computational aeroelasticity using a pressure-based solver_kamakoti

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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 OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2004


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Copyright 2004

by

Ramji Kamakoti


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This dissertation is dedicated to my parents and sister.


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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.


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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.........................................1Problem Statement........................................................................................................4

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

Aerodynamic Models..................................................................................................10

Physical Models...................................................................................................10

Reduced-Order Models .......................................................................................12Review of Coupled Computational Aeroelasticity (CAE) Models ............................13

Fully coupled Analysis ........................................................................................14Loosely and Closely Coupled Analysis...............................................................16

Loosely coupled analysis .............................................................................16

Closely coupled analysis ..............................................................................17Review 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...............................................................................32Transformation to curvilinear coordinates ...................................................33

Geometric Conservation Law..............................................................................37

Turbulence Modeling ..........................................................................................38

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


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Filter-based turbulence model for unsteady Reynolds-Averaged Navier-

Stokes (RANS) computations.................................................................42Boundary conditions ....................................................................................43

Wall treatment..............................................................................................43

Structural Dynamics Model.................................................................................44

Moving Grid Module...........................................................................................48Overview of Algorithm...............................................................................................50

Discretized Form of Equations............................................................................50Evaluation of Contravariant velocities on Non-staggered Grid ..........................52

Pressure-Based Flow Solver (Semi-Implicit Method for Pressure-Linked

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

computations....................................................................................................58

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

First order Implicit Scheme:.........................................................................61First-order time-averaged scheme:...............................................................62

Second order implicit scheme ......................................................................62Second order time-averaged evaluation of Jacobian....................................63Newmark Integration Method for Structure Solver.............................................64

4 COMPUTATIONAL PROCEDURE AND CODE VALIDATION.............................66

Computational Procedure ...........................................................................................66

Geometry definition and Computational Grids ..........................................................67Geometry Definition............................................................................................67

Computational Grids ...........................................................................................68

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

Coupling and Interfacing Procedure...........................................................................70

Code Validation..........................................................................................................74Steady-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 84Geometric Conservation Law..............................................................................88

Two-dimensional channel flow: First order backward Euler.......................89

Two-dimensional channel flow: PISO algorithm.........................................94Three-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


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Time Scales and Choice of Time Step Size for the Coupled Problem..............113

Flutter Boundary Prediction for AGARD Wing at a Transonic Mach Number116Flutter 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


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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

usingBackward Eulermethod .................................................................................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-Dwing 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


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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


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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 imposedboundary 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 ofstages 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 locationin 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 differentmomentum 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 (15111) usingBackward Eulermethod...............................................91

4-27. Error norm versus grid velocity using various schemes for channel flow for

15111 grid usingBackward Eulermethod ............................................................91


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4-28. Velocity profile for channel flow with Re=100 at different time instants for fine

grid (30121) usingBackward Eulermethod..........................................................93

4-29. Error norm versus grid velocity using various schemes for channel flow for30121 grid usingBackward Eulermethod ............................................................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

usingBackward Eulermethod .................................................................................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,FACMINand , 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 angleof 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 differentangles of attack for grid configuration I.................................................................110

5-8. Spanwise displacements at three different time instants to compare PISO andSIMPLE 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


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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 CFLnumbers..................................................................................................................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-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


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Abstract of Dissertation Presented to the Graduate School

of the University of Florida in Partial Fulfillment of theRequirements 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- 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


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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.


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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.


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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

Aerodynamic

Forces

Elastic

Forces

Inertial

Forces

Static Aeroelasticity

Load distribution Divergence

Control system reversal

Dynamic Aeroelasticity

Flutter Buffeting Dynamic response


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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


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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


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(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 interpolationmethod 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


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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


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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 numbe r

Flutterspeedi

ndex

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


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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.


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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.


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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


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pointx, y and some time ton 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 U

t x

= +

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

symbols, Uand Mrepresent 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 Bernoullis 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 Greens 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).


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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


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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


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

{ } { }1

N

i i

i

w( x, y,z,t ) q ( t ) ( x, y,z )=

= (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 i are the normal

modes of the structure, withNbeing 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


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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 directcoupling using time-integration method

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

bilinear interpolation and by virtualsurface methods

CFD grid (151 x 30 x 35) FEM grid (36 plate elements) Fighter type wing with M=0.854 and=1 deg

Validity of coupling plateelements with Euler

equation

Virtual surface methodtransfers loads moreaccurately than bilinear

interpolation technique

Garcia,Guruswamy(1999)

Model for coupled nonlinear beam FEMmodel with N-S solver for static

aeroelastic analysis of high AR wings

Flow solver: ARC3D fluids module ofENSAERO-WING code

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

M=0.85 and =2

FEM results are accurateexcept for deflections which

were smaller than modalresults

Nonlinear and linear beammodels predicted similarpressure coeff results

Geometrical nonlinearitywas found to be negligible


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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.


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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 equationsAsymptotic expansion

Boundary LayerFull Navier-Stokes

other

Function matching

Infinite plate splineThin plate splineother

Conservative interfaces Infinite plate splineThin plate splineConservative/ consistentother

Modal analysis

Influence coefficient

Linear FEMNonlinear EM

other

Fluids Module Interface Methodology Structural Module


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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-


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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 CFL3DAEAuthor (s)(year)


Cunningham,

Batina, Bennett

(1988)

Computational scheme for transonicaeroelastic analysis to perform flutter

analysis

Flow: Transonic small disturbanceformulation

Structure: Lagrange Equations of motionbased on the natural vibrational modes

AGARD configuration with 45 deg

sweep angle and M=0.338-1.141

Aerodynamic forces andflutter characteristics

obtained using linearformulation compared well

with expt.

Non-linear flutter resultscompared well with exptbut not so with linear

resultsCan treat configurations

with arbitrary liftingsurfaces

Lewis and

Smith

(1998)

External aeroelastic simulation forinternal aerodynamics and shell

structures

Flow: ENS3DPredictor-corrector scheme for structural

integration

Tested on an engine liner to study flutter

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

Results showed the engineliner to be dynamically

stable

Inner flow Mach no. hadlittle effect on aeroelasticresponse

Effect of pressure loadings

on the shell structures werenot considered in thismethod


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Table 2-2. Continued

Author (s)

(year)


Schuster,

Vadyak,

Atta(1990)

A 3-D flow solver coupled with linearstatic structural model to study

aeroelastic response of aircraftGrid deflection method is used to update

the grid after each time step.

Flow solver: ENS3DSwept, tapered wing with constant cross-

section with M=0.9 and =9 deg wasused

Wing mesh: 92 x 32 x 32 points

Aeroelastic analysiscompared well with

experiment with respect topressure coefficient and

twist

Flexible wing/bodyconfiguration gave betterresults compared to rigid

body configuration

Separation on the uppersurface was not predicted

Lee Rauschand Batina

(1993, 1995,1996)

Navier-Stokes aerodynamics to computeAGARD 445.6 wing flutter

Flow: Implicit upwind Euler/N-S solverStructure: Modal analysisMoving mesh: Spring analogyGrid: 193 x 41 x 65 C-H typeM=0.96, Re=364,600 per foot of chord

Difference in flutter speedindex and frequency index

between Euler and N-Ssolver was pointed out

Hartwich,

Dobbs, Arslanand Kim

(2000)

Study LCO for a B-1 configuration usingN-S equations

Flow: CFL3D a 3D N-S solverStructure: Lagranges equations of

motion

Moving mesh: Spring analogy and TFIusing master/slave concept

Grid: 281 x 137 x 65 C-O typeM=0.975, =7.38 deg and Re=5,900,000

Predicted aerodynamicdamping matched well with

experimental trends

Fell short of predicting atrue 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)


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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)


Liu, Cai, Zhu,

Wong and Tsai(2000)

AGARD 445.6 Wing flutter using acoupled CFD-CSD

Flow: Parallel multi-block EulerStructure: Modal dynamic equationsMoving mesh: Arc-length based TFI and

spring analogy

Interface: Transformation spline matrixGrid: 176,601 points (32 blocks)M=0.338-1.141

Flutter speed/frequency ingood agreement withexperiment

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-SStructure: Static elastic equationsMoving mesh: Spring analogy and TFIM=0.85 and =5 deg

Convergence was sped-upusing relaxation technique.

Difference in solutionsbetween 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.


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

Author (s)

(year)


Farhat, Pierson

and Degand

(2000)

Computational method to simulatetransient aeroelastic response of flexible

aircraft during high-G maneuversFlow: Arbitrary Lagrangian-Euler

equations are incorporated into the

unstructured flow solver (Euler)

Structure: Corotational formulationM=0.901 and =1 deg on Langley

fighter

Qualitative validation ofresults was done

Geometric conservationlaw was incorporated

Viscous effects wereneglected

Farhat and

Lesoinne

(2000)

Serial and Parallel methodologies fornonlinear transient aeroelastic problems

Flow: ALE formulationMoving mesh: Dynamic mesh equations

coupled with the flow equationsM=0.901 on an AGARD 445.6 wing

Partitioned algorithms werefound to be efficient than

monolithic schemes

Geuzine,

Brown andFarhat (2002)

Three-field formulation for flutteranalysis of F-16 configuration

Flow: ALE formulationStructure: Elastodynamic equationsMoving 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 conservativeexchange of aerodynamicand elastodynamic data was

shown

Method was found to beeffective in the transonicregime 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


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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,


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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)

FastMay not preserve original

grid qualityGordons TFI based method

(Wong et al., 2000)

Erikssons TFI basedmethod (Hartwich and

Agrawal, 1997)

Perturbation method

(Reuther et al., 1996)

Faster and Preserves grid

quality

May encounter crossover

near the moving boundary

Moving mesh partialdifferential 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


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

thatfor 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.


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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-

basedfinite 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-basedfinite 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-basedALE finite volume as well asfinite element

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

second-order time-accurate density-based ALEfinite 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 onfinite 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


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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 ws 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 F

Map pressure

to FEM grid

Interpolate to

CFD grid

CSD

Move

Grid

Fluid/Structure

Interface

W

NAVIER-

STOKES

LINEAR

ANALYTICAL

EULER

FULL

POTENTIAL

TRANSONIC

SMALLDISTURBANC

SHAPE

FUNCTIONS

MODAL

APPROACH

3-D FINITE

ELEMENTS

2-D FINITE

ELEMENTS

EQUIVALENT

BEAM

FLUID STRUCTURE

INTERFACIN

G

COMPLEXITYINPHYS

ICS

COMPLEXITYINGEOMETRY


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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 onsuperposition 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): Characterizesan irregular surface by using functions

that minimize an energy functional

Finite plate spline (FPS): Uses platebending elements to represent aplanform by a number of quadrilateral or

triangular elements

Non-uniform B-splines (NUBS): usesthe fact that a 3-D surface can berepresented 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 requiredNoncoincident points are requiredExtrapolations are linearNo minimum number of grid points

required but 3 are preferred for accuracy

No minimum number of grid pointsrequired but 3 are preferred for accuracy

Only 2-D application was looked at

Four curves and four data points requiredPoints cannot be coincident

Valid for 2-D interpolations onlyNo extrapolation possible


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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


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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

Authors 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 (avector) to go back and forth betweenCFD/CSD data

Exterior BEM solver for CFD gridregeneration

Code used: ENS3DAEAGARD 445.6 at M=0.95 and =2CFD grid (63 x 26 points)Structure grid (121 points)

Performs force transferalwith good accuracy

Performs accuratedisplacement extrapolation

CSD grid points should lieinside CFD surface grid.

Boundary element nearleading/trailing edge causes

instability.


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Table 2-7. Continued

Authors 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 structuresAGARD 445.6CFD grid (145 X 37 points)Structure grid (121 points)

Gives very goodextrapolation results on theCFD grid

Eliminates edge effectsfound in the direct BEM

solver

Deals with complexconfigurations


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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 x

+ =

(3-1)

Momentum:

( ) ( ) iji j ij i ij

pu u u

t x x x

+ = +

(3-2)

Energy:

( ) ( ) ( )jj i ijj i j

qpH u H u

t x t x x

+ = +

(3-3)


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wherexi is the position vector, tis time,is density, ui is velocity vector,p is pressure, ij

is viscous stress tensor, qj is heat flux vector, obtained from Fouriers law, given by

j

j L j

T hq

Pr x

= =

(3-4)

whereis the molecular viscosity, is the thermal conductivity, andPrL is the laminar

Prandtl number defined as:

p

L

CPr

=

His stagnation enthalpy given by

1

2i 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:

2

3

ji l

ij ij

j i l

uu u

x x

= +

(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 (,,) where =(x,y,z,t),

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

computational domain (,,) 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 f

f f f J

f f f

=

(3-7)


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where the metricsfijs 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 x

f z y y z f z y y z f z y y z

f y z z y f z x x z f x y y x

= = =

= = =

= = =

(3-8)

andJis the Jacobian g

computational aeroelasticity using a pressure-based solver_kamakoti

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