroelasticity.

Static aeroelasticiy: includes phenomena such as load distribution, divergence,

and system reversal. Load distribution occurs because static deformation influences the

AerodynamicForces

Elastic Forces

InertialForces

Static Aeroelasticity • Load distribution • Divergence • Control system reversal

Dynamic Aeroelasticity• Flutter • Buffeting • Dynamic response

3

distribution of aerodynamic pressures over the structure. Divergence is another static

instability (it occurs at a speed called divergence speed) in which the elasticity of the

lifting surface plays a critical role in producing the instability. Another static aeroelastic

phenomenon is control system reversal (it occurs at control reversal speed) in which the

effects of structure displacements are cancelled by elastic deformations of the structure

itself.

Almost every flight vehicle (manned or unmanned) that flies through the

atmosphere undergoes some degree of aeroelasticity. Catastrophic phenomena such as

flutter must be avoided at all costs, and all vehicles must be cleared of such phenomena

before they are put to use. Flight test and wind-tunnel testing are two ways to test for

such phenomena, but they are both expensive and occur late in the design process. Hence,

computational techniques are used first, to assess the aeroelastic characteristics of these

flight vehicles.

While computational methods that study different aspects of aeroelastic response

have been studied for some time, numerous open research issues remain to be resolved.

For example, many approaches in computational aeroelasticity seek to synthesize

independent computational approaches for the aerodynamic and the structural dynamic

subsystems. This strategy is known to be fraught with complications associated with the

interaction between the two simulation modules. Some of the issues arise from the fact

that CFD and CSD mesh systems are quite different. Frequently, the former uses a

Eulerian or spatially fixed-coordinate system, while the latter uses a Lagrangian or

material fixed-coordinate system. Hence, care must be taken to develop a suitable

interfacing technique between the two modules. Also, the time scales can be very

4

different for the two modules, hence one must be careful while performing unsteady

calculations.

There are three major classifications for CAE: fully coupled, closely coupled, and

loosely coupled analyses. In loosely coupled analysis, the fluid and structure modules are

treated as two separate modules, with only external interaction between them. This kind

of methodology can be seen as a multi-disciplinary problem. This method is limited to

small perturbations with moderate linearity. In fully coupled analysis, the governing

equations for fluids and structures are combined into one set of equations, and these

equations are solved and integrated simultaneously. Since the matrices associated with

structures are orders of magnitude stiffer than those associated with fluids, it is virtually

impossible to solve the entire system using a single numerical scheme. Methods have

been developed using fully coupled methods, but they are restricted to two-dimensional

problems and small-scale three-dimensional problems. In the closely coupled approach,

fluids and structures are modeled in separate domains, but are coupled into one module

by an interface technique. The exchange of information between these modules takes

place at the interface or the boundary. The coupling is integrated, thereby allowing the

two modules to exchange information at the boundaries in an efficient manner. Our study

emphasizes this kind of approach.

Problem Statement

The objective of our study was to develop a computational model that is capable of

performing fluid-structure interaction computations on three-dimensional geometries.

Our model was based on a three-dimensional, multi-block, structured CFD solver for the

Navier-Stokes equations. Structural modal dynamic equations were solved

simultaneously and were strongly coupled with the flow equations using fully implicit

5

(iterative) and semi-explicit (non-iterative) time-marching methods. Since the structure

deformation is usually small, a linear structure model was found to be sufficient. To

address the unsteady flow around deforming structures, since the flow can be complex

because of compressibility, existence of shock waves, and effects of viscosity and

turbulence, a more complex model was required. The flow solver addresses the full 3-D

Reynolds-averaged Navier-Stokes (RANS) equations with well-validated turbulence

models. The solver also has the capability to include effects for multi-block moving

boundary treatment. Robust interfacing techniques were also embedded in the coupled

solver to account for transfer of information between the two modules.

Our study aimed to expand a well-validated CFD approach to coupled aeroelastic

models and consider the complexity of coupling procedures in 3-D wing models. A non-

iterative flow solver was used for flow computations, greatly reducing the overall cost of

computations (as the fluids module is the most time consuming among all the modules).

In developing this model, the following issues were addressed:

• efficient moving boundary technique for multi-block structured grids

• preservation of geometric conservation law

• choice of time step of fluid and structure solvers

• accurate computation of contravariant velocities using momentum interpolation method for collocated grids.

The focus of this work was to study the fluid-structure interaction problem for 3-D

wing geometries. The flutter boundary was predicted for a transonic Mach number case.

The AGARD 445.6 wing (Yates 1987) was used to demonstrate the methodology. This

configuration was chosen because extensive research has been done in the field of

aeroelasticity using this model (thus experimental and numerical results are readily

6

available). Several flow solvers, ranging from transonic small disturbance models to full

three-dimensional Navier-Stokes solver and its thin layer approximations have been

coupled to the normal modes of the structure to determine the flutter boundary for the

AGARD wing geometry. Particularly, we are interested in the predicting the transonic dip

observed at transonic mach numbers. This dip is important in determining the minimum

velocity at which flutter can occur across the flight envelope of the vehicle; hence

predicting this dip is critical. Figure 1-2 shows the measured and computed flutter

boundary for the AGARD wing using several numerical methods. The solid line

represents the measured flutter boundary originally published by Yates (1963). Both

linear and nonlinear aerodynamic models have been employed to determine the flutter

boundary. Linear analysis using transonic small disturbance model (CAP-TSD; Bennett

et al. 1989) was found to predict the flutter boundaries accurately at subsonic and

supersonic speeds but failed to predict the dip, accurately, at transonic Mach numbers.

Specifically, linear analysis was found to be unconservative in the transonic speed

regime, where it predicted a significantly higher flutter speed. This is attributed to the

highly nonlinear effects arising from formation and disappearance of shock waves at this

Mach number regime as the aircraft undergoes unsteady, flexible motion. Inclusion of

viscous effects to the transonic small disturbance model (CAP-TSDV; Robinson et al.

1991) increased the predictive capability of the model at transonic Mach number, as seen

from the figure. Within nonlinear models, one can use both inviscid as well as viscous

analysis to determine the flutter boundary. The flutter boundary obtained by solving the

unsteady Euler aerodynamics equation of motion coupled to the normal modes of the

structure (CFL3D-Euler; Lee-Rausch and Batina 1995) is shown in Figure 1-2. The result

7

from this method was found to be overconservative, predicting a significantly lower

flutter speed. Viscous effects such as boundary layer thickening and/or flow separation

due to shock waves were found to be important factors in determining the transonic dip

accurately (Schuster et al. 2003). The inclusion of viscous effects was found to improve

the prediction of transonic dip (Lee-Rausch and Batina 1996; Gordnier and Melville

2001; Liu et al. 2003).

Figure 1-2. Flutter speed index prediction for AGARD 445.6 wing using several methods

Lee-Rausch and Batina (1996) coupled an unsteady thin-layer approximation of the

Navier-Stokes equations with the normal modes of structure. A moving mesh method

based on spring analogy was incorporated to account for grid movement after each time

step. Gordnier and Melville (2000) coupled an unsteady compressible Navier-Stokes

0.25

0.5

0.75

0.4 0.6 0.8 1 1.2

Mach number

Flut

ter s

peed

inde

x

CFL3D (Euler), Lee-Rausch et al. (1995)

CFL3D (N-S), Lee-Rausch et al. (1996)

CAP-TSD; Bennett et al. (1989)

CAP-TSDV, Robinson et al. (1991)

Liu et al. (2003)

Gordnier and Melville (2001)

Experiment

8

model with normal modes of structure using a Beam-Warming type implicit time

marching scheme with sub-iterations. An overset grid approach with algebraic mesh

deformation method was used to account for grid movement. The geometric conservation

law, which takes care of certain geometric quantities associated with mesh movement,

was invoked as well. Liu et al. (2003) coupled an unsteady RANS model with normal

modes of structure to predict the flutter boundary for the AGARD wing. Spring analogy

along with transfinite interpolation technique was used to move the multi-block mesh. An

implicit time stepping scheme using sub-iterations was employed to march in time.

Flutter boundary obtained using these methods are summarized in Figure 1-2. Significant

improvement was also seen, while using nonlinear viscous models, at supersonic speed

regimes. Not all of the above-mentioned models incorporated all the features essential in

producing a robust CAE model. Some of the limitations that the previous CAE models

face can be listed as follows

• Fixed-grid computations (Bennett et al. 1989; Robinson et al. 1991) – this is often the case while using transonic small disturbance model

• Use of inviscid flow solvers (Bennett et al. 1989; Lee-Rausch and Batina 1995) – failed to predict viscous effects such as boundary layer growth and/or flow separation

• Implicit time-marching schemes with sub-iterations (Gordnier and Melville 2001; Liu et al. 2003 – being an iterative scheme, it can be computationally expensive. Although such an implicit scheme is unconditionally stable, the choice of time step is limited by the frequency of oscillations of structure.

• Failure to include geometric conservation law (Lee-Rausch and Batina 1995, 1996; Liu et al. 2003)– essential while solving problems on moving mesh systems

We aim at addressing all of the above-mentioned issues and to develop a robust

CAE model capable of predicting the flutter boundary of three-dimensional wing

geometries accurately.

9

A review of the various existing methods in the field of computational

aeroelasticity is given in Chapter 2. The governing equations used by the different models

are addressed in Chapter 3. Chapter 4 discusses the computational procedure and setup

involved with a CAE model. Individual module-validation results along with coupled

simulation results and discussions are given in Chapter 5. Conclusions and thoughts on

future directions are given in Chapter 6.

10

CHAPTER 2 LITERATURE REVIEW

Next we review various aspects and modules related to the field of computational

aeroelasticity (Bennett and Edwards 1998; Friedmann 1999; Huttsell et al. 2001). First,

the various models associated to unsteady aerodynamics are presented. Then we review

various classes of CAE: fully coupled analysis (or unified fluid-structure interaction),

closely coupled aeroelastic analysis, and loosely coupled analysis. We discuss advances

in the field of moving mesh methods for re-meshing purposes and interfacing techniques

for exchanging information between different modules used in some coupled aeroelastic

models. The various formulations of Geometric Conservation Law (GCL) are also

reviewed.

Aerodynamic Models

To understand the fluid-structure interaction problem, we need to model both the

structure and the fluid efficiently. However, since our emphasis was on the fluid (rather

than structure) models, we first review some physical models from the fluids perspective

undergoing time-dependent motion. Different classes of coupled CAE models (explained

earlier) and the issues associated therein are discussed later in this chapter.

Physical Models

Physical models used for treating fluid-structure interaction problems can vary

enormously in their complexity, based on the applications. One of the simplest models is

based on piston theory (Dowell and Hall, 2001), which expresses the pressure, p, at some

11

point x, y and some time t on the oscillating body, as a simple function of the motion at

the same point and instant. It can be expressed as follows

U w wp UM t xρ ∂ ∂ = + ∂ ∂

where w is a function of x, y and t and it is the instantaneous deflection of the body. The

symbols ρ, U and M represent free-stream density, velocity, and Mach number,

respectively. This simple method is only useful for a limited set of flow conditions, and is

usually used to verify more complex models in the appropriate limit. An improved model

to the piston theory is the full-potential flow theory, which works under the assumption

that the flow is inviscid and irrotational. The potential flow model solves the nonlinear

wave equation for the velocity potential, from which the velocity (and thereby the

pressure) can be obtained using Bernoulli’s equation. If the body profile is assumed to be

thin, the nonlinear equation can be cast into a linear convected-wave equation, which has

found uses for many fluid-structure interaction problems such as flutter and gust response

analysis (Bisplingoff et al. 1955; Fung 1955). The linear convected-wave equation has

trouble satisfying the boundary conditions (Dowell and Hall 2001) because in the

boundary condition, both the velocity potential and its gradient over different portions of

the fluid domain are unknown (leading to a mixed-boundary problem). This is resolved

by reducing the convected-wave equation (partial differential equation) to an integral

equation using Green’s theorem or Fourier transform. This is also referred to as the

boundary element approach. Another well-known model is based on small perturbation

theory (Bisplingoff et al. 1955; Fung 1955), but it was found to fail when the flow is

transonic (when shock waves may appear and disappear).

12

Another class of models is the time-linearized or dynamically linear model, in

which a steady-state nonlinear solution is used as a starting point; then a small dynamic

perturbation about this steady flow is considered, and all subsequent flow variables and

shock motion are assumed to vary in a linear fashion. This model leads to an order of

magnitude reduction in computer resources compared to the nonlinear model, and was

found to be sufficient for many problems. However, this method was found to be less

useful for turbomachinery problems. This approach can be extended to determine a full

dynamically nonlinear solution, which involves solving a nonlinear convected-wave

equation for potential flow or Euler or Navier-Stokes models. Either finite-difference or

finite-volume schemes in spatial variables can be used to convert the system of partial

difference equations to ordinary differential equations, which forms the basis for CFD.

Additional models must be developed to account for turbulence flow features, and for

transition from laminar to turbulent flows. Another class of models beginning to gain

interest in the field of fluid-structure interaction is reduced-order modeling (ROM)

techniques, discussed next.

Reduced-Order Models

For the past several decades, researchers have worked in the field of CFD to

develop models for complex unsteady flows. The computational cost for high

dimensionality model, especially for aeroelastic problems, has limited the use of full CFD

models for such applications. Recently, advances are being made to develop a novel

technique for unsteady flows based on the modal character of flows, which can be termed

reduced-order models. In the structural dynamics world, over the years, finite element

models for structural dynamics have been reduced in size by using the normal or

eigenmodes of the structure, thereby reducing the model to a few degrees of freedom

13

from thousands of degrees of freedom (Dowell and Hall 2001). This reduces the

computation time for solving such problems, while maintaining the accuracy of the

physical phenomena. This method has also gained interest in the field of fluid dynamics,

because such an approach gives us great benefits (saving computational costs and giving

insight into the dynamics of the fluid models by considering their different modal

structures). This method involves constructing a computational aerodynamic model

using the dominant eigenmodes of unsteady aerodynamic flows. Combining such a

reduced-order aerodynamic model with a structural modal model is an efficient way to

form an aeroelastic modal model with a modest number of degrees of freedom.

Extracting the dominant eigenmodes for large dimensional systems can be potentially

difficult. Hence another modal approach that seeks to include more information on the

flow response to enhance the accuracy of the reduced model has been developed and it is

called the proper orthogonal decomposition (POD) method (Ahlman et al. 2002; Zhang et

al. 2003). It is a much simpler approach than the eigenmode approach, and it uses a

methodology based on nonlinear dynamics and signal processing. One disadvantage of

this method is that determining the POD modes can be computationally expensive

compared to determining the eigenmodes. Extensive research is being done to construct

nonlinear aerodynamic ROMs and to use the eigenmode ROM approach to develop better

turbulence models. However, it is still unknown whether ROM or POD approach can

accurately predict all the length scales associated with the turbulence models.

Review of Coupled CAE Models

Before looking at the various CAE models, the generalized equations of motion

(Schuster et al. 2003) are given to explain CAE methodologies

14

[ ] [ ] [ ] M q( t ) C q( t ) K q( t ) F( t )+ + =!! ! (2-1)

1

N

i ii

w( x, y,z,t ) q ( t ) ( x, y,z )φ=

=∑ (2-2)

where w( x, y,z,t ) is the structural displacement at any time instant and position and

q(t) is the generalized displacement vector. The matrices [M], [C], [K] are the

generalized mass, damping, and stiffness matrices; respectively and φi are the normal

modes of the structure, with N being the total number of modes of the structure. The term

on the right-hand side of Eq. (2-1), F(t), is the generalized force vector (which is

responsible for linking the unsteady aerodynamics and inertial loads with the structural

dynamics). Equations (2-1) and (2-2) show that the distinct terms representing the

structures, aerodynamics, and dynamics disciplines give us the flexibility in choosing

different methods for a given system. For example, linear structural models can be

coupled with a 3-D unsteady RANS model, to develop a CAE model without actually

changing the overall formulation of the equations of motion. This example of a closely

coupled model is the emphasis of our study. However, fully coupled models and loosely-

coupled or uncoupled models have been developed. Some of these models are discussed

next.

Fully coupled Analysis

In this method, the governing equations are reformulated by combining fluid and

structural equations of motion to obtain a unified set of equations, which are then solved

and integrated in time simultaneously. While using a fully coupled procedure, one must

deal with fluid equations in a Eulerian reference system, and structural equations in a

Lagrangian system. This leads to the matrices being orders of magnitude stiffer for

15

structure systems as compared to fluid systems, thereby making it virtually impossible to

solve the equations using a monolithic computational scheme for large-scale problems.

Initially, Guruswamy and Byun (1993, 1994) combined Euler flow equations with plate

finite-element structures; and later combined the Navier-Stokes equations with shell

finite-element structure to perform fluid-structure calculations. They used a domain

decomposition method, wherein fluids and structures are solved in separate modules. On

the same note, Garcia and Guruswamy (1999) computed the transonic aeroelastic

response of 3-D wings by coupling a nonlinear-beam finite-element model with Navier-

Stokes equations. This kind of fully coupled method has limitations on grid size, and is

currently limited to 2-D problems as they are computationally expensive. These models

and the test cases used to study them are shown in Table 2-1.

Table 2-1. Description and key results of a few fully-coupled analysis methods

Author (s) (year)

Description of work Major Results

Guruswamy, Byun (1993, 1994)

• Compute aeroelasticity by direct coupling using time-integration method

• Fluid: Euler equations • Structure: Plate finite elements • Aerodynamic loads are transferred by

bilinear interpolation and by virtual surface methods

• CFD grid (151 x 30 x 35) • FEM grid (36 plate elements) • Fighter type wing with M=0.854 and

α=1 deg

• Validity of coupling plate elements with Euler equation

• Virtual surface method transfers loads more accurately than bilinear interpolation technique

Garcia, Guruswamy (1999)

• Model for coupled nonlinear beam FEM model with N-S solver for static aeroelastic analysis of high AR wings

• Flow solver: ARC3D fluids module of ENSAERO-WING code

• Structural code: Nonlinear beam FEM • Aeroelastic research wing (ARW-2) @

M=0.85 and α=2

• FEM results are accurate except for deflections which were smaller than modal results

• Nonlinear and linear beam models predicted similar pressure coeff results

• Geometrical nonlinearity was found to be negligible

16

Loosely and Closely Coupled Analysis

In this class of methodologies, unlike the fully coupled analysis, the structural and

fluid equations are solved using two separate solvers. This can result in two different

computational grids (structured or unstructured), which are not likely to coincide at the

boundary. This calls for an interfacing technique to be developed, to exchange

information back and forth between the two modules. This is true for both loosely and

closely coupled approaches. We now review each of these methods separately.

Loosely coupled analysis

The loosely coupled approach has only external interaction between the fluid and

structure modules; or the information is exchanged after partial or complete convergence

(Smith et al. 1996a). This approach is like a multidisciplinary computing environment

(MDICE) (Seigel et al. 1998), where one effectively controls the interaction between two

commercial codes for each of the modules by means of interfacing techniques. This gives

us the flexibility of choosing different solvers for each of the modules but the coupling

procedure loses accuracy as the modules are updated only after partial or complete

convergence. A typical block diagram of MDICE is shown in Figure 2-1. Here, the

interface methodology has been divided into two categories: function matching interface

and conservative interface. Function matching interfaces provide the closest match

between data on the two computational grids. Conservative interfaces aim at conserving

relevant properties (such as forces and momentum) during the transfer process.

17

Figure 2-1. Sample MDICE environment for aeroelastic simulation. (Seigel et al., 1998)

Closely coupled analysis

This method will be the main focus of this thesis. Here, the fluid and structure

equations are solved separately using different solvers but are coupled into one single

module with exchange of information taking place at the interface or the boundary via an

interface module. The information exchanged here are the surface loads, which are

mapped from CFD grid onto CSD grid, and displacement field, which are mapped from

CSD grid onto CFD grid. The transfer of surface displacement back to the CFD module

implies deformation of the CFD boundary mesh and this calls for a moving boundary

technique to enable re-meshing the entire CFD domain for further computations as we

march in time. This can cause potential problems for multi-block grids with complex

geometries and will be looked at in-depth shortly.

Several models have been combined for individual modules to arrive at a coupled

model. From the fluids perspective, models ranging from simple potential flow models to

complex 3-D RANS models have been used. On the other hand, models ranging from

linear beam finite elements to nonlinear solid finite elements have been used for structure

module. These models are interlinked via necessary interfacing techniques, the

Panel methods Parabolized Navier-Stokes Euler equations Asymptotic expansion Boundary Layer Full Navier-Stokes other

Function matching • Infinite plate spline • Thin plate spline • other Conservative interfaces • Infinite plate spline • Thin plate spline • Conservative/ consistent • other

Modal analysis Influence coefficient Linear FEM Nonlinear EM other

Fluids Module Interface Methodology Structural Module

18

complexity of which depends on what two models are used for the individual modules. A

brief summary of some of the models that have been developed in the past will be shown

next.

Cunningham et al. (1988) developed a computational scheme to perform transonic

aeroelastic analysis by coupling transonic small disturbance (TSD) potential flow

equations (CAP-TSD) with the natural vibrational modes of the structure. Viscous effects

were later incorporated into the flow solver by including an inverse integral boundary

layer model. The equations of motion were solved on a sheared cartesian grid where the

lifting surfaces were modeled as thin plates. This kind of approach simplified the task of

generating grids and no moving boundary algorithm was required as the surface velocity

boundary condition was applied at a mean plane. This technique of using TSD

formulation failed in the presence of a strong shock or when viscous effects are

dominant.

To overcome this, Schuster et al. (1990) came up with a model that uses a 3-D flow

solver coupled with a linear structure model to study the aeroelastic analysis of a fighter

aircraft (ENS3DAE). Thin layer approximations to the full three-dimensional

compressible RANS equations were used. A three-dimensional implementation of the

Beam-Warming implicit scheme was employed for temporal integration. The equations

were solved on multi-block curvilinear grids. The linear generalized mode shapes were

used to model the structure. A grid motion algorithm that uses an algebraic shearing

technique was used to account for the grid movement.

A similar method (CFL3DAE), developed by Lee-Rausch and Batina (1995, 1996),

couples a linear, normal mode structural dynamics model with the thin-layer three-

19

dimensional compressible RANS model. Time marching was accomplished by means of

a second order accurate backward time differencing scheme. A pseudo time sub-iteration

method was introduced to expedite the convergence at each time step. A moving mesh

algorithm based on spring analogy was used here. This model was used to predict the

wing flutter boundary. An overview of the above-mentioned models, namely, CAP-TSD,

ENS3DAE and CFL3DAE, have been given by Bennett and Edwards (1998) and Huttsell

et al. (2001). The main features and results of these methods are shown in Table 2-2.

Table 2-2. Description of CAE simulations using CAP-TSD, ENS3DAE and CFL3DAE Author (s) (year)

Description of work Major Results

Cunningham, Batina, Bennett (1988)

• Computational scheme for transonic aeroelastic analysis to perform flutter analysis

• Flow: Transonic small disturbance formulation

• Structure: Lagrange Equations of motion based on the natural vibrational modes

• AGARD configuration with 45 deg sweep angle and M=0.338-1.141

• Aerodynamic forces and flutter characteristics obtained using linear formulation compared well with expt.

• Non-linear flutter results compared well with expt but not so with linear results

• Can treat configurations with arbitrary lifting surfaces

Lewis and Smith (1998)

• External aeroelastic simulation for internal aerodynamics and shell structures

• Flow: ENS3D • Predictor-corrector scheme for structural

integration • Tested on an engine liner to study flutter

with M=0.7 in inner region and M=0.4 in the annular region

• Results showed the engine liner to be dynamically stable

• Inner flow Mach no. had little effect on aeroelastic response

• Effect of pressure loadings on the shell structures were not considered in this method

20

Table 2-2. Continued Author (s) (year)

Description of work Major Results

Schuster, Vadyak, Atta (1990)

• A 3-D flow solver coupled with linear static structural model to study aeroelastic response of aircraft

• Grid deflection method is used to update the grid after each time step.

• Flow solver: ENS3D • Swept, tapered wing with constant cross-

section with M=0.9 and α=9 deg was used

• Wing mesh: 92 x 32 x 32 points

• Aeroelastic analysis compared well with experiment with respect to pressure coefficient and twist

• Flexible wing/body configuration gave better results compared to rigid body configuration

• Separation on the upper surface was not predicted

Lee Rausch and Batina (1993, 1995, 1996)

• Navier-Stokes aerodynamics to compute AGARD 445.6 wing flutter

• Flow: Implicit upwind Euler/N-S solver • Structure: Modal analysis • Moving mesh: Spring analogy • Grid: 193 x 41 x 65 C-H type • M=0.96, Re=364,600 per foot of chord

• Difference in flutter speed index and frequency index between Euler and N-S solver was pointed out

Hartwich, Dobbs, Arslan and Kim (2000)

• Study LCO for a B-1 configuration using N-S equations

• Flow: CFL3D – a 3D N-S solver • Structure: Lagrange’s equations of

motion • Moving mesh: Spring analogy and TFI

using master/slave concept • Grid: 281 x 137 x 65 C-O type • M=0.975, α=7.38 deg and Re=5,900,000

• Predicted aerodynamic damping matched well with experimental trends

• Fell short of predicting a true LCO phenomenon

Liu et al. (2000, 2003) presented an integrated CFD-CSD code for flutter

calculations based on a parallel, multi-block, multigrid flow solver for solving the full

Navier-Stokes equations. The flow solver is strongly coupled with the structural modal

dynamics equations. A dual time-stepping scheme was introduced to enable simultaneous

integration of flow and structural equations without a time delay. A moving mesh method

based on transfinite interpolation (TFI) (Eriksson, 1981) and spring analogy (Hartwich

and Agrawal, 1997) was also incorporated in the code. Message passing interface (MPI)

21

was used to enable data transfer between the two modules. The method was tested to

perform the static aeroelastic analysis and the wing flutter on the AGARD 445.6 wing.

The key results from this model are shown in Table 2-3.

Table 2-3. Summary of work with a moving mesh algorithm

Author (s) (year)

Description of work Major Results

Liu, Cai, Zhu, Wong and Tsai (2000)

• AGARD 445.6 Wing flutter using a coupled CFD-CSD

• Flow: Parallel multi-block Euler • Structure: Modal dynamic equations • Moving mesh: Arc-length based TFI and

spring analogy • Interface: Transformation spline matrix • Grid: 176,601 points (32 blocks) • M=0.338-1.141

• Flutter speed/frequency in good agreement with experiment

• Transonic dip captured

Cai, Liu and Tsai (2001)

• Static aeroelasticity of AGARD 445.6 wing using Euler/N-S equations

• Flow: Parallel multi-block N-S • Structure: Static elastic equations • Moving mesh: Spring analogy and TFI • M=0.85 and α=5 deg

• Convergence was sped-up using relaxation technique.

• Difference in solutions between rigid and flexible wing were spotted

A three-field formulation for solving transient nonlinear aeroelastic problems was

suggested by Farhat et al. (2000) where they used an Arbitrary Lagrangian and Eulerian

(ALE) method for solving the equations on a deforming mesh. In fact, most CAE

problems can be formulated as a three-field problem: the fluid, the structure and the

moving mesh. In the case of ALE formulation, separate set of equations are specified for

grid movement that are directly coupled with the ALE flow equations. The fluid and

structure equations are coupled by the interface conditions. Unstructured meshes were

used for both fluid and structure solver. Farhat and Lesoinne (2000) improved upon the

existing serial and parallel algorithms for nonlinear transient aeroelastic problems. A

review of some of these methodologies is presented in Table 2-4.

22

Table 2-4. Summary of work related to ALE formulation Author (s) (year)

Description of work Major Results

Farhat, Pierson and Degand (2000)

• Computational method to simulate transient aeroelastic response of flexible aircraft during high-G maneuvers

• Flow: Arbitrary Lagrangian-Euler equations are incorporated into the unstructured flow solver (Euler)

• Structure: Corotational formulation • M=0.901 and α=1 deg on Langley

fighter

• Qualitative validation of results was done

• Geometric conservation law was incorporated

• Viscous effects were neglected

Farhat and Lesoinne (2000)

• Serial and Parallel methodologies for nonlinear transient aeroelastic problems

• Flow: ALE formulation • Moving mesh: Dynamic mesh equations

coupled with the flow equations • M=0.901 on an AGARD 445.6 wing

• Partitioned algorithms were found to be efficient than monolithic schemes

Geuzine, Brown and Farhat (2002)

• Three-field formulation for flutter analysis of F-16 configuration

• Flow: ALE formulation • Structure: Elastodynamic equations • Moving mesh: Dynamic mesh equations

combined with flow eqns. • M=0.7-1.4 on F-16 wing • Grid size: 403,919 (63,044 on wing

surface)

• Energy conservative exchange of aerodynamic and elastodynamic data was shown

• Method was found to be effective in the transonic regime and not as effective in the subsonic and supersonic regime

Review of Moving Boundary Models

Having reviewed the various developments in the field of computational

aeroelasticity as far as coupling procedure, our focus shift towards one of the most key

aspects of computational aeroelasticty, which is the deforming mesh method. Since the

structure movement needs to be accounted for in the fluid domain, we need to ensure that

the entire flow domain is re-meshed appropriately. Also, an efficient moving boundary

module is very important for performing unsteady flow calculations such as flutter

simulation of wings and turbo-machinery blades. Since the grid needs to be updated

frequently in unsteady computations, a fast and automatic grid deformation procedure is

23

an essential feature. Several models have been developed over the past decade and we

will review some of the methods in this section and point out the advantages and

disadvantages, if any.

Initially, a spring analogy method, originally proposed by Batina (1989) for

unstructured grids and later expanded by Robinson et al. (1991) to structured grids, was

used to generate dynamic grids for structured and unstructured solvers. This method can

handle large deformations but, being an iterative method resembling an elliptic grid

generator, it was found to be computational expensive for larger grid sizes.

Schuster et al. (1990) and Bhardwaj et al. (1998) used a simple algebraic shearing

technique to deform the grid by redistributing the grid points along grid lines that are in

the direction normal to the surface. This method can cause potential problems when the

geometry becomes complex when it becomes difficult to locate the radial direction

normal to the surface. Also, this method is limited to small deformations and large

deformations may lead to poor grid quality and crossover of grid lines.

A transfinite interpolation (TFI) method (Eriksson, 1982) is typically used for

regenerating individual blocks in multi-block meshes. Hartwich and Agrawal (1997)

combined the spring analogy method with the TFI method for regenerating multi-block

grids. Spring analogy was used to move the boundary edges of the blocks whereas TFI

was used to re-mesh the surface and interior volume of each block. A point-by-point

match was enforced between two abutting blocks. Potsdam and Guruswamy (2001)

improved the above method and incorporated parallelization for mesh regeneration.

Another class of methods for re-meshing purposes is solving the moving mesh

partial differential equations (Huang et al., 1994; Huang and Russell, 1999; Huang,

24

2001). In this method, a mesh equation is formulated and solved to move the nodes in a

consistent fashion by accounting for clustering of nodes in regions of large solution

variation. A monitor function was incorporated into the equation to enable mesh

smoothing. This method can be computationally expensive for complex 3-D problems. A

comparison of some of the above-mentioned methods is shown in Table 2-5.

Table 2-5. Comparison of moving mesh algorithms Method Advantage Disadvantage Spring analogy (Robinson et al., 1991) Robust Needs more Memory and

CPU Transfinite interpolation (Erikkson, 1982) Fast May not preserve original

grid quality Gordon’s TFI based method (Wong et al., 2000) Eriksson’s TFI based method (Hartwich and Agrawal, 1997) Perturbation method (Reuther et al., 1996)

Faster and Preserves grid quality

May encounter crossover near the moving boundary

Moving mesh partial differential equation (MMPDE) (Huang, 2001)

Easy to implement and accounts for grid quality near regions or large gradients

Computationally expensive

Review of Geometric Conservation Law

A key aspect of solving problems on a deforming grid is to ensure that the

Geometric Conservation Law (GCL) is preserved. It takes care of certain geometric

quantities associated with the deformed grid or the new grid. In the numerical

perspective, it is called the discrete geometric conservation law (DGCL). The DGCL

states that the computation of the geometric quantities associated with a moving grid

should be computed in such a way that, independent of the mesh movement, the

numerical scheme used for integrating the flow equations must preserve a uniform flow

field (Guillard and Farhat 2000). This is in conjunction with the fact that preserving

25

uniform field implies first order accuracy. In addition, Guillard and Farhat (2000) showed

that for a p-order time-accurate scheme on a fixed mesh, satisfying the corresponding p-

order DGCL is a sufficient condition for the scheme to be at least first order time

accurate on a moving mesh. They established the requirement that preserving the uniform

flow field on moving grids is related to a consistency condition. It has also been proven

that not satisfying the DGCL introduces a weak instability in the numerical solution on

moving grids (Lesoinne and Farhat, 1996).

Substantial evidence exists showing that not satisfying the geometric conservation

law leads to erroneous solutions or spurious oscillations in the solution (Guillard and

Farhat 2000; Lesoinne and Farhat, 1996; Farhat et al., 2001 & 2003). For example, Shyy

et al. (1996) demonstrated that without explicitly enforcing GCL, O(1) error could be

induced in the computation simply due to the grid movement effect. It has also been

shown that satisfying the DGCL can improve the time-accuracy of computations on

moving grids (Koobus and Farhat, 1999). One of the widely used methods for fluid-

structure interaction problems is the ALE formulation. It formulates the Navier-Stokes

equations in three co-ordinate systems namely, material or Lagrangian (for structure

motion), spatial or Eulerian (for fluid motion) and referential (for grid movement). Farhat

et al. (2001, 2003) showed that for ALE schemes, satisfying the DGCL leads to a

necessary and sufficient condition for the numerical scheme to preserve non-linear

stability on a fixed grid. However, there have been a few cases where satisfying or not

satisfying the GCL produced the same results (Morton et al., 1998).

It should be noted that since GCL arises due to the numerical procedures devised

based on grid movement, its implications are expected to be scheme dependent.

26

Alternative forms of the GCL have been implemented over the years to study its impact

on solution accuracy. Thomas and Lombard (1979) implemented the GCL for density-

based finite difference schemes on structured meshes by updating the value of the

Jacobian at each time step. Shyy et al. (1996, 2001) implemented the GCL along the lines

of Thomas and Lombard for pressure-based finite volume schemes by updating the

Jacobian values after every time step using a first order backward Euler time-integration

scheme. Lesoinne and Farhat (1996) developed a first order, time accurate scheme

preserving the GCL using the density-based ALE finite volume as well as finite element

schemes on unstructured grids. Koobus and Farhat (1999) proposed a GCL scheme for

second-order time-accurate density-based ALE finite volume schemes. Farhat et al.(2001)

summarized six different time-integration schemes based on ALE formulation, some of

them preserving the DGCL and some of them that did not, and showed the impact the

different schemes have on solution accuracy. In this effort, we assess selected approaches

for multi-block structured grids based on finite volume formulation and do a comparative

study on these methods. Most previously conducted studies employed the density-based

fluid flow solver; in the present effort, the pressure-based fluid flow solver (Shyy, 1994;

Shyy et al., 1997 and Thakur et al., 2002) is utilized. The implications of different

implementation of GCL and the fluid flow solver are of main interest. Together with the

previously cited references, the present work offers a more complete assessment of the

GCL.

Review of Interfacing Techniques

Having looked at the three major modules required for aeroelastic computations,

namely, fluid, structure and moving mesh modules, we now take a look at the interfacing

technique that links these individual modules in an efficient manner. For coupled

27

analysis, the exchange of information between the fluid and structure models takes place

at the common boundaries. A typical coupled fluid structure analysis diagram is shown in

Figure 2-2. The interfacing module is highlighted here for convenience. As can be seen

from the figure, for every time step, we need to map the surface loads, P, from the CFD

grid system onto the structural grid to obtain the forces, F, on the CSD grid system,

which are then used to obtain the displacements, w, on the CSD grid. These w’s need to

be interpolated onto the CFD grid to obtain the CFD surface grid.

Figure 2-2. Coupled fluid-structure flow diagram. (Guruswamy, 2002)

Figure 2-3. Varying levels of complexity in modeling for fluids and structures

(Guruswamy, 2002)

CFD P FMap pressure to FEM grid

Interpolate to CFD grid

CSD

Move Grid

Fluid/Structure Interface

W

NAVIER-STOKES

LINEAR ANALYTICAL

EULER

FULL POTENTIAL

TRANSONIC SMALL

DISTURBANCE

SHAPE FUNCTIONS

MODAL APPROACH

3-D FINITE ELEMENTS

2-D FINITE ELEMENTS

EQUIVALENT BEAM

FLUID STRUCTURE

INTER

FAC

ING

CO

MPLEX

ITY IN

PHY

SICS

CO

MPLEX

ITY IN

GEO

METR

Y

28

Since the fluid and structural module can be modeled at different levels of

complexity, the fidelity of the interfacing technique depends on how the fluid and

structure are modeled. This has been depicted in Figure 2-3. Maintaining accuracy in the

data exchange process is very important in order to obtain correct aeroelastic results.

Often times, the structural grid is unstructured or coarser than the CFD grid, thereby

demanding accurate interpolation techniques to transfer surface loads from the CFD grid

on to the structural grid. We will now review a few interpolation/extrapolation techniques

employed in the recent years to accomplish this data exchange.

Table 2-6. Summary of representative interface techniques Interface method Limitations • Infinite plate spline (IPS): based on

superposition of the solutions for the PDE of equilibrium for an infinite plate

• Multi-quadratic-biharmonic (MQ): interpolation technique that represents an irregular surface – makes use if quadratic basis functions

• Thin plate spline (TPS): Characterizes an irregular surface by using functions that minimize an energy functional

• Finite plate spline (FPS): Uses plate bending elements to represent a planform by a number of quadrilateral or triangular elements

• Non-uniform B-splines (NUBS): uses the fact that a 3-D surface can be represented by a tensor product of 2 splines

• Inverse isoparametric mapping (IIM): based on FEM scheme where an isoparametric element uses shape functions to perform interpolation

• Minimum of 3 grid points required • Noncoincident points are required • Extrapolations are linear • No minimum number of grid points

required but 3 are preferred for accuracy • No minimum number of grid points

required but 3 are preferred for accuracy • Only 2-D application was looked at • Four curves and four data points required • Points cannot be coincident • Valid for 2-D interpolations only • No extrapolation possible

29

Smith et al. (1996b, 2000) reviewed six interpolation methods: Infinite-plate

splines (IPS), finite-plate splines (FPS), multiquadric-biharmonics (MQ), thin-plate

splines (TPS), Non-Uniform B-Splines (NUBS) and Inverse Isoparametric Mapping

(IIM). Moyroud et al. (2000) demonstrated a technique based on parent volume grid and

child surface grid concept to perform interpolation on three-dimensional unstructured

triangular grids. A brief description along with the limitations of some of these methods

is given in Table 2-6.

Guruswamy (2002) reviewed interfacing techniques based on specific finite

element techniques employed for the structural model. The flow solver used was the

Euler/Navier-stokes solver. The FE models considered were modal model, beam finite

elements, plate/shell finite elements, wing-box FE model and the detailed FE model. For

the modal analysis, where the structural modes are evaluated using the Raleigh-Ritz

approach, a simple bilinear interpolation method proved to be an accurate method for

structured mesh systems. For the case when the structure mesh had irregular meshes, an

area coordinate approach was used. When beam structures are employed, load vectors

were used along with the shape functions to output transverse displacement, twist and

bending along the elastic axis for different span-wise locations. When plate or shell

elements are used as the finite element structures, a node-to-element approach was used

where shape functions were used to define the coordinates and planar displacements of

the element. Another method found to be effective for plate/shell elements was the virtual

surface method where a mapping matrix is used to exchange information between the two

grids. More details of this approach can be found in Guruswamy (2002). When the wing

is modeled as a wing-box, where only the components between the spars and ribs are

30

considered for modeling purposes, a discrepancy might occur as there is a discontinuity

in surface at the leading and trailing edges. In such cases, forces are lumped onto

structural nodes and bending and twisting moment conservation is enforced. Deflection at

the FEM nodes were obtained by using transformation functions by assuming that the

wing is chordwise rigid. Brown (1997) proposed a method that combines the node-to-

element approach used for plate/shell FE and the lumped method for wing-box structures.

For detailed FE models, where the interior of the FE grid could be irregular and the

surface elements could take both triangular and quadrilateral elements, the area

coordinate method of the virtual surface method was found to be an efficient one.

A different approach called the boundary element method was proposed by Chen

and Gao (2001), Chen and Jadic (2000) and Chen and Hill (1999) to perform

displacement interpolations between the two grid systems. In this method, a universal

spline matrix is generated to transform the structural displacement, us, to aerodynamic

displacement, ua. It is given by [ ] a Su B u= , where [B] is the spline matrix. Brief

description of this method is demonstrated in Table 2-7.

Table 2-7. Summary of Boundary element methods Author’s Name Description of work Major Results Chen, Jadic (1998) (2-D case) Chen, Hill (1999) (3-D case)

• Direct boundary element method (BEM) solver for CFD/CSD interfacing

• Generation of universal spline matrix (a vector) to go back and forth between CFD/CSD data

• Exterior BEM solver for CFD grid regeneration

• Code used: ENS3DAE • AGARD 445.6 at M=0.95 and α=2 • CFD grid (63 x 26 points) • Structure grid (121 points)

• Performs force transferal with good accuracy

• Performs accurate displacement extrapolation

• CSD grid points should lie inside CFD surface grid.

• Boundary element near leading/trailing edge causes instability.

31

Table 2-7. Continued Author’s Name Description of work Major Results Chen, Gao, (2001)

• Indirect boundary element method (IBEM) solver for CFD/CSD interfacing

• Multi-block BEM method to handle discontinuous structures

• AGARD 445.6 • CFD grid (145 X 37 points) • Structure grid (121 points)

• Gives very good extrapolation results on the CFD grid

• Eliminates edge effects found in the direct BEM solver

• Deals with complex configurations

32

CHAPTER 3 GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM

In this chapter, we discuss the formulation of the governing equations of various

modules associated with aeroelasticity and then look at the numerical schemes associated

with these modules. We will categorize them into different categories and describe each

module in detail.

Governing Equations

We first take a look at the governing equations associated with the various modules

used in our computations starting with the flow solver.

Flow Module

Navier-Stokes equations

We use a full 3-D compressible Navier-Stokes solver as our CFD model. The

equations written in cartesian coordinates, using indicial notations, read as follows

Continuity:

( ) 0jj

ut xρ ρ∂ ∂+ =

∂ ∂ (3-1)

Momentum:

( ) ( ) iji j i

j i ij

pu u ut x x x

τρ ρ

∂∂ ∂ ∂+ = − +∂ ∂ ∂ ∂

(3-2)

Energy:

( ) ( ) ( )jj i ij

j i j

qpH u H ut x t x x

ρ ρ τ∂∂ ∂ ∂ ∂+ = − +

∂ ∂ ∂ ∂ ∂ (3-3)

33

where xi is the position vector, t is time, ρ is density, ui is velocity vector, p is pressure, τij

is viscous stress tensor, qj is heat flux vector, obtained from Fourier’s law, given by

jj L j

T hqx Pr x

µκ ∂ ∂= − = −∂ ∂

(3-4)

where µ is the molecular viscosity, κ is the thermal conductivity, and PrL is the laminar

Prandtl number defined as:

pL

CPr

µκ

=

H is stagnation enthalpy given by

12 i iH h u u= + (3-5)

with h being the specific enthalpy.

The constitutive relation between stress and strain rate for Newtonian fluid is used

to relate the components of the stress tensor to velocity gradients:

23

ji lij ij

j i l

uu ux x x

τ µ µ δ ∂∂ ∂= + − ∂ ∂ ∂

(3-6)

Transformation to curvilinear coordinates

For arbitrary-shaped geometries, it is efficient to use body-fitted curvilinear

coordinates. We denote the curvilinear coordinates as (ξ,η,ς) where ξ=ξ(x,y,z,t),

η=η(x,y,z,t) and ς=ς(x,y,z,t). The transformation of the physical domain (x,y,z) to the

computational domain (ξ,η,ς) is achieved via transformation metrics, which are related to

the physical, coordinates as follows.

11 12 13

21 22 23

31 32 33

1x y z

x y z

x y z

f f ff f f

Jf f f

ξ ξ ξη η ηζ ζ ζ

=

(3-7)

34

where the metrics fijs are defined as follows

11 12 13

21 22 23

31 32 33

f y z z y f z x x z f x y y xf z y y z f z y y z f z y y zf y z z y f z x x z f x y y x

η ζ η ζ η ζ η ζ η ζ η ζ

ξ ζ ξ ζ ξ ζ ξ ζ ξ ζ ξ ζ

ξ η ξ η ξ η ξ η ξ η ξ η

= − = − = −= − = − = −

= − = − = −

(3-8)

and J is the Jacobian given by

J x y z x y z x y z x y z x y z x y zξ η ζ ζ ξ η η ζ ξ ξ ζ ξ ζ η ξ η ξ ζ= + + − − − (3-9)

The governing equations can be re-written in generalized body-fitted coordinates as

follows:

Continuity equation

( ) ( ) ( )( ) 0J U V Wtρ ρ ρ ρ

ξ η ζ∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂ (3-10)

u-momentum equation

( ) ( ) ( ) 11 21 31

11 12 13 21 22 23

31 32 33

( )J u p p pUu Vu Wu f f ft

u u u u u uq q q q q qJ J

u u uq q qJ

ρ ρ ρ ρξ η ζ ξ η ζ

Γ Γξ ξ η ζ η ξ η ζ

Γζ ξ η ζ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + ∂ ∂ ∂ ∂

(3-11)

v-momentum equation

( ) ( ) ( ) 12 22 32

11 12 13 21 22 23

31 32 33

( )v p p pUv Vv Wv f f ft

v v v v v vq q q q q qJ J

v v vq q qJ

ρ ρ ρ ρξ η ζ ξ η ζ

Γ Γξ ξ η ζ η ξ η ζ

Γζ ξ η ζ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + ∂ ∂ ∂ ∂

(3-12)

35

w-momentum equation

( ) ( ) ( ) 13 23 33

11 12 13 21 22 23

31 32 33

( )w p p pUw Vw Ww f f ft

w w w w w wq q q q q qJ J

w w wq q qJ

ρ ρ ρ ρξ η ζ ξ η ζ

Γ Γξ ξ η ζ η ξ η ζ

Γζ ξ η ζ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + ∂ ∂ ∂ ∂

(3-13)

Energy equation

( ) ( ) ( ) 11 12 13

21 22 23 31 32 33

11 12 13

( ) h

h h

k

H h h hUH VH WH q q qt J

h h h h h hq q q q q qJ J

k k kq q qJ

Γρ ρ ρ ρξ η ζ ξ ξ η ζ

Γ Γη ξ η ζ ζ ξ η ζ

Γξ ξ η ζ η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + ∂ ∂ ∂ ∂ ∂

21 22 23

31 32 33

k

k

k k kq q qJ

k k kq q qJ

Γξ η ζ

Γ Φζ ξ η ζ

∂ ∂ ∂+ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + ∂ ∂ ∂ ∂

(3-14)

36

11 11 12 11 13 11

21 21 22 21 23 21

31 31 32 31 33 31

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

u u uq u f U q u f V q u f WJ

u u uq u f U q u f V q u f WJ

u u uq u f U q u f V q u f WJ

Γφξ ξ η ζ

Γη ξ η ζ

Γζ ξ η ζ

ξ

∂ ∂ ∂ ∂= + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂

∂∂ 11 12 12 12 13 12

21 22 22 22 23 22

31 32 32 32 33 32

1

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(

v v vq v f U q v f V q v f WJ

v v vq v f U q v f V q v f WJ

v v vq v f U q v f V q v f WJ

qJ

Γξ η ζ

Γη ξ η ζ

Γζ ξ η ζ

Γξ

∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂

∂∂ 1 13 12 13 13 13

21 23 22 23 23 23

31 33 32 33 33 33

) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

w w ww f U q w f V q w f W

w w wq w f U q w f V q w f WJ

w w wq w f U q w f V q w f WJ

ξ η ζ

Γη ξ η ζ

Γζ ξ η ζ

∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + ∂ ∂ ∂ ∂

where

t tt

L t kPr Prh k µ µµΓ µ µ Γ Γ µσ

= + = + = + (3-15)

2 2 211 11 12 13 12 21 11 21 12 22 13 23

2 2 222 21 22 23 13 31 11 31 12 32 13 31

2 2 233 31 32 33 23 32 31 21 32 22 13 23

+ +

+ + + +

q f f f q q f f f f f fq f f f q q f f f f f fq f f f q q f f f f f f

= = = + +

= = = + +

= = = + +

(3-16)

Here, U, V and W are the components of the contravariant velocity, which are the

scalar products of velocity vector and area vector at a control volume interface. They can

be interpreted as the volume flux normal to control volume interfaces; specifically, U is

the local volume flux along the ξ coordinate, V along the η coordinate and W along the ς

coordinate. They are given by

37

11 12 13

21 22 23

31 32 33

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

U f u x f v y f w zV f u x f v y f w zW f u x f v y f w z

= − + − + −= − + − + −= − + − + −

! ! !! ! !! ! !

(3-17)

where x! , y! and z! are the grid velocities which are approximated by a first order

backward time difference scheme given by

o o ox x y y z zx y z

t t t− − −= = =∆ ∆ ∆

! ! ! (3-18)

Here, ∆t is the time step size of the flow solver and the superscript o refers to the

previous time step. Since the flow equations are solved on a moving mesh system, care

must be taken to preserve the geometric conservation law (GCL) as originally formulated

by Thomas and Lombard (1972).

Geometric Conservation Law

The GCL is derived from the conservation of mass by setting ρ=1 and v=0. It can

be written as follows:

.V S

d dV dSdt

=∫ ∫ sw (3-19)

where ws is local velocity of cell boundary. The GCL can also be states as the change in

volume of each control volume between two time instants, tn and tn+1, must be equal to

the volume swept by the cell boundary during that time ∆t= tn+1-tn.

The above expression is referred to as the integral form of GCL. A differential

statement of the GCL can be derived from the integral statement of GCL. We first

perform a transformation from the cartesian coordinate system (x,y,z) to the body-fitted

coordinate system ( , ,ξ η ζ ), which leads to the following form of the integral statement:

38

( )ξ η ζ = ∇ ξ η ζ∫ ∫V V

d Jd d d Jd d ddt s.w (3-20)

Here, J represents the volume element in the transformed coordinate system hence

each node is associated with a particular value of J. Therefore, the computed value of J

must be consistent with the value of ∆V implied by the numerical scheme used for

solving the flow equations. Earlier, arbitrary procedures were used to compute J, for e.g.,

instantaneous mesh distribution at a given time instant was used to evaluate J at that

particular time, which lead to an erroneous solution.

Expanding the right hand side of Eq. (3-20) and after performing necessary

manipulations, we arrive at the following form for the differential statement of GCL.

( ) ( ) ( ) 0t t t tJ ξ η ζξ η ζ+ + + = (3-21)

where, , ,t t tξ η ζ are the metric terms given by

( )( )( )

( ) ( )

( ) ( )

( ) ( )

t

t

t

x y z y z y z x z x z x y x y

x y z y z y z x z x z x y x y

x y z y z y z x z x z x y x y

η ζ ζ η η ζ ζ η η ζ ζ η

ζ ξ ξ ζ ζ ξ ξ ζ ζ ξ ξ ζ

ξ η η ξ ξ η η ξ ξ η η ξ

ξ

η

ζ

= − − + − + − = − − + − + − = − − + − + −

! ! !

! ! !

! ! !

(3-22)

Equation (3-21) is solved numerically to update the Jacobian values at each time

step. The numerical solution for Eq. (3-21) requires only an initial condition, which is

obtained from the initial fixed grid and is given by Eq. (3-9).

Turbulence Modeling

The conservation equations (3-1) - (3-3) hold good only for laminar flows. For

turbulent flows, we need to re-construct the equations based on an averaging process in

the context of RANS methods.

39

Two kinds of averaging can be used for the averaging process: (1) Reynolds

averaging, or (2) Favre averaging. In solving for compressible flows, the use of Reynolds

averaging alone introduces correlations involving density fluctuations and these can be

difficult to model. Hence, a combination of both Reynolds and Favre averaging is used to

overcome this. This also leads to a set of governing equations that resemble the

incompressible counterpart. The averaging process of different flow properties is shown

below.

"

"

"

#

"

''

''

''

''

''

'

'

i i i

ij ij ij

j j j

u u u

T T T

h h h

q q q

p p p

τ τ τ

ρ ρ ρ

= +

= +

= +

= +

= +

= +

= +

Mass weighted Favre averaging is used for ui, τij, T, h and qj (tilde and double

prime denote Favre-averaged mean and fluctuating components respectively) whereas

Reynolds averaging is used for ρ and p (bar and prime denote Reynolds-averaged mean

and fluctuating components respectively) to recast the conservation equation. The mean-

flow governing equations now take the form:

"( ) 0ij

ut xρ ρ∂ ∂+ =

∂ ∂ (3-23)

"( ) ""( ) ( )'' ''j j

iji j ij i j

pu u u u ut x x x

ρ ρ τ ρ∂ ∂ ∂ ∂+ = − + −∂ ∂ ∂ ∂

# (3-24)

40

" "" " #( )" ( ) ( )

'' '' '' '' '' ''

'' '' '' '' '' ''

1 12 2

1 2

i i j j i i jjj j

ijj i j i ij j i ij j

pH u u u H u u u q u ht x t x

u u u u u u ux x

ρ ρ ρ ρ ρ

τ ρ τ ρ

∂ ∂ ∂ ∂ + + + = − + ∂ ∂ ∂ ∂

∂ ∂ + − + − ∂ ∂ #

(3-25)

As can be seen from the equation, the averaging process leads to additional

unknowns in the form of Reynolds stresses, '' ''i ju uρ− , which needs to be modeled. This

leads to the well-known closure problem for turbulent flow computations. In order to

determine these quantities, turbulence models are required. The two major approaches to

model these terms are eddy viscosity models and Reynolds-stress models. The former

approach will be used in the present study. The eddy viscosity model makes use of the

Boussinesq’s eddy viscosity hypothesis, which is based on the assumption that the

Reynolds stresses are a local property of the mean flow and are related to the mean flow

gradients via a turbulent viscosity given by:

# # #'' '' 2 2

3 3i j l

i j t t ij ijj i l

u u uu u kx x x

ρ µ µ δ ρ δ ∂ ∂ ∂− = + − ⋅ − ⋅ ∂ ∂ ∂

(3-26)

where µt is the turbulent viscosity that needs to be modeled.

k-εεεε transport equations

The transport equations for the turbulent kinetic energy, k, and eddy viscosity, ε,

can be written by means of one equation given by:

1 2( ) ( ) tj

j j

u R Rt t x xφ

µ φρφ ρ φ µσ

∂ ∂ ∂ ∂+ = + + + ∂ ∂ ∂ ∂ (3-27)

where φ = k or ε with the following expressions for R1 and R2

41

1 1

for the k equation

for the equation

t

t

RR C R

kε

µµ ε ε

=

(3-28)

2 *

22 *

2 2 *

for the k equation

for the equation

t

C kk

R

C Ck k

µ

ε ε

ρρε

µ

ε ερ ρ ε ε

− ≡ − =

− ≡ −

(3-29)

with

2 2 22 2 2u v w u v u w v wRx y z y x z x z y

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (3-30)

where 1 2, , , and kC C Cµ ε ε εσ σ are constants, the values of which, as suggested by

Launder and Spalding (1972) are as follows

1 20.09, 1.44, 1.92, =1.0 and 1.3kC C Cµ ε ε ε= = = σ σ = (3-31)

The above-mentioned governing equations for k and ε are transformed to body-

fitted coordinates, integrated over the control volume and discretized in a manner similar

to the momentum equation, which will be described later.

The turbulent viscosity, µt, is the medium through which the time and length scale

effects of turbulent flows are introduced into the equations. Thus, modeling µt requires

specification of local length and time scales. The k–ε models provide these scales via the

modeled turbulent kinetic energy (k) and the rate of viscous dissipation of turbulent

kinetic energy (ε). Dimensional analysis yields the following expression for turbulent

viscosity:

2

t

C kµρµ

ε= (3-32)

42

where Cµ is a dimensionless constant to be defined later.

Filter-based turbulence model for unsteady RANS computations

One of the fundamental problems with unsteady RANS model is that it is based on

steady state mean flow data whereas the main turbulent scales returned by the model are

the macro or the most energetic length scales. Hence, these models are found to have

difficulties in resolving flow structures resulting from multiple length scales. The model

is also known to over-predict turbulent production and hence effective viscosity in

stagnation flow regions. A filter-based model (Johansen et al., 2003) is investigated here

to improve upon the predictive capability of the standard k−ε two-equation model. In

RANS computations, the true resolution is not only dictated by the mesh size but also by

the magnitude of eddy viscosity. These in turn affect the local Reynolds number, which

needs to be O(1) magnitude in order to resolve the flow structure satisfactorily. Also, one

should note that an excessive eddy viscosity can smear out the flow structures within the

reach of a grid resolution. In such cases, the effective viscosity in the model should be

reduced to resolve the structures satisfactorily. To achieve this, a filter is imposed on the

turbulence model via eddy viscosity, which does not resolve structures smaller than the

filter size. The filter size is chosen based on the maximum grid size in the domain. The

filter-based viscosity model can be summarized as follows

2

3/ 21,tkC Min

kµεµ ρ

ε∆ ⋅ = ⋅

(3-33)

This choice of the function, f= 3/ 21,Mink∆ ⋅ε

, assures that in near wall nodes the function

will always return a value of 1.0 and hence wall functions can still be applied at near wall

regions. More details about the filter-based model can be found in Johansen et al. (2003).

43

Boundary conditions

Boundary condition needs to be specified due to the elliptic nature of the k-ε

equations. All the variables at the inlet are either specified or estimated. If the inlet

velocity is specified, k is estimated based on a fraction of the square of the inlet velocity

and ε is estimated from k and a characteristic length scale representing the size of the

turbulent eddies (usually a fraction of the inlet dimension). The estimates for k and ε are

given by

( )232 ik U T∞= (3-34)

3/ 4 3/ 2

0.07C k

Lµε =

where iT is the turbulence intensity usually set in the range of 0.02 and 0.05 and L is a

typical inlet dimension length scale. The outlet boundary conditions are determined via a

zero gradient condition based on the assumption that the flow is convection-dominated.

Wall treatment

Near wall boundaries, the local Reynolds number is of the order of one and hence

viscous effects are more dominant, hence the k-ε model cannot be used as it is formulated

based on the assumption of high Reynolds number. Two approaches have been proposed

to handle near-wall effects, one being the low Reynolds number model and the other

being wall-function method. The former method requires a very fine grid resolution near

the wall and hence makes the computations expensive. The latter method is based on the

assumption that there exists local equilibrium between production and dissipation of

turbulent kinetic energy. It was proved to be an accurate and robust approximation and it

does not require a fine grid near the wall or boundary.

44

For a fully developed turbulent flow near a no–slip wall, the normalized near–wall

tangential velocity, assuming a two–layer structure (viscous sublayer followed by log

layer), can be written as follows

+

+

for laminar sublayer (y 11.63)1 ln( ) for law of the wall layer (y 11.63)

yu

Ey

+

++

<=

>κ (3-35)

where κ is the von Karmen constant which has a value of 0.42 and E is a factor that

depends on wall roughness, streamwise pressure gradient, etc. It is assigned a value of

9.793 for a smooth wall. Here, +y is the local Reynolds number given by

+ nu ly τρµ

= (3-36)

and u+ is the non-dimensional velocity given by

tuuu

+

τ

= (3-37)

where uτ is the friction velocity at the wall given by

walluττ

ρ= (3-38)

The expression for the shear stress at the wall, wallτ , expressed in terms of turbulent

kinetic energy, in the log layer, is given by

1/ 4 1/ 2

ln( )

t

wall

C k uEy

µρ κ

τ += (3-39)

Structural Dynamics Model

For the current structural modeling of the wing, only linear effects will be

considered (Kamakoti et al., 2002). This simplification allows for a good description of

45

the motion of the wing, without being computationally hampered by complex nonlinear

effects. Since the wing is to be modeled as a linear structure, it is possible to model the

deformations as a summation of different modes of deformation without looking at the

complex interaction of the modes. To this end, we choose to model the wing as a linear

finite element structure that can undergo bending and torsion. It is also important to note

that we choose to keep the cross sections of the wing rigid, further uncoupling the

bending and torsional displacements.

The linear finite element that we choose to model the wing is a beam that has mass,

stiffness, and damping matrices of the actual wing. Thus, the deformations become that

of a Bernoulli-Euler beam bending and torsion, the equations for which reads as

2 2

2 2

d d wEI pdx dx

=

(3-40)

where p is the distributed loading (force per unit length) acting in the same direction as

the out-of-plane displacement (w), E is the Young's modulus of the beam, and I is the

area moment of inertia of the beam's cross section. Since, the cross sections of the wing

are assumed to be rigid, there is a point at which the vertical displacement of the beam is

a result of only the bending of the wing. This point is where the elastic axis of the wing

intersects the cross section. The generalized displacements from bending and torsion are

measured from this point. This has been depicted in Figure 3-1.

46

Figure 3-1. Displacements Measured with respect to the Elastic Axis

To find the equations of motion, Lagrange’s equation was used. The equations take

the form given by

Qq q q q

d T T V Fdt ∂ ∂ ∂ ∂− + = − + ∂ ∂ ∂ ∂ ! !

(3-41)

where q represents the generalized displacements, vertical and torsional displacements, F

represents the Rayleigh dissipation function, and Q represents the generalized forces.

The kinetic energy and the potential energy of the wing are given by T and V,

respectively. The generalized coordinates for the wing are functions of the position of

the cross section along the span of the wing and time. Here, the generalized coordinates

are referred to as w, representing the classical generalized coordinates of bending, and θ,

representing the classical generalized coordinates of torsion. The equations of motion

that govern the structural dynamics of the wing take the well-known form given by

( )+ + =M C K R!! !q q q t (3-42)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, R(t) is a

vector containing the generalized forces associated with aerodynamic loads, and q, q!!!! ,

and q!! are the displacement, velocity and acceleration vectors of the finite element

assembly, respectively. The above equation can be solved using modal approach by

47

composing the solution with the eigenvectors of the vibration problem, i.e., the

displacement vector can be written as

; ;= = =q Z q Z q Z! !!! !!Φ Φ ΦΦ Φ ΦΦ Φ ΦΦ Φ Φ (3-43)

where ΦΦΦΦ is the modal matrix containing the eigenvectors, orthonormalized with the mass

matrix, and Z is the generalized displacement. The eigenvectors are orthogonal to both

mass and stiffness matrices and if we assume Rayleigh damping, it is also orthogonal to

the damping matrix. Pre-multiplying Eq. (3-42) by ΦΦΦΦΤ, we get

t + t + t = t2T TZ( ) Φ CΦZ( ) Ω Z( ) Φ R( )!! ! (3-44)

where

2 , 1Φ Φ = Φ Φ =T TK Ω M (3-45)

The initial conditions on Z(t) are obtained using Eq. (3-43) at time 0, as follows

0

0

0

0

=

=

Φ

Φ

T

T

Mq

Mq

ZZ !!

Equation (3-44) can be written as n individual equations, one for each mode, as follows

2( ) ( ) ( ) ( )

( ) ( )

2Ti

i i i i i i i

i

z t t t t i = 1, 2, . . . , n

t tz z r

rξ ω ω

= Φ

+ + =R

!! !! (3-46)

with initial conditions

0

0

|

|

Ti t=0 i

Ti t=0 i

xx

=

=

ΦΦ

Mq

Mq!!

Here ωi is the natural frequency for the ith mode and ξi is the corresponding damping

parameter for that mode. The solution to the above equation can be obtained for each

mode using direct integration algorithms. The structural solver integrates these equations

of motion in time for one time step given the time step size, the pressures on structural

48

nodes at the initial time, and the initial geometry of the wing. It uses an implicit

Newmark scheme for temporal calculations and hence is not limited by a time step size

unlike other explicit methods.

Moving Grid Module

For fluid/structure interaction problems, we must account for grid movement along

the deformed surface. Since the structure moves after every time step, we need to

accommodate this movement in the CFD domain. This is usually done with some type of

dynamics related mesh algorithm. In our approach, we use the perturbation method for

single block grid, which is a TFI-like method as proposed by Eriksson (1982). A main

feature of the perturbation method is that it regenerates the computational grid based on

the original grid, hence it has the capability to preserve the quality of the original grid.

For a one-dimensional problem, it uses a simple formula based on spring analogy,

originally proposed by Batina (1989) for tetrahedral cells and later extended by Robison

et al. (1991) for hexahedral cells, given by

( )new old old new oldi i i s sx x S x x= + − (3-47)

where xi represents the location of interior grid point, xs the location of grid point on a

boundary, and S the normalized arc length along the radial mesh line measured from the

outer domain. Specifically,

2 2 21 1 1 1

2 2 21 1 1 1

( ) ( ) ( )

( ) ( ) ( )

il l l l l l l

i nl l l l l l l

x x y y z zS

x x y y z z= + + +

= + + +

Σ − + − + −=

Σ − + − + − (3-48)

To use this one-dimensional perturbation method, we only need to know the

displacements of the end points. The positions of the interim points are solely determined

based on the displacements of these end points. Here, we consider such displacements as

49

the source of perturbation. By design, a perturbation will spread throughout the whole

domain, but exerts more effect on points near the boundary perturbed. Based on Eq.

(3-47), a more general 3-stage perturbation method is proposed for three-dimensional

problems treated by the single-block approach.

As far as the multi-block grid movement is concerned, we use the master/slave

strategy to establish a relationship between the moving surface points (master points) and

vertices located at the other blocks (slave points). The movement of the master points is

based on the displacements obtained from the structure solver. The movement of the

slave points follows its master point. A slave point, which has the coordinate of xs, moves

when its master point moves from mx# to xm. A simple but effective formula suggested by

Hartwich and Agrawal (1997), based on spring analogy, is expressed as follows:

s s m mx =x +θ(x -x )# # (3-49)

where the subscripts m and s represent master and slave, respectively, and tilde (~)

indicates the new position. θ is the decay function; here, we use the Gaussian distribution

function, as suggested by Hartwich and Agrawal (1997), given by

exp min[ , /( )]FACMIN dv dm= − +θ β ε (3-50)

where

2 2 2( ) ( ) ( )v m v m v mdv x x y y z z= − + − + − (3-51)

2 2 2( ) ( ) ( )m m m m m mdm x x y y z z= − + − + −# # # (3-52)

and ε is an arbitrary small number to eliminate division by zero.

In Eq. (3-50), the coefficient β affects the stiffness. A larger β causes the block to

behave more like a rigid body and a smaller value makes the body behave like a softball.

50

The factor, FACMIN, in Eq. (3-50) plays an important role in the re-meshing part when

the displacement of the master nodes is small. This will be explained with an example

later on.

Overview of Algorithm

The numerical procedure employed by the different modules will be discussed in

this section.

Discretized Form of Equations

The governing equations presented in Eq. (3-10)-(3-14) are discretized on a

structured grid. The velocity components and the scalar variables (pressure, density,

kinetic energy, etc.) are located at the cell-center of the control volume whereas the

contravariant components of velocity (U, V, W) are located at the center of the control

volume surfaces as shown in Figure 3-2.

Figure 3-2. Location of variables u, v and p on a 2-D non-staggered grid for the pressure–based algorithm.

The discretized form of the continuity equation is written as:

[ ] [ ] [ ] 0o o

e n t

w s b

J J U V Wt

ρ ρ ρ ρ ρ− + + + =∆

(3-53)

o

o

o

o oW E

N

S

Pw

s

e

n

o

x

x x

x •

•

•

•

U

V

U

V

u,v,w,p.ρ

51

The discretized form of the momentum equations can be obtained in a similar

manner. Here, we present the details for the u–momentum equation based on first order

backward Euler method. The v– and w–momentum equations follow similar derivation:

11 12 13 11

21 22 23 21

31 32 33 31

( )

( )

( ) 0

eo o o

wn

st

b

J u J u w w wUu q q q f pt J

w w wVu q q q f pJ

w w wWu q q q f pJ

ρ ρ Γρ∆ ξ η ζ

Γρξ η ζ

Γρξ η ζ

− ∂ ∂ ∂+ − + + + + ∂ ∂ ∂

∂ ∂ ∂− + + + + ∂ ∂ ∂

∂ ∂ ∂− + + + = ∂ ∂ ∂

(3-54)

The above equation is comprised of three flux terms namely, convection fluxes,

pressure fluxes and diffusion fluxes. A second order difference operator is used for the

flux terms. The final form of the discretized u-momentum equation takes the form:

11 12 13( )u u u u u u up p E E W W N N S S T T B B up

a u a u a u a u a u a u a u f p f p f p Sξ η ζ= + + + + + + + + + (3-55)

which can also be written as

u up p nbr nbr ua u a u S= +∑ (3-56)

where ‘nbr’ represents the six neighboring points to the point P. The various coefficients

in the Eq. (3-55) are given by

11 11

22 22

33 33

2 2

2 2

2 2

u uE W

e w

u uN S

n s

u uT B

t b

u u u u u u uP E W N S T B

U Ua q a qJ J

U Ua q a qJ J

U Ua q a qJ J

Ja a a a a a at

ρ Γ ρ Γ

ρ Γ ρ Γ

ρ Γ ρ Γ

ρ∆

= − + = − +

= − + = − +

= − + = − +

= + + + + + +

(3-57)

and the source term is given by

52

12 13 21 23 31 32

11 21 31( )

e n t

uw s b

e n tp w s b

u u u u u uS q q q q q qJ J J

+ p f f f

Γ Γ Γξ ζ ξ ζ ξ η

∂ ∂ ∂ ∂ ∂ ∂= + + + + + ∂ ∂ ∂ ∂ ∂ ∂

+ +

(3-58)

Evaluation of Contravariant velocities on Non-staggered Grid

While performing computations on non-staggered grids, the velocity field is

evaluated at the cell centers whereas the contravariant components of the velocity are

evaluated at the cell faces, hence necessary interpolation technique need to be employed

to evaluate the contravariant components of velocity in a consistent fashion. We use the

momentum interpolation technique originally proposed by Rhie and Chow (1983) to

accomplish this. We rewrite the u-momentum equation (Eq. (3-56)) as follows:

*11

11 u

u up p nbr nbr

u

a u a u S f p

H f pξ

ξ

= + −

= −∑

(3-59)

The source term S* contains the same terms as S except the component of the pressure

gradient term in the ξ-direction. Using indicial notation, the above equation can be

written as:

( ) ( ) 11 111 1 2 1 1 2

1

i

u

i / /P Pi

Hu f p f pa a + −

= − −

(3-60)

A similar equation can be written for the east neighbor

( ) ( ) 1

1 11 111 3 2 1 1 21

1

i

u

i / /P Pi

Hu f p f pa a

+

+ + ++

= − −

(3-61)

We use momentum interpolation method to evaluate interface velocity based on the nodal

pressure at the interface. The expression for interface velocity is given by

( )111 2 1

1 2 1 2

u

i / i iP Pi / i /

fHu p pa a+ +

+ +

= − −

(3-62)

53

The first time on the right is evaluated using Eq. (3-60) and (3-61) as follows

( )1 2 1

1u u u

i iP P Pi / i i

H H HI Ia a a

+ +

= + −

(3-63)

[ ] ( ) ( ) ( ) ( ) ( ) 1

1 1

11 11 11 111 1/ 2 1 1/ 2 1 3/ 2 1 1/ 2(1 )

1i i i i

i i

ii

P P

I u I uII f p f p f p f p

a a+

+ +

+ − + += + − +

−− + −

Here, Ii is the interpolation coefficient along the x-coordinate direction. Substituting the

above expression into Eq. (3-62), we get

( )1/ 2

1

1/ 2 1/2 11 1 1/2 1/ 2 3/ 2 1/ 21/2

11 ( ) ( ) ( )i

i i

iii i i i i i i i

P P Pi

u uIIf p p p p p p

a a a+

+

+ + + + − + ++

= − − − − − − −

(3-64)

where the over bars represent interpolation using the nodal values on either side of the

interface. Similar expressions can be written for vi+1/2 and wi+1/2. Finally, the expression

for contravariant velocities at the cell face for a stationary grid (grid velocities are zero)

can be evaluated using Eq. (3-17) as follows

[ ]

1

1/ 2 11 1/ 2 12 1/ 2 13 1/ 2

2 2 211 12 13 1/ 2 1 3/ 2 1/ 2 1/ 2 1/ 2

1/ 2

(1 )1( ) ( ) ( ) ( )i i

i i i i

i ii i i i i i i

P P Pi

U f u f v f w

I If f f p p p p p pa a a

+

+ + + +

+ + + + + −+

= + + −

− + + − − − − −

(3-65)

For unsteady computations, situations may arise where we might be forced to use a

small time step size based on the CFL type stability condition. It has been shown

previously (Choi, 1999; Yu et al., 2002a and Yu et al., 2002b) that a small time step leads

to minor oscillations in pressure and velocity field while using the original Rhie-Chow

momentum interpolation method. They proposed modified momentum interpolation

schemes to calculate cell-face velocity to eliminate the effect of time step size on the

solution. One such proposed scheme to calculate the cartesian u-velocity component at

the cell face is shown here. Similar expressions follow for v and w-components.

54

( )

1/ 2

1

1

1/ 2 1/ 2 11 1 3/ 2 1/ 2 1/ 2 1/ 21/ 2

1/ 2 1 11/ 2

1/ 2

(1 )1 ( ) ( ) ( )

(1 )1

i

i i

i i

i ii i i i i i i i

P P Pi

l l l l l li ii i i i i

iP P Pi

I Iu u f p p p p p pa a a

I Ia u a u a ua a a

+

+

+

+ + + + + + −+

+ + ++

+

− = + − − − − −

− + − −

(3-66)

where l ii

Jt

a ρ=

∆. The superscript l denotes previous time step. In the above equation, the

terms inside the first box bracket constitute the original Rhie-Chow momentum

interpolation scheme and the term inside the second bracket is the additional term that

was proposed to eliminate the time dependency effect. The above formulation was

developed for cartesian grids. We extend the above scheme to make it suitable for

curvilinear grids, which means calculating contravariant velocities in an appropriate

manner from the cartesian velocity components at the cell faces. The expression for u-

component of contravariant velocity reads as

[ ]

1

1/ 2

1

1/ 2 11 1/ 2 12 1/ 2 13 1/ 2

2 2 211 12 13 1/ 2 1 3/ 2 1/ 2 1/ 2 1/ 2

1/ 2

1 11 1 12 1 131/ 2

(1 )1( ) ( ) ( ) ( )

(1 )1

i i

i

i

i i i i

i ii i i i i i i

P P Pi

l l l lii i i i

P Pi

U f u f v f w

I If f f p p p p p pa a a

IU a f u f v f wa a

+

+

+

+ + + +

+ + + + + −+

+ + + ++

= + + −

−+ + − − − − −

−+ − + +

( ) ( )1 11 12 13i

l l l l lii i i i

P

I a f u f v f wa

− + +

(3-67)

Similar expressions can be derived for V and W components of contravariant

velocities. The terms in the last brackets are the additional terms stemming from the

unsteady terms added to the original Rhie-Chow momentum interpolation method in Eq.

(3-66). We will carry out test cases on a uniform cartesian grid (cavity flow) as well as a

curvilinear non-uniform grid (cylinder flow) to study the impact of these terms for

different time step sizes and the results will be presented at a later stage.

55

Pressure-Based Flow Solver (SIMPLE)

We use a pressure–based flow solver for computing both compressible and

incompressible flows as well as laminar and turbulent flows. The flow solver is based on

the SIMPLE (Semi–Implicit Method for Pressure–Linked Equations) algorithm originally

developed by Patankar and Spalding (1972). This method was originally developed for

incompressible flows and it has been extended to solve compressible flows by modifying

the pressure correction equation to include the effects of density on pressure (Shyy &

Braaten 1988). Additionally, the algorithm has been extended to body–fitted curvilinear

coordinates in order to handle arbitrarily–shaped flow boundaries. This approach can

handle flows at all speeds without any fundamentally different treatment for any

particular flow regime.

The SIMPLE algorithm can be divided into two steps, the prediction and correction

steps. In the prediction step, we start with a guessed or intermediate solution of the

pressure field, p* and the velocity fields, u*, v* and w* and the velocity and pressure

fields are obtained by adding a correction term as follows:

' *

' *

' *

' *

u u uv v vw w wp p p

= +

= +

= += +

(3-68)

where the primed quantities denote the correction terms.

To obtain the corrected quantities, we start with the u-momentum equation for the

intermediate solution i.e., by substituting the predicted quantities into the momentum

equations. The expression for u* is given here. Expressions for v* and w* follow similar

procedures.

56

* * * * * * * * * *11 12 13( )

p E W N S T B

u u u u u u up E W N S T B up

a u a u a u a u a u a u a u f p f p f p Sξ η ζ

= + + + + + + + + + (3-69)

Subtracting Eq. (3-69) from Eq. (3-55) and neglecting all the terms on the right

hand side related to velocity (this is the main approximation of the SIMPLE algorithm),

we get the following expressions for the velocity corrections.

' ' ' '11 12 13

' ' ' '21 22 23

' ' ' '31 32 33

u f p f p f p

v f p f p f p

w f p f p f p

ξ η ζ

ξ η ζ

ξ η ζ

= + +

= + +

= + +

(3-70)

The corresponding corrections for the contravariant velocities can be written as

' ' ' '11 12 13

' ' ' '21 22 23

' ' ' '31 32 33

U f u f v f wV f u f v f wW f u f v f w

= + +

= + +

= + +

(3-71)

Substituting Eq. (3-70) into Eq. (3-71) and after necessary approximations, we

obtain the following expressions for U, V and W.

22 2* '1311 12

22 2* '2321 22

2 2 2* '31 32 33

u v wp p p

u v wp p p

u v wp p p

ff fU U pa a a

ff fV V pa a a

f f fW W pa a a

ξ

η

ζ

= + + +

= + + +

= + + +

(3-72)

where

* * * *11 12 13

* * * *21 22 23

* * * *31 32 33

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

U f u x f v y f w zV f u x f v y f w zW f u x f v y f w z

= − + − + −

= − + − + −

= − + − + −

! ! !! ! !! ! !

(3-73)

57

We now obtain the pressure correction equation by substituting the above

expressions for the contravariant velocities into the continuity equation. The pressure

correction equation can be written as

'' ' ' ' ' ' '

p p E E W W N N S S T T B B pa p a p a p a p a p a p a p S= + + + + + + (3-74)

where the coefficients are given by

22 21311 12

22 22321 22

2 2 231 32 33

E u v wp p p e

N u v wp p p n

T u v wp p p t

ff faa a a

ff faa a a

f f faa a a

= + +

= + +

= + +

22 21311 12

22 22321 22

2 2 231 32 33

W u v wp p p w

S u v wp p p s

B u v wp p p b

ff faa a a

ff faa a a

f f faa a a

= + +

= + +

= + +

(3-75)

'* * * * * *( )

p E W N S T B

o o

w e s n b tp

a a a a a a a

J JS U U U U U Ut

ρ ρρ

= + + + + +

−= − + − + − +∆

(3-76)

In this method, pressure correction is used to update contravariant velocities and

the cartesian components of the velocity are used to update cartesian velocity and

pressure fields. This procedure is iterated until the convergence criterion is met. An

overview of the SIMPLE algorithm is shown in Figure 3-3.

58

Figure 3-3. Overview of the SIMPLE algorithm

PISO Algorithm for unsteady computations

For unsteady flow computations we use a predictor-corrector type method to avoid

the iterative procedure associated with the SIMPLE family of methods. One such method

is the PISO algorithm, originally developed by Issa (1985). It is modified to suit multi-

block, curvilinear, structured grids. First, we will look at the discretized equation for fully

implicit scheme based on SIMPLE algorithm.

1 1

11 21 31( ) ( ) ( ) ( )n n n

n u n

P

n n ne w n s t b

J u Ja u H u

t tf p p f p p f p p+ ++ = +

∆ ∆

− − − − −

−ρ ρ (3-77)

In the above equation, ap is the nodal coefficient containing the special discretization

terms and Hu contains all the convective and diffusive flux discretization terms as well as

any source term that may be present excluding the pressure gradient terms. The

superscript n denotes the old time step and n+1 the new time step. The implicit solution

START

Solve discretized momentum equations

Solve pressure correction equation

Correct pressure and velocity

Converged?

STOP

No

Yes

Iterate

u*, v*

p’

u, v, p

59

for above equation requires several iterations at each time step, which can be

computationally expensive. In order to eliminate such an iterative procedure, we use the

PISO algorithm, which uses a series of predictor and corrector stages to arrive at the

converged solution for a given time step. The intermediate solution at the predictor stage

is denoted by * and the solution at the corrector stages are denoted by **, ***, etc. The

solution at the last corrector stage is considered the final solution at new time step, n+1.

Predictor step: In the predictor step, the u, v, w-momentum equations are solved

implicitly but the pressure field is provided by the previous time step. The u-momentum

equation is shown below. The v and w-momentum equations follow a similar pattern

* *

11 21 31( ) ( ) ( ) ( )n n n

u

P

n n ne w n s t b

J u Ja u H u

t tf p p f p p f p p+ = +

∆ ∆

− − − − − −

ρ ρ (3-78)

Corrector step: In the first corrector stage, a new velocity (u**, v**, w**) is sought

along with a corresponding pressure field (p*). The velocity field obtained from the

predictor step (u*, v*, w*) is used to arrive at the desired quantities. The u** momentum

equation is shown here.

** * * * *11 21 31( ) ( ) ( ) ( )

n n nu

P e w n s t bJ u J

u H ut t

a f p p f p p f p p+ = +∆ ∆

− − − − − −

ρ ρ (3-79)

To determine p* in the above equation, we combine the above equation with the

continuity equation to formulate an equation for pressure correction, p, which is defined

by

p=p*-pn

Once we solve for p, we correct the velocities u*, v*, w* to obtain u**, v**, w**.

Subsequent corrector steps are carried out in a similar fashion by assuming u**,

v**, w**, p* as the intermediate velocity field to obtain u***, v***, w***, p** and so on.

60

A total of four corrector stages were used to arrive at the final solution at the new time

level, n+1. The PISO algorithm, being a semi-implicit method, is limited by the choice of

time step based of CFL number, which is defined as follows

u tCFLx

∆=∆

where u is the free stream velocity, ∆x is the grid spacing. Time step size is chosen based

on free stream velocity and grid spacing so as to obtain a CFL number with an O(1)

magnitude. An overview of the PISO algorithm is shown in Figure 3-4. A detailed

procedure of the algorithm can be found in Thakur et al. (2002).

Updating Jacobian values for moving boundary treatment

While formulating the above-mentioned solver for the moving boundary problem,

we need to make sure that the geometric conservation law is satisfied. This is ensured by

updating the Jacobians in a consistent fashion. The equation that needs to be satisfied is

re-written here for convenience

( ) ( ) ( ) 0t t t tJ ξ η ζξ η ζ+ + + = (3-21)

We will now look at how to calculate the Jacobian values in a consistent fashion. In this

regard, we will implement four different time-integration schemes for evaluating the

Jacobian values: first order implicit scheme, first order time-averaged scheme, second

order implicit scheme and a second order time-averaged scheme. The formulation for the

different GCL schemes is as follows.

61

Figure 3-4. Overview of PISO algorithm for unsteady flow computations

First order Implicit Scheme:

This scheme was initially suggested by Thomas and Lombard (1972) for density-

based finite difference schemes and later was implemented using a pressure-based finite

volume method by Shyy et al. (1996 & 2001). The difference equation for the fully

implicit scheme is given by:

( ) ( ) ( )1 1 11 n n nn nt t tJ J t

ξ η ζξ η ζ+ + ++ = − ∆ + +

(3-80)

where the metric coefficients are calculated from Eq. (3-22) at the (n+1)th time step and

the grid velocities are calculated as follows

1 1 11 1 1, ,

n n n n n nn n nx x y y z zx y z

t t t

+ + ++ + +− − −= = =

∆ ∆ ∆! ! ! (3-81)

Predictor step

Corrector step 1

Corrector step 2

START

Solve pressure correction equation

Correct pressure and velocities

STOP

Correct pressure and velocities

Solve second pressure correction equation

Solve discretized momentum equations with pressure field

from previous time step

u**, v**, p*

u*, v*

p

p’

u***, v***, p**

62

First-order time-averaged scheme:

It was suggested (Farhat et al., 2001) that a time-averaged evaluation of the metric

terms leads to a more consistent evaluation of the Jacobian especially when the time step

is large. Hence, we re-formulate the governing equation (Eq. (3-21)) by time-averaging

the evaluation of metrics over more than one mesh configuration as opposed to

evaluation of metrics on a single mesh configuration (Eq. (3-80)). It can be written as

follows:

( ) ( ) ( )1 1 1 1

2 2 2 0n n

n n nt t t

J Jt ξ η ζ

ξ η ζ+

+ + +− + + + = ∆ (3-82)

The metric terms are computed as follows:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 12

1 12

1 12

121212

n n nt t t

n n nt t t

n n nt t t

ξ ξ ξ

ξ ξ ξ

ξ ξ ξ

ξ ξ ξ

η η η

ζ ζ ζ

+ +

+ +

+ +

= +

= +

= +

As can be seen from the above expression that the metrics are evaluated at a mesh

configuration, which is in between the mesh configurations at tn and tn+1. The grid

velocities for the metric term are evaluated as follows:

1 1 11 1 12 2 2, ,

n n n n n nn n nx x y y z zx y zt t t

+ + ++ + +− − −= = =∆ ∆ ∆

! ! ! (3-83)

Second order implicit scheme

The difference equation for a second order implicit scheme (3-point backward), as

suggested by Koobus and Farhat (1999) is used here. For structured meshes, it can be

written as

63

( ) ( ) ( )1 1

1 1 1

3 122 2 0

n n n

n n nt t t

J J J

t ξ η ζξ η ζ

+ −

+ + +− +

+ + + = ∆

(3-84)

Here, the metrics are evaluated in a similar fashion to the one used for first order implicit

scheme at the (n+1)th grid configuration. The grid velocity is also calculated in a similar

fashion as given by Eq. (3-81).

Second order time-averaged evaluation of Jacobian

Here we use a time-averaged procedure to evaluate the fluxes. Koobus and Farhat

(1999) employed such a procedure for unstructured meshes and we extend such a scheme

to a structured mesh here. The scheme can be written as follows.

( ) ( ) ( )

( ) ( ) ( )

1 11 1 12 2 2

1 1 12 2 2

3 12 12 221 02

n n n

n n nt t t

n n nt t t

J J J

t ξ η ζ

ξ η ζ

ξ η ζ

ξ η ζ

+ −

+ + +

− − −

− + + + + ∆ + + + =

(3-85)

where

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 11 12 2

1 11 12 2

1 11 12 2

1 1;2 21 1;2 21 1;2 2

n n n n n nt t t t t t

n n n n n nt t t t t t

n n n n n nt t t t t t

ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

ξ ξ ξ ξ ξ ξ

η η η η η η

ζ ζ ζ ζ ζ ζ

+ + − −

+ + − −

+ + − −

= + = +

= + = +

= + = +

(3-86)

Here, the averaging is done in between configurations [tn, tn+1] and [tn, tn-1] and the mesh

velocities for each of these metric terms are evaluated as follows:

1 1 11 1 12 2 2

1 1 11 1 12 2 2

, ,

, ,

n n n n n nn n n

n n n n n nn n n

x x y y z zx y zt t t

x x y y z zx y zt t t

+ + ++ + +

− − −− − −

− − −= = =∆ ∆ ∆− − −= = =∆ ∆ ∆

! ! !

! ! ! (3-87)

64

As can be seen from the above equations that we need three different mesh

configurations to arrive at this scheme. All of the above-mentioned schemes will be

implemented using several test cases to arrive at the appropriate scheme for aeroelastic

computations.

Newmark Integration Method for Structure Solver

In this section, the numerical procedure used to solve the structure equations of

motion is presented. The equations of motion of the structure, written for the individual

modes, can be written in a generalized form for any particular mode as follows

22z z z rξω ω+ + =!! !! (3-88)

Equation (3-88), written at a time t+∆t, reads as follows

22t t t t t t t tz z z rξω ω+∆ +∆ +∆ +∆+ + =!! !! (3-89)

In the above expression, ξ, ω, and t tr +∆ are known before hand. The task is to evaluate the

displacement, velocity and acceleration at the time t+∆t. The following expressions for

velocity and displacement are formulated at time t+∆t first as a function of acceleration at

t+∆t and displacement, velocity and acceleration from previous time level t.

[ ](1 )t t t t t tz z tz zδ δ+∆ +∆= ∆+ − +! ! !! !! (3-90)

212( )t t t t t t tz z z t tz zα α+∆ +∆

= + ∆ ∆ + − +! !! !! (3-91)

where, α and δ are parameters that are chosen based on desired stability and accuracy.

For the Newmark scheme to be unconditionally stable, values of 0.25 and 0.5 are chosen

for α and δ, respectively.

65

Substituting Eqs. (3-90) and (3-91) into Eq. (3-89) and solving for t tz +∆!! and then

using Eqs. (3-90) and (3-91) to calculate t tz +∆! and t tz +∆ , the following relation can be

established for displacement, velocity and acceleration.

t t t

t t t t t

t t t

z zz z rz z

+∆

+∆ +∆

+∆

= +

A L!! !!! ! (3-92)

where

[ ][ ] ( )

12 2

12

2 1 12 2

1 1( ) 2(1 ) ( 2 ) ( )

11 ( ) 2(1 ) 1 2 ( )

( ) 2(1 ) 1 2 (1 )

t t

tt

t t

α β δ κ β κ β

δ α δβ δ δκ βδ δκ δβ

α α α β δ ακ αβ ακ α β

− − − − − − − ∆ ∆ = ∆ − − − − − − − − ∆ ∆ − − − − − ∆ − − −

A (3-93)

2 2

2

2

t

t

βωβδ

ωαβω

∆ = ∆

L (3-94)

1

2 2

1 2 ; t t t

ξδ ξββ α κω ω ω

− = + + = ∆ ∆ ∆

66

CHAPTER 4 COMPUTATIONAL PROCEDURE AND CODE VALIDATION

In this chapter, the computational procedure for performing fluid-structure

interaction computations will be discussed in depth. The computational setup for the test

case that is used to carry out computations will be shown as well. We also present

validation results associated with several modules of the computational aeroelasticity

model.

Computational Procedure

The overall procedure for carrying out computational aeroelastic computations can

be divided into the following major steps.

1. Constructing the geometry for aeroelastic computations and also to supply appropriate boundary conditions and initial conditions.

2. Perform steady state CFD computation to obtain initial guess for starting coupled computations.

3. Perform unsteady CFD computations using steady state result as initial guess and obtain necessary aerodynamic forces on the surface of the wing.

4. Interpolate aerodynamic forces onto the structural mesh.

5. Perform CSD computation to obtain the deformation of the geometry

6. Extrapolate the displacements onto CFD surface grid.

7. Re-mesh CFD grid based on the deformation obtained from the CSD calculations using the moving boundary module.

8. Repeat steps 3-7 using current solution as the initial guess for the subsequent steps.

Now we will take a closer look at the above-mentioned steps along with the grid

generation details, interpolation/extrapolation schemes used.

67

Geometry definition and Computational Grids

Geometry Definition

The geometry used for all of our test cases for aeroelastic computations is the well-

documented AGARD (Advisory Group for Aerospace Research and Development) 445.6

wing (Yates, 1987). This is the first AGARD standard aeroelastic configuration. It was

first tested in the Transonic Dynamics Tunnel (TDT) at the NASA Langley Research

Center (Yates et al. 1963). The AGARD 445.6 wing is a swept back wing with a quarter-

chord sweep angle of 45o with a NACA 65A004 airfoil (4% thickness) cross-section. It

has a panel aspect ratio of 1.65 and a taper ratio of 0.66. The root chord of this model is

1.833 feet with the semi-span of 2.5 feet. The wing tested at NASA Langley was a semi-

span, wall-mounted model made with laminated mahogany. A schematic of the AGARD

wing is shown in Figure 4-1. The computational grids used by the fluid and structure

solvers are described next.

Figure 4-1. Schematic of the AGARD 445.6 wing used in the wind tunnel (Yates, 1963)

68

Computational Grids

CFD grid

A CFD mesh is generated around the AGARD 445.6 wing by placing the wing in

the middle of the computational domain, which has dimensions of 18x9x9 units. The

geometry could be generated by using the CAD module of any commercial mesh-

generating software such as ICEMCFD or PATRAN, the latter being used for this case.

ICEMCFD was used to construct the CFD mesh around the wing. The entire

computational domain with all the blocks is shown in Figure 4-2. As can be seen from the

figure, it is a multi-block domain comprising of 10 blocks. An O-grid was employed

around the wing to preserve orthogonality of grid near the wing. Since it is a very thin

wing, care must be taken while generating mesh around the wing tip and trailing edge to

avoid any negative cell volumes. The initial coarse gird had a total of 4838 points

distributed over the wing surface (118 points in the chordwise direction and 41 points in

the spanwise direction). The entire CFD domain had a total of 322,622 points. The CFD

surface grid along with the meshing system at the leading and trailing edges are shown in

Figure 4-3 for clarity.

69

Figure 4-2. Overview of the Multi-block CFD grid

Figure 4-3. CFD surface grid along with grid distributions at the leading and trailing edges

CSD grid

For the structure solver, since we use beam elements, a ten-element beam mesh

spanning the semi-span of the wing was used. However, in order to make the

interpolation and extrapolation procedures between the CFD and structure mesh efficient

Y

X

Z

70

and easier, an intermediate surface mesh was generated with QUAD4-type elements.

Equal width spanwise elements, four per beam element, along the spanwise direction

were used to generate this intermediate mesh. The QUAD4 elements, however, were non-

uniform along the chordwise direction to comply with the geometry. This intermediate

surface mesh on the AGARD wing was generated using PATRAN. It is comprised of

2400 elements surrounding the wing (60 along the chordwise direction and 40 equal

width elements along the spanwise direction). The intermediate or temporary mesh is

shown in Figure 4-4.

Figure 4-4. Schematic of the FEM grid on the AGARD wing

Coupling and Interfacing Procedure

Grid generation is only the preliminary step towards performing coupled

aeroelastic analysis. The next step is to formulate an efficient and robust interface

technique that links the two different grid systems (CFD and CSD) mentioned in the

previous section. This is essential while formulating a closely-coupled CAE model. The

interface technique as well as the coupling procedure will be discussed at length in this

section.

X

YZ

71

Once the aerodynamic pressures are obtained on the CFD surface grid of the wing,

they need to be interpolated onto the CSD grid to act as the external force for the

structural equations of motion. A bilinear interpolation procedure was used to transfer the

scalar pressures onto the structure grid. To do this, the top and bottom surface of the wing

are treated as two-dimensional surfaces. The pressures are then mapped from the CFD

surface grid onto the corresponding intermediate CSD surface grid. This is done by

locating the four CFD grid points engulfing a given CSD grid point and employing a

straightforward bilinear interpolation procedure. This is demonstrated in Figure 4-5. Once

we locate the CFD grid points engulfing the FE grid point (p, q), we calculate

intermediate pressures at points (1) and (2) by a linear interpolation procedure and these

intermediate pressures are then used to evaluate pressure at point (p, q) by performing

another linear interpolation. Such a scheme gives an order of accuracy between one and

two.

Figure 4-5. Schematic to demonstrate interpolation technique

Now that the surface pressure distribution on the CSD grid is obtained, they have to

be converted into pressure forces. This is accomplished by computing the unit normal and

the surface area for each QUAD4 element. The scalar pressures are then multiplied to the

area of the element and the unit normal to obtain the pressure forces for each element. As

mentioned before, the intermediate CSD grid is divided into 40 equal-width elements

• •

• •

×××× (p,q)

(i,j)

(i,j+1)

(i+1,j)

(i+1,j+1)

(1)

(2)

CFD grid lines FEM grid lines (i, j) : CFD grid locations (p,q) : FEM grid location

72

along the spanwise direction. A set of 4 such elements along with all the corresponding

chordwise elements are treated as a “super element”, which constitutes a single beam

element. Each super element has 4 equal width surface elements in the spanwise direction

and 60 surface elements in the chordwise direction as shown in Figure 4-6. The net

pressure force for each super element is then calculated by summing up all the pressure

forces for the elements contributing to the super element. This is evaluated for each of the

super elements and the resulting net force calculated acts as the external force for the

beam element. The mass and stiffness matrices for the structure are obtained from full

NASTRAN models.

Figure 4-6. Schematic of a super element: Portion of the entire structure

The structural equations of motion are now solved using the interpolated forces and

the matrices to obtain the displacements. The displacements are output in terms of a

vertical deflection, w and torsional angle, θ. These are used to obtain the new shape of the

wing and hence the new location of the CFD surface grid location. Since Bernoulli-Euler

beam theory is assumed, the cross-section of the wing remains unchanged. The new

location of the CFD surface grid points are determined by assuming a rigid link

connecting each CFD grid point to the beam element. The links are assumed to be

perpendicular to the elastic axis as shown in Figure 4-7.

73

Figure 4-7. Sample CFD mesh superimposed on the discretized beam structure

The state of the beam at any instant along the spanwise direction is given by

1 2 3 1 2 3 | Tsw w w w= θ θ θ ,

where w represents the vertical deflection and θ the twist at each spanwise section and

subscripts 1, 2, 3 denote displacements in the x, y and z directions respectively. In the

present study, there is no deflection in the x and y directions and no rotation about the x

and z axes. In other words, the deflection of a CFD grid point P can be written as

T RP P Pw w w= + ,

where TPw is the translation component and

RPw is the rotational component. The

translation and rotation component corresponding to CFD spanwise grid locations are

obtained by performing a linear interpolation using CSD spanwise grid points. Once the

translation and rotation component for each CFD spanwise grid location is obtained, an

extrapolation technique is used to obtained the new locations of the CFD surface grid.

This is demonstrated through Figure 4-8. In the schematic, quantities with subscript o

denote the original location of the CFD surface grid points and subscript 1 denotes the

new location. By knowing w and θ at each CFD grid spanwise location, the new location

is obtained using equations (3-95) and (3-96).

• • • • • ••

• • • • • •

• • • • • ••

Finite element nodes

CFD element nodes

Rigid links

74

x1=xocosθ+zosinθ (3-95)

z1=zocosθ-xosinθ+w (3-96)

Figure 4-8. Schematic to demonstrate the extrapolation procedure

Once the new CFD surface grid is obtained, the CFD domain needs to be re-

meshed. This is accomplished using the moving boundary module explained earlier. The

new CFD surface grid acts as the source of perturbation needed to enable re-meshing as

explained in chapter 3. The next step is to ensure that the grid velocities are updated in a

consistent fashion on the boundary and also, care should be taken to preserve the

geometric conservation law mentioned in the previous chapter. This entire procedure is

then repeated for subsequent time steps to arrive at desired aeroelastic results.

Code Validation

One of the most important aspects of developing a computational tool is to validate

the model with theory or experiment or prior computational results. Since there are

numerous issues associated with a CAE model, this section aims at looking each of the

modules individually and to ensure consistency with previously published results for

these modules. First, steady state CFD results will be shown, which will be followed by

various issues that one may experience with unsteady flow computations, for e.g.,

accurate computation of contravariant velocities (for small time step sizes) and Jacobians

Z

Xw(y,t)

Θ (y,t)

(xo,zo)

(x1,z1)

Elastic Axis

75

(for moving mesh problems). Finally, results from the simplified structure model are then

compared with experimental results as well.

Steady-state CFD Computations

First, we validate our CFD code using the AGARD 445.6 wing defined in the

previous chapter and present some steady state results. The top view of the computational

domain along with the boundary conditions is shown in Figure 4-9. We perform a viscous

compressible flow calculation based on a Reynolds number of 366,400 (based on a unit

chord length at the root) and a transonic Mach number of 0.96. The steady state pressure

contours on the top surface of the wing is shown in Figure 4-10. Pressure coefficients at

different spanwise locations at the top surface of the wing are shown in Figure 4-11.

Results from both figures are in excellent agreement with prior numerical calculations

performed by Lee Rausch and Batina (1993).

Figure 4-9. Top view of the CFD domain showing the type of boundary conditions specified at different surfaces

X

Y

NO SLIP WALL

NO SLIP WALL

INLET

OUTLET

76

Figure 4-10. Steady state surface pressure contours on the AGARD wing

Figure 4-11. Steady state pressure coefficient distribution at different spanwise locations on the top surface

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

η=0.25

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

η=0.50

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

η=0.96

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.1

0

0.1

0.2

0.3

0.4

η=0.75

X

Y

0 1 20

0.5

1

1.5

77

Unsteady Computations using PISO Algorithm

To validate the pressure-based algorithm for time-dependent computations, the

unsteady problem of flow around a square cylinder is studied. Computations are

performed at a Reynolds number of 215, for which experimental results are readily

available. The flow is laminar and exhibits the well-known vortex shedding phenomenon

at this Reynolds number. A perturbation is given to the initial flow field to quickly reach

the periodic oscillatory flow pattern. The computational domain and the boundary

conditions employed for performing the unsteady computations are shown in Figure 4-12.

Figure 4-12. Computational domain for flow past square cylinder along with imposed boundary conditions

On the left boundary, the steady-state inlet condition with constant horizontal

velocity is used. On the right boundary, the outlet condition employing zero gradient

extrapolation of the velocity and specified pressure is invoked. The top and bottom

boundaries are treated as no-slip boundaries, which sets all the velocity components to

zero. The plot of the periodic behavior of the cross-stream component of the velocity (v)

NOSLIP BOUNDARY

NOSLIP BOUNDARY

INLE

T

OU

TLE

T

78

obtained by probing at a single point downstream of the cylinder is shown in Figure 4-13.

The v-component of the velocity alternates due to the vortex shedding behind the

cylinder. Three different time step sizes are used to perform unsteady computations using

the PISO algorithm. The time step sizes chosen were 0.002, 0.001 and 0.0005 which

correspond to CFL numbers of 0.27, 0.13 and 0.07 respectively, which are all well within

O(1) magnitude ensuring stability and accuracy of PISO algorithm As can be seen from

the figure, minute differences are seen in the amplitude of oscillations for different time

step sizes. Specifically, the amplitude of the oscillations was found to increase with

decrease in time step size. The well-known Karmen vortex street is plotted for two

choices of time step sizes to visualize the differences, if any, in the vortex structure. This

is shown in Figure 4-14. It can be seen from the plot that the frequency of vortex

shedding is identical for the two choices of time steps; however, the vortex structure itself

looks a little different in terms of the rotation angle of some of the structures. For

verification purposes, identical computations were run using the SIMPLE family of

methods using the first order backward Euler time marching scheme. The plot of v-

velocity is shown in Figure 4-15. It can be seen from the figure that similar amplitude

increase pattern is seen using SIMPLE method as well. Such a difference in amplitude

and vortex structure associated with different time step sizes can be attributed to the first

order time marching scheme used by the PISO and SIMPLE family of algorithms.

79

Figure 4-13. Periodic oscillation of the cross-stream (v) component of velocity using PISO algorithm for square cylinder case at Re=215.

Figure 4-14. Vordex shedding past a square cylinder using PISO algorithm for Re=215. A) ∆t=0.001, B) ∆t=0.0005

X

Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.2

-0.1

0

0.1

0.2

Vortex shedding past square cylinder, Dt=0.0005

X

Y

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Vortex shedding past square cylinder, Dt=0.002A

B

80

Figure 4-15. Periodic oscillation of the cross-stream (v) component of velocity using SIMPLE algorithm for square cylinder case at Re=215.

Effect of number of stages on accuracy and stability of PISO algorithm

The number of momentum and energy corrector stages can have a significant

impact on the stability of the PISO algorithm. The original PISO algorithm, proposed by

Issa (1985), included two momentum corrector steps to go with one energy corrector

step. This method was found to be unstable for skewed meshes due to the omission of

cross-derivative p terms, which is necessary when solving problems on non-orthogonal

grids (Thakur and Wright 2004). A method proposed by Thakur and Wright (2004),

which includes tow additional corrector stages, is used for performing unsteady

computations here. As a first step to this, we study the impact of the number of corrector

stages on stability and accuracy using two test cases.

81

First, we perform unsteady computations over a square cylinder, identical to the

one mentioned in the previous section. All computations were run using a time step size

of 0.002, corresponding to a CFL number of 0.27. Number of stages of 2, 4 and 8 were

looked at to study the accuracy and stability. The residual (pressure) history for the

different number of stages is shown in Figure 4-16. It can be seen from the figure that

using 4 or 8 corrector stages produced much better convergence, compared to using 2

corrector stages. Plot of v-velocity at a point downstream of the cylinder, using different

number of stages, is shown in Figure 4-17. It can be inferred from the plot that the

number of stages for this case did not have a significant impact on solution accuracy.

Since a Cartesian type gird is used for these computations, the number of stages did not

have a significant impact on the stability or the solution accuracy of the PISO algorithm.

Figure 4-16. Pressure residual history for unsteady flow over a square cylinder (Re=215)

82

Figure 4-17. Periodic oscillation of Cross-stream velocity (v) using different number of stages for PISO algorithm

As a second test case, unsteady, laminar flow computations, over a circular cylinder

is carried out. Computations were run at a Reynolds number of 100 with a time step size

of 0.002 seconds, corresponding to a CFL number of 0.27. This case is chosen to test the

impact of number of stages on a curvilinear grid. The grid, along with the boundary

condition, used for these computations is shown in Figure 4-18. Using 2 corrector stages

for this case was found to be unstable due to the curvature in the geometry. The residual

(pressure) plot using 4 and 8 corrector stages are shown in . Minor differences were

observed in the convergence tolerance obtained using the different number of stages;

however, the convergence rate is in comparable range for both cases. Similar v-velocity

plot at a point downstream of the cylinder is shown in Figure 4-20. Both cases produced

identical results and the Strouhal number evaluated using either 4 or 8 stages were found

to be 0.164, which is in excellent agreement with experiment (Williamson 1989). Similar

results were observed for compressible, turbulent flow over a three-dimensional wing as

83

well. Based on the above results, it was found that using 4 stages was necessary for

performing computations on curvilinear mesh systems, as well as skewed geometries.

Using 8 corrector stages increased the computational cost by two-fold, compared to using

4 stages, and did not affect the accuracy significantly. Hence, we chose 4 corrector stages

for all further computations. Some cases may require additional stages depending on the

complexity of the problem.

Figure 4-18. Computational domain and boundary conditions imposed for flow over a circular cylinder

Figure 4-19. Pressure residual history for unsteady flow over a circular cylinder (Re=100)

OUTLET

OUTLET

INLE

T

OU

TLE

T

84

Figure 4-20. Periodic oscillation of cross-stream velocity (v) for different number of corrector stages

Momentum Interpolation Techniques for Computing Contravariant Velocities

The impact of different momentum interpolation methods to compute contravariant

velocities, specifically the effect of time step size, on solution accuracy will be

demonstrated here. Prior work in this field showed in-depth analysis for various Reynolds

numbers and grid resolutions using the 2-D cavity flow case (Choi, 1999). We chose a

similar case to validate the proposed momentum interpolation scheme and then extend

this method to study the flow around a cylinder employing curvilinear mesh. Schematic

of the cavity is shown in Figure 4-21 along with the boundary conditions. The cavity flow

case was run at a Reynolds number of 100 based on cavity height of 1 unit on a 21x21

uniform grid. Three different time step sizes of 1, 0.1 and 0.01, corresponding to CFL

numbers of 40, 4 and 0.4, were chosen to study the impact of time step size on solution

accuracy. The plot of u-velocity and pressure using both the original Rhie-Chow

85

momentum interpolation scheme as well as the modified momentum interpolation

scheme is shown in Figure 4-22 for different time step sizes along with steady state

solution for comparison. The velocity plots using both the original and modified schemes

produced almost identical results, whereas noticeable deviation from steady state solution

can be seen in the pressure plot for the original momentum interpolation scheme for 0.01

time step size. This is in agreement (Choi, 1999) with the finding that smaller time step

size leads to pressure fluctuations for the original momentum interpolation scheme. This

deviation is not observed with the modified scheme.

Figure 4-21. Schematic of Cavity flow grid along with boundary conditions X

Y

0 10

1CAVITY LID (Re=100)

NO SLIP WALL

NO

SLI

PW

ALL

NO

SLI

PW

ALL

86

Figure 4-22. Velocity and pressure contours for cavity flow at Re=100 using different

momentum interpolation schemes for various time step sizes at y=0.5 location in the cavity.

Next, the results for the various momentum interpolation schemes on a curvilinear

mesh using unsteady flow around a cylinder example are shown. The overall

computational domain along with the grid is shown in Figure 4-23. The cylinder is placed

in the middle of a circular domain as can be seen in the figure. An orthogonal O-grid with

increasing grid density towards the cylinder is used for this test case. There are 201 points

in the circumferential direction and 121 points in the radial direction. Unsteady

computations were performed until steady state is achieved, which occurs at a non-

dimensional time of about 250. Two different time step sizes were chosen (0.1 and 0.01)

X

P

10 20 300

0.1

0.2Steady statedt=0.1dt=0.01

Original Rhie-Chow scheme

X

U

10 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Steady statedt=0.1dt=0.01

Original Rhie-Chow scheme

X

U

10 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Steady statedt=0.1dt=0.01

Modified Rhie-Chow scheme

X

P

10 20 300

0.1

0.2

Steady statedt=0.1dt=0.01

Modified Rhie-Chow scheme

X

P

0.2 0.4 0.6 0.8

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Steadydt=1dt=0.1dt=0.01

Original Rhie-Chow schemeX

U

0.2 0.4 0.6 0.8

-0.2

-0.15

-0.1

-0.05

Steadydt=1dt=0.1dt=0.01

Original Rhie-Chow scheme

X

U

0.2 0.4 0.6 0.8

-0.2

-0.15

-0.1

-0.05

Steadydt=1dt=0.1dt=0.01

Modified Rhie-Chow scheme

X

P

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Steadydt=1dt=0.1dt=0.01

Modified Rhie-Chow scheme

87

to compare the solutions with the steady state solution, corresponding to CFL numbers of

10 and 1, respectively.

Figure 4-23. Schematic of computational domain surrounding a cylinder

Plots of u-velocity and pressure along the centerline downstream of the cylinder for

the different time step sizes were compared with steady state solutions in Figure 4-24.

From the velocity plots, it can be seen that the original Rhie-Chow scheme seems to

perform slightly better than the modified scheme. The modified scheme seems to deviate

from the steady state solution towards the exit. This can be attributed to the grid

resolution at the exit, which is much coarser than near the cylinder location thereby

leading to larger metric terms (fij) while computing contravariant velocities and

subsequently cartesian velocities. However, while comparing the pressure plots, we can

see that the original momentum interpolation scheme produces significant fluctuations

towards the exit. We can also see that these fluctuations tend to increase with decrease in

time step size, which is in agreement with theory. These fluctuations are not observed for

the modified momentum interpolation scheme as seen from the plot. Thus we can say that

X

Y

-40 -20 0 20 40-40

-30

-20

-10

0

10

20

30

40

88

the addition of extra terms to the original momentum interpolation scheme does have a

significant impact in reducing pressure fluctuations observed for smaller time steps. Even

though the fluctuations seem small for cartesian type grids, they are quite significant for

curvilinear grids and hence needs to be addressed while performing unsteady

computations with smaller time steps.

Figure 4-24. Velocity and pressure plot for flow around a cylinder at Re=40 using different momentum interpolation schemes for various time step sizes at the symmetry line downstream of the cylinder

Geometric Conservation Law

In this section, the various formulations of GCL (both first and second order fully

implicit and semi-implicit methods) will be tested using two different kinds of grids.

First, the schemes were tested on 2-D Cartesian mesh using channel flow example with

X

P

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Steadydt=1dt=0.1dt=0.01

Modified Rhie-Chow schemeX

U

0.2 0.4 0.6 0.8

-0.2

-0.15

-0.1

-0.05

Steadydt=1dt=0.1dt=0.01

Modified Rhie-Chow scheme

X

U

0.2 0.4 0.6 0.8

-0.2

-0.15

-0.1

-0.05

Steadydt=1dt=0.1dt=0.01

Original Rhie-Chow scheme

X

P

0.2 0.4 0.6 0.8

-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Steadydt=1dt=0.1dt=0.01

Original Rhie-Chow scheme

X

P

10 20 300

0.1

0.2Steady statedt=0.1dt=0.01

Original Rhie-Chow scheme

X

P

10 20 300

0.05

0.1

0.15

0.2

Steady statedt=0.1dt=0.01

Modified Rhie-Chow scheme

X

U

10 20 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Steady statedt=0.1dt=0.01

Modified Rhie-Chow scheme

X

U

10 20 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Steady statedt=0.1dt=0.01

Original Rhie-Chow scheme

89

arbitrary grid movement. Unsteady flow computations were performed using both

SIMPLE and PISO methods. Next, unsteady flow computations on a 3-D AGARD wing

was used to demonstrate the impact of the different GCL schemes on solution accuracy.

Two-dimensional channel flow: First order backward Euler using SIMPLE algorithm

A 2-D channel flow with fixed dimensions of 15×1 is used for this case. An

incompressible, laminar flow calculation with a Reynolds number of 100 based on inlet

length was used. The original grid for the channel is a uniform grid with 151 points in the

x-direction and 11 points in the y-direction as shown in Figure 4-25A. Calculations were

performed by moving the grid, arbitrarily, towards the center with varying stiffness after

each time step. We ensure convergence of fully developed solution at every time step. In

other words, steady state is reached at every time step in spite of the artificial grid

movement. The grid is moved in the following manner.

( ) ( ) ( 1) ( )x i a t i b t∆ = + −

where ∆x(i) represents the grid movement for the ith node; a(t) and b(t) are time varying

quantities representing the spacing between the first 2 nodes and the spacing between the

last 2 nodes for the 1st half of the grid. Similar procedure is followed for the 2nd half of

the grid. Such a gird movement produces different grid velocities at each time step. The

grid snapshot at different time steps is shown in Figure 4-25B-D.

90

Figure 4-25. Computational grids for channel flow. A) Original grid for the channel, B)-D) Modified grid distribution at different time instants

The velocity profile at different time instants for each of the schemes is shown in

Figure 4-26. The steady state velocity profile is shown as a solid line and as can be seen

from the figure, deviation from steady state solution is observed for higher order schemes

as well as for the first order time-averaged scheme. The first order fully implicit scheme

was found to produce the most accurate solution among all GCL schemes considered.

The error norm was also computed by assuming the steady state solution for channel flow

as the exact solution. The plot of error norm versus grid velocity for each of the four

schemes is shown in Figure 4-27. As can be seen from the plot, the error norm is within

comparable ranges for all cases except for the second order time-averaged scheme. This

is consistent with the velocity profile plot shown in Figure 4-26. It can clearly be seen

that first order implicit scheme produces the least error norm compared to all other

schemes.

X

Y

0 5 10 150

0.5

1

1.5

2 (b)

X

Y

0 5 10 150

0.5

1

1.5

2 (d)

X

Y

0 5 10 150

0.51

1.5

2 (c)

XY

0 5 10 1500.5

1

1.52 (a)A

B

C

D

91

Figure 4-26. Velocity profile for channel flow with Re=100 at different time instants for coarse grid (151×11) using Backward Euler method

Figure 4-27. Error norm versus grid velocity using various schemes for channel flow for 151×11 grid using Backward Euler method

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT ime step 1T ime step 2T ime step 3

F irst order T ime-averaged

u-velocity

Y

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT im e step 1T im e step 2T im e step 3

S econd order T ime-averaged

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lTim e step 1Tim e step 2Tim e step 3

S econd order Implciit

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lTim e step 1Tim e step 2Tim e step 3

F irst order Implicit

Error Norm vs Grid Velocity

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

Grid Velocity

Erro

r Nor

m

1st Implicit1st time avg2nd Implicit2nd time avg

92

A grid refinement study was done on the same test case by doubling the number of

grid points in both directions. A similar grid movement pattern was used here. The plot of

velocity profiles is shown in Figure 4-28. The error norm versus grid velocity is shown in

Figure 4-29 and as can be seen from the figure, the pattern is very similar as compared to

the coarse grid case; however the magnitude of error is less for fine grid case than coarse

grid case as expected. Thus we can infer that there is not much of an advantage of using a

higher order GCL scheme as compared to a first order implicit scheme. This observation

can be understood with the view that GCL could be related to Guillard and Farhat’s

(2000) statement that for a p-order time-accurate scheme on a fixed mesh, satisfying the

p-order accurate DGCL is a sufficient condition for the scheme to be first order time

accurate on a moving mesh. The above statement coupled with the fact that our flow

solver is first order implicit in time leads to the observation that the first order GCL

scheme performs the best among the schemes considered.

93

Figure 4-28. Velocity profile for channel flow with Re=100 at different time instants for fine grid (301×21) using Backward Euler method

Figure 4-29. Error norm versus grid velocity using various schemes for channel flow for 301×21 grid using Backward Euler method

0

0.005

0.01

0.015

0.02

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Grid Velocity

Erro

r Nor

m

1st Implicit

1st time avg

2nd Implicit

2nd time avg

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT ime step 1T ime step 2T ime step 3

F irst order Implicit

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT ime step 1T ime step 2T ime step 3

F irst order T ime-averaged

u-velocity

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT ime step 1T ime step 2T ime step 3

S econd order Implicit

u-velocity

Y

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Analytica lT ime step 1T ime step 2T ime step 3

S econd order T ime-averaged

94

Two-dimensional channel flow: PISO algorithm

The impact of different schemes for GCL on solution accuracy for the PISO

algorithm is discussed here. Both first and second order schemes as well as fully implicit

and time-averaged schemes, like the ones developed for Euler algorithm were

implemented for the PISO algorithm as well. The main reason for testing GCL for the

PISO algorithm is due to the non-iterative nature of the PISO algorithm, which makes it

much faster than the fully implicit Euler method. However, there is a limitation in the

choice of time step since the PISO algorithm is a semi-implicit algorithm, hence confined

to smaller CFL numbers. The test case for the channel flow case using the PISO flow

solver is identical to the Euler case except for the magnitude of grid movement since CFL

number needs to be preserved.

As can be seen from the above expression, both time step size and grid spacing has

an effect in deciding the CFL number. Too small a grid spacing will lead to undesirably

high CFL numbers, hence the reason for smaller magnitude grid movement for the PISO

algorithm test case. Other than the magnitude, the grid movement follows the identical

pattern as mentioned for the Euler case. Similar velocity profile plot at different time

instants for each of the GCL schemes is shown in Figure 4-30. The first order fully

implicit scheme was found to produce the most accurate solution among all GCL

schemes considered even though it is not very clear from the plot. However, the error

norm shown in Table 4-1 give a better comparison of the four schemes looked at here.

From the table, it can be seen that first order implicit scheme for GCL produces the least

error norm compared to other cases. This again shows that the first order implicit scheme

to evaluate the Jacobians performs best in the context of flow solver under consideration.

95

Figure 4-30. Velocity profile for channel flow at different time instants for 151x11 grid using PISO method

Table 4-1. Error Norm versus Grid velocity for the four GCL schemes for 3-D wing case using Backward Euler method

Time step # 1st order Implicit 1st order time-averaged 2nd order Implicit 2nd order time-averaged1 0.0184 0.0183 0.0185 0.0184 2 0.0228 0.0227 0.0227 0.0231 3 0.0223 0.0224 0.0225 0.0228

Three-dimensional elastic wing: AGARD 445.6

As a next case study for validating the GCL, a uniform flow test case for a 3-D

wing was used. A flow over a 3-D AGARD 445.6 wing, mentioned earlier, is used to

illustrate the effect of the various schemes. Similar to the channel flow case, two different

examples are presented. First, only the wing geometry is employed to define the mesh

system. The fluid flow is considered to be uniform and the solid boundary is ignored. In

this example, again, the interest lies in examining the impact of grid movement and the

V 2

Y

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ana lyticalT im e step 1T im e step 2T im e step 3

S econd order T im e-averaged

V 2

Y

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ana lyticalT im e step 1T im e step 2T im e step 3

F irst order T im e-averaged

V 2

Y

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1AnalyticalT im e step 1T im e step 2T im e step 3

F irst order Im plicit

V 2

Y

0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Ana lyticalT im e step 1T im e step 2T im e step 3

S econd order Im plicit

96

implementation of GCL on the numerical outcome of a trivial analytical solution. Second,

a time dependent, turbulent flow computations around a solid, elastic wing is used to

demonstrate the various GCL schemes on solution accuracy.

For the first case, the mesh on the surface of the wing was arbitrarily moved in a

direction normal to the surface with a linear variation in the spanwise direction with the

root being fixed. A plot of the spanwise deflections at different time instants is shown in

Figure 4-31. Uniform flow calculation was performed after moving the entire grid after

each time instant and error norm calculated based on the trivial solution for pressure on

the surface of the wing. All four schemes were tested and the corresponding error norm

tabulated in Table 4-2. A plot of these error norms versus grid velocity is shown in Figure

4-32. As can be seen from the figure, there is hardly any difference between the four

schemes implemented to preserve GCL. Although the plots look indistinguishable, it can

be seen from the Table 4-2 that there are minute differences between each schemes,

however, they are insignificant. This is in contrary to the results obtained by Farhat et al.

(2001), where they showed that considerable differences were obtained between various

schemes employed to preserve GCL. This could be due to the choice of first order fully

implicit fluid flow solver used to perform these computations. Apparently, depending on

the fluid flow solver, different outcome can be realized in terms of the performance of the

implementation of GCL.

97

Figure 4-31. Plot depicting the arbitrary movement of the wing in the spanwise direction

Figure 4-32. Error norm versus grid velocity for the various schemes for AGARD wing using Backward Euler method

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

0.015 0.020 0.025 0.030 0.035 0.040

Grid Velocity

Erro

r Nor

m

1st Implicit

1st time-avg

2nd Implicit

2nd time-avg

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 2 4 6 8 10 12Spanwise Indices

Def

lect

ion

timestep 1

timestep 2

timestep3

timestep 4

timestep 5

timestep 6

98

Table 4-2. Error Norm versus Grid velocity for the four GCL schemes for 3-D wing case Error Norm

Grid Velocity 1st order implicit 1st order time averaged 2nd order implicit 2nd order time averaged

0.016 2.583E-05 2.583E-05 2.584E-05 2.584E-05

0.023 1.475E-04 1.479E-04 1.474E-04 1.478E-04

0.034 1.991E-04 1.992E-04 1.992E-04 1.986E-04

0.038 1.907E-04 1.907E-04 1.907E-04 1.907E-04

It has been shown (Lesoinne and Farhat, 1996; Farhat et al., 2001 & 2003) that the

choice of GCL scheme does affect the accuracy of coupled fluid-structure solutions.

Therefore, a 3-D aeroelastic case was used to examine this. Time dependent, turbulent

aeroelastic calculation based on the aforementioned AGARD wing geometry was

adopted. The grid movement is taken care of using the moving boundary module

explained earlier. Calculations were performed for a turbulent Reynolds number of

107,000 per foot, based on root chord, with a 5 deg angle of attack. The spanwise

deflection at the first four time instants, for the first order fully implicit GCL scheme, is

shown in Figure 4-33. Since the results are very similar for the remaining schemes, they

are not shown in the same plot; instead, the tip deflection at two different time instants is

tabulated in Table 4-3. As can be seen from the table, the values are virtually identical.

This leads us to believe that the choice of GCL scheme does not really affect the

aeroelastic solution as well in our case. Such an observation for the 3-D wing case can be

explained as follows. Comparing Figure 4-28and Table 4-2 in the context of grid

velocity, it can be seen that the grid velocity associated with the 3-D wing movement is

an order of magnitude less than that for the channel flow case. It has been proved by

previous authors (Farhat et al., 2001) that, for smaller grid velocity, the impact of GCL is

99

minimal on solution accuracy. Hence, it seems clear that both the detailed schemes

employed by the flow solver and the magnitude of grid movement directly influence the

performance of the GCL scheme for moving boundary computations.

Figure 4-33. Spanwise deflection of AGARD wing at four different time instants

Table 4-3. Tip deflection at two different time instants for different GCL schemes for 3-D wing case

Scheme Time instant # 2 Time instant # 4

1st Implicit 7.613E-02 1.168E-01

1st Semi-implicit 7.616E-02 1.170E-01

2nd Implicit 7.621E-02 1.171E-01

2nd Semi-implicit 7.620E-02 1.171E-01

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 2 4 6 8 10 12

Spanwise Indices

Def

lect

ion

(ft)

Time instant # 1

Time instant # 2

Time instant # 3

Time instant # 4

100

Moving Boundary Module

The effect of some of the parameters associated with the moving grid module is

demonstrated using a 3-D multiblock grid shown in Figure 4-34.

Figure 4-34. Schematic of multi-block grid used to validate moving mesh module

As can be seen from the figure, there are 2 moving boundary surfaces (the bottom

surfaces of block 1 and 2). These boundaries are given an arbitrary sinusoidal

perturbation and the different parameters (β and FACTOR) are changed to study how

these parameters affect the re-meshing of the rest of the domain. The perturbation and the

final mesh after the entire mesh has been re-grid is shown in Figure 4-35. The overall

grid quality is preserved for all choice of parameters, but the rigidity of movement is

different based on the choice of parameters.

101

Figure 4-35. Figure depicting the effect of the 2 parameters, FACMIN and β on the re-meshing

Here, the master nodes are the nodes in the bottom surface of the domain, which is

arbitrarily perturbed. Note that all four figures have the same amount of perturbation. The

slave nodes are the vertices of all the other blocks.

It can be seen from the Figure 4-35 that smaller values of both β and FACTOR

correspond to a movement like a rigid body. This is desirable near the boundary where

there are sharp corners and the grid quality needs to be preserved. This will automatically

take care of the value of y+, which is an important factor while employing turbulent flow

X

Y

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Factor=500, beta=1/256

X

Y

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Factor=50, beta=1/64

X

Y

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Factor=500, beta=1/64

X

Y

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Factor=50, beta=1/256

102

models with wall-functions. Hence, this was chosen as the appropriate for all further

calculations.

Structure Solver

In this section, the structure solver used will be validated. As mentioned earlier, a

simplified beam model with 10 elements along the spanwise direction is used. This is a

really simplistic model compared to other models such as plate models or full 3-D finite

element models. Hence, it is imperative to ensure that this model is suitable for fluid-

structure interaction computations. Since a modal approach is used to solve the structural

equations of motion, only the first few modes, or the dominant modes, need to be

compared with existing finite element models. In this case, the first four modes (first two

bending and first torsional modes) are compared with a 120 element plate model

published by Yates (1987). The comparison of the modal frequencies of the first four

modes is shown in Table 4-4. As can be seen from the table, modal frequencies of the

first two modes are in excellent agreement with the results published by Yates and the

remaining two modes are in reasonable agreement as well considering the simplicity of

the structure model used in the present study.

Table 4-4. Comparison of wing mode shapes between 10 element beam model (present study) and 120 element plate model (Yates, 1987)

68.9 72.7 2nd bending

122.3 142.7 2nd torsion

50.9 52.2 1st torsion

14.1 14.1 1st bending

Yates (1987) (120 plate elements)

Present Beam model (10 beam elements)

Mode Number

103

Once the finite element model is validated, care must be taken to ensure that the

integration scheme employed for this structure model is an accurate and stable one. To

accomplish this, an arbitrary load on the beam is applied and the tip deflection of the

beam is studied with respect to number of time steps. Calculations are performed with

and without damping to study the impact of time step size on accuracy of solution. Two

different time step sizes of 0.0001 and 0.001 were chosen for this study. The plots of tip

deflection for these time steps are shown in Figure 4-36 and Figure 4-37, respectively.

The static deflection was calculated using the applied load and the stiffness matrix, and it

was found to be 1.75. As can be seen from the plots, when damping is present, the

solution does converge towards the static deflection as expected. When there is no

damping, it can be seen that the deflection oscillates to twice the value of static deflection

for ∆t=0.0001 case, which is expected as well and it can be seen that it does not show any

semblance of convergence. However, for the ∆t=0.001 case, when damping is not

present, the oscillations seem to damp or die down with increasing time. This can be

attributed to the amplitude decay and period elongation observed with the Newmark

family of methods as ∆t increases (Bathe, 1996). It has been proven that Newmark

method performs best for ∆t/T < 0.01. Here T is the time period of vibrations. For the

present beam model, T is 0.075, which was evaluated from the eigenvalue problem;

hence any ∆t < 0.00075 leads to inaccuracy in the solution. This explains the period

elongation and amplitude decay observed for the ∆t=0.001 case. Hence a time step size

less than 0.00075 must be chosen for integrating the structural equations of motion using

Newmark scheme in order to obtain an accurate and stable solution.

104

Figure 4-36. Tip deflection of AGARD wing versus number of time steps for ∆t=0.0001

Figure 4-37. Tip deflection of AGARD wing versus number of time steps for ∆t=0.001

105

CHAPTER 5 RESULTS AND DISCUSSION

This chapter will be broken into two different sections. First, coupled results over

an AGARD wing using for an incompressible flow condition will be presented.

Aerodynamic parameters for different angles of attack will be presented for this case.

Secondly, coupled simulation for a compressible flow condition using the AGARD wing

to compute the flutter boundary for a transonic Mach number case will be presented. This

case, in particular, will be compared to prior experimental and numerical result.

Coupled Simulation for Incompressible Flow Conditions

An unsteady, incompressible, viscous flow calculation on the AGARD wing will be

conducted first for a Reynolds number of 3.66x105 per foot of root chord. A SIMPLE

type of algorithm was used to solve the flow equations, which is based on an iterative

procedure for each time step. A first order implicit backward Euler time stepping scheme

was used to integrate the flow equations in time. Such a scheme enabled us to use a larger

time step size as compared to the PISO method as will be discussed in the next section.

For the structure solver, all modes of the structure were used and the equations were

integrated using an implicit Newmark method. Two different girds were considered for

this case. Both grids are multi-block structured grids with a total of 10 blocks. An O-grid

is used around the wing for both cases. The first of the grids had a total of 330,000 points

in the entire computational domain with a total of 4800 on the surface of the wing (120 in

the chordwise direction and 40 in the spanwise direction). We call this grid as

Configuration I. The second of the 2 grids had a total of 800,000 points in the entire

106

domain, with a total of 12,000 points distributed on the surface of the wing (200 points

along the chordwise direction and 60 points along the spanwise direction). We call this

grid as Configuration II. All computations pertaining to this case were run on one

processor of a 4 processor 833 Mhz machine, which had a memory of 12 GB. The

computational time for coarse grid computations were about 5-6 minutes per step

whereas it was about 30-35 minutes for the fine grid computations. More than 90% of the

computational time was attributed to the flow solver.

Figure 5-1. Spanwise wing shapes at different time instants (Grid configuration I)

First, computations are performed on the coarser of grids (configuration I), which is

later compared with the finer of the two grids (configuration II). An angle of attack of 1

degree was chosen initially. The spanwise variation of wing shape for different time steps

is shown in . Time-varying deflection at 3 spanwise locations of the wing is shown in

Figure 5-2. It can be seen from the figure that the displacements tend to converge

asymptotically to a steady state value after a certain time. The small deviation seen close

Increasing time

107

to 0.007 seconds can be attributed to the point when the 2nd torsion mode occurs at a

frequency of approximately 142 Hz. Consequently, this value was very close to the

corresponding mode obtained while doing an eigenvalue analysis on the structure.

Figure 5-2. Time varying displacement of wing at different spanwise locations (Grid configuration I)

Plots of aerodynamic parameters such as the lift coefficient and lift/drag ratio with

respect to time are shown next. Plot of time history of lift coefficient for an angle of

attack of 1 degree is shown in Figure 5-3. Results from both grid configurations are

shown in the plot for comparison. Plot of lift/drag ratio for both grids is shown in Figure

5-4. We do see that the lift coefficient and lift/drag ratio converge asymptotically as well

to the same value. Although grid II plot shows some oscillations initially, it seems to

converge to a steady state value after a certain time. We also observe an inflection point

in these plots at the same time where the 2nd torsion mode was found to occur. This

108

phenomenon is observed for both and it will be seen that this is independent of angles of

attack as well.

Figure 5-3. Time history of lift coefficient for AGARD 445.6 wing subject to 1-degree angle of attack for both grid configurations

Figure 5-4. Time history of lift/drag ratio for AGARD 445.6 wing subject to 1-degree angle of attack for both grid configurations.

The pressure contour on the surface of the wing at steady state on both the original

and deformed geometry for both grids is shown in Figure 5-5. We see that there is not

much of a difference between the contours for the two grid configurations, however, we

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0 0.01 0.02 0.03 0.04 0.05Time (sec)

Lift

Coe

ffici

ent

Grid config IGrid config II

-12

-10

-8

-6

-4

-2

0

2

4

6

0 0.01 0.02 0.03 0.04 0.05Time (sec)

Lift/

Dra

g

Grid config I

Grid config II

109

do see a notable difference in both pressure coefficient and contours for the deformed

shape as compared with the original shape. This can be attributed to the shape change of

the wing, which continuously keeps changing as we march in time.

Figure 5-5. Pressure contour on the surface of the wing at steady state

As a next step, variation of aerodynamic parameters for different angles of attack is

shown. Angles of attack of 3 degrees and 5 degrees were chosen in addition to the initial

1-degree angle of attack for this study. All computations for these two angles of attack

were performed on grid configuration I. Plot of lift coefficient comparing different angles

of attack is shown in Figure 5-6. It can be seen that lift coefficient converges

X

Y

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Coarse grid (Deformed Geometry)

X

Y

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fine grid (Deformed Geometry)

X

Y

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Fine grid (Original Geometry)

X

Y

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Coarse grid (Original Geometry)Grid config I (Original Geometry) Grid config II (Original Geometry)

Grid config I (Deformed Geometry) Grid config II (Deformed Geometry)

110

asymptotically to a finite value for all three angles of attack. We also observe that a

higher value of lift coefficient is obtained for larger angle of attack, as expected. The

lift/drag ratio for these angles of attack is also computed and is shown in Figure 5-7. The

lift/drag ratio for 3 and 5 deg seem to converge to an almost identical value but are

significantly larger than the 1-degree case.

Figure 5-6. Comparison of lift coefficient time history for AGARD wing subject to different angles of attack for grid configuration I

Figure 5-7. Comparison of lift coefficient time history for AGARD wing subject to different angles of attack for grid configuration I

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.004 0.008 0.012 0.016 0.02 0.024Time (sec)

Lift

coef

ficie

nt

5 deg

3 deg

1 deg

-13

-9

-5

-1

3

7

11

0 0.004 0.008 0.012 0.016 0.02 0.024Time (sec)

Lift/

Dra

g

5 deg

3 deg

1 deg

111

Comparison of PISO and SIMPLE Algorithms

Since the flow solver consumes most of the CPU time during a coupled simulation,

a non-iterative method based on PISO algorithm was proposed to perform flow

computations instead of the SIMPLE algorithm, which was an iterative method as

mentioned before. As a first step towards developing this capability for fluid-structure

interaction computations, the PISO algorithm was compared with the SIMPLE algorithm

to ensure similar results between the two schemes when identical time step size is used

and to also compare the computational cost between the solvers. Coupled simulations

were carried out using the above-mentioned example for a Reynolds number of 3.64x105

based on unit chord length. Grid configuration I is used for these computations. A plot

comparing the spanwise wing displacement at three consecutive time instants is shown in

Figure 5-8. As can be seen from the figure, the agreement is very good between these two

algorithms. The computational cost, however, by using the PISO algorithm was greatly

reduced compared to the SIMPLE algorithm. The ratio of the CPU time requirement

between the SIMPLE and PISO approach is about 10 per time step for this grid. Even

though the implicit nature of SIMPLE algorithm allows us to choose a larger time step

and still produce a stable solution, we still need to choose a small time step to obtain

good accuracy as the time marching scheme is first order in nature. Also, the frequencies

pertaining to the structure limits the choice of a higher time step size.

112

Figure 5-8. Spanwise displacements at three different time instants to compare PISO and SIMPLE methods using incompressible flow around an AGARD wing example at a Re=3.64x105 based on unit root chord.

Coupled Simulation for Compressible Flow Conditions

In this section, coupled simulation is demonstrated with compressibility effects

included in addition to the viscous effects mentioned in the previous sections.

Specifically, the flutter boundary is predicted for the AGARD 445.6 wing at subsonic,

supersonic and transonic Mach numbers. The emphasis, however, will be on the transonic

Mach number case since this is the region where a transonic dip is seen in the flutter

boundary and complicated phenomenon are bound to occur. Also, extensive numerical

data are available for this case and hence will be focused upon. The Mach numbers

corresponding to subsonic, supersonic and transonic flows are 0.67, 1.072 and 0.96,

respectively. Steady state solution at an angle of attack of 0 degrees was used as the

initial condition to start the unsteady computations. The wing is started with either an

initial displacement and zero initial vibrational velocity or zero displacement and nonzero

initial vibrational velocity. The wing then starts oscillating in time resulting in either a

0.00E+00

1.00E-06

2.00E-06

3.00E-06

4.00E-06

5.00E-06

6.00E-06

1 3 5 7 9 11

Spanwise location

Dis

plac

emen

t

Euler, tstep=1PISO, tstep=1Euler, tstep=2PISO, tstep=2Euler, tstep=3PISO, tstep=3

113

damped, or diverging, or neutral vibrations. An initial vibrational velocity was provided

in our case to study the oscillations of the wing.

Time Scales and Choice of Time Step Size for the Coupled Problem

The PISO algorithm, being a semi-implicit algorithm in time, is limited by a

stability condition regarding the choice of time step size required. The structure solver,

which uses an implicit time marching scheme, does not have a stability limit on the

choice of time step size. Hence, the choice of time step was determined solely based on

the stability condition of the flow solver. For time accurate resolution of unsteady

computations, the specified time step must be of the same order of magnitude as the

smallest characteristic time scale. Three major time scales can be defined for the problem

under consideration: diffusive time scale, convective time scale, and time scale due to

vibration of structure. They are defined as follows

2

dtLρδ

∆ =Γ

1s

s

t f∆ =

*cLt

uδ∆ =

Here, δL is the local mesh size, u* is the local characteristic speed, fs is the modal

frequency of the structure and Γ is the diffusion coefficient. The time step size is chosen

based on the smaller of the above-mentioned time scales. Plots of the diffusive and

convective time scales for the transonic Mach number case are shown in Figure 5-9.

Time scale based on structural vibration is not shown here as they are not grid dependent

and are constant for practical purposes. It can be seen from the figure that the convective

114

time scales are orders of magnitude smaller than diffusive time scales and hence time step

size should be determined based on the convective time scales.

Figure 5-9. Diffusive (left) and convective (right) time scales near wing tip region for different time step sizes and grids for M=0.96

The criteria for choosing the time step size can also be stated in terms of

dimensionless parameters associated with the corresponding time scales. The parameter

based on the convective time scale is the CFL (or Courant) number, which is defined as

*u tCFL L∆= δ

In curvilinear coordinates, the CFL number can be written as

max( , , )tCFL U V WJ∆=

X

Z

1.5 2 2.5 3 3.5-0.5

-0.25

0

0.25

0.5

2.0E-021.6E-021.1E-026.5E-032.0E-03

X

Z

1.5 2 2.5 3 3.5-0.5

-0.25

0

0.25

0.5

4.0E-053.3E-052.5E-051.8E-051.0E-05

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.56.0E-054.8E-053.5E-052.3E-051.0E-05

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

4.0E-053.3E-052.5E-051.8E-051.0E-05

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

2.0E-021.6E-021.1E-026.5E-032.0E-03

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

2.0E-021.6E-021.1E-026.5E-032.0E-03

Grid I (350K), ∆t=1x10-5 Grid I (350K), ∆t=1x10-5

Grid I (350K), ∆t=5x10-5 Grid I (350K), ∆t=5x10-5

Grid I (800K), ∆t=1x10-5 Grid I (800K), ∆t=1x10-5

115

where J is the Jacobian and U, V, W are the contravariant velocities. The corresponding

dimensionless parameter for the diffusion time scale is given by

2D

tLρδ

=Γ∆

which, written in curvilinear coordinates, reads as follows

The stability condition for unsteady flows can be stated in terms of these

dimensionless parameters CFL and D, i.e., the larger of these parameters should be of the

order of magnitude 1 for stability purposes. This CFL number needs to be in the O(1)

magnitude range for the PISO algorithm to be stable. Computations are performed using

two grid configurations, grid I (350K points) and grid II (800K points), mentioned in the

previous section. Two time step sizes of 5x10-5 and 1x10-5 seconds were chosen for grid

configuration I to study the effect of time step size on solution. Only time step of 1x10-5

seconds was used for performing computations on grid II. Contour plots of CFL numbers

for the 3 cases (grid I with time step sizes of 1x10-5 and 5x10-5 seconds and grid II with

time step size of 1x10-5 seconds), for M=0.96, are shown in Figure 5-10. Only the region

surrounding the wing tip is shown, as this is the region with minimum grid spacing. The

CFL numbers in all other regions are well below O(1) magnitude. It can be seen from the

plot that CFL numbers for all cases are in the O(1) magnitude range. The CFL numbers

for the higher time step case for grid I resulted in higher CFL numbers as expected. It can

also be seen that the order of magnitude of CFL numbers for grid I with time step 5x10-5

and grid II with time step size 1x10-5 are in comparable ranges. This is arises because of

3311 22max( , , )qq qtDJ J J Jρ

Γ∆=

116

the fineness in grid spacing for grid II compared to grid I, which allows for a higher

choice of time step size for this configuration.

Figure 5-10. Diffusive (left) and convective (right) nondimensional parameter at wing tip spanwise location for different grids and time step sizes for M=0.96 case

Flutter Boundary Prediction for AGARD Wing at a Transonic Mach Number

Having verified the choice of time step sizes for the coupled problem, our focus

shifts towards determining the flutter speed index (FSI) for different Mach numbers.

First, we show the evaluation of the FSI for 0.96 Mach number case and then summarize

the FSI for the subsonic and supersonic Mach number cases for verification purposes.

The flutter speed index is defined as

fU Ub

∞=αω µ

X

Z

1.5 2 2.5 3 3.5-0.5

-0.25

0

0.25

0.5 1.00.80.70.50.4

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.487542

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

1.21.00.90.70.6

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.0060.0050.0040.0030.002

X

Z

1.5 2 2.5 3 3.5-0.5

-0.25

0

0.25

0.5 0.0050.0040.0030.0020.001

X

Z

1.5 2 2.5 3 3.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.0250.0200.0150.0100.005

Grid I (350K), ∆t=1x10-5Grid I (350K), ∆t=1x10-5

Grid I (350K), ∆t=5x10-5

Grid II (800K), ∆t=1x10-5 Grid II (800K), ∆t=1x10-5

Grid I (350K), ∆t=5x10-5

117

where U∞ is the freestream velocity, b is the half root chord, ωα is the first torsional mode

frequency and µ is the mass ratio of the wing. It represents the condition at which the

oscillation is neither growing nor decaying. If the speed index U is smaller than Uf, the

motion is stable and if U is greater than Uf, then the motion is unstable. Unsteady

computations were run for a series of dynamic pressures to determine the flutter point.

The freestream density and Mach number are held constant as the dynamic pressure is

varied to determine the flutter point, thereby leading to a variation in Reynolds number

and temperature in order to provide a consistent set of flow conditions. The Reynolds

number was in the range 5.96x105 to 6.72x105 for this particular Mach number for the

various dynamic pressures considered. This change in Reynolds number did not produce

a significant effect on the flow solutions.

The growth and decay of the second mode of the structure is studied to arrive at the

flutter point. The flutter point was obtained by running a solution for a significantly long

period of time to arrive at a neutrally stable solution, where the amplitude of oscillations

of the generalized displacement of the structure is neither growing nor decaying.

As a first step, we show computations performed using the grid configuration I

(350 K points). The generalized displacements for three different choices of dynamic

pressure ratios (1.03, 0.98 and 0.905) using a time step size of 5x10-5 seconds are shown

in Figure 5-11. The dynamic pressure ratio is measured with respect to the experimental

dynamic pressure at which neutrally stable flutter was predicted, which occurs at a value

of 2935 N/m2 (61.3 lbf/ft2) for a Mach number of 0.96. It can be seen from the plot that

the amplitude ratio decreases with decrease in dynamic pressure. Amplitude growth is

seen for two of the dynamic pressure ratios (1.03 and 0.98) whereas amplitude decay is

118

seen for a dynamic pressure ratio of 0.905. Critical flutter speed is the speed at which

neutrally stable oscillations are produced or the amplification factor is 1. Thus, from the

above findings, the critical flutter speed is bound to occur at a dynamic pressure ratio

between 0.98 and 0.905, which was found to be 0.96 by doing a linear interpolation.

Computations were carried out at this choice of dynamic pressure to verify the neutrally

stable flutter condition does occur at this interpolated value and as expected, an

amplification factor close to 1 was obtained for a dynamic pressure ratio of 0.96, which

confirms that a linear interpolation between the two points produces an accurate estimate

for the flutter boundary.

Figure 5-11. Generalized displacement versus time for three different dynamic pressures for ∆t=5x10-5

Similar computations are performed using a time step size of 1x10-5 for grid

configuration I. Plots of the generalized displacements are shown in Figure 5-12. The

amplification factor was found to be greater than one for both choices of dynamic

x10-5

119

pressure ratios considered (1.03 and 0.98). A linear extrapolation was done to predict the

flutter point that produced an amplification factor of 1. This gave a dynamic pressure

ratio of 0.95 for the critical flutter condition. Computations were performed for this

choice of dynamic pressure for verification purposes. It was found that the amplification

ratio for this case is 1.004, which is close to the expected value but still offset from the

actual flutter point. Further extrapolation was done based on the above choice of dynamic

pressure to arrive at an amplification ratio of 1.001 for a dynamic pressure ratio value of

0.945.

Figure 5-12. Generalized displacement versus time for three different dynamic pressures for ∆t=1x10-5

Next, computations were performed for grid configuration II (800K) for a dynamic

pressure ratio of 0.98. A time step size of 1x10-5 seconds was used in this case to preserve

CFL stability condition. The generalized displacement for this case is shown in Figure

5-13 and is compared with the generalized displacement of grid I with 5x10-5 seconds

x10-5

120

time step case. Both amplitude and frequency were slightly different from the grid I case

but are in good comparison. The amplification factor for this case was found to be 0.99.

Hence, refining the grid increases the computed flutter speed, although grid independence

has not yet been achieved. Similar findings were reported in Gordnier and Melville

(2001) as well. They showed that a finer mesh resolution improved the flutter boundary

prediction, which is in agreement with our findings as well. Difference in frequency for

different grids was reported in their work as well. The frequency increased with grid

refinement, which is also in agreement with our results. The flutter points for all of the

above-mentioned cases are shown in Figure 5-14 for clarity to better visualize the

interpolated and extrapolated values of dynamic pressure and also to demonstrate the

variation of amplification factor with dynamic pressure.

Figure 5-13. Effect of grid resolution on generalized displacements using similar CFL numbers

121

Figure 5-14. Flutter points for M=0.96 using different time step sizes (1x10-5 and 5x10-5 for grid I (350 K points) and grid configurations (350 K grid and 800 K grid)

The impact of wing shape change on flow features is demonstrated next. The tip

deflection of the wing versus time for grid configuration I with time step 5x10-5 seconds

is shown in . Similar wing tip deflections were observed by using a 1x10-5 seconds time

step and for grid II with minor changes in magnitude. Two points, the point of maximum

and minimum tip deflection, are chosen to show the wing shape and the differences in

flow features. The wing shapes at these locations, which occur at 0.01 and 0.02 seconds,

are shown in Figure 5-16. In particular, the supersonic region and pressure contours at

these two time instants are shown for the two grid configuration I and grid II using time

step sizes of 5x10-5 and 1x10-5 respectively. The Mach number contours at the mid-span

plane of the wing, corresponding to supersonic flow in the entire domain for the two grids

at points of maximum and minimum deflection, is shown in . The maximum tip

deflection point produced a slightly thicker region of supersonic for both grid

configurations. A much thicker supersonic region is seen for the finer grid compared to

the coarser grid configuration. The pressure coefficient contour is plotted to highlight this

aspect. Specifically, the critical pressure region is plotted to highlight the supersonic

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

0.88 0.92 0.96 1 1.04

q/qe

Am

plifi

catio

n fa

ctor

Grid I (350 K points), Dt=5E-05Grid I (350 K points), Dt=5E-05Interpolated flutter point (Grid I, Dt=5x10-5)Extraoikated points (Grid I, Dt=1E-5)Grid II (800K points), Dt=1E-05

122

region on the surface of the wing subject to wing shape. The critical pressure coefficient

is defined as follows

( )12

2

112 2 111

2

crp

MC

M

− − + = − − +

γ γγ

γγ

where M is the freestream Mach number and γ is the specific heat ratio. Based on

M=0.96, the critical pressure coefficient was found to be -0.0697. A pressure coefficient

below this critical pressure coefficient constitutes the supercritical region or the region of

supersonic flow on the surface of the wing. The pressure coefficient contours for both

grid configurations at the two deflection points are shown in Figure 5-18. Only the

supercritical region is shown in the plot. Slight differences in pressure contours are seen

between the two deflection points for both grids. The finer mesh resolution enhances the

low-pressure region over the wing better. This explains a larger region of supersonic flow

visualized from the Mach number contour plots shown in .

Figure 5-15. Tip deflection of wing versus time for grid I (350 K points) using a time step size of 5x10-5

Minimum deflection

point

Maximum deflection

point

123

Figure 5-16. Wing shapes at two time instants corresponding (left) corresponds to maximum tip deflection, which occurs at around 0.01 seconds (right) corresponds to minimum tip deflection, which occurs at around 0.02 seconds

Figure 5-17. Mach number contours representing supersonic region in the flow domain at mid-span plane for transonic Mach number of 0.96.

X

Z

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

4

5Maximum displacement (350 K grid)

X

Z

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

4

5Minimum displacement (350 K grid)

X

Z

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

4

5Minimum displacement (800 K grid)

X

Z

-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

-4

-3

-2

-1

0

1

2

3

4

5Maximum displacement (800 K grid)

124

Figure 5-18. Surface pressure contours indicating supercritical region or region of supersonic flow on the surface of the wing (M=0.96).

While examining the Mach number contours for the two different time step choices

considered for grid configuration I, minor differences in region of supersonic flow were

observed. With the limitation in both spatial and temporal resolutions, the results

obtained are insightful but not quantitatively conclusive in all aspects. Nevertheless,

broad agreement between the present and other reported approaches are observed,

indicating that the combined fluid and structure models employed here perform

satisfactorily.

Flutter Computations Using a Filter-Based Turbulence Model (M=0.96)

In this section, we use the filter-based turbulence model to predict the flutter

boundary for similar conditions explained earlier. This model was found to have an effect

while performing unsteady RANS computations. The eddy viscosity is modeled using a

X

Y

0 1 2 3 40

1

2

-0.086-0.102-0.118-0.134

Minimum deflection (350 K grid)

X

Y

0 1 2 3 40

1

2

-0.086-0.102-0.118-0.134

Maximum deflection (350 K grid)

-1

0

1

2

3

4

5

X

Y

0 1 2 30

1

2

-0.100-0.130-0.160-0.190

Maximum deflection (800 K grid)

Z

-1

0

1

2

3

4

5

X

Y

0 1 2 30

1

2

-0.100-0.130-0.160-0.190

Minimum deflection (800 K grid)

125

filter, whose size is dependent on the grid size under consideration. The eddy viscosity

model is written here for convenience.

2

3/ 21,tkC Min

kµεµ ρ

ε∆ ⋅ = ⋅

Here, ∆ is the filter size, which has a lower bound as the maximum grid size dimension

so that all scales are numerically resolvable. The blending function, f= 3/ 21,Mink∆ ⋅ε

,

needs to return a value of 1 near the wall regions to enable the use of wall functions

approach. Due to this limitation, we cannot choose too small a filter size. Three different

filter sizes (0.1, 0.15 and 0.2) were chosen initially to determine the distribution of the

blending function in the entire domain. In particular, we wanted to verify whether the

blending function did return a value of 1 near the wing region. A contour plot of blending

function, where it takes a value of 1, at midspan on the wing is shown in Figure 5-19. The

region near the wing cross-section is zoomed in as well to verify that wall functions can

be used in this region. It can be seen from the plot that, for a filter size of 0.1, there are

some regions around the wing, especially near the leading edge, where the blending

function is not 1 and hence not suitable for wall functions approach. Both filter size 0.15

and 0.2 did reproduce the original eddy viscosity model near the wing boundary. A filter

size of 0.15 was chosen as a suitable choice of filter size for all of our computations.

Unsteady computations were performed for only two choices of dynamic pressures

(q/qe=0.905 and 0.98). Computations are carried out using a time step size of 5 x 10-5

seconds on grid configuration I. Generalized displacement plot for q/qe=0.98 is compared

with the standard k-ε model with wall functions in Figure 5-20. The difference between

the two cases was found to be minimal. The flutter points obtained using filter-based

126

model and k-ε model is compared in Figure 5-21. Performing a linear interpolation

between the two values of dynamic pressures, the value at which the amplification factor

reaches zero was found out to be 0.961. It can be seen from the plot that the filter-based

model does not vastly improve upon the flutter boundary prediction as compared to the

standard k-e model for this Mach number.

Figure 5-19. Blending function plot at mid-span. (left) Blending function for entire domain (right) Near wing region zoom. Red contours indicate a blending function value of 1.0.

XZ

0.5 0.75 1 1.25 1.5

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

∆=0.1

X

Z

0.6 0.8 1 1.2 1.4 1.6

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

∆=0.15

X

Z

0.6 0.8 1 1.2 1.4 1.6

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

∆=0.2

X

Z

-2 0 2 4 6 8

-2

-1

0

1

2

3 ∆=0.2

X

Z

-2 0 2 4 6 8

-2

-1

0

1

2

3 ∆=0.15

X

Z

-2 0 2 4 6 8

-2

-1

0

1

2

3 ∆=0.1

127

Figure 5-20. Comparison of filter-based model and standard k-e model for q/qe=0.98 with ∆t=5x10-5 and filter size, ∆=0.15.

Figure 5-21. Flutter boundary comparison between filter-based turbulence model and standard k-ε model using grid I and ∆t=5x10-5

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

0.88 0.92 0.96 1 1.04

q/qe

Am

plifi

catio

n fa

ctor

Standard model

Filter-based model

x10-5

128

Summary of Flutter Boundary Prediction for AGARD Wing

In this section, we briefly demonstrate the FSI prediction using our model for

subsonic and supersonic Mach numbers of 0.678 and 1.072, respectively and summarize

the results along with the cases presented for a transonic Mach number of 0.96. Similar

approach, like the one used for the transonic Mach number case, was used to arrive at the

FSI. The FSI for a subsonic Mach number of 0.678 was found to be 0.419, which is in

good agreement with both experiment (0.4174) and numerical (0.417) results. However,

for the supersonic Mach number case, the FSI evaluated 0.41 was found to be

significantly different from the experimental results (0.32). Prior researchers have also

observed this. However, the present computations lie well within the range of computed

results presented by other authors. The reason for a difference between experiment and

numerical results, for one, can be attributed to the absence of fuselage modeling in our

computations. Also, the experiment included boundary-layer transition, which was not

modeled in our computations. A plot of the spanwise wing shape at maximum and

minimum tip deflection points for Mach numbers 0.678 and 1.072 is shown in Figure

5-22. It can be seen from the plot that the magnitude of deflection for supersonic case is

higher than that of the subsonic Mach number case. The plot of surface pressure contours

over the wing indicated that the flow remains entirely supersonic over the surface of the

wing for supersonic Mach number of 1.072. An oblique shock was observed for the

supersonic Mach number case, as seen from the Mach number plot shown in Figure 5-23.

129

Figure 5-22. Spanwise wing shape at maximum and minimum tip deflection for (left) M=0.678 and (right) M=1.072

Figure 5-23. Mach number contours at (left) maximum and (right) minimum tip deflection points for supersonic Mach number of 1.072

A summary of the flutter points evaluated using our model for different choice of

grids and time step sizes, along with comparison to experiment and numerical results, is

shown in Table 5-1. The comparison is also demonstrated via Figure 5-24. It can be

observed from the figure that the present approach compared reasonable well with both

experimental as well as prior numerical results in spite of the simplicity of the structure

model used and the first order accuracy of our flow solver.

X

Z

0 5 10

-4

-2

0

2

4

1.411.391.371.351.33

Maximumdeflectionpoint

X

Z

0 5 10 15

-4

-2

0

2

4

1.411.391.371.351.33

Minimumdeflectionpoint

130

Table 5-1. Comparison of critical flutter speed and dynamic pressure with experiment and other numerical results

Mach number Model Flutter Speed Index

Standard k-ε model (350 K points, ∆t=1x10-5)

0.294

Standard k-ε model (350 K points, ∆t=5x10-5)

0.297

Standard k-ε model (800 K points, ∆t=1x10-5)

0.293

Filter-based model (350 K points, ∆t=5x10-5) 0.297

Experiment 0.308

Gordnier and Melville (2001) 0.329

0.96

Lee Rausch and Batina (1996) 0.294

Standard k-ε model (350 K points, ∆t=5x10-5) 0.419

Experiment 0.417 0.678

Lee-Rausch and Batina (1996) 0.417

Standard k-ε model (350 K points, ∆t=5x10-5) 0.410

Experiment 0.320 1.072

Liu et al. (2003) 0.460

131

Figure 5-24. Summary of flutter speed index prediction for AGARD 445.6 wing

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2Mach number

Flut

ter s

peed

inde

x

Experiment

Lee-Rausch and Batina (1996)

Liu et al. (2003)

Present (Grid 1)

Present (Grid II)

132

CHAPTER 6 CONCLUSIONS AND FUTURE WORK

Conclusions

This work is motivated by our interest in developing a suitable computational

capability to account for fluid and structure interactions to perform computational

aeroelastic analysis on three-dimensional wing geometries. The computational capability

is applied to study both incompressible as well as compressible flow computations

around three-dimensional wing geometries. The aeroelastic phenomenon looked at is the

flutter characteristics for the AGARD 445.6 wing at a transonic Mach number. The

Reynolds-averaged Navier-Stokes equations, cast in their conservation form, along with a

linear structure solver based on beam finite elements and appropriate moving boundary

and interfacing procedures, are employed. The two-equation k-ε model with wall

functions method for near wall treatment is adopted for turbulence closure.

A multi-block, structured flow solver that uses finite volume formulation based on

a pressure-based method was employed to integrate the RANS equations. It is coupled

with linear structure model with beam finite elements that is based on Bernoulli-Euler

beam theory. A predictor-corrector based PISO algorithm was proposed to perform

unsteady computations for the flow solver. This method, being a non-iterative algorithm,

greatly reduces the computational cost of flow computations, which constitutes the major

part of the computational time of the coupled solver. Since it is a semi-implicit method, it

is controlled by the CFL-type stability condition and hence the time step size needed to

be chosen carefully based on the CFL number. A fully implicit Newmark method is used

133

to integrate the modal equations of motion of the structure and hence is not limited by a

time step size. The time step size chosen for the coupled simulation is entirely based on

the flow solver and time-synchronization is ensured between the two modules to prevent

any phase lag.

The two modules are integrated at the interface or boundary using interfacing

techniques. Particularly, a bilinear interpolation technique was used to transfer surface

pressure from flow domain to structure domain and a linear extrapolation is used to

transfer displacement field from structure domain to fluid domain. A robust moving

boundary module based on master/slave concepts and transfinite interpolation method

was embedded into the coupled solver to account for re-meshing of the entire flow

domain once the new surface or geometry under consideration is known. Impact of two

different parameters on the rigidity of grid movement was studied to preserve the

distance of the first node from the wall to enable the use of wall functions approach

associated with our turbulence model. A filter-based model is introduced as well to

improve upon the predictive capability of the standard k-ε two-equation model.

Various issues were addressed while developing this computationally methodology.

Since the computations are performed on a dynamically moving mesh, the geometric

properties need to be conserved as the mesh configuration changes with time. This is

ensured via the geometric conservation law. This, when implemented in curvilinear

coordinates, calls for updating the Jacobians associated with the cell volumes in a

consistent fashion. Four different schemes for updating Jacobian values were examined in

an attempt to arrive at a robust choice for performing moving grid computations. Both

first and second order fully implicit as well as time-averaged evaluation of metrics were

134

considered. Three test cases, two using the 2-D channel flow one using the 3-D elastic

wing, with different Reynolds number, time dependency, and geometry movement were

carried out to assess the performance of the different GCL schemes implemented. The

first order fully implicit method was found to produce the least error norm for all test

cases. The results have demonstrated that the impact of GCL on the solution accuracy is

not simply governed by the formal order of accuracy of the discretization scheme. The

results also show that the choice of fluid flow solver and magnitude of grid movement

strongly affect the GCL scheme for performing aeroelastic computations. Since the first

order fully implicit GCL scheme simplifies the data structure in the code development

effort, we reached a conclusion that this scheme is appropriate in the context of the fluid

flow solver employed.

Another aspect that was probed into was the time dependency effect associated

with the momentum interpolation method, which is required to compute the contravariant

velocity components when using a collocated or non-staggered grid. The original Rhie-

Chow momentum interpolation method was developed with steady state computations in

mind. However, while performing unsteady computations, this scheme was found to

produce spurious oscillations in the pressure and velocity fields especially when a smaller

time step is used. Hence, this scheme needed to be modified to include time dependency

effects to accurately compute the contravariant velocities. The proposed scheme

eliminated any time dependency effect observed with the original scheme and hence was

used for all our future computations.

Coupled simulations are carried out using the AGARD 445.6 wing after taking care

of the above-mentioned issues. Both incompressible as well as compressible flow

135

computations were carried out to study the coupling procedure and to compute the flutter

boundary for the AGARD wing at various Mach numbers. For the incompressible,

turbulent flow computation, the full structure model with all the modes was used. Wing

shapes at different time instants were shown and it was seen that the deflection reaches

steady state after a certain time due to the presence of damping in the structure. We also

showed the time history of aerodynamic parameters and showed that they converged to a

steady state value as well. A grid refinement study was done as well and the aerodynamic

parameters compared with the coarse grid solution. Oscillations in the aerodynamic

parameters were observed before converging towards steady state value. The effect of

angle of attack on the aerodynamic parameters was also looked at for three different

values of angle of attack. Results were found to be in good agreement with theory. An

iterative flow solver based on SIMPLE family of methods was used for all of the above

computations.

The main objective of this thesis was to study flutter characteristics around a three-

dimensional wing. Computations around the AGARD wing were performed at three

different Mach numbers of 0.678, 0.96 and 1.072. All flow computations were performed

using the non-iterative PISO algorithm. The PISO algorithm was validated first by

comparing it with the SIMPLE algorithm using the AGARD wing example for

incompressible flows. Excellent agreement was observed with one another, however, the

computational cost, by using PISO method was improved by ten folds. Only one mode of

the structure is used here to predict the flutter boundary of the wing. The flutter boundary

is predicted by varying the dynamic pressure and by studying the generalized

displacement of the structure. Two grid configurations, one with 350,000 points and

136

another with 800,000 points were considered to study the transonic Mach number case.

Two different time steps were used for the coarser grid to determine the impact of

different time step sizes on solution accuracy. The coarser of the two grids predicted

flutter boundary accurately for subsonic and transonic Mach number cases. but were

found to deviate for supersonic Mach number case. Prior researchers have made such an

observation as well. Computations for transonic Mach number case using the finer grid

showed a closer agreement with experiment compared the coarser grid. The two choice of

time steps on the coarser grid produced some differences in flutter prediction, although

the differences were small, which can be attributed to the first order time marching

scheme employed by the flow solver. Differences in flow features, particularly

supersonic region in the flow domain, were highlighted based on the wing shape

observed at different time instants. This gave insight into the flow physics associated with

deformation in the structure.

The filter-based turbulence was employed as well to perform unsteady flow

computations as the RANS model is tuned by steady state mean flow data and hence was

found to have some shortcomings when dealing with time dependent computations. The

eddy viscosity term was modified using a filter to overcome this. The filter size was so

chosen that wall functions could still be applied near wall regions. Flutter computations

performed using the filter-based model did not show any significant improvement over

the standard k-e model but the results were consistent when compared with the k-ε

model.

To conclude, we have shown that the computational methodology has exhibited

capability to predict flutter characteristics in an accurate manner in spite of coupling a

137

simple linear structure solver with a more complex flow solver. The simulations have

given insight into the flow physics associated with performing a fluid-structure

interaction computation over elastic bodies. It has also given insight into several

numerical issues encountered while carrying out these computations and what needs to be

done to overcome this.

Future Directions

The current research can be extended in the following direction:

• Extend the methodology to investigate nonlinear structural dynamics models to address issues related to larger and more complicated deformation characteristics. Issues such as limit cycle oscillations, buffeting, etc, can be investigated in detail.

• Refine both spatial and temporal resolutions, including possibly adopting higher order time marching schemes. Coupled fluid and structure simulations are very time consuming. Priority should be given to help reduce the computational cost, including higher order schemes, parallel computational capabilities, and adaptively updated grid distributions.

138

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144

BIOGRAPHICAL SKETCH

Ramji Kamakoti was born in 1976, in the historic city of Pondicherry in India

(although he was brought up in Chennai). He received his bachelor’s degree in

mechanical engineering from the Birla Institute of Technology and Science, located in a

place called Pilani, in the state of Rajasthan. He then decided to come to the US for

graduate studies in the field of engineering mechanics at the University of Florida, where

he obtained his master’s degree in 2000. For the previous 6 years, he did research in the

field of fluid-structure interaction (in the computational thermo-fluids group under the

guidance of professor Wei Shyy).