Loosely Coupled Time Integration of Fluid- Thermal-Structural Interactions in Hypersonic Flows

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Brent Adam Miller, M.S.

Graduate Program in Aeronautical and Astronautical Engineering

The Ohio State University 2015

Dissertation Committee: Jack J. McNamara, Advisor Thomas Eason III Datta Gaitonde Sandip Mazumder S. Michael Spottswood Manoj Srinivasan Copyright by

Brent Adam Miller

2015 Abstract

This dissertation describes time integration strategies for ﬂuid-thermal-structural interactions in hypersonic and high speed (Mach 1) ﬂow using separate solvers for each discipline.

The development of reusable hypersonic cruise vehicles requires analysis capability that can capture the coupled, highly-nonlinear interactions between the

fluid flow, structural mechanics, and structural heat transfer. This coupled analysis must also be performed over significant portions of the flight trajectory due to the long-term thermal evolution of the vehicle. The fluid and structural physics operate at significantly smaller time scales than the thermal evolution, requiring time marching that can capture the small time scales for time records that encapsulate the longer time scale of the thermal response. This leads to extreme computational times and motivates research that seeks to maximize efficiency of the time integrations for the coupled problem.

The goal of this dissertation is to develop time integration procedures that sig- niﬁcantly improve computational efﬁciency while also maintaining time accuracy

ii and stability for ﬂuid-thermal-structural analysis. This is achieved using carefully designed loosely coupled schemes for the ﬂuid, thermal, and structural solvers.

Here, different time integrators are used for the solvers of each physical field, and boundary conditions are exchanged at most once per time step. Coupling schemes for both time-accurate and quasi-steady flow models are considered. Computa- tional efficiency and time accuracy are improved through the use of both extrapolating predictors and interpolation during the exchange of boundary conditions; the latter of which enables the use of different sized time steps between the solvers, known as subcycling.

The developed coupling procedures are compared to several other schemes, including a basic one that does not use the predictors, and a subiteration-based strongly coupled scheme. Response predictions of multiple configurations of a panel in two dimensional supersonic and hypersonic flow are performed. Using second order implicit time integrators for the individual solvers, the predictor- based and strongly coupled schemes are demonstrated to retain the second order accuracy with and without subcycling, while the others reduced to first order.

Simulations of two panel configurations are performed to investigate the performance of the coupling schemes in predicting a stable response for one configuration, and an unstable flutter response for the other. From the stable response analysis, the predictor-based schemes are found to be the least computationally expensive compared to the strongly coupled and basic loosely coupled schemes.

In the second conﬁguration, the panel is predicted to undergo snap-through and

iii ultimately a dynamically unstable limit cycle response with each scheme. The predictor-based and strongly coupled schemes converge to a consistent limit cycle at larger time steps compared to the others. The predictor-based schemes are shown to provide a 3-10 times reduction in computational cost compared to the other schemes.

Finally, a 30 second response of the panel with a flutter instability using time- accurate CFD is performed and compared to the response using quasi-steady surrogate aerothermodynamic and quasi-steady analytical aerothermodynamic models. The unsteady CFD and surrogate based responses have excellent agreement throughout the response, with differences under 1%. The time to flutter predicted from the analytical aerothermodynamic models is 15% higher than that predicted by the CFD and surrogate models. However, the post-flutter response shows good agreement between the three responses, indicating that the quasi-steady assumption can accurately capture the dynamics of the flow response.

iv Dedication

To my family for their love and support.

v Acknowledgments

I would like to thank my advisor, Prof. Jack McNamara, for providing me with guidance and support throughout my graduate studies. Professor McNamara’s drive for high quality research in ﬂuid-thermal-structural interactions has been vi- tal to my academic and professional growth, and to this work. Without his passion for the ﬁeld and support for my development as a researcher, I surely would not have stayed to pursue a doctoral degree. I would also like to thank Profs. Datta

Gaitonde, Sandip Mazumder, and Manoj Srinivasan for serving on my dissertation and candidacy committees, and Drs. Thomas Eason and S. Michael Spottswood for serving on my dissertation committee.

I am deeply grateful for the ﬁnancial support I have received for this work.

This research was performed from government awards from the AFRL-University

Collaborative Center in Structural Sciences (Cooperative Agreement FA8650-13-2-

2347), with Dr. Ravi Penmetsa as program manager; an HPCMPO Frontier PETTT

Project Grant, under the direction of David Bartoe; the DoD Science, Mathematics, and Research for Transformation (SMART) Scholarship for Service Program; and

vi the AFRL/DAGSI Ohio Student-Faculty Research Fellowship Program. I would also like to thank the Ohio Supercomputer Center for providing computational resources, and the NASA Langley Research Center for providing CFL3D.

Technical contributions to this work have been made by researchers at the US

Air Force Research Laboratory, as well as past and current members of Professor

McNamara’s research group at the university. I appreciate and thank Drs. Ravi

Chona, S. Michael Spottswood, and Tom Eason of the AFRL/RQSS Structural Sci- ences Center for their technical insights. I also gratefully acknowledge Drs. Adam

Culler, Andrew Crowell, and Abhijit Gogulapati for their significant contributions that enabled this work. I would like to thank Dr. Culler for providing the original quasi-steady fluid-thermal-structural models and motivating a great deal of this work; Dr. Crowell for providing me with the surrogate fluid models and helping me learn CFL3D; and Dr. Gogulapati for providing routines to use the surrogate models in my solvers and for general guidance.

I am very grateful to all the other graduate students that I have had the pleasure to work with in the ofﬁce. Thank you Dr. Krista Kecskemety, Dr. Andrew Crowell,

Steve Nogar, Zach Riley, Rohit Deshmukh, Zach Witeof, Joe Connolly, Jonathen

LaFontaine, Kirk Brouwer, Kelsey Shaler, and everyone else for your support and friendship. Finally I would to thank my parents, Robert and Robyn Miller, and my siblings, Shawna Harvey and Bobby Miller, for always being there for me and encouraging me all the way. Without your continued support I would not have made it this far.

vii Vita

2009 ...... B.S. Aeronautical and Astronautical Engineering, The

Ohio State University

2010 ...... M.S. Aeronautical and Astronautical Engineering, The

Ohio State University

2009-2010 ...... DAGSI Fellowship, The Ohio State University

2010-2014 ...... SMART Scholarship, The Ohio State University

2014 - 2015 ...... Graduate Research Assistant, The Ohio State Univer-

sity

Publications

Journal Publications

Miller, B. A. and McNamara, J. J., ”Time Marching Considerations for Response

Prediction of Structures in Hypersonic Flows,” AIAA Journal, Accepted for publi- cation.

viii Miller, B. A., McNamara, J. J., Culler, A. J., and Spottswood, S. M., ”The Impact

of Flow Induced Loads on Snap-Through Behavior of Acoustically Excited, Ther-

mally Buckled Panels,” Journal of Sound and Vibration, Vol. 330, 2011, pg. 5736-5752.

Crowell, A. R., Miller, B. A., and McNamara, J. J, ”Robust and Efﬁcient Treat-

ment of Temperature Feedback in Fluid-Thermal-Structural Analysis”, AIAA Jour-

nal, Vol. 52, No. 11, 2014, pg. 2395-2413.

Crowell, A. R., McNamara, J. J., and Miller, B. A., ”Hypersonic Aerothermoe-

lastic Response Prediction of Skin Panels Using Computational Fluid Dynamics

Surrogates,” Journal of Aeroelasticity and Structural Dynamics, Vol. 2, No. 2, June

2011, pg. 3-30.

Deshmukh, R., Culler, A. J., Miller, B. A., and McNamara, J. J., ”Response of

Skin Panels to Combined Self and Boundary Layer Induced Fluctuating Pressure,”

Journal of Fluids and Structures. In review.

Conference Publications

Miller, B. A., and McNamara, J. J., Loosely Coupled Time-Marching of Fluid-

Thermal-Structural Interactions with Time-Accurate CFD, 56th AIAA/ASCE/AHS/

ASCStructures, Structural Dynamics, and Materials Conference, January 5-9, Kissim- mee, FL, AIAA 2015-0686, 2015.

ix Miller, B. A. and McNamara, J. J., Efﬁcient Time-Marching of Fluid-Thermal-

Structural Interactions, 55th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dy-

namics, and Materials Conference, January 13-17, National Harbor, MD, AIAA-2014-

0337, 2014.

LaFontaine, J. H., Gogulapati, A., Miller, B. A., and McNamara, J. J., Develop-

ment of a Framework for Study of Fluid-Thermal-Elastic-Plastic Interactions, 55th

AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Confer-

ence, January 13-17, National Harbor, MD, AIAA-2014-1516, 2014.

Miller, B. A., Crowell, A. R., and McNamara, J. J., Loosely Coupled Time-

Marching of Fluid-Thermal-Structural Interactions, 54th AIAA/ASME/ASCE/AHS/

ASCStructures, Structural Dynamics, and Materials Conference, April 8-11, Boston,

MA, AIAA-2013-1666, 2013.

Miller, B. A., Crowell, A. R., and McNamara, J. J., Modeling and Analysis

of Shock Impingements on Thermo-Mechanically Compliant Surface Panels, 53rd

AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Confer- ence, April 23-26, Honolulu, HI, AIAA-2012-1548, 2012.

Miller, B. A., McNamara, J. J., Culler, A. J., and Spottswood, S. M., The Impact

x of Flow Induced Loads on Snap-Through Behavior of Acoustically Excited, Ther- mally Buckled Panels, 51st AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dy- namics, and Materials Conference, April 12-15, Orlando, FL, AIAA-2010-2540, 2010.

LaFontaine, J. H., Gogulapati, A., Miller, B. A., and McNamara, J. J., Effects of Strain Hardening on Fluid-Thermal-Structural Interactions, 56th AIAA/ASCE/

AHS/ASCStructures, Structural Dynamics, and Materials Conference, January 5-9, Kissim- mee, FL, AIAA 2015-1629, 2015.

LeVett, M. A., Liang, Z., Miller, B. A., and McNamara, J. J., Investigation into

Parallel Time Marching of Fluid-Thermal-Structural Interactions, 56th AIAA/ASCE/

AHS/ASCStructures, Structural Dynamics, and Materials Conference, January 5-9, Kissim- mee, FL, AIAA 2015-1632, 2015.

Riley, Z. B., Deshmukh, R., Miller, B. A., and McNamara, J. J., Characteriza- tion of Structural Response to Hypersonic Boundary Layer Transition, 56th AIAA/

ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, January

5-9, Kissimmee, FL, AIAA 2015-0688, 2015.

Crowell, A. R., Miller, B. A., and McNamara, J. J., Robust and Efﬁcient Treat- ment of Temperature Feedback in Fluid-Thermal-Structural Analysis, 54th AIAA/

ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, April

xi 8-11, Boston, MA, AIAA 20143-1663, 2013.

McNamara, J. J., Crowell, A. R., Miller, B. A., Riley, Z. B., Deshmukh, R., and

Culler, A. J., Structural Response Prediction of Surface Panels in Hypersonic Flow,

International Forum on Aeroelasticity and Structural Dynamics, Bristol, UK, IFASD-

2013-29C, 2013.

Crowell, A. R., Miller, B. A., and McNamara, A. J., Computational Modeling for Conjugate Heat Transfer of Shock-Surface Interactions on Compliant Skin Pan- els, 52nd AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materi- als Conference, April 4-7, Denver, CO, AIAA 2011-2017, 2011.

Crowell, A. R., McNamara, J. J., and Miller, B. A., Surrogate Based Reduced-

Order Aerothermodynamic Modeling for Structural Response Prediction at High

Mach Numbers, 52nd AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, April 4-7, Denver, CO, AIAA 2011-2014, 2011.

Fields of Study

Major Field: Aeronautical and Astronautical Engineering

xii Table of Contents

Abstract ...... ii

Dedication ...... v

Acknowledgments ...... vi

Vita ...... viii

List of Tables ...... xvii

List of Figures ...... xix

List of Symbols ...... xxiv

Chapter

1 Introduction and Objectives ...... 1 1.1 Introduction ...... 1 1.2 Literature Review ...... 6 1.2.1 Time Integration Frameworks for Multi-Physics Problems 6 1.2.2 Time Integration of Fluid-Structure Interactions ...... 10 1.2.3 Time Integration of Fluid-Thermal Interactions ...... 13 1.2.4 Time Integration of Fluid-Thermal-Structural Interactions 15 1.2.5 Summary of the State of the Art for Fluid-Thermal-Structural Interactions ...... 18

xiii 1.3 Objectives of this Dissertation ...... 18 1.4 Key Novel Contributions of this Dissertation ...... 20

2 Governing Equations and Time Discretization ...... 22 2.1 Governing Equations ...... 22 2.2 Fluid Time Discretization ...... 27 2.2.1 Quasi-Steady Fluid Modeling ...... 28 2.2.2 Geometric Conservation Law ...... 29 2.2.3 Time-Accurate Fluid Modeling ...... 31 2.3 Structural Time Discretization ...... 33 2.3.1 Newmark-β Time Integration ...... 34 2.3.2 Fourth Order Runge Kutta Time Integration ...... 36 2.4 Thermal Discretization ...... 38 2.4.1 Crank-Nicolson Time Integration ...... 38 2.4.2 Forward Euler Time Integration ...... 40

3 Coupling Procedures ...... 41 3.1 Quasi-Steady Aerothermodynamics ...... 42 3.1.1 Predictor Implicit Scheme ...... 44 3.1.2 Predictor-Corrector Implicit Scheme ...... 49 3.1.3 Additional Quasi-Steady Coupling Schemes ...... 50 3.1.3.1 Basic Implicit Scheme ...... 50 3.1.3.2 Conventional Explicit Scheme ...... 52 3.1.3.3 Strong Implicit Scheme ...... 55 3.2 Time-Accurate Aerothermodynamics ...... 57 3.2.1 Predictor Implicit Scheme ...... 59 3.2.2 Predictor-Corrector Implicit Scheme ...... 64 3.2.3 Additional Coupling Schemes for Time-Accurate Aerother- modynamics ...... 65

xiv 3.2.3.1 Basic Implicit Scheme ...... 66 3.2.3.2 Strong Implicit Scheme ...... 67 3.3 Summary of Schemes ...... 69

4 Conﬁgurations and Computational Models ...... 71 4.1 Quasi-Steady Analytical Fluid Modeling ...... 73 4.1.1 Piston Theory ...... 74 4.1.2 Eckert’s Reference Enthalpy ...... 74 4.2 Quasi-Steady CFD-Based Surrogate Fluid Modeling ...... 77 4.3 Time-Accurate CFD Fluid Modeling ...... 77 4.4 Thermal Modeling ...... 79 4.5 Structural Modeling ...... 82 4.6 Summary of Conﬁgurations ...... 87

5 Order of Accuracy Analysis ...... 88 5.1 Quasi-Steady Aerothermodynamics ...... 89 5.1.1 Without Subcycling ...... 90 5.1.2 Structural Subcycling ...... 93 5.1.3 Summary of Accuracy Analysis ...... 96 5.2 Time-Accurate Aerothermodynamics ...... 100 5.2.1 Without Subcycling ...... 101 5.2.2 Structural Subcycling ...... 103 5.2.3 Fluid Subcycling ...... 103 5.2.4 Fluid and Structural Subcycling ...... 106 5.2.5 Summary of Accuracy Analysis ...... 109

6 Fluid-Thermal-Structural Response: Quasi-Steady Aerothermodynam- ics ...... 111 6.1 Response Study: Conﬁguration 1 ...... 111 6.2 Response Study: Conﬁguration 2 ...... 118

xv 6.3 Computational Expense ...... 126

7 Fluid-Thermal-Structural Response: Time Accurate Aerothermodynam- ics ...... 130 7.1 Initial Stable Response ...... 131 7.2 Flutter Response with Elevated Initial Temperature ...... 135 7.2.1 Initialization: Static Aeroelastic Equilibrium ...... 136 7.2.2 Flutter Analysis ...... 141 7.2.3 Fluid-Structure Work Balance ...... 146 7.3 Long Time Record Analysis ...... 148 7.4 Computational Cost ...... 155

8 Concluding Remarks ...... 161 8.1 Principal Conclusions Obtained in This Study ...... 161 8.2 Recommendations for Future Research ...... 167

Bibliography ...... 171

Appendix

A Least Squares Based Linear Fitting ...... 182

B Temperature-Dependent Material Properties ...... 185

C CFD Mesh Convergence Study ...... 189

D Thermal Solver Veriﬁcation ...... 193

E Structural Solver Veriﬁcation ...... 195

xvi List of Tables

3.1 Summary of coupling schemes for quasi-steady aerothermodynamics...... 69 3.2 Summary of coupling schemes for time-accurate aerothermodynamics...... 70

4.1 Post-shock flight conditions for each configuration...... 72 4.2 Panel properties of each configuration...... 72 4.3 Rayleigh Damping parameters for each configuration...... 86 4.4 Summary of configurations for quasi-steady aerothermodynamics. 87 4.5 Summary of Configuration 2 for time-accurate CFD...... 87

5.1 Computed order of accuracy at smallest time step for Conﬁgura- tion 1...... 99 5.2 Computed order of accuracy at smallest time step for Conﬁgura- tion 2...... 100 5.3 Computed orders of accuracies with time-accurate aerothermodynamics...... 110

6.1 Characteristic behavior of responses for Configuration 1...... 118 6.2 Post-flutter behavior of Configuration 2...... 125 6.3 Computational cost parameters (milliseconds) ...... 128 6.4 Computation time for 10 seconds of response for Configuration 1 129 6.5 Computation time for 35 seconds of response for Configuration 2 129

xvii 7.1 Relative errors of mid-panel displacement compared to benchmark. 136 7.2 Post-ﬂutter behavior for each coupling scheme at different time steps...... 146 7.3 Computational cost parameters, in seconds...... 157 7.4 Walltime for one second of stable response, in minutes...... 158 7.5 Walltime for three seconds of ﬂutter response, in hours...... 159

B.1 Coefﬁcient of thermal expansion and Young’s modulus for Al-7075 186 B.2 Thermal capacitance and thermal conductivity for Al-7075 . . . . 187 B.3 Coefﬁcient of thermal expansion and Young’s modulus for Ti-6Al- 2Sn-4Zr-2Mo ...... 187 B.4 Thermal capacitance and thermal conductivity for Ti-6Al-2Sn-4Zr- 2Mo ...... 188

xviii List of Figures

1.1 Falcon HTV-3X Blackswift concept vehicle [14, 17]...... 3 1.2 Schematic of Fluid-Thermal-Structural Interactions...... 4

2.1 Fluid and Structural Domains...... 23

3.1 Sequence of coupling scheme...... 44 3.2 Quasi-Steady Predictor Implicit (PI) coupling flowchart...... 46 3.3 Quasi-Steady Conventional Explicit (CE) coupling flowchart. . . . 54 3.4 Quasi-Steady Strong Implicit (SI) coupling flowchart...... 56 3.5 Generalized sequence for FTSI of time-accurate aerothermodynamics...... 58 3.6 Time-Accurate Predictor Implicit (PI) coupling flowchart...... 61 3.7 Time-Accurate Strong Implicit (SI) coupling flowchart...... 68

4.1 Detail of wedge surface...... 71 4.2 Flow computational grid (85,860 cells)...... 79 4.3 Thermal model domain...... 80 4.4 Structural model domain...... 83

5.1 Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 1 over one second...... 91 5.2 Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 2 over one second...... 92

xix 5.3 Order of Accuracy of panel displacement, velocity, and temperature for both configurations without subcycling (Quasi-steady aerothermodynamics)...... 94 5.4 Order of Accuracy of fluid pressure and heat flux for both configurations without subcycling (Quasi-steady aerothermodynamics). . 95 5.5 Order of Accuracy of panel displacement, velocity, and temperature for both configurations with 25 structural steps per thermal step (Quasi-steady aerothermodynamics)...... 97 5.6 Order of Accuracy of fluid pressure and heat flux for both configurations with 25 structural steps per thermal step (Quasi-steady aerothermodynamics)...... 98 5.7 Average temperature, mid-panel displacement, and mid-panel velocity over 0.112 seconds with time-accurate CFD...... 102 5.8 Order of Accuracy of panel displacement, velocity, temperature,

pressure, and heat ﬂux with ∆tT = ∆tS = ∆tF (Time-accurate aerothermodynamics)...... 104 5.9 Order of Accuracy of panel displacement, velocity, temperature,

pressure, and heat ﬂux with ∆tT = 4∆tS = 4∆tF (Time-accurate aerothermodynamics)...... 105 5.10 Order of Accuracy of panel displacement, velocity, temperature,

pressure, and heat ﬂux with ∆tT = ∆tS = 4∆tF (Time-accurate aerothermodynamics)...... 107 5.11 Order of Accuracy of panel displacement, velocity, temperature,

pressure, and heat ﬂux with ∆tT = 4∆tS = 16∆tF (Time-accurate aerothermodynamics)...... 108

6.1 Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 1 over 10 seconds...... 113

xx 6.2 Profiles of temperature,displacement, pressure, and heat flux over panel at t = 0 and t = 10s for Configuration 1...... 114 6.3 Examples of stable responses of mode 2 compared to benchmark solution (red dashed line) for Configuration 1...... 116 6.4 Examples of stable responses of mode 2 compared to benchmark solution (red dashed line) for Configuration 1...... 117 6.5 Average temperature (red) and normalized mid-panel displacement (blue) of Configuration 2 over 35 seconds...... 119 6.6 Profiles of temperature,displacement, pressure, and heat flux over panel at t = 0 and t = 24.5s (immediately prior to snap-through) for Configuration 2...... 120 6.7 Center displacement during initial snap-through and final flutter response for Configuration 2...... 121 6.8 Displacement and power spectral density at 3/4 chord length. . . 123 6.9 x-t diagram of post-flutter limit cycle oscillations...... 124

7.1 Average temperature, mid-panel displacement, and mid-panel velocity of panel with time-accurate aerothermodynamics from the benchmark solution...... 132 7.2 Relative errors of the mid-panel displacement compared to benchmark solution for SI, BI, PI, and PCI schemes over select time step sizes...... 134 7.3 Prescribed temperature distribution from quasi-steady analytical aerothermodynamics at t = 17.5 sec...... 138 7.4 Initial panel solution with prescribed temperature distribution in aeroelastic equilibrium, compared to quasi-steady analytical aerothermodynamics...... 139 7.5 Flowﬁeld pressure of static aeroelastic solution ...... 140

xxi 7.6 Temperature rise and panel displacement during snap-through and subsequent flutter...... 142 7.7 Flutter displacement envelope and PSD at 3/4 chord location of SI, BI, PI, and PCI schemes at ∆t = 62.5µs...... 144 7.8 Space-time contours of normalized displacement over .05 seconds for SI (benchmark), BI, PI, and PCI schemes, ∆t = 62.5µs...... 145 7.9 Work violation over one second, normalized by work done by structural solver, for the BI, PI, PCI, and SI schemes...... 149 7.10 Normalized work violation at one second at each time step size for the BI, PI, PCI, and SI schemes...... 150 7.11 Average temperature, mid-panel displacement, and 3/4 chord panel displacement of long-time record analysis with time-accurate CFD. 151 7.12 Power Spectral density of flutter between long time record simulation and short time record simulation (SI scheme, ∆t = 62.5µs). . 152 7.13 Temperature, flutter displacement envelope, and power spectral density of panel with unsteady CFD aerothermodynamics (blue), quasi-steady surrogate aerothermodynamics (red), and quasi-steady analytical aerothermodynamics (black)...... 153 7.14 Displacement, temperature, pressure, and heat flux across panel immediately prior to flutter with unsteady CFD (20.8 seconds), quasi- steady surrogates (20.8 seconds), and quasi-steady analytical aerothermodynamics (24.5 seconds)...... 156

C.1 Deformation profiles used to compare mesh convergence...... 190 C.2 Pressure and heat flux comparisons for both deformation profiles. 191 C.3 Generalized aerodynamic force (GAF) over 1.25 cycles of forced oscillation...... 192

D.1 Heat ﬂux proﬁle prescribed to panel upper surface...... 194

xxii D.2 Comparison of current thermal model to Abaqus using a constant heat ﬂux proﬁle...... 194

E.1 Uniform in space temperature loading over time...... 196 E.2 Comparison of current structural solver to Abaqus with a rising temperature loading...... 197 E.3 Random in time, uniform in space, pressure loading applied to panel for transient response veriﬁcation...... 198 E.4 Comparison of current structural solver to Abaqus (linear and nonlinear) with a random pressure loading...... 198

xxiii List of Symbols

An Modal weight for mode n a Least square fit for offset b Least square fit for slope BI Basic Implicit scheme (quasi-steady or time-accurate fluid) [C] Structural damping matrix

[CT] Thermal capacitance matrix C Computational cost per time step CFD Computational Fluid Dynamics CE Conventional Explicit scheme (quasi-steady fluid) c Specific heat cf Local skin friction coefficient D Plate stiffness e Internal energy per unit volume E Young’s Modulus Fa Structural applied load vector FT Structural thermal load vector FTSI Fluid-thermal-structural interactions GCL Geometric Conservation Law H Total enthalpy h Panel thickness B HA Mapping from domain A to domain B

xxiv HCV Hypersonic cruise vehicle J Metric Jacobian [K] Structural stiffness matrix

[KT] Thermal conductivity matrix L2 L2 norm L Panel length M Mach Number [M] Structural mass matrix

M∞ Freestream Mach number

MT Thermal moment ~n Outward unit normal N Number of time steps

NT In-plane thermal load

Nx In-plane load O Order of PCI Predictor-Corrector Implicit scheme (time-accurate ﬂuid) PI Predict Implicit scheme (quasi-steady or time-accurate ﬂuid) P , p Fluid pressure P r Prandtl Number Q Fluid conserved variable vector

QT Thermal discrete heat load vector q Heat flux q2 Post-shock dynamic pressure R Discretized divergence of fluid fluxes

RGCL Discretized Geometric Conservation Law residual r Recovery factor

Rex Reynolds number of panel location x SI Strong Implicit scheme (quasi-steady or time-accurate ﬂuid)

xxv St Stanton Number T Temperature T Discretized temperature

Tinit Initial panel temperature

Tw Wall temperature

T∞ Freestream temperature t Time

t0 Overhead time

U2 Post-shock ﬂuid velocity u Displacement

u0 In-plane displacement u Discretized structural displacement ~v Velocity w Transverse structural displacement W Work violation across ﬂuid-structure interface

WF Work done by ﬂuid

WT Work done by structure X Reference domain ~x Spatial coordinates

α Coefﬁcient of thermal expansion

αR Mass-proportional Rayleigh coefﬁcient

βR Stiffness-proportional Rayleigh coefﬁcient

ΓF −S Fluid-Structure interface γ Ratio of speciﬁc heats (1.4) ∆t Time step

δx, δz Linear shape function Error

xxvi tol Error tolerance κ Thermal conductivity µ Dynamic viscosity φ Solution quantity of interest ρ Density σ Stress tensor ν Poisson’s ratio

φn Mode shape n

ΩF Fluid domain

ΩS Solid/structural domain ω Frequency

Subscripts

0 Stagnation property

2 Post-shock property

∞ Freestream property aw Adiabatic wall property e Boundary layer edge property

F Of the ﬂuid domain/model

I Interpolated between time steps ls Evaluated using least-squares ﬁt

P Predictor

S Of the structural domain/model

T Of the thermal domain/model w Evaluated at the wall

Superscripts ∗ Eckert’s Reference enthalpy condition

xxvii i Subcycling counter between thermal steps j Subcycling counter between structural steps k Fluid step counter m Structural time step counter n Thermal time step counter

xxviii Chapter 1

Introduction and Objectives

1.1 Introduction

Sustained ﬂight at hypersonic speeds has been an elusive goal for the aerospace community for the past half century [1]. As early as the 1960s programs were started to develop such capabilities, among the most ambitious being the National

Aerospace Plane (NASP) in the 1980s, which was ultimately canceled before any prototypes were built [2]. Today, there is a continued interest in hypersonic vehicles from National Aeronautical and Space Administration (NASA) and the De- partment of Defense [3, 4]. This is evident in successful vehicle programs such as: the X-43 Hyper-X scramjet demonstrator from NASA [2]; the X-51 WaveRider scramjet demonstrator from the Air Force Research Laboratory (AFRL) [5–7]; and the Hypersonic International Flight Research and Experimentation (HIFiRE) hypersonic testbed from the US Air Force (USAF) and Australian Defence Science and Technology Organisation (DSTO) [8, 9]. These programs are less ambitious

1 than the older NASP program, each seeking to demonstrated speciﬁc technologies or providing experimental ﬂight testbeds.

The AFRL is particularly interested enabling the development of a HCV that: operates in the Mach 5 - 7 range for prolonged periods of time; ﬂies 2000 nau- tical miles without refueling; operates with a takeoff weight in excess of 300,000 pounds; and is reusable [10–12]. In order to meet these goals, a vehicle would be designed to operate with a lifting body and an air-breathing scramjet engine in an integrated airframe-propulsion framework [2, 4, 13, 14]. Additionally, instead of ablation-based thermal protection systems used on previous vehicle concepts to reduce the heat load of the vehicle structure [2], a thin-gauge metallic structure capable of operating at elevated temperatures is desired in pursuit of minimizing weight and increasing reusability [12, 15]. As an example of such a vehicle, the

Falcon HTV-3X ”Blackswift” concept is illustrated in Fig. 1.1 [14, 16].

To enable the development of such HCVs, there is still an extensive set of technical challenges that must be addressed [1, 10, 11, 15–17]. These challenges partially arise from the extreme loading environment of hypersonic ﬂight and sur- vivability and reusability requirements of the HCV. Any vehicle operating in the hypersonic environment is subjected to prolonged intense heating and severe pressure loading, including high-magnitude and high-frequency turbulent boundary layer loads and engine noise excitation [10, 11, 15]. These high loadings, combined with thin metallic structures, are expected to produce compliance of the structure to the aerothermodynamics in a coupled manner.

2 Figure 1.1: Falcon HTV-3X Blackswift concept vehicle [14, 17].

3 Fluid Domain Solid Domain

Aerodynamic Temperature Heat Transfer Aerothermal Heating Heat Flux

Fluid Temperature Dynamics

Deformations Aeroelastic Aerodynamic Structural Pressure Mechanics Pressure

Figure 1.2: Schematic of Fluid-Thermal-Structural Interactions.

This coupling results in the fluid-thermal-structural interactions (FTSI) schemat- ically depicted in Fig. 1.2. The fluid domain and vehicle structure domain interact at the vehicle surface. Aerothermal coupling occurs between the fluid heat flux and structural temperatures, the fluid pressure loading and structural deformations are aeroelastically coupled. Both the fluid pressure and heat flux are intrinsically linked within the fluid dynamics. There is a direct dependency on the deformations to the thermal state through thermal stress and material property changes; while heat generation due to deformation is typically negligible. However, note that there is an indirect feedback coupling from the deformation to the thermal state through the modification in the heat flux. Hence, all interactions between the

ﬂow around the vehicle, its thermal state and deformations may be important.

Because of the high aerothermodynamic heating over the ﬂight trajectory, the thermal state of the vehicle evolves over signiﬁcant portions of the trajectory [12,

4 18]. The long-term thermal evolution, combined with the mutual interactions between the fluid loadings and thermo-structural vehicle state, results in a path- dependence of the structural response [19]. This path dependence requires that accurate simulation of the vehicle be done over the full trajectory instead of specific flight conditions as traditionally done. Due to conflicting similarity requirements, testing scaled prototypes to capture these multidisciplinary interactions is impractical [13, 20]. Hence, computational modeling to predict the structural state is critical to the design of HCVs.

A fundamental challenge in computational modeling of FTSI is the inherent differences in time scales between the fluid, thermal, and structural physics. Gen- erally, characteristic times of the fluid and the structure are orders of magnitude smaller than the thermal response [21–24]. The characteristic times of the fluid are the smallest, and numerical stability issues generally limit the maximum time step size of computational solvers [25]. The structural time scales are based on the nat- ural frequencies of the structural modes of interest, which can range from O(1) Hz to O(1000) Hz [10]. The thermal time scale is typically much larger than the fluid and structural scales due to characteristic time scales for typical properties, and can change for large durations of a trajectory (O(seconds) to O(minutes)) [12, 18].

This disparity in time scales can lead to conﬂicting requirements for time step siz- ing of the computational models to accurately capture the dynamics of the ﬂuid and structural responses, over the long-time records that encapsulate the slower thermal response [10, 11]. Time integration of these models must be done in an

5 efﬁcient manner to enable simulation of the FTSI over long time records.

1.2 Literature Review

This dissertation focuses on the time-integration problem for simulating FTSI over

long time records. The current state-of-the-art for this problem is established by

reviewing the following areas: 1) general frameworks for time-marching multi-

physics systems; 2) time integration of ﬂuid-structural interactions; 3) time in-

tegration of ﬂuid-thermal interactions; and 4) time integration of ﬂuid-thermal-

structural interactions.

1.2.1 Time Integration Frameworks for Multi-Physics Problems

A fundamental issue of multi-physics analysis is how to model and integrate the

different physics together. For FTSI, the problem can be thought of as having two

spatial domains and three physical fields [26]. The two domains consist of the fluid and solid (vehicle) domains. The three fields consist of the fluid, thermal, and structural dynamics.

The physical ﬁelds may be coupled together through a monolithic or partitioned approach. In the monolithic approach, the equations are solved simulta- neously as a single integrated solver using a consistent time integration algorithm

[27, 28]. For explicit time integrators the solutions are exchanged at each time step, and for implicit integrators the cross-coupling terms of the Jacobian matrix

6 are computed and exchanged at each time step. This consistency can provide exact conservation of the state variables as well as ensuring energy is maintained through the fluid-structure interface, which are ideal for ensuring convergence and stability [28, 29]. However, because of the tightly integrated nature of monolithic solvers, the approach is less flexible compared to partitioned methods[30]. The monolithic approach requires a dedicated solver for each set of multiphysics interactions; e.g. a fluid-structure interactions solver could not be directly used with a thermal solver for fluid-thermal-structural interactions without extensive modi-

ﬁcation. Furthermore, for large industry-level simulations monolithic solvers can become overly complex to maintain, and computationally overly-burdensome [31–

33].

In contrast, the partitioned approach is used to couple the physical fields together from separate solvers, each with potentially different time integration algorithms [27, 30]. The individual solvers are coupled through a careful exchange of boundary conditions at the interfaces of the domains, and an exchange of state variables for multifield interaction within a single domain (e.g. thermal-structural coupling). Here, only information at the interface conditions and of the time integration approach are required, leading to a non-intrusive means of coupling the different fields [27, 34–36]. Thus, the partitioned approach is attractive because it more easily allows for the integration of different solvers for each physical field, rendering the incorporation of new solver technologies relatively straight forward.

Because of these advantages, this approach is suitable for larger and more complex

7 problems [27, 31, 33], and is more favored over monolithic approaches [27]. How-

ever because each solver has a different time integration algorithm and associated

lags in boundary conditions at interfaces, ensuring convergence and stability is a

challenge [27].

In the partitioned approach, the time integrators for the individual solvers are

generally implicit or a mix of implicit and explicit algorithms [27]. For implicit time

integrators, at a given solution time t the solver requires the solution at time t + ∆t to march forward in time. Within a single solver, this is achieved by inverting the

Jacobian matrix of the system, but a Jacobian for the global system is not available in the partitioned approach. Because of this, each solver cannot directly determine the t + ∆t solution from the other solvers, thus this must be accounted for in the coupling scheme..

Two general classes of coupling procedures for the partitioned approach are strongly coupled schemes and loosely coupled schemes.1 Strongly coupled schemes make use of subiterations between the solvers at each time step to converge the coupled t + ∆t solution [27, 39]. By converging the solutions between solvers, the order of accuracy of the individual solvers is retained, but the use of subiterations make the analysis more computationally expensive. Like monolithic schemes, strongly coupled schemes can maintain a conservation of energy across the ﬂuid-

1There exist some variations of terminology in the literature. Strongly coupled as used here may also be referred to as tightly coupled [27]. Loosely coupled as used here may also be referred to as weakly coupled or staggered [37]. Some texts even refer to monolithic coupling as tight and partitioned coupling as loose [38]. For the purposes of this dissertation, monolithic/partitioned and strongly/loosely will be used exclusively to avoid confusion.

8 structure interface.

In contrast to strong coupling, loosely coupled schemes neglect the subitera-

tion loop between solvers, exchanging information between solvers just once per

time step. These schemes are less computationally expensive per time step com-

pared to strongly coupled schemes, however, they can reduce the accuracy and

stability of the global solution. This is partially due to the fact that loosely coupled

schemes are incapable of exactly conserving energy across the ﬂuid-structure inter-

face. [40, 41] Because of these restrictions, loosely coupled schemes must be care-

fully designed to maintain the order of accuracy of the individual solvers and max-

imize numerical stability. Even with these restrictions, loosely coupled schemes are

attractive since they offer potentially signiﬁcant computational savings and imple-

mentation simplicity compared to a strongly coupled scheme [42].

If the time scale of one of the physical ﬁelds is small enough compared to the

other physical ﬁelds, the transients may be negligible on the coupled response.

For high-speed ﬂow, this can be used to avoid time integration, and reduce com-

putational expense, of the fluid solver using the quasi-static or quasi-steady flow assumptions. Both of these assumptions use instantaneous boundary conditions from the other solvers to compute fluid loads with a time history of the flow. For quasi-static flow, the flow is assumed to be completely steady, neglecting motion of surfaces. For quasi-steady flow, the impact of the velocity of the surface motion is included. Note that for pressure, the quasi-steady assumption includes the effects of aerodynamic damping, while the quasi-static assumption neglects it.

9 1.2.2 Time Integration of Fluid-Structure Interactions

Loosely coupled partitioned time integration schemes are used extensively for

ﬂuid-structure interactions. Due to the vast body of work in this ﬁeld, only a representative subset is reviewed here.

Even before considering the coupling of fluid and structural solvers together, a significant challenge of fluid-structure interactions using CFD is the deformation of the fluid mesh. Traditional CFD solvers cannot be used for unsteady flow without modification because the mesh motion introduces additional dynamics that must be included, requiring an Arbitrary Lagrangian-Eulerian (ALE) or similar moving mesh framework [43]. However, time integration schemes suitable for marching the discretized Navier-Stokes equations from an Eulerian framework may not retain their accuracy and stability in an ALE framework with mesh motion

[44, 45]. A signiﬁcant source of this loss of accuracy is from non-conservation of the state variables in the presence of a moving mesh from the computation of time derivatives on changing mesh volumes. The Geometric Conservation Law (GCL),

first coined in [46], is a condition that is satisfied if a flow solver can exactly repro- duce uniform flow in the presence of arbitrary mesh motion. Violating the GCL has been found to reduce the accuracy and stability of the flow solver [47, 48]. The

GCL can be satisﬁed through a variety of ways including: computing the moving mesh metrics to satisfy a discrete form of the GCL [45, 48]; reformulating the

Navier-Stokes equations to avoid computing time derivatives on changing cells

10 [49]; or adding source terms to the discretized Navier-Stokes equations that auto-

matically satisfy the GCL [50, 51].

For loosely coupled schemes, the use of extrapolating predictors for the struc-

ture or aerodynamic loads have been found to improve the solution accuracy. The

surface displacement must be predicted at the next time step for the ﬂuid solver

if it is updated ﬁrst; conversely the aerodynamic loads must be predicted if the

structure is updated ﬁrst. Structural predictors will conserve forces across the

ﬂuid-structure boundary exactly; however the displacement and velocity of the

ﬂuid-structure boundary cannot be exactly matched [42]. The converse is true for

ﬂuid load predictors.

Edwards et al. [52] compared several coupling schemes coupling a linear modal

structure to a transonic nonlinear ﬂow solver, and the choice of a linear extrapola-

tion of the ﬂuid loads as a predictor were advocated in [53, 54]. Later studies [50]

added a corrector step to the formulation in [53] to update the structure with the

aerodynamic loads computed at the end of the time step. Aerodynamic predictors

have also been used more recently in [55] using linear and cubic pressure extrap-

olations with an implicit multi-step structural time integrator and an explicit ﬂuid

solver.

In a series of studies [40, 42, 56–59], several coupling schemes were developed

with structural predictors for ﬂuid-structure interaction. The generalized serial stag-

gered (GSS) scheme was speciﬁcally designed to maintain the second order temporal accuracy of the individual ﬂuid and structural solvers using a second order

11 extrapolation of the structural displacement as a prediction for the ﬂuid solver.

A modified version of the GSS, the Improved Serial Staggered (ISS) scheme offsets the structural solver by half a time step compared to the fluid solver, which was shown to have superior conservation of energy transfer with the time integrators used. The schemes were also formulated to satisfy the geometric conservation law with a specific fluid time integrator [42]. These schemes were contrasted with a conventional serial staggered (CSS) scheme that did not make use of such predictors and suffered in accuracy and stability compared to the GSS and ISS schemes. The schemes were also found to have better energy conservation across the interface than the CSS scheme [40, 42], which can be a significant factor in numerical stability.

Another type of loosely coupled scheme was developed in [60] based on a combined interface boundary condition (CBIC), in which an additional differential equation is solved along the fluid-structure interface to predict a higher-order correction to the interface velocity and fluid pressure. The CIBC scheme was compared favorably to a CSS scheme in both relative accuracy and stability using an implicit structural time integrator and explicit fluid time integrator for low speed

ﬂow.

One potentially useful feature of loosely coupled partitioned approaches for the class of ﬂuid-thermal-structural interactions considered in this work is the ability to use different time step sizes for the individual solvers. Denoted as subcycling, the disparity of time scales between solvers may be exploited, reducing computational

12 expense by calling the thermal or structural solver and exchanging information between solvers less frequently [61]. Loosely coupled schemes with subcycling have also been successfully applied to high-order DNS flow solvers modeling panel response in supersonic flow [62] using a fourth order explicit fluid time integrator and a second order implicit structural integrator; however the scheme used to couple the solvers resulted in first order time accuracy.

1.2.3 Time Integration of Fluid-Thermal Interactions

In fluid-thermal interactions, the thermal state of the structure is coupled to the aerothermodynamics through the heat transfer and temperature at the fluid-structure boundary. Unlike fluid-structure interactions, mesh motion is generally not required, and existing CFD codes can be used without significant modification.

Schemes for partitioned fluid-thermal analysis have been investigated by Birken et al. for loosely and strongly coupled time integration using an implicit first order scheme [63] and a second order implicit Runge-Kutta scheme [64]. In [63], a basic loosely coupled scheme was found to predict the cooling of a panel exposed to transonic flow with similar error to a strongly coupled scheme that was twice as expensive. In [64], the same analysis is performed using second order implicit schemes with loose and strong coupling with adaptive time stepping. The loosely coupled scheme reduced to first order while the strongly coupled scheme maintained second order accuracy, with computational expense favoring the strongly

13 coupled scheme with adaptive time stepping.

Combined interface boundary condition methods have also been used for ﬂuid- thermal analysis by Roe et al. in [65], and the accuracy and stability was compared favorably to the ﬂuid-thermal analogue of the CSS scheme. The scheme was shown to retain the stability boundary of the individual solvers. However, the individual solvers required low CFL numbers for stability, limiting the size of the global time step.

Predictors has been used successfully in fluid-thermal analysis of rocket nozzle heating in hypersonic flows in [66]. The coupling scheme was a based on a predictor and corrector of the thermal state, in which an extrapolation of the fluid loads was used to compute the thermal predictor before updating the fluid. Note this is similar to the predictor-corrector scheme used in [50] for fluid-structural coupling.

Because the fluid response time is orders of magnitude smaller than the thermal response time for high-speed vehicles, the quasi-static flow assumption may be used to neglect the time integration of the fluid solver [22, 24, 67]. With this assumption, the thermal problem is still treated as transient, but the instantaneous heat flux is computed from steady-state flow solutions. In [24] the laminar heating of a spherical dome in hypersonic flow was modeled using a transient thermal solver and a high-order flow solver marched to steady state. In [67], a similar analysis was performed, except the thermal solver was marched forward in time for several steps before the quasi-static flow solution was updated at a coarser step.

An adaptive time stepping scheme was used to determine the coarseness the ﬂow

14 solution stepping based on a tolerance criteria. In [22], a similar scheme without adaptive time stepping was used to analyze the heating of metallic panels in hypersonic flow. Instead of applying the quasi-static heat flux directly, a heat transfer coefficient was applied between the fluid solution updates to provide a linear change in heat flux in response to temperature changes. In general the coupling used in these analyses are formally first order accurate in time because the fluid solutions used for the next thermal time step are computed using the previous thermal step.

1.2.4 Time Integration of Fluid-Thermal-Structural Interactions

A number of studies have emerged on fluid-thermal-structural interactions (FTSI) of vehicles in high-speed flows, with varying levels of simplification to the problem. Many studies on aerothermoelasticity model the thermal field as a prescribed temperature rise decoupled from the fluid-structural interactions [68–72]. How- ever, the mutual coupling between the thermal and fluid-structural systems can have a significant influence on the temperature distribution and ultimately the structural state [19, 73, 74].

In order to capture the prolonged thermal evolution coupled with the fluid and structural response, the quasi-static assumption can be applied to both the fluid and structural solvers. With this assumption, the transients of the structural heat transfer is modeled, but the time histories of the structural and fluid solvers are

15 neglected. In [75, 76] a partitioned approach to couple finite element flow thermal and structural models was used in a quasi-static analysis for panels [75] and leading edges [76]. The thermal evolution was modeled in [75] using steady-state solutions from the flow, and updates to the static structural state at set intervals. In

[76] the ﬂow and thermal solvers were integrated together in a transient analyses with a static structural response; a similar method was also used in [77]. Due to the static structure assumption, any dynamic response or instabilities of the structure cannot be captured with these frameworks. Note that the procedures in these studies did not make use of predictors, and the global solution is generally only

ﬁrst order accurate in time.

Quasi-steady or quasi-static flow can also be modeled using simplified aerothermodynamic models or CFD-based reduced order models in conjunction with transient structural and thermal solvers. Simplified quasi-steady flow models were used in [73, 78] to couple fluid-thermal-structural interactions of the cylindrical bending of simply supported panels [78] and a stiffened composite panel from the

NASP program [73]. In [73], the long-term thermal evolution was modeled using static structural deformations, and the transient aeroelastic stability was tested at discrete points in time. In [78], the thermal and structural solvers were both time integrated using a loose coupling algorithm. Structural subcycling was also used to decrease the computational cost, with boundary conditions from the structural solver passed to the thermal solver in a time-averaged sense. A fourth order explicit time integrator was used for the structure, however the scheme is only for-

16 mally ﬁrst order accurate due to the coupling with the thermal solver. This loosely

coupled time integration framework was used in several other studies for CFD-

based surrogate models [79], reduced order thermal and structural models [80],

and stochastic analysis based on uncertainty in the turbulent heating [81]. A simi-

lar methodology was also used in [82] to model the aerothermoelastic response of

a composite panel. In general these coupling schemes are only ﬁrst order accurate.

Limited studies have also been performed on the coupling of unsteady ﬂuid,

thermal, and structural solvers. Lohner¨ et al. [83] discussed coupling CFD, computational thermal dynamics, and computational structural dynamics codes in a monolithic or partitioned manner. Tran and Farhat [84] considered the aerothermoelastic response of a flat panel by coupling a CFD solver with finite element thermal and structural solvers using a CSS-like scheme. This scheme was not predictor-based, and was formally first order accurate in time. However only small time records (0.1 seconds) were considered. In [85] a framework for full aerothermoelastic coupling in a monolithic solver using finite elements was developed. However, this restricted the fluid, thermal, and structural problems to be discretized with the same time integrator and marched at the same time step.

17 1.2.5 Summary of the State of the Art for Fluid-Thermal-Structural

Interactions

From the literature, it is evident that loosely coupled schemes offer promise in maintaining accuracy in fluid-structural analysis and fluid-thermal analysis. While such an approach is used almost exclusively for fluid-thermal-structural analysis, there has not been a rigorous evaluation for approaches used. The loose coupling techniques already in use for FTSI generally degrade the accuracy to first order because they do not make use of the accuracy-preserving techniques from fluid- structure and fluid-thermal coupling procedures. This loss of accuracy can also lead to a loss in stability, requiring smaller time steps, and increasing the computational expense of coupled simulations. Furthermore, studies of FTSI have almost exclusively used the quasi-steady and quasi-static flow assumptions, with only very limited studies with unsteady flow. While the quasi-steady/quasi-static assumptions are attractive to avoid the small fluid time scales for computational efficiency, the capability to perform simulations with time-accurate aerothermodynamics is important to test the limits of these assumptions in a coupled analysis.

1.3 Objectives of this Dissertation

The primary goal of this dissertation is the development and application of loosely coupled time integration schemes for ﬂuid-thermal-structural interactions in hy-

18 personic and high speed (M1) ﬂow. These schemes are to be designed to maintain the second order accuracy used in individual solvers through the use of extrapolating predictors, and exploit the disparities of time scales through the use of subcycling. Schemes for both time-accurate and quasi-steady ﬂow are considered.

The speciﬁc objectives are:

1. Develop second-order accurate loosely coupled time integration procedures

for ﬂuid-thermal-structural interactions using either time-accurate CFD or

quasi-steady ﬂow models.

2. Assess the accuracy and stability of the coupling procedures through numer-

ical studies.

3. Benchmark the various coupling schemes at different time step levels against

a strongly-coupled solution for panels operating in supersonic/hypersonic

ﬂows.

4. Compare the computational expense of the various time marching proce-

dures.

The remainder of this dissertation is arranged as follows: the governing equations and time discretization of the individual solvers are outlined in Chapter 2; the different time marching schemes developed and examined are detailed in Chap- ter 3; the computational conﬁgurations for the panels considered are described in

Chapter 4; order of accuracy comparisons are provided in Chapter 5; a study of

19 panel responses to quasi-steady ﬂow models and comparison of computational expense are discussed in Chapter 6; a study of panel responses to time-accuracy

CFD and comparison of computational expense are discussed in Chapter 7; and the principal conclusions and suggested future work are provided in Chapter 8.

1.4 Key Novel Contributions of this Dissertation

The principal contributions to the state-of-the-art made in this dissertation are summarized below:

1. Development of two second-order accurate loosely coupled schemes for ﬂuid-

thermal-structural interactions for quasi-steady or quasi-static aerothermo-

dynamics. These schemes are based on predictors of the ﬂuid loads with

structural subcycling.

2. Extensions of the two second-order accurate loosely coupled schemes to in-

clude time-accurate aerothermodynamics with both structural and ﬂuid sub-

cycling.

3. A 30 second transient analysis of the heating and deformation of a panel in

high-speed ﬂow undergoing thermally-induced ﬂutter using a time-accurate

CFD solver.

4. Assessment of the impact of the quasi-steady ﬂow assumption on a panel

in high-speed ﬂow compared to unsteady ﬂow, including both CFD-based

20 quasi-steady modeling strategies and lower ﬁdelity analytical aerothermodynamic models.

21 Chapter 2

Governing Equations and Time

Discretization

2.1 Governing Equations

Fluid-thermal-structural interactions consist of two spatial domains, as illustrated

in Fig. 2.1. Although only two spatial domains exist, the problem is divided into

the three physical ﬁelds [26]: ﬂuid dynamics; structural dynamics; and thermal

dynamics. The ﬂuid domain, denoted as ΩF , is made up of the physical ﬂuid

field. The structural domain, denoted as ΩS, is shared between the structural and thermal fields. The two domains interact at the fluid-structure interface, ΓFS.

The ﬂuid, structural, and thermal ﬁelds are all governed by the equations of continuity of mass, momentum, and energy. Without loss of generality, these

ﬁelds can be fundamentally represented as the same equations using an Arbitrary

22 ΓF-S Fluid-Structure Interface

ΩF Fluid Domain (Fluid field)

ΩS Structural Domain (Thermal and Structural Fields)

Figure 2.1: Fluid and Structural Domains.

Lagrangian-Eulerian (ALE) frame of reference [43]:

1 ∂ h i (Jρ) + ∇~ · ρ ~v − ~x˙ = 0 (2.1a) J ∂t X

1 ∂ h i (Jρ~v) + ∇~ · ρ~v ⊗ ~v − ~x˙ − σ = ~0 (2.1b) J ∂t X

1 ∂ h i (Je) + ∇~ · e ~v − ~x˙ − σ · ~v + ~q = 0 (2.1c) J ∂t X

where X is a reference domain, J is the metric Jacobian with respect to the reference

domain, ~x˙ is the motion of the reference domain relative to the spatial domain, ρ is the density, ~v is the velocity of the medium with respect to the spatial domain, e is the speciﬁc internal energy, σ is the stress tensor, and ~q is the heat ﬂux.

The compressible Navier-Stokes equations can be represented by Eqs. 2.1 di-

23 rectly. If mesh motion is used, the reference domain X is the initial undeformed

ﬂuid domain, or the initial undeformed mesh when discretized. If mesh motion is

not used, X represents the unchanging spatial domain, J = 1 and ~x˙ = ~0, reducing

Eqs. 2.1 to:

∂ρ + ∇~ · (ρ~v) = 0 (2.2a) ∂t ∂ (ρ~v) + ∇~ · (ρ~v ⊗ ~v − σ) = ~0 (2.2b) ∂t ∂e + ∇~ · (e~v − σ · ~v + ~q) = 0 (2.2c) ∂t

The structural ﬁeld in ΩS consists of the conservation of mass and momentum in

Eqs. 2.1(a-b). Generally the Lagrangian frame of reference is used for structural mechanics, except for very large deformations. Using the Lagrangian framework, the frame of reference X is the material itself, so ~v = ~x˙. The mass and momentum

equations reduce to:

∂(Jρ) = 0 (2.3a) ∂t

ρ~u¨ − ∇~ · σ = ~0 (2.3b)

where ~u is the deformation of the material from the initial conﬁguration. Note that the density ρ is usually assumed constant in time for most structural applications, reducing J to 1, and making Eq. (2.3)(a) trivial. Equation (2.3) represents the structural dynamics equations of motion.

24 The thermal field in ΩS consists of the conservation of energy in Eq. 2.1(c). The internal energy of a solid is defined in Eq. (2.4) [86], and the heat flux by Cauchy’s law in Eq. 2.5 [87]:

de ≡ cρdT (2.4)

~q ≡ −κ∇~ T (2.5) where c is the temperature-dependent speciﬁc heat, T is the temperature, and κ is the temperature dependent thermal conductivity. Using these deﬁnitions, the thermal heat transfer equation is reduced to:

ρcT˙ − ∇~ · σ · ~u˙ + κ∇~ T = 0 (2.6)

Note the second term with the stress tensor in Eq. (2.6) is the heat generated from internal strain rates. This term is generally negligible and can be neglected in aerospace applications [23], resulting in the familiar heat transfer equation:

ρcT˙ − ∇~ · κ∇~ T = 0 (2.7)

The ﬂuid domain ΩF and structural domain ΩS interact at the at the ﬂuid-

25 structure interface ΓFS:

σF · ~n = σS · ~n (2.8a)

~ ~qF · ~n = −(κ∇T )S · ~n (2.8b)

~xF = (~u + ~x)S (2.8c)

v˙F = ~u˙ S (2.8d)

TF = TT (2.8e)

where ~n is the unit normal of ΓFS in the direction of ΩF . The subscripts T , S, and F represent variables from the thermal, structural, and fluid fields, respectively. Equation (2.8a) specifies that the fluid stresses must be equal to the structural stresses at the fluid-structure boundary. This requirement is usually relaxed by considering only the pressure for the fluid stresses, since shear stresses generally do not have a significant impact on the structural response. Equation (2.8b) specifies that the heat flux between the structure and fluid must be equal at the boundary. Equations (2.8c-d) specify that the structural displacements and velocities must be equal to the boundary motion of the fluid. Equation (2.8e) specifies that the fluid and thermal field temperatures must be equal at the boundary.

In addition to the interactions at ΓFS, the thermal and structural ﬁelds interact

26 in ΩS via:

TT = TS (2.9a)

(σ · ~u˙)T = (σ · ~u˙)S (2.9b)

If the strain-induced heating is neglected and Eq. (2.7) is used, Eq. (2.9b) is not needed, and the thermo-structural coupling reduced to a matching of the internal temperature ﬁeld.

In practice, the coupling is applied through the use of prescribed Dirichlet boundary conditions and Neumann-type boundary loadings. The fluid stresses are applied to the structure as loading, satisfying Eq. (2.8a). The fluid heat flux is applied to the thermal field as a heat loading for Eq. (2.8b). The fluid mesh motion and fluid velocities at the surface are defined by the structural deformations in

Eqs. (2.8c-d). The fluid temperature at the boundary is set from the thermal field for Eq. (2.8e). Finally, the structural temperatures are treated as a field loading from the thermal field to satisfy Eq. (2.9b).

2.2 Fluid Time Discretization

The fluid may be modeled as unsteady flow, solving Eqs. 2.1 directly, or as quasi- steady flow, ignoring the history effects of the fluid.

27 2.2.1 Quasi-Steady Fluid Modeling

The quasi-steady assumption for ﬂuid modeling has been used with success in

ﬂuid-thermal-structural analyses of structures subject to high speed ﬂows[24, 73,

75, 76, 78, 88]. Common examples of this assumption in high speed ﬂows are sim- pliﬁed theoretical models (e.g., Piston Theory[89]), semi-empirical models (e.g.,

Eckert’s Reference Enthalpy [90]), or CFD-based models (e.g., CFD surrogates [79,

91]). Quasi-steady aerothermodynamic models are expected to play an extensive role in the design and life prediction of structures in high-speed flow due to the potential for significantly smaller computational expense compared to time- integrating an unsteady flow solver. Thus, this study explores these types of models in the analysis of the different time marching procedures.

With quasi-steady ﬂow, transient mesh motion is not considered, and the Eule- rian framework for the Navier-Stokes equations from Eq. 2.2 can be used instead of the more general ALE framework in Eq. (2.1). Furthermore, since the time- history is not modeled, the temporal derivatives are all assumed to be zero, reducing Eqs. (2.2) to:

∇~ · (ρ~v) = 0 (2.10a)

∇~ · (ρ~v ⊗ ~v − σ) = ~0 (2.10b)

∇~ · (e~v − σ · ~v + ~q) = 0 (2.10c)

28 These equations can then be discretized in space using CFD-based techniques, or approximated using simpliﬁed theoretical models. Because there are no time- history effects, any model based on the quasi-steady assumption can be represented as a function of the instantaneous boundary conditions. For FTSI applications, these models are functions of the instantaneous surface displacements, velocities, and temperatures, with outputs being the surface pressure and heat ﬂux:

p (xw, t) = p uw, ˙uw, Tw (2.11)

q (xw, t) = q uw, ˙uw, Tw (2.12)

where p is the pressure, q is the heat ﬂux, ~xw is the surface location, ~uw is the surface

˙ displacement, ~uw is the surface velocity, and Tw is the surface temperature. Note that although an ALE framework is not required, the instantaneous velocity of the surface motion still impacts the ﬂow with the quasi-steady assumption through

˙ the velocity boundary condition in Eq. (2.8d). If ~uw is neglected, the ﬂuid becomes quasi-static. In the context of ﬂuid-structure interactions, a quasi-static pressure does not account for aerodynamic damping.

2.2.2 Geometric Conservation Law

For unsteady ﬂow, enforcement of a Geometric Conservation Law (GCL) on the

ﬂuid discretization in the presence of mesh motion has been found to improve numerical stability [46, 47]. The GCL states that for a uniform ﬂow condition, the

29 fluid solver should exactly predict the flow to be uniform when arbitrary mesh motion is prescribed. If uniform flow is assumed, Eqs. (2.1) simplifies to:

1 ∂J − ∇~ · ~x˙ = 0 (2.13) J ∂t X

Equation (2.13) represents the GCL in differential form. In general, this condition is not automatically satisfied by the spatial discretization or time marching scheme, resulting in numerical errors even for uniform flow. These errors are from an in- consistency in the computed change in volume of a cell and the volume that the cell face sweeps through over the time interval, resulting in a loss of conservation of the cell volume. Note that if the cells in a mesh do not change volume (e.g. pure mesh translation or rotation), the GCL will automatically hold for flux-preserving schemes.

A simple way to satisfy the GCL is to add the terms from Eq. (2.13) to the conservation equations in Eqs. (2.1) as a source term [50, 51]:

1 ∂ h i 1 ∂J (Jρ) + ∇~ · ρ ~v − ~x˙ = ρ − ∇~ · ~x˙ (2.14a) J ∂t X J ∂t X 1 ∂ h i 1 ∂J (Jρ~v) + ∇~ · ρ~v ⊗ ~v − ~x˙ − σ = ρ − ∇~ · ~x˙ (2.14b) J ∂t X J ∂t X 1 ∂ h i 1 ∂J (Je) + ∇~ · e ~v − ~x˙ − σ · ~v + ~q = e − ∇~ · ~x˙ (2.14c) J ∂t X J ∂t X

Note that the GCL terms on the right hand side of Eqs. (2.14) vanish analytically, so these equations are analytically equivalent to the original equations in

30 Eq. (2.1). The ﬂuid solver in this work is assumed to use this form of the conservation equations to satisfy the GCL.

2.2.3 Time-Accurate Fluid Modeling

For time-accurate ﬂow, the nonlinear set of partial differential equations in Eqs. (2.1) in spatially-discretized form may be represented as [50]:

1 ∂Q = R(Q) (2.15) J ∂t

Q ≡ [ρ, ρ~v, e]T (2.16)

∂ [ξ, η, ζ] J ≡ (2.17) ∂ [x, y, z]

∂F ∂G ∂H R(Q) ≡ − + + + QRGCL (2.18) ∂ξ ∂η ∂ζ

! ! ∂ 1 ∂ ξ˙ ∂ η˙ ∂ ζ˙ RGCL ≡ + + + (2.19) ∂t J ∂ξ J ∂η J ∂ζ J where F, G, and H are the discretized ﬂuxes in the ξ, η, and ζ directions, respectively. Note that ξ, η, and ζ represent the coordinates in the reference domain, and x, y, and z are the spatial coordinates of the current mesh conﬁguration. The Ja-

31 cobian J is the ratio of the reference cell volume to the current cell volume. The

added source term GGCL is used to satisfy the GCL, and vanishes for decreasing

time step.

Equation (2.15) is marched in time using the second order backwards Euler

scheme. The second order backward Euler, also known as the three-point back-

ward difference scheme, is a standard implicit time integration scheme for ﬂow

solvers that is unconditionally stable for linear systems [92]. The temporal deriva-

tive of the conserved variables is discretized as:

3Qk+1 − 4Qk + Qk−1 Q˙ k+1 ≈ + O ∆t2 (2.20) 2∆t

where ∆t is the time step and k is the ﬂuid time step counter. As indicated, this is

a second order approximation of the derivative. Discretizing Eq. (2.15) at the k + 1

time step yields:

3Qk+1 − 4Qk + Qk−1 = Rk+1 + O ∆t2 (2.21) 2J∆t

Solving for Qk+1 yields:

2J∆tRk+1 + 4Qk − Qk−1 Qk+1 = + O ∆t3 (2.22) 3

Note that Eq. (2.22) is third order accurate between two successive time steps.

This is known as the local time accuracy. The order of accuracy from the beginning

32 of the solution is known as the global accuracy; the global accuracy is generally one order less than the local accuracy. To show this, consider the response for a

0 speciﬁed time period t0 and initial condition Q . The solution starting from the initial condition can be expressed as the sum of the incremental changes between time steps:

t0/∆t X k+1 k 3 t0/∆t 2 Q − Q + O ∆t = Q + t0O ∆t (2.23) k=1 where t0/∆t is the number of time steps required to reach t0. Because of the number of time steps scales with the inverse of ∆t, the third order local errors in O (∆t3) are multiplied by 1/∆t, changing the global errors to O (∆t2) and reducing the accuracy to second order.

Any external loadings that affect the ﬂux terms in Eq. (2.18) must be at least second order accurate in time in order for the O (∆t2) term in Eq. (2.21) to remain true. Finally, note that Eq. (2.21) has nonlinearities in the ﬂux terms, so an iterative scheme must be used to maintain the accuracy.

2.3 Structural Time Discretization

The structural governing equation in Eq. (2.3)(b) is spatially discretized in the form:

[M] ¨u + [C (u, T)] ˙u + [K (u, T)] u = FT (T) + Fa (2.24)

33 where [M] is a mass matrix, [C] is the damping matrix, [K] is the stiffness matrix,

Fa is the mechanical load vector, FT is the thermal load vector, u is the structural deformation vector, and T is the temperature vector. Note that [M] is constant in

Lagrangian frames, based on Eq. (2.3)(b), while both [C] and [K] may be dependent on displacement and temperature. The stiffness matrix may be nonlinear due to either geometric or material nonlinearities.

Two time integration methods are used for the coupling schemes considered: the implicit Newmark-β method, and the explicit 4th order Runge Kutta method.

The Newmark-β is a standard implicit time integration scheme for structural dynamics that can maintain second order accuracy and remain unconditionally stable

[93]. The 4th order Runge-Kutta method is a popular explicit general-purpose time integrator [94].

2.3.1 Newmark-β Time Integration

The Newmark-β (N-β) method is a popular family of implicit time integrators that may be ﬁrst, second, or fourth1 order accurate.[93, 95] The displacement and velocity are discretized according to:

1 um+1 ≈ um + ∆t ˙um + ∆t2 γ¨um+1 + (1 − γ) ˙um (2.25a) 2

˙um+1 ≈ ˙um + ∆t 2β¨um+1 + (1 − 2β)¨um (2.25b)

1The fourth order accurate version (Fox-Goodwin) is conditionally stable. The fourth order accuracy only occurs if no damping is present in the system[93].

34 where γ and β are numerical parameters, and m is the structural time step counter.

By setting γ = 1/2 and β = 1/4 the equations reduce to a constant, averaged acceler-

ation across the time step. This classic scheme is unconditionally stable for linear

systems [95]. Equations (2.25) simplify to:

1 üm+1 + üm um+1 ≈ um + ∆t ˙um + ∆t2 + O ∆t3 (2.26a) 2 2 üm+1 + üm ˙um+1 ≈ ˙um + ∆t + O ∆t3 (2.26b) 2

Both the velocity and displacement are locally third order accurate, and globally second order accurate. Solving for the velocity and acceleration at the new time step in terms of the displacement at the new time step results in:

2 ˙um+1 = um+1 − um − ˙um (2.27a) ∆t 4 4 ¨um+1 = um+1 − um − ˙um − ¨um (2.27b) ∆t2 ∆t

Substituting Eqs. (2.27) into Eq. (2.24) results in:

m+1 h i m+1 m+1 2 Ke u = Fes + O ∆t (2.28)

35 where

m+1 h i 4 2 m+1 m+1 Ke ≡ [M] + [C] + [K] (2.29a) ∆t2 ∆t

m+1 Tm+1 am+1 m m Fes ≡ F + F + [M] ¨ue + [C] e˙u (2.29b)

m 4 4 ¨ue ≡ ¨um + ˙um + um (2.29c) ∆t ∆t2 m 2 e˙u ≡ ˙um + um (2.29d) ∆t

Because the of the nonlinearity of the damping and stiffness matrices, Eq. (2.28) must be computed iteratively at each time step to maintain accuracy. Any thermal or applied mechanical loadings must be computed using second order accurate method to maintain the accuracy as well.

2.3.2 Fourth Order Runge Kutta Time Integration

The 4th order Runge-Kutta (RK4) time integrator is an explicit, 4 stage procedure.

Each stage of the scheme is based on the instantaneous acceleration of the structure, which can be computed by inverting Eq. (2.24):

¯a(t, u, ˙u) ≡ ¨u(t) = [M]−1 FT(T(t)) + Fa(t) − [C (u, F(t))] ˙u − [K(u, T(t))] u (2.30)

Each stage is computed using Eqs. (2.31) for the velocities, and Eqs. (2.32) for

36 the accelerations:

1 m kv = ˙u (2.31a)

2 m 1 1 kv = ˙u + /2∆tka (2.31b)

3 m 1 2 kv = ˙u + /2∆tka (2.31c)

4 m 3 kv = ˙u + ∆tka (2.31d)

1 n n n ka = a(t , u , ˙u ) (2.32a)

2 n 1 m 1 1 m 1 1 ka = a(t + /2∆t, u + /2∆tkv, ˙u + /2∆tka) (2.32b)

3 n 1 m 1 2 m 1 2 ka = a(t + /2∆t, u + /2∆tkv, ˙u + /2∆tka) (2.32c)

4 n m 3 m 3 ka = a(t + ∆t, u + ∆tkv, ˙u + ∆tka) (2.32d)

The displacement and velocity at the new time step is computed using a weighted

average of each stage:

∆t um+1 = um + k1 + 2k2 + 2k3 + k4 + O ∆t5 (2.33a) 6 v v v v ∆t ˙um+1 = ˙um + k1 + 2k2 + 2k3 + k4 + O ∆t5 (2.33b) 6 a a a a

The local accuracy of Eqs. (2.33) is of order 5, however the global accuracy is of order 4. The thermal and mechanical loading used in Eq. (2.30) must be 4th order

37 accurate to maintain the global accuracy.

2.4 Thermal Discretization

The heat conduction equation in Eq. (2.7) can be represented in spatially discretized

form as:

[CT(T)] T˙ + [KT(T)] T = QT (2.34)

where T, [CT], [KT], and QT are the temperature vector, thermal capacitance matrix, thermal conductivity matrix, and heat load vector, respectively. Note that the thermal capacitance matrices and thermal conductivity matrices may be temperature- dependent, resulting in a nonlinear system of equations.

For the coupling procedures presented in Chapter 3, two time integration methods are considered: the implicit Crank-Nicolson method, and the explicit forward

Euler method. The Crank-Nicolson scheme is a popular second order scheme that is unconditionally stable [93].

2.4.1 Crank-Nicolson Time Integration

The Crank-Nicolson (C-N) is a trapezoidal rule based time integration scheme designed to be second order accurate and unconditionally stable for linear systems.

The temperature derivative is discretized as:

38 Tn+1 − Tn T˙ n+1/2 ≈ + O ∆t2 (2.35) ∆t

where n is the thermal time step counter. Discretizing Eq. (2.34) about the n + 1/2 time step results in:

n+1 n n+1 n n+1/2 T − T n+1/2 T + T n+1/2 2 [CT] + [KT] = QT + O ∆tT (2.36) ∆tT 2 where

Tn+1 + Tn ≈ Tn+1/2 + O ∆t2 (2.37) 2

Solving for Tn+1:

−1 n+1/2 1 n+1 h i n+ /2 3 T = CfT RgT + O ∆t (2.38)

where

n+1/2 h i n+1/2 ∆t n+1/2 CfT ≡ [CT] + [KT] (2.39a) 2

n+1/2 n+1/2 ∆t n+1/2 n n+1/2 RgT ≡ [CT] − [KT] T + ∆tQT (2.39b) 2

The new temperature Tn+1 is locally third order accurate. Again, similar to

Eq. (2.23), the global accuracy is formally second order. The thermal heat load vec-

39 1 tor QT must be computed at the n+ /2 with at least a second order approximation.

Similarly, if [CT] and [KT] are temperature-dependent, they must be computed at

the n + 1/2 time step, using a second order approximation or through subiterations.

2.4.2 Forward Euler Time Integration

The forward Euler (FE) method is a simple ﬁrst order explicit scheme. The temper-

ature derivative is discretized as:

Tn+1 − Tn T˙ n ≈ + O (∆t) (2.40) ∆t

Equation (2.34) is discretized at the n time step:

n+1 n n T − T n n n [CT] + [KT] T = QT + O (∆t) (2.41) ∆t

Solving for Tn+1 yields:

n+1 n n −1 n n n 2 T = T + ∆t(([CT] ) (− [KT] T + QT ) + O ∆t (2.42)

Again, Tn+1 is locally second order accurate, but the global accuracy is only ﬁrst order. Because of the explicit nature of the scheme, all quantities on the right-hand side of Eq. (2.42) are computed at the n time step, so approximations for the n + 1 step are not required. However, the scheme is only conditionally stable, requiring small time steps.

40 Chapter 3

Coupling Procedures

Two loosely coupled time-integration procedures are developed for ﬂuid-thermal- structural interactions. One is designed for quasi-steady aerothermodynamics, so that only the thermal and structural solvers are marched in time. The second is intended for use with unsteady aerothermodynamics, and includes time marching of the thermal, structural, and ﬂuid solvers. In addition to these, several other schemes are described that are used for comparisons in the later chapters.

Regardless of the time integration procedure used, interpolation of the discrete quantities must be performed if the meshes of the various solvers are not coinci-

41 dent:

F uF = HS (uS) (3.1a)

F ˙uF = HS ( ˙uS) (3.1b)

F TF = HT (TT ) (3.1c)

T qT = HF (qF ) (3.1d)

S pS = HF (pF ) (3.1e)

S TS = HT (TT ) (3.1f)

B where HA is an interpolation function from mesh A to mesh B, corresponding to the ﬂuid (F ), structural (S), or thermal (T ) meshes. Equations (3.1a-e) are applied to the ﬂuid-structure boundary, and Eq. (3.1f) is applied throughout the structural domain. Note that the deformation velocity ˙u has the same interpolation function

as the deformation u. Alternatively, the velocity on the ﬂuid solver ˙uF can be

computed from a ﬁnite difference of the deformations as seen by the ﬂuid solver

uF , consistent with the mesh deformation scheme.

3.1 Quasi-Steady Aerothermodynamics

A generalized framework for partitioned coupling of FTSI, using quasi-steady

ﬂuid models, is shown in Fig. 3.1. The sequence shows the framework over a

single thermal time step, with increasing time represented by the horizontal axis.

42 The thermal time integration is represented by the red line, and the structural time marching by the black line. Separate pressure and heat-ﬂux based quasi-steady models are represented by the blue boxes. This framework includes the possibility of multiple structural time steps over this period for structural subcycling. Fig- ure 3.1 depicts two structural steps over the thermal step, however the framework is generalizable for any integer number of structural steps per thermal step.

Before the ﬁrst step, the solvers are assumed to be synchronized at the same

m n 1 point in time, i.e. t = tS = tT . The sequence for a single thermal time step is:

1. Displacement, velocity, and surface temperature are passed from the thermal

solver to the quasi-steady heat ﬂux model.

2. The instantaneous heat ﬂux is computed using Eq. (2.12) and passed to the

thermal solver.

3. The thermal solution is updated to time t + ∆tT .

4. The updated thermal solution is passed to the structural solver.

5. Displacement, velocity, and temperature are passed from the structural solver

to the quasi-steady heat ﬂux solver.

6. Pressure is computed from Eq. (2.11) and passed to the structural solver.

7. The structural solution is updated to time t + ∆tS.

1Consistent with Chapter 2, n is used for the thermal time step counter and m is used for the structure.

43 Quasi-Steady Heat Flux

1 2 Thermal 3

4 9

Structure 7 8 5 6

Quasi-Steady Pressure

t t+ΔtS t+ΔtT Time

Figure 3.1: Sequence of coupling scheme.

8. Steps 5-7 are repeated to time t + ∆tT .

9. The updated displacements and velocities are passed to the thermal solver

for the next cycle.

Note that the pressure, heat flux, displacements, and temperatures may be modified before being passed between the solvers, depending on which specific coupling scheme is used.

3.1.1 Predictor Implicit Scheme

The coupling scheme developed here, termed the Predictor Implicit (PI) scheme, is

a loosely coupled scheme based on implicit time integrators, predictions for the

44 pressure and heat flux, and subcycling of the structural solver. The scheme makes use of predictors of the fluid loads and temporal interpolations of the temperature to maintain second order accuracy. The Newmark-β and Crank Nicolson schemes from Ch. 2 are considered for the structural and thermal solvers, respectively. Note that this scheme does not strictly require these specific time integrators; it can be used for any implicit time integrators with minimal modification to the framework.

The algorithm for the scheme is represented as a ﬂowchart in Fig. 3.2. There are four discrete components: the thermal solver (in red); the structural solver

(in blue); separate heat flux and pressure quasi-steady solvers (both in blue); and the interface between each solver (green). Communication between the various solvers is achieved through the interface, where the discrete values are interpolated to and from each solver using the mappings defined in Eq. (3.1). Note that the subscript T is used for quantities defined in the thermal solver, S for those defined in the structural solver, and F for values defined in either fluid model. Fi- nally, recall that n is used for data defined at the thermal time step, and m for those at the structural time step.

Starting at a new thermal time step n, the structural deformations and velocities, as well as the surface temperatures, are mapped to the quasi-steady heat flux solver. The instantaneous heat flux is computed, and mapped back to the thermal solver. Since the Crank-Nicolson time integrator is used, the heat flux is expected at the time step n + 1/2 as specified by Eq. (2.36). Simply using the heat flux at

45 Thermal Solver Heat Flux Model

(CN)

(Nβ)

Structural Solver Interface Pressure Model

Figure 3.2: Quasi-Steady Predictor Implicit (PI) coupling ﬂowchart.

46 n the current time step (qT ) would result in the introduction of first order truncation error into Eq. (2.36), reducing the global accuracy to first order. Instead, a simple linear prediction of the heat flux is be used to maintain the order of accuracy:

1 1 qn+1/2 ≈ qn+ /2 = qn + qn − qn−1 + O ∆t2 (3.2) P 2 T

1 n+ /2 1 where qP is the heat ﬂux predicted at the n + /2 step. Because the truncation error from Eq. (3.2) is equivalent to that in (2.36), the order of accuracy is not reduced. The temperature at the new time step n + 1 can now be computed using

(2.38).

With the thermal solution updated, the new temperature is mapped to the structural mesh, and control is passed to the structural solver. The instantaneous pressure is computed by mapping the surface displacement, velocity and temperature, to the quasi-steady pressure model. The pressure is then mapped back to the structural solver. From the Newmark-β time integrator, both temperature and pressure are expected at the m + 1 structural time step, as shown in Eqs. (2.28)-

(2.29). Using the same argument as the heat ﬂux, the pressure at m + 1 is predicted using:

m+1 m+1 m m m−1 2 p ≈ pP = p + p − p + O ∆tS (3.3)

m+1 where pP is the pressure predicted at the m + 1 time step. Again, the second order truncation error does not decrease the global accuracy.

47 The temperature must also be properly applied to maintain accuracy. For the

case of no subcycling (∆tS = ∆tT ), the temperature at the m + 1 step is trivially deﬁned by Tm+1 = Tn+1. However, with subcycling the temperature must be

interpolated over the thermal time step. This is done using a simple linear inter-

polation:

n+1 n m+i m+i n T − T 2 T ≈ TI = T + (i∆tS) + O ∆tS (3.4) ∆tT

m+i n where i is a counter between thermal time steps such that tS = tT + i∆tS, and

m n tS = tT . The maximum value of i is the ratio of time steps, ∆tT /∆tS. The linear interpolation represents a second order approximation, maintaining the solution accuracy.

After the structure is subcycled up to the thermal solution time, control is then passed back to the thermal solver for the next thermal time step. Before this, however, the displacements and velocities of the structure used for computing the heat

ﬂux are averaged over the previous thermal time step. Note that the averaging is not required to maintain the order of accuracy of the solution, instead, it is used to

ﬁlter out higher frequency oscillations of the structure over the thermal time step.

This averaging must be second order accurate or the heat flux will contain first order truncation errors. A simple arithmetic mean between n − 1 and n is second order accurate only at the n − 1/2 time step, and first order at the n step where it is required. Instead, a least-squares linear fit of the displacement and velocity used:

48 n+1 n+1 2 u ≈ ˜u = au + ∆tT bu + O ∆tS (3.5a)

n+1 ˜n+1 2 ˙u ≈ ˙u = au˙ + ∆tT bu˙ + O ∆tS (3.5b)

where a is the least-square fit at the n step, and b is the least-square fit of the rate of change between the n and n + 1 steps. Note that the least-squares fit reduces to using the instantaneous quantities if subcycling is not used. Details of the least squares fit and the definitions of a and b are given in Appendix A.

3.1.2 Predictor-Corrector Implicit Scheme

The Predictor-Corrector Implicit (PCI) scheme is a modification of the PI scheme that adds a correction step to the structural and thermal solvers. It follows the same algorithm as the PI scheme in Fig. 3.2, except for the addition of a correction step at the end of the thermal and structural time steps. Immediately prior to updating the structural time step to m = m+1, the updated fluid pressure pm+1 is applied to the structure, and the structural solution um+1 is recomputed as a correction to the pressure predictor. Similarly for the thermal solver, the updated heat flux qn+1 is used to recompute the temperature Tn+1 as a correction to the heat flux predictor.

Note that the PCI scheme is more expensive than the PI scheme per time step due to the additional updates to the thermal and structural solver. The cost associated with the quasi-steady pressure and heat ﬂux models is the same though, since the

49 ﬂuid loads are still computed just once per time step.

For structural subcycling, the PCI scheme uses the same linear interpolation of the temperature defined in Eq. 3.4. Note that the thermal correction is applied at the end of the structural subcycling steps, after the heat flux is updated with the averaged displacement and velocity as defined in Eq. 3.5.

3.1.3 Additional Quasi-Steady Coupling Schemes

In addition to the PI and PCI schemes developed here, three other quasi-steady coupling schemes are considered for comparison. One is a basic coupling scheme that does not include predictors or interpolations, and another is a conventional scheme using explicit time integrators. Finally, a strongly coupled scheme is used as a benchmark.

3.1.3.1 Basic Implicit Scheme

The Basic Implicit (BI) scheme represents the simplest method of loose coupling with implicit time-integrators, in which data is passed between solvers when the solutions are updated and used directly. The implicit time integrators Newmark-β and Crank-Nicolson are used for the structural and thermal solvers, respectively.

The scheme also allows for structural subcycling.

The BI scheme follows the same basic strategy as the PI scheme as shown in

Fig. 3.2, however there are several key differences. Instead of using predictors and interpolations to maintain second order accuracy, the data exchanged between

50 solvers is used directly. For the heat ﬂux, the Crank-Nicolson time integrator in

Eq. (2.36) requires the heat load at the n + 1/2 step, however, the current heat ﬂux

at step n is passed instead:

n+1/2 n q ≈ q + O (∆tT ) (3.6)

Similarly, for the Newmark-β scheme the pressure is deﬁned at the m + 1 step

in Eq. (2.29), and the pressure at step m is used:

m+1 m p ≈ p + O (∆tS) (3.7)

Both of these approximations are only ﬁrst order accurate since the ﬁrst order

truncation error propagates through Eq. (2.36) and (2.29), increasing the global

truncation error from O (∆t2) to O (∆t).

The temperature is also held constant over the thermal time step for the structural solver. If subcycling is not used, the m time step from the thermal solver and n from the structural solver are equivalent, so Tm+1 = Tn+1 is automatically achieved. If subcycling is used, the structural temperature is simply set to the most recent thermal solution:

m+i n+1 T ≈ T + O (∆tS) (3.8) where i is a counter between thermal time steps. This approximation also intro-

51 duces ﬁrst order truncation errors into the structural solution.

The BI scheme does not make the same averaging of the displacement and velocity for the heat ﬂux as the PI scheme in Eqs. (3.5). Instead the instantaneous solutions un and ˙un are used. This does not degrade the order of accuracy of the system, but higher frequency oscillations over the thermal time step are not ﬁltered out.

Finally, note that while the first order errors are introduced via the fluid-structural coupling, fluid-thermal coupling, and thermal-structural coupling (for ∆tT 6= ∆tS), the introduction of first order errors at any point will render the entire global solution first order accurate. For example, if the pressure estimate is first order, but the heat flux and temperature estimates are both second order, the structural truncation error will be first order. This error propagates to the heat flux solver since u and ˙u are inputs. The first order errors in the heat flux then affect the thermal solution, and the entire system reduces to first order. Therefore, it is vitally important to maintain accuracy at every coupling interface to maintain the global solution accuracy.

3.1.3.2 Conventional Explicit Scheme

The conventional explicit (CE) scheme has been used in several previous studies of

fluid-thermal-structural interactions [78, 88, 96]. This scheme is based on explicit time integrators, specifically the first order forward Euler method for the thermal solver, and the 4th order Runge-Kutta method for the structural solver, each de-

52 tailed in Chapter 2. Like the PI and BI schemes, the CE scheme also allows for

structural subcycling. The ﬂowchart of the CE algorithm is detailed in Fig. 3.3.

The most signiﬁcant difference the CE scheme has from the PI and BI schemes

is the explicit nature of the time integrators. The heat ﬂux from the Forward Euler

integrator in Eq. (2.41) is expected at the n time step, which is readily available.

However, the Forward Euler scheme is only ﬁrst order accurate in time, so ﬁrst order truncation errors are present in the temperature.

Due to the four stages of the 4th order Runge-Kutta integrator, deﬁned in Eqs. (2.31)-

(2.33), the structural portion of the algorithm is signiﬁcantly different in Fig. 3.3 from the PI and BI algorithms. The quasi-steady pressure model is invoked four times at each structural time step. Using i as the Runge-Kutta stage number, the

i displacement ˆui and velocity ˆ˙u at each stage are computed using:

1 ˆu1 = um ˆ˙u = ˙um (3.9a)

2 2 m 1 1 ˆ m 1 1 ˆu = u + /2∆tSkv ˙u = ˙u + /2∆tSka (3.9b)

3 3 m 1 2 ˆ m 1 2 ˆu = u + /2∆tSkv ˙u = ˙u + /2∆tSka (3.9c)

4 m 4 ˆ4 m 4 ˆu = u + ∆tSkv ˙u = ˙u + ∆tSka (3.9d)

i i where the coefﬁcients kv and ka are deﬁned in Eqs. (2.31) and (2.32), respectively.

Like the BI scheme, the temperature is held constant over the thermal time step as deﬁned in Eq. (3.8). This affects not only the structural loads, but the pressure computations at each stage. Even without structural subcycling, the ﬁrst order

53 Thermal Solver Heat Flux Model

(FE)

(RK4)

Structural Solver Interface Pressure Model

Figure 3.3: Quasi-Steady Conventional Explicit (CE) coupling ﬂowchart.

54 errors in the temperature solution degrade the 4th order Runge Kutta structural solution to formal ﬁrst order accuracy.

Finally, similar to the PI scheme, the CE scheme takes an average of the displacement and velocity for the heat flux computation. Instead of a fit of a first order polynomial, a simple arithmetic mean is used:

N n n+1 1 X m+j u + 1 ≈ ¯u = u + O (∆tT ) (3.10a) N j=0 N n+1 ¯n+1 1 X m+j ˙u ≈ ˙u = ˙u + O (∆tT ) (3.10b) N j=0 where j is a counter between thermal time steps, and N is the ratio of time steps.

This is a ﬁrst order approximation, though the coupling scheme is ﬁrst order any- way.

3.1.3.3 Strong Implicit Scheme

Lastly, a strongly-coupled implicit (SI) scheme is considered as a benchmark solution. The ﬂowchart of the algorithm is illustrated in Fig. 3.4. This scheme is based on the Crank-Nicolson and Newmark-β time integrators for the thermal and structural solvers, similar to the BI and PI schemes. Unlike the previous schemes, the thermal and structural solvers are subiterated between time steps to converge the heat ﬂux and pressure. Note that subcycling is not considered here because the strong coupling subiterations occur within a single time step, and require both the thermal and structural solvers to be updated as part of the subiteration process. A

55 Thermal Solver Heat Flux Model

(CN)

(Nβ)

Structural Solver Interface Pressure Model

Figure 3.4: Quasi-Steady Strong Implicit (SI) coupling ﬂowchart. single time step counter n is used in Fig. 3.4.

Because of the subiteration loop, predictors for the heat ﬂux and pressure are not needed. The solution values at the n + 1 time step from the previous subiteration are used to compute the loads. The second order truncation errors are maintained as the solution converges between subiterations. However, because the thermal solver expects the heat ﬂux at step n + 1/2 the heat load is averaged between time steps:

56 qn+1 + qn qn+1/2 ≈ + O ∆t2 (3.11) 2 T

This averaging preserves the second order accuracy. Convergence criteria are set based on the relative differences in heat ﬂux and pressure between subiterations relative to the change between time steps:

n+1 p − pn+1 = s+1 s p n+1 n (3.12a) ps+1 − p n+1 q − qn+1 = s+1 s q n+1 n (3.12b) qs+1 − q

where p is the error norm of the pressure, q is the error norm of the heat ﬂux, and s

is the subiteration index. This error represents the difference between subiterations

compared to the change over a time step. The solution is considered sufﬁciently

converged when:

max (p, q) = < tol (3.13)

−3 where tol is the convergence tolerance, set to 10 .

3.2 Time-Accurate Aerothermodynamics

A generalized framework for partitioned coupling of FTSI using time-accurate

aerothermodynamics is shown in Fig. 3.5. Similar to Fig. 3.1 for quasi-steady ﬂuid

57 To Fluid 1 3 Thermal 2

Structure 5

4 6 9

Fluid 7 8 1 3 To Thermal

t t+ΔtF t+ΔtS t+ΔtT Time

Figure 3.5: Generalized sequence for FTSI of time-accurate aerothermodynamics.

models, the demonstrative sequence of the framework is for a single thermal time

step, with time as the horizontal axis. The thermal, structural, and ﬂuid time inte-

grations are represented by the red, gray, and blue lines, respectively. Like the

quasi-steady framework, this includes the possibility of subcycling the solvers.

However, there are multiple levels of subcycling now, typically with ∆tF < ∆tS <

∆tT . Figure 3.5 depicts two structural steps and four ﬂuid steps over the thermal time step, though the framework is general for any integer ratio of time steps.

Before step 1 in Fig. 3.5, the solvers are assumed to be at the same point in time: t = tm = tn = tk. Recall that n, m, and k are used for the thermal, structural, and

ﬂuid solver time step counters, respectively. The sequence for one thermal time

step is:

58 1. Heat ﬂux is passed from the ﬂuid solver to the thermal solver.

2. The thermal solution is updated to time t + ∆tT .

3. The updated temperature is passed to both the ﬂuid and structural solvers.

4. Pressure is passed from the ﬂuid solver to the structural solver.

5. The structural solution is updated to time t + ∆tS.

6. Structural displacements and velocities are passed from the structural solver

to the thermal solver.

7. The ﬂuid solution is updated to time t = t + ∆tF .

8. Step 7 is repeated to time t = t + ∆tS (ﬂuid subcycling).

9. Steps 4-8 are repeated to time t = t + ∆tT (ﬂuid-structural subcycling).

3.2.1 Predictor Implicit Scheme

The Predictor Implicit (PI) scheme using quasi-steady aerothermodynamic models is generalized here for time-accurate aerothermodynamics. In addition to the

Crank-Nicolson and Newmark-β time integrators for the thermal and structural solvers, the 2nd order Backwards Euler integrator discussed in 2.2.3 is used for the

ﬂuid solver. Again, note that the PI scheme is not fundamentally tied to these time integrators; the procedure is general enough to be used with most implicit time integrators with only minimal modiﬁcation.

59 The algorithm for the scheme is illustrated as a ﬂowchart in Fig. 3.6. Unlike the

quasi-steady version, the ﬂuid is represented here as a single uniﬁed domain as

opposed to two separate pressure and heat ﬂux solvers. Starting at a new thermal

time step n = n+, the thermal solver requests the heat ﬂux from the ﬂuid solver at

the n+ 1/2 time step as dictated by Eq. (2.36). The heat ﬂux at the n+ 1/2 is predicted

using a linear extrapolation. Because the ﬂuid may have a smaller time step than

the thermal solver, the prediction is performed in a least squares sense:

n+1/2 n+1/2 3 2 q ≈ q = aq + ∆tT bq + O ∆t (3.14) P 2 T

n−1 where aq is the least-squares ﬁt of q , and bq is the change in heat ﬂux over the

2 time step n − 1 to n. Because this estimate produces truncation errors of O (∆tT ), the order of accuracy of the Crank-Nicolson integrator is preserved. Note that

Eq. 3.14 reduces to Eq. 3.2 if subcycling is not used. The heat flux prediction is computed within the fluid solver before being passed to the thermal solver, in contrast to the quasi-steady version. This is done to minimize the transfer of data between solvers; if the thermal solver performed the prediction, the heat flux at every fluid time would need to be mapped to the thermal solver instead of just once per thermal time step.

Once the heat ﬂux prediction is mapped to the thermal solver, the thermal solution is marched forward in time. The surface temperatures are then mapped to the

ﬂuid solver, and the internal temperatures to the structural solver. The structural

60 Thermal Solver Fluid Solver

(CN)

(Nβ)

(2BE)

Structural Solver Interface

Figure 3.6: Time-Accurate Predictor Implicit (PI) coupling ﬂowchart.

61 solver then requests the pressure at the m + 1 step. The ﬂuid solver predicts the pressure using a least-squares based linear extrapolation:

m+1 m+1 2 p ≈ pP = ap + 2∆tSbp + O ∆tS (3.15)

m−1 where ap is the least-squares ﬁt of p , and ap is the change in pressure over the

m − 1 to m structural time step. Like the heat ﬂux prediction, this is performed

within the ﬂuid solver, and then passed to the structural solver. This prediction

also reduces to Eq. 3.3 if ﬂuid subcycling is not used.

Before the structural solution can be updated, the temperature at the m + 1 step

must be set. If thermal-structural subcycling is not used, the proper temperature

is simply the newest thermal solution. When subcycling is used, the same linear

interpolation of the temperature from the quasi-steady version in Eq. (3.8) is used.

With the temperature applied, the structural solution is updated to step m + 1, and

the solution is passed to the ﬂuid solver.

The ﬂuid solver requires the surface temperature, displacement, and velocity

computed at the k+1 time step. If no subcycling is used, the newest solutions from

the thermal and structural solvers are already at the k + 1 step. With subcycling,

62 the values are interpolated:

n+1 n k+i k+i n T − T 2 T ≈ TI = T + (i∆tF ) + O ∆tF (3.16a) ∆tT m+1 m k+j k+j m ü + ü 2 ˙u ≈ ˙u = ˙u + (j∆tF ) + O ∆t (3.16b) I 2 F m+1 m k+j k+j m m 1 2 ü + ü 2 u ≈ u = u + (j∆tF ) ˙u + (j∆tF ) + O ∆t (3.16c) I 2 2 F

where i is a ﬂuid counter between thermal time steps, and j is a ﬂuid counter be-

tween structural time steps. The temperature and structural velocities are linearly

interpolated, while the structural displacement quadratically interpolated to main-

tain consistency with the velocity. Note that the interpolations for displacement

and velocity are dependent on the Newmark-β scheme’s deﬁnition of acceleration;

the interpolation must be adjusted for other time integrators.

With the surface motion and temperature deﬁned, the ﬂuid solver does not

require any predictions and the solution is marched to the next ﬂuid time step.

If ﬂuid subcycling is used, the interpolations from Eq. 3.16 are updated and the

solution is marched forward until tk = tm+1. Control is then passed back to the structural solver. If ﬂuid-structural subcycling is used, the entire structural and

ﬂuid solver process is repeated until tk = tm = tn+1. Control is passed to the thermal solver to being the next thermal time step.

63 3.2.2 Predictor-Corrector Implicit Scheme

The Predictor-Corrector Implicit (PCI) scheme for quasi-steady aerothermodynam-

ics is extended here for time-accurate aerothermodynamics. Like the quasi-steady

version, the PCI scheme is a modiﬁcation of the PI scheme in Fig. 3.6 to include cor-

rection steps for the thermal and structural solvers. Once the ﬂuid is updated to

the new structural time step, the structural solution over that time step is recom-

puted with the updated pressure. Similarly, the thermal solution is recomputed

once the ﬂuid is updated to the new thermal time step. There is not a correction

step associated with the ﬂuid solver since it does not require any extrapolating

predictors.

These correctors allow the ﬂuid loads to be exactly matched between the three

solvers at each time step, with the tradeoff that the temperatures and displace-

ments are no longer exactly matched at each time step. The ﬂuid displacements

are based on the initial update with the ﬂuid load predictors, while the structural

displacement is based on the actual ﬂuid load. The temperature between the ther-

mal solver and structural/ﬂuid solvers are similarly mismatched. However, if the

ﬂuid load predictors are good estimates, the mismatch is minimal.

Note that for ﬂuid subcycling, the displacement interpolation in Eq. (3.16b-c)

cannot be used. Equation. (3.16b-c) preserves continuity of displacement and ve-

locity only if the acceleration terms are computed from a single Newmark-β integration step; however the corrector step violates this assumption. Instead, a cubic

64 polynomial interpolation is applied to ensure continuity of displacement and ve-

locity:

2 3 k+j m m j∆tF j∆tF uF ≈ uF + ˙uF (j∆tF ) + u2 + u3 (3.17a) ∆tS ∆tS 2 k+j m u2 j∆tF u3 j∆tF ˙uF ≈ ˙uF + 2 + 3 + (3.17b) ∆tS ∆tS ∆tS ∆tS

where j is a ﬂuid counter between structural steps, and u2 and u3 are deﬁned as:

m+1 m m+1 m u2 ≡ 3 uF − uF + ∆tS ˙uF + 2 ˙uF (3.18a)

m+1 m m+1 m u3 ≡ −2 uF − uF + ∆tS ˙uF + ˙uF (3.18b)

The fluid subscript F is used in Eqs. (3.17)-(3.18) to differentiate the displacements used in the fluid solver from the displacements in the structural solver. Fi- nally, note that if the pressure predictor is exact, the cubic interpolation reduces to the constant averaged acceleration interpolation defined in Eq. (3.16b-c).

3.2.3 Additional Coupling Schemes for Time-Accurate Aerother-

modynamics

As with the quasi-steady coupling, several other coupling schemes are considered here for comparison to the PI scheme, including an analogue for the quasi-steady

BI scheme, and a strongly coupled scheme as a benchmark.

65 3.2.3.1 Basic Implicit Scheme

The Basic Implicit (BI) scheme for time-accurate aerothermodynamics uses the

same types of simpliﬁcations as the quasi-steady BI scheme. It represents a “naive”

version of the PI scheme in which predictors and interpolations are not used. It fol-

lows the same general logic as the PI scheme in Fig. 3.6.

Like the quasi-steady BI scheme, the heat ﬂux and pressure used for the thermal

and structural solvers are simply based on the most recent ﬂuid solution, which

introduces ﬁrst order truncation errors to the thermal and structural solutions. The

temperature is also held constant over the thermal time step for both the ﬂuid and

structural solver, as is the displacement and velocity over the structural time step.

For the structure, Eq. (3.8) is used, and similarly for the ﬂuid:

k+i n+1 T ≈ T + O (∆tF ) (3.19a)

k+j m+1 ˙u ≈ ˙u + O (∆tF ) (3.19b)

k+j m+1 u ≈ u + O (∆tF ) (3.19c)

where i is a fluid counter between thermal time steps, and j is a fluid counter between structural time steps. If the internal mesh velocities of the fluid solver are computed using finite difference formulations of the mesh locations, these velocity terms must be explicitly frozen in time between structural time steps. Otherwise the mesh velocities will be computed as zero over a structural time step, and erro-

66 neously large between structural steps. This introduces non-convergent behavior of the ﬂuid solution with decreasing time step size.

If subcycling is not used, then Eqs. (3.8) and (3.19) do not introduce first order truncation errors. However, the first order errors from the pressure and heat flux loads will propagate to the fluid solver, and the global solution of the system will only be first order accurate.

3.2.3.2 Strong Implicit Scheme

A strongly-coupled implicit (SI) scheme is also considered for time-accurate aerothermodynamics. The ﬂowchart of the algorithm is shown in Fig. 3.7. The scheme is based on the same time integrators as the PI and BI schemes, and uses subiterations between all three solvers to achieve convergence at the coupling interface. Like the quasi-steady version, subcycling is not considered because all quantities must be subiterated and converged at each time step, and a single time step counter n is used.

At the start of a new time step, the heat flux at step n + 1/2 is again computed using an averaging defined in Eq. (3.11). The heat flux is mapped to the thermal solver, the temperatures are updated, and mapped to the fluid and structural solvers. The structural solver then receives the pressures from the fluid and updates the displacements and velocities, which are passed to the fluid solver. The

ﬂuid solver updates its solution and checks the convergence of the heat ﬂux and

−2 pressure using Eqs. (3.12)-(3.13). A tolerance tol = 10 is used. If the tolerance

67 Thermal Solver Fluid Solver

(CN)

(Nβ) (2BE)

Structural Solver Interface

Figure 3.7: Time-Accurate Strong Implicit (SI) coupling ﬂowchart.

68 is met, the thermal solver moves to the next time step. Otherwise, the heat ﬂux is

averaged again using the new ﬂuid solution, and the step is restarted. Note that

the tolerance set here is higher than that used for the quasi-steady SI scheme to

ensure that the time-accurate ﬂow solver can reach the tolerance at each time step

without an overly-burdensome number of subiterations.

3.3 Summary of Schemes

The coupling schemes for quasi-steady aerothermodynamics are summarized in

Table 3.1 for the Predictor Implicit (PI), Predictor-Corrector Implicit (PCI), Basic

Implicit (BI), Conventional Explicit (CE), and Strong Implicit (SI) schemes. The

schemes may use Newmark-β (N-β) or 4th order Runge-Kutta (RK4) for the structural time integration, and either Crank-Nicolson (CN) or ﬁrst order forward Euler

(FE) for the thermal time integration.

Table 3.1: Summary of coupling schemes for quasi-steady aerothermodynamics.

Structural Time Pressure Thermal Time Heat Flux Temperature Integrator Loading Integrator Loading Interpolation PI N-β Predictor CN Predictor Linear PCI N-β Pred/Corr1 CN Pred/Corr1 Linear BI N-β Direct CN Direct None CE RK4 Each Stage FE Averaged None SI N-β Iterative CN Iterative N/A2 1 Predictor for ﬂuid loading, additional corrector step for thermal/structural solution. 2 Subcycling is not used in the SI scheme.

For time-accurate aerothermodynamics, the PI, PCI, BI, and SI schemes are

69 summarized in Table 3.2. All three schemes use 2nd order backward Euler for the

ﬂuid time integration, Newmark-β for the structural time integration, and Crank-

Nicolson for the thermal time integration.

Table 3.2: Summary of coupling schemes for time-accurate aerothermodynamics.

Pressure Heat Flux Temperature Displacement Loading Loading Interpolation Interpolation PI Predictor Predictor Linear Const. Accel.1 PCI Pred/Corr Pred/Corr Linear Cubic2 BI Direct Direct None None SI Iterative Iterative N/A3 N/A3 1 Constant, averaged acceleration. 2 Cubic interpolation reduces to constant, averaged acceleration if the pressure predictor is exact. 3 Subcycling is not used in the SI scheme.

70 Chapter 4

Conﬁgurations and Computational Models

In this study, the various coupling schemes are applied to model the heating and deformation of a compliant panel on the surface of a vehicle ﬂying at 30 km as shown in Fig. 4.1. The ﬂow is inclined 5 degrees from the vehicle surface, and is assumed to transition to fully turbulent 1 meter upstream of the panel leading edge.

30 km

M∞= 4 - 9.5 M2 T∞ = 227 K T2 p∞ = 1200 Pa 2 p2 μ∞ =14.8 μN∙s/m μ2 Turbulent Trip 5° Compliant Panel

1.0m 1.0m L

Figure 4.1: Detail of wedge surface.

Two separate conﬁgurations with different panel properties and ﬂight condi-

71 tions are considered for analysis. Conﬁguration 1 is a 1.5m titanium (Ti-6Al-2Sn-

4Zr-2Mo) panel in Mach 9.5 ﬂow, and Conﬁguration 2 is a 1.0m aluminum (Al-

7075) panel in Mach 4 ﬂow. Both conﬁgurations are based on previous studies

[78, 96]. The post-shock ﬂow conditions and structural properties for the two con-

ﬁgurations are detailed in Table 4.1 and Table 4.2, respectively. Note the material properties are temperature dependent. Tables of the properties across a range of temperature are given in Appendix B.

Table 4.1: Post-shock ﬂight conditions for each conﬁguration.

Conﬁguration 1 2 M∞ 9.5 4.0 M2 7.95 3.64 2 P2 (N/m ) 3446 1938 T2 (K) 317.0 261.0 2 µ2 (µ · s/m ) 19.4 16.6

Table 4.2: Panel properties of each conﬁguration.

Conﬁguration 1 2 L (m) 1.5 1.0 h (m) 0.005 0.0025 3 ρp (kg/m ) 4539 2770 ∗ cp (J/kgK) 4630 851 κ (J/m2K)∗ 6.88 132 ν 0.32 0.325 α (µm/mK)∗ 7.09 22.2 E (GP a)∗ 113.47 71.3 Tinit (K) 300 300 *Function of temperature (Shown at 300K)

Analysis of Conﬁguration 1 is performed with quasi-steady aerothermody-

72 namic models, and Configuration 2 with both quasi-steady and time-accurate aerothermodynamics. For Configuration 1, surrogate-based models to capture the pressure and heat flux are used. For Configuration 2, engineering-level analytical models are used for the quasi-steady analysis. These two configurations are chosen to compare the performance of the coupling schemes for a dynamically stable response (Configuration 1), and a dynamically unstable flutter response (Configu- ration 2). The two types of aerothermodynamic models are included to compare the performance of the coupling schemes using fluid models that dominate the computing time (surrogates), or models that are relatively cheap compared to the thermal-structural solvers (analytical). Configuration 2 is also used for the the analysis with time-accurate aerothermodynamics to study the unstable flutter response, performed with an unsteady CFD flow solver. Both configurations use a mode-based structural solver, and a 2D finite element thermal solver. Details of the aerothermodynamics, structure, and thermal models are provided next.

4.1 Quasi-Steady Analytical Fluid Modeling

The quasi-steady analytical fluid models are used for Configuration 2. Pressure is computed with third order Piston Theory, and the heat flux is computed with the

Eckert’s Reference Enthalpy method.

73 4.1.1 Piston Theory

Piston theory provides a simple point-function relationship between the unsteady

pressure and surface motion [97, 98]. Third order Piston theory has been shown in

previous studies [78] to provide a reasonable representation of the pressure for a

hypersonic similarity parameter1smaller than one. The third-order expression for piston theory is given by:

" q2 1 ∂w ∂w p(x) = p∞ + 2 + M2 U2 ∂t ∂x γ + 1 1 ∂w ∂w2 + M2 + (4.1) 4 U2 ∂t ∂x 3# γ + 1 2 1 ∂w ∂w + M2 + 12 U2 ∂t ∂x

where γ = 1.4 is the ratio of speciﬁc heats, q2 is the post-shock dynamic pressure,

U2 is the post-shock ﬂow velocity, and w is the transverse displacement into the

ﬂow.

4.1.2 Eckert’s Reference Enthalpy

Eckert’s reference enthalpy method [90] uses boundary layer relations from incom-

pressible ﬂow theory with ﬂow properties evaluated at a reference condition to ac-

count for the effects of compressibility. Reference methods have been used exten-

1The hypersonic similarity parameter is deﬁned as Mδ, where δ is the slope of the surface relative to the ﬂow direction [89].

74 sively in approximate analyses to efﬁciently model convective heating of aerospace vehicles [99–104]. Eckert’s reference enthalpy is deﬁned by

∗ H = He + 0.50(Hw − He) + 0.22(Haw − He) (4.2a)

Haw = r(H0 − He) + He (4.2b)

2 Ue H0 = He + (4.2c) 2

r = (P r∗)1/3 (4.2d)

∗ where H , He, Hw, Haw, and H0, are the reference, boundary layer edge, wall, adiabatic wall, and total enthalpies, respectively. Ue is the local boundary layer edge velocity, r is the recovery factor, and P r is the Prandtl Number at the reference condition. Note that the boundary layer edge velocity and enthalpy are determined using the pressure from piston theory, Eq. (4.1), in conjunction with isentropic flow relations [105] to compute the local boundary layer edge Mach number and temperature. Thus, flow effects due to the deformed panel shape and the panel velocity are included in the aerodynamic heating predictions. Note that flow over the panel is assumed to be fully-turbulent using the defined recovery factor in

Eq. (4.2d).

Using ﬂow properties evaluated at the reference enthalpy, the aerodynamic heat ﬂux is computed using:

75 ∗ ∗ q = St ρ Ue(Haw − Hw) (4.3a)

∗ cf 1 St∗ = (4.3b) 2 (P r∗)2/3

∗ 0.370 cf = ∗ 2.584 (4.3c) (log10 Rex) ∗ ρ Ue(x + x0) Re∗ = (4.3d) x η∗

where q is the heat ﬂux, St is the Stanton number, cf is the local skin friction coef-

ficient, Rex is the Reynold’s number for a given panel position x, and x0 is the dis- tance from the transition location to the panel leading edge. The Stanton number is determined using the Colburn-Reynolds analogy in Eq. (4.3b), and the local skin friction coefficient is calculated using the Schultz-Grunow formula in Eq. (4.3c) with the local Reynolds number defined in Eq. (4.3d).

In addition to Eqs. (4.2) – (4.3), temperature-enthalpy relations are needed to determine the values of enthalpy at the wall and at the edge of the boundary layer.

These relations are also required to determine a temperature from the reference enthalpy in order to evaluate reference density and reference viscosity using the ideal gas law and Sutherland’s law, respectively [106].

76 4.2 Quasi-Steady CFD-Based Surrogate Fluid Model-

ing

For Configuration 1, the heat flux and pressure are computed using CFD-based surrogate models. The surrogates are data-driven models built from 1500 steady- state CFD solutions combined with theoretical corrections. The steady state CFD solutions are varied with eight parameters: the freestream Mach number; the first six free-vibration modes of the panel; and an isothermal wall temperature. The surrogate functions provide a response surface over this parameter space for steady- state heat flux and pressure. The pressure surrogate uses a piston-theory based correction to account for aerodynamic damping from the velocity of the panel deformation. The heat flux response surface is computed assuming isothermal surface temperatures, thus a correction is incorporated into the surrogate to account for spatially varying wall temperatures. This correction is derived from compressible boundary layer theory. Details of the surrogate construction, the corrections, and their verification are presented in [91, 107].

4.3 Time-Accurate CFD Fluid Modeling

The panel response with time-accurate ﬂow is computed with an unsteady CFD solver. For this work a modiﬁed version of the NASA Langley CFL3D solver [108,

77 109] is used. CFL3D uses an implicitly time-marched, ﬁnite-volume algorithm based on upwind-biased spatial differencing to solve the Reynolds-Averaged Navier-

Stokes (RANS) equations. The algorithm, which is based on a cell-centered scheme, uses upwind-differencing based on either flux-vector splitting or flux-difference splitting, and can sharply capture shock waves. The Menter k − ω SST [110] turbulence model is used in this study. The solution is marched in time using the second-order backwards difference scheme shown in Eq. (2.22). Mesh deformation and linear modal-based structural representation is also included for aeroelastic computations. The geometric conservation law is satisfied through the addition of source terms as detailed in Sec. 2.2.2.

Since CFL3D’s native capabilities do not allow for nonlinear structural modeling or fluid-thermal coupling, it has been extensively modified to incorporate these effects. Before each time step, the fluid solver is updated with new temperature boundary conditions and mesh deformations. After the time step, the pressures and heat flux are extracted for the structural and thermal solvers.

The computational grid for the ﬂuid domain is illustrated in Fig. 4.2. The shock generated from ﬂow inclination to the initially undeformed surface is not modeled directly. Instead, the post-shock conditions from Table 4.1 are applied as the

“freestream” conditions in the CFD solver. The fluid domain extends 2 meters upstream of the panel, 0.5 meter downstream, and 1.5 meters into the flow. Note that maximum panel deflections are on the order of 2-3 cm, two orders of magnitude smaller than the domain size. The bottom boundary is a no-slip wall condition

78 1.5m 140 cells

2.0m 1.0m 0.5m 80 cells 500 cells 32 cells

Figure 4.2: Flow computational grid (85,860 cells).

with a constant temperature of 300K, except over the panel where the temperature

is coupled to the response. The grid contains 612 cells in the streamwise direction

and 140 cells in the normal direction, for a total of 85,680 cells. The highest concen-

tration of cells are in the region of the panel, with 500 cells located along the length

of the panel. The grid is divided into 35 blocks of equal size for parallelization,

with each block containing 2,448 cells. A convergence study for this mesh is given

in Appendix C.

4.4 Thermal Modeling

The thermal domain is illustrated in Fig. 4.3. Two dimensional heat transfer along

the length and through the thickness of the panel is considered from the two-

dimensional form of Eq. (2.7):

79 z q(x)

h x L Figure 4.3: Thermal model domain.

∂T (x, z, t) ∂ ∂T (x, z, t) ρc(x, z, T ) − κ(x, z, t) − ∂t ∂x ∂x (4.4) ∂ ∂T (x, z, t) κ(x, z, t) = 0 ∂z ∂z with boundary conditions:

∂T (x, z, t) = 0 (4.5a) ∂x x=0,L

∂T (x, z, t) = 0 (4.5b) ∂z z=−h/2

∂T (x, z, t) κ(x, z, t) = q(x) (4.5c) ∂z z=h/2

where q(x) is a positive heat ﬂux from the ﬂow. Note that both c and κ are functions

of temperature, and thus functions of space and time. The panel is insulated on all

sides except the upper surface exposed to the ﬂow.

Equation (4.4) is discretized using Galerkin’s method with a ﬁnite element for-

mulation. Linear shape functions of the temperature are used for the elements:

80 T (x, z, t) ≈T (xi, zj, t)(1 − δx)(1 − δz)+

T (xi+1, zj, t)(δx)(1 − δz)+ (4.6)

T (xi, zj+1, t)(1 − δx)(δz)+

T (xi+1, zj+1, t)(δx)(δz) where i and j are the discrete counters in x and z, and:

x − xi δx = (4.7a) xi+1 − xi

z − zj δz = (4.7b) zj+1 − zj

The system of equations is then represented using Eq. (2.34). Because of the temperature-dependent properties, the system is weakly nonlinear. For the Crank-

Nicolson time integration, the thermal capacitance and conductivity matrices are computed at the n+ 1/2 time step. A second-order extrapolation of the temperature can be used to compute these matrices:

3 1 Tn+1/2 ≈ Tn − Tn−1 + O ∆t2 (4.8) 2 2 T

This extrapolation preserves the order of accuracy of the Crank-Nicolson scheme, and avoids the need for subiterations of the thermal solver.

For the quasi-steady analysis, the thermal solver is discretized with 1000 ele-

81 ments along the panel length, and 4 elements through the thickness. For the time- accurate fluid analysis, the discretization is 500 elements through the panel length, and 4 elements through the thickness. In both cases, the discretization along the length is finer than required, but they are chosen to be consistent with the surrogate discretization (1000 elements) and CFD discretization (500 elements). Note that the surrogate model is discretized at a finer resolution than the CFD model because the higher Mach numbers the surrogate is based on requires a finer grid.

Veriﬁcation of the thermal solver compared to the commercial ﬁnite element solver

Abaqus R is located in Appendix D.

4.5 Structural Modeling

The structure is represented as a panel in cylindrical bending as illustrated in

Fig. 4.4. The aerodynamic pressure acts on the upper surface of the panel, while a speciﬁc backpressure pb is applied to the bottom surface. The backpressure is deﬁned as pb = pw for an undeformed, unheated panel, so that the undeformed panel is initially in aeroelastic equilibrium. The panel is pinned at each end.

The panel is modeled using nonlinear von Karm´ an´ strains [95], and includes thermal strains due to a non-uniform temperature distribution through the length and thickness of the panel [87]:

82 z p(x)

h x

pb(x) L

Figure 4.4: Structural model domain.

∂2 ∂2w(x, t) ∂w(x, t) ∂2 ∂2w(x, t) mw(x, t) − I + C¯ + D(x, t)) ∂t2 ∂x2 ∂t ∂x2 ∂x2 (4.9) ∂2w(x, t) ∂2M (x, t) −N (t) + T + q (x, t) = 0 x ∂x2 ∂x2 A

E(x, t)h3 D(x, t) ≡ (4.10a) 12(1 − ν2) " 2# E(x, t)h ∂u0 1 ∂w Nx ≡ + − NT (t) (4.10b) 1 − ν2 ∂x 2 ∂x E(x, t)α(x, t) Z h/2 NT ≡ [T (x, z, t) − Tinit]dz (4.10c) 1 − ν −h/2 E(x, t)α(x, t) Z h/2 MT = [T (x, z, t) − Tinit]zdz (4.10d) 1 − ν −h/2

where w is the transverse displacement, u0 is in-plane displacement, m is the mass

¯ per unit length, Im is the cross-sectional mass moment of inertia of the panel, C is the material damping, E is the Young’s modulus, ν is Poisson’s ratio, α is the coefﬁcient of thermal expansion, and T0 is the initial temperature. Nx represents

83 the total in-plane force on the panel, with nonlinear coupling between the trans-

verse and in-plane displacements. NT is the thermal contribution to the in-plane forces, and MT is the thermal moment from variations of the temperature through the thickness.

The boundary conditions at the panel supports are given by Eq. (4.11).

w(0, t) = w(L, t) = 0 (4.11a)

u0(0, t) = u0(L, t) = 0 (4.11b)

2 ∂ w(x, t) MT (x, t) 2 = − (4.11c) ∂x x=0,L D(x, t) x=0,L

The in-plane displacement term, ∂u/∂x, is eliminated from the expression for

Nx by solving Eq. (4.10b) for u0:

( ) Z x 1 − ν2 1 ∂w (ξ, t)2 u0 (x, t) = [Nx (t) + NT (ξ, t)] − dξ (4.12) 0 E (ξ, t) h 2 ∂ξ

At x = L the in-plane displacement is constrained to be zero by the immovable supports, and Nx can be solved from Eq. (4.10b) to yield:

−1 Z L 1 Z L h ∂w2 Nx = dx 2 dx − NT (t) (4.13a) 0 E(x, t) 0 2(1 − ν ) ∂x −1 Z L 1 Z L α(x, t) Z h/2 NT = dx [T (x, z, t) − Tinit]dzdx (4.13b) 0 E(x, t) 0 1 − ν −h/2

84 where Eq. (4.13b) represents an averaging of the thermal force.

Equation (4.9) is discretized using Galerkin’s method. The transverse displace-

ments w(x, t) are approximated using a series of global mode shapes:

N X w(x, t) = An(t)φn(x) (4.14) n=1

where φ(x) are the free-vibration modes of the undeformed panel:

nπx φn(x) = sin (4.15) L

The discretized equations of motion can then be represented using Eq. (2.24).

Six mode shapes are included for Configuration 1, and fifteen modes for Configu- ration 2. The number of modes were chosen based on convergence of the responses given in Chapter 6 for both configurations. The integrations for Galerkin’s method is performed over 1001 points the quasi-steady analyses, and 501 points for the time-accurate fluid analysis to keep a consistent discretization with the fluid and thermal solvers. The temperatures are also integrated with 5 points through the thickness to be consistent with the thermal solver. The numerical integration is performed using the composite Simpson’s 1/3 method, which is 4th order accurate in space [94].

The damping matrix [C] is deﬁned using Rayleigh damping, as a combination

of the mass and stiffness matrices [93]:

85 [C] = αR [M] + βR [KT ] (4.16)

where α and β are the Rayleigh coefﬁcients, and [KT ] is the tangent stiffness matrix.

The mass proportional damping (α) applies damping to lower frequencies, while the stiffness proportional damping (β) applies damping to higher frequencies. The parameters are chosen to apply 2% critical damping to the fundamental frequency of the panel, and at 1000 Hz, so a small but nonzero damping exists over the frequency ranges of interest. Table 4.3 lists the fundamental frequency of the panel

(ω0) and the corresponding values for α and β for the two panel conﬁgurations.

Table 4.3: Rayleigh Damping parameters for each conﬁguration.

Conﬁguration 1 2 ω0 (Hz) 5.32 6.09 α (1/s) 1.33 1.52 β (µs) 6.33 6.33

For the Newmark-β method, Eqs. (2.28)-(2.29) are nonlinear. Subiterations are used to converge the system, until the relative maximum residual of the displacement and force vectors are under 10−10. Veriﬁcation of the structural solver compared to Abaqus R is given in Appendix E.

86 4.6 Summary of Conﬁgurations

A summary of the two conﬁgurations for quasi-steady aerothermodynamics is given in Table 4.4, and the conﬁguration for time-accurate aerothermodynamics in Table 4.5.

Table 4.4: Summary of conﬁgurations for quasi-steady aerothermodynamics. Conﬁguration 1 2 Panel Material Titanium Aluminum Mach Number 9.5 4.0 Panel Length 1.5 m 1.0 m Panel Thickness 5 mm 2.5 mm # Mode Shapes 6 15 # points along length 1001 1001 # points through thickness 5 5 Pressure Model Surrogate Piston Theory Heat Flux Model Surrogate Eckert’s Ref. Enthalpy

Table 4.5: Summary of Conﬁguration 2 for time-accurate CFD. Panel Material Aluminum Mach Number 4.0 Panel Length 1.0 m Panel Thickness 2.5 mm # Mode Shapes 15 # points along length 501 # points through thickness 5

87 Chapter 5

Order of Accuracy Analysis

In this study, the order of accuracy of the coupling schemes are numerically demon-

strated. For the quasi-steady aerothermodynamics analysis, the Predictor Implicit

(PI), Predictor-Corrector Implicit (PCI), Basic Implicit (BI), Conventional Explicit

(CE), and Strong Implicit (SI) schemes are compared for both Conﬁguration 1 and

2. From the time-accurate CFD analysis, the PI, PCI, BI, and SI schemes are com-

pared based on Conﬁguration 2. Both 1-1 time stepping and subcycling are con-

sidered.

The order of accuracy is based on relative differences between solutions at dif-

ferent time steps. The relative difference is computed in terms of an L2 norm in

both space and time:

v u N M u 1 X X 2 L2 = t (φi − φi+1) (5.1) MN mn mn n=1 m=1 where φmn is the solution at location m and time step n, M is the number of points

88 in space, and N is the number of points in time. Using this metric, the order of

accuracy can be determined based on the slope of the logarithm of the L2 versus

time-step size. A second order accurate solution will trend towards a slope of two

for decreasing time step sizes, while a ﬁrst order solution will trend to a slope of

one. The order of accuracy is investigated for all of the solution quantities: dis-

placement and velocity of the structural response; temperature from the thermal

response; and heat ﬂux and pressure from the ﬂuid response.

5.1 Quasi-Steady Aerothermodynamics

The temporal order of accuracy is investigated for each scheme, with and without

structural subcycling. Simulations are performed for 7 time step levels, each half

the size of the previous level:

1i−1 ∆ti = 10−3 × (5.2) S 2 where i is the time step level. The time steps range from a maximum size of 10−3 seconds to a minimum of 1.5625 × 10−5 seconds. Analysis is performed over a period of one second, both with 1-1 time stepping (no subcycling), and with 25 structural steps per thermal step.

The average temperature, mid-panel displacement, and mid-panel velocity is shown in Fig. 5.1 for Configuration 1, and Fig. 5.2 for Configuration 2. Both configurations have a qualitatively similar response over this time period. The tempera-

89 ture immediately rises at a nearly constant rate, causing the panel to buckle almost immediately. The panel buckles into the flow from the positive thermal moment biasing the panel upwards. As the temperature rises, the panel continues to bow further into the flow. The buckling of the panel leads to initial transients which eventually damp out. Note that while the responses are similar, Configuration 1 has a larger temperature rise and more significant vibrations than Configuration 2.

5.1.1 Without Subcycling

2 The order of accuracy is first investigated without subcycling (∆tS = ∆tT ). L norms of the panel displacement, velocity, and temperature are shown in log-log format in Fig. 5.3, and fluid pressure and heat flux in Fig. 5.3 for both configurations. All five quasi-steady coupling schemes are shown. The solid gray line has a slope of 1 and represents an exact first order trend. The dashed black line has a slope of 2 and represents an exact second order trend. These lines are fit to each data set as a reference for a true first or second order rate of convergence. Note that the L2 are not normalized; their magnitudes are dependent on the magnitude of the physical quantity itself.

In general, the CE and BI schemes exhibit a ﬁrst order convergence, while the

PI and SI schemes are second order, as expected. Both conﬁgurations show similar trends in both the rate of convergence and relative size of convergence. In every case, the PI, PCI and SI schemes have nearly identical L2 norms, and both follow

90 3 330

324

2 318 ave w/h T 312 1 Displacement Temperature 306

0 300 0 0.2 0.4 0.6 0.8 1 Time (sec) (a) Average temperature (red) and normalized mid-panel displacement (blue) 20

5 wh(1/sec) ˙w/h 0

-5

-10 0 0.2 0.4 0.6 0.8 1 Time (sec) (b) Normalized mid-panel velocity.

Figure 5.1: Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 1 over one second.

91 3 305

304

2 303 ave w/h T 302 1 Displacement Temperature 301

0 300 0 0.2 0.4 0.6 0.8 1 Time (sec) (a) Average temperature (red) and normalized mid-panel displacement (blue) 10

4 wh(1/sec) ˙w/h 2

-2 0 0.2 0.4 0.6 0.8 1 Time (sec) (b) Normalized mid-panel velocity.

Figure 5.2: Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 2 over one second.

92 the second order trend lines closely through the range of time steps considered.

The BI scheme generally has the largest L2 norms, except with temperature where the CE scheme is highest since it uses the ﬁrst order forward Euler time integrator.

The CE scheme has significantly smaller L2 norms of displacement, velocity, pressure and heat flux compared to the BI scheme. The velocity L2 norms for CE are even smaller than the PI, PCI and SI schemes at the largest time steps. This is because of the fourth order Runge-Kutta time integration for the structure; while the globally first order accurate due to the thermal time integration and the coupling, the smaller errors introduced the structural time integrator reduces the error magnitudes. However, the displacement and temperature L2 norms for the CE scheme is higher than the PI and SI schemes throughout the entire range of time steps considered. Finally, note that the L2 norms of the CE scheme for Configuration 2 at the largest time steps are significantly beyond of the upper bound shown.

5.1.2 Structural Subcycling

2 The order of accuracy is next investigated with subcycling (∆tT = 25∆tS). The L norms of the panel displacement, velocity, and temperature are shown in Fig. 5.5, and fluid pressure and heat flux in Fig. 5.6 for both configurations. Note the SI scheme is not included here because the strong coupling prohibits subcycling.

The results are similar to the case without subcycling. The PI and PCI schemes follows the second order trend over the range of time steps, while in general the CE

93 -4 -5

-5 -6 ) ) 2

-6 2 (L

(L -7

10 PI PI 10 -7 PCI log PCI log -8 BI BI CE CE -8 SI SI -9 1st Order 1st Order 2nd Order -9 2nd Order -10 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Displacement (Conﬁguration 1) (b) Displacement (Conﬁguration 2)

-3 -2 -3.5

-4 -3 ) ) -4.5 2 2 (L -4 (L PI

10 -5 PI 10 PCI PCI log log -5.5 BI BI -5 CE CE -6 SI SI 1st Order 1st Order -6.5 -6 2nd Order 2nd Order -7 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Velocity (Conﬁguration 1) (d) Velocity (Conﬁguration 2)

-2 -4

-3 -5 -4 ) )

2 2 -6 -5 (L PI (L PI 10 10 -6 PCI PCI

log log -7 BI BI -7 CE CE SI -8 SI -8 1st Order 1st Order 2nd Order 2nd Order -9 -9 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (e) Temperature (Conﬁguration 1) (f) Temperature (Conﬁguration 2)

Figure 5.3: Order of Accuracy of panel displacement, velocity, and temperature for both conﬁgurations without subcycling (Quasi-steady aerothermodynamics).

94 0

1 -1 0 ) 2 ) -2 2

-1 (L

(L PI

PI 10 10 PCI PCI

log -3 -2 BI log BI CE CE -3 SI SI -4 1st Order 1st Order -4 2nd Order 2nd Order -5 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Pressure (Conﬁguration 1) (b) Pressure (Conﬁguration 2)

3 1

2 0

1 ) )

2 2 -1 (L 0 (L PI 10 PI 10 PCI PCI

log log -2 BI -1 BI CE CE SI SI -3 -2 1st Order 1st Order 2nd Order 2nd Order -3 -4 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Heat Flux (Conﬁguration 1) (d) Heat Flux (Conﬁguration 2)

Figure 5.4: Order of Accuracy of fluid pressure and heat flux for both configurations without subcycling (Quasi-steady aerothermodynamics).

95 and BI schemes follow a ﬁrst order trend. The BI and CE schemes have a similar

magnitude in L2 norms for displacement, velocity, and pressure. The CE scheme does have lower L2 norms for the heat ﬂux in Conﬁguration 1 compared to the BI

scheme, however the BI scheme has lower L2 norms for temperature. Note that

for Conﬁguration 2 the CE scheme diverged at the three largest time steps because

the thermal time step was larger than the stability boundary of the forward Euler

time integration scheme. Interestingly, the relative error of the temperature for

Conﬁguration 2 in Fig. 5.5(f) has a slope of almost two for the BI scheme in addition

to the PI and PCI schemes. However, this is not a general trend, since the accuracy

for the BI scheme is clearly order 1 for Conﬁguration 1 in Fig. 5.5(c), and the other

solution quantities remain ﬁrst order. The temperature L2 norm is also consistently

higher for the BI scheme than the PI or PCI schemes in Fig. 5.5(f), and the BI scheme

starts to shift from the second order trend line at the smallest time steps.

5.1.3 Summary of Accuracy Analysis

The summary of the computed order of accuracy for displacement, velocity, pres-

sure, temperature, and heat ﬂux is listed in Table 5.1 for Conﬁguration 1, and Ta-

ble 5.1 for Conﬁguration 2. The value given with each scheme is the ratio of ther-

mal to structural time steps (e.g. ”1” or 25”). The order of accuracy is computed as

the slope of the L2 norms on a logarithmic scale at the two smallest time steps:

96 -3 -4 -4 -5 -5 ) )

2 2 -6 (L -6 (L 10 10 PI -7 PI log PCI log PCI -7 BI BI CE -8 CE -8 1st Order 1st Order 2nd Order -9 2nd Order -9 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Displacement (Conﬁguration 1) (b) Displacement (Conﬁguration 2)

-1 -1

-2 -2

-3 -3 ) ) 2 2 (L -4 (L -4 10 10 PI PI

log PCI log PCI -5 -5 BI BI CE CE -6 1st Order -6 1st Order 2nd Order 2nd Order -7 -7 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Velocity (Conﬁguration 1) (d) Velocity (Conﬁguration 2)

0 -2

-2.5 -1 -3 -2 ) ) -3.5 2 2 (L -3 (L

10 -4 10 PI PI log PCI log -4.5 -4 PCI BI BI CE -5 CE -5 1st Order 1st Order -5.5 2nd Order 2nd Order -6 -6 -3.5 -3 -2.5 -2 -1.5 -1 -3.5 -3 -2.5 -2 -1.5 -1 log (∆ t) log (∆ t) 10 10 (e) Temperature (Conﬁguration 1) (f) Temperature (Conﬁguration 2)

Figure 5.5: Order of Accuracy of panel displacement, velocity, and temperature for both conﬁgurations with 25 structural steps per thermal step (Quasi-steady aerothermodynamics).

97 2 1 1 0 0

) ) -1 2 2 (L -1 (L 10 10 PI -2 PI

log PCI log PCI -2 BI -3 BI CE CE -3 1st Order -4 1st Order 2nd Order 2nd Order -4 -5 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Pressure (Conﬁguration 1) (b) Pressure (Conﬁguration 2)

4 2 3 1 2 ) ) 2 2

(L (L 0 1 10 10 PI PI log PCI log -1 PCI 0 BI BI CE CE -2 -1 1st Order 1st Order 2nd Order 2nd Order -2 -3 -3.5 -3 -2.5 -2 -1.5 -1 -3.5 -3 -2.5 -2 -1.5 -1 log (∆ t) log (∆ t) 10 10 (c) Heat Flux (Conﬁguration 1) (d) Heat Flux (Conﬁguration 2)

Figure 5.6: Order of Accuracy of fluid pressure and heat flux for both configurations with 25 structural steps per thermal step (Quasi-steady aerothermodynamics).

98 log (L2/L2) accuracy ≈ 6 7 (5.3) log(2)

2 2 2 where L6 and L7 are the L norms at the sixth (second smallest) and seventh (smallest) time step levels, respectively.

The PI, PCI and SI schemes have a computed order of accuracy at or very close to two for all variables in both conﬁgurations. The BI and SI schemes are generally at or slightly above one. Note that the seemingly second order accurate temperature trend observed for Conﬁguration 2 with subcycling in Fig. 5.5(f) is only about

1.6 at the smallest time step. This is consistent with the observed trend in Fig. 5.5(f) that the BI L2 norms shift away from the second order trend line at the smallest time steps. The accuracy is expected to drop closer to one as even smaller time steps are taken.

Table 5.1: Computed order of accuracy at smallest time step for Conﬁguration 1.

Case Displacement Velocity Temperature Pressure Heat Flux PI 1 2.00 2.00 1.99 2.00 2.00 PCI 1 2.00 2.00 2.00 2.00 2.00 SI 1 2.00 2.00 2.00 2.00 2.00 BI 1 1.04 1.00 1.04 1.04 1.04 CE 1 1.00 1.00 1.00 1.00 1.00 PI 25 2.01 2.00 2.00 2.01 2.00 PCI 25 1.97 1.96 2.00 1.96 2.00 BI 25 1.01 1.02 1.01 1.02 1.02 CE 25 1.00 1.00 1.00 1.00 1.05

99 Table 5.2: Computed order of accuracy at smallest time step for Conﬁguration 2.

Case Displacement Velocity Temperature Pressure Heat Flux PI 1 2.00 2.00 2.00 2.00 2.00 PCI 1 2.00 2.00 2.00 2.00 2.00 SI 1 2.00 2.00 2.00 2.00 2.00 BI 1 1.01 1.01 1.00 1.01 1.01 CE 1 1.00 1.00 1.00 1.00 1.00 PI 25 2.00 1.99 2.05 2.00 2.01 PCI 25 2.00 1.99 2.05 2.00 2.01 BI 25 1.01 1.02 1.61 1.01 1.01 CE 25 1.00 1.05 1.16 1.00 1.01

5.2 Time-Accurate Aerothermodynamics

The temporal order of accuracy for the schemes with a time-accurate CFD solver is also investigated, with and without subcycling. Similar to the quasi-steady analysis, simulations are performed for a seven time step levels with the step sized reduced by a factor of two between levels:

1i−1 ∆ti = 10−3 × (5.4) F 2 where i is the time step level. Note this is the same as Eq. (5.2) for the quasi-steady analysis, however the time steps are deﬁned as the ﬂuid time steps in Eq. (5.4).

The ﬂuid time step sizes range from 10−3 seconds at the largest to 1.5625 × 10−5 at the smallest. Solutions both with and without subcycling are considered over a time period of 0.112 seconds. Because there are two levels of subcycling (ﬂuid subcycling and structural subcycling), the impact of each level of subcycling is

100 investigated. The following cases are considered: structural subcycling where

∆tT = 4∆tS = 4∆tF ; ﬂuid subcycling where ∆tT = ∆tS = 4∆tF ; and ﬂuid subcycling combined with structural subcycling where ∆tT = 4∆tS = 16∆tF .

The average temperature, mid-panel displacement, and mid-panel velocity over the time period is shown in Fig. 5.7. The average temperature rises nearly linearly half a degree above the initial temperature. The center displacement is initially in aeroelastic equilibrium, but due to the rising temperature and the associated thermal moment, bows into the ﬂow. The mid-chord panel velocity rises over the time period but also exhibits higher-frequency variations in the panel motion.

5.2.1 Without Subcycling

The L2 norms of panel displacement, velocity, temperature, ﬂuid pressure and heat

ﬂux are plotted in Fig. 5.8 for the analysis without subcycling (∆tT = ∆tS = ∆tF ).

As with the analysis with quasi-steady aerothermodynamics, each data set is ac- companied by a solid gray line for a perfect ﬁrst order trend or a dashed black line for a perfect second order trend.

In Fig. 5.8 both the PI, PCI and SI schemes have a clearly second order trend, while the BI scheme is ﬁrst order. The PI, PCI and SI schemes have similar error magnitudes for each variable over the range of time steps. The PI and PCI schemes have noticeably higher errors in temperature compared to the SI scheme at larger time steps, however at smaller time steps the difference is marginal. The BI scheme

101 0.6 300.5

0.5 300.4

0.4 300.3 0.3 w/h 300.2 0.2

300.1 0.1 Displacement Temperature 0 300 0 0.02 0.04 0.06 0.08 0.1 Time (sec) (a) Average temperature (red) and normalized mid-panel displacement (blue) 7

wh(1/sec) ˙w/h 2

-1 0 0.05 0.1 0.15 Time (sec) (b) Normalized mid-panel velocity.

Figure 5.7: Average temperature, mid-panel displacement, and mid-panel velocity over 0.112 seconds with time-accurate CFD.

102 has the largest L2 norms for every variable at each time step size.

5.2.2 Structural Subcycling

Order of accuracy using structural subcycling is also investigated for the PI, PCI

and BI schemes. The ﬂuid and structural solvers march on the same time step,

while the thermal solver has a time step that is four times larger (∆tT = 4∆tS =

4∆tF ). This represents a case without interpolations of displacement in the ﬂuid

solver, but with interpolations of temperature in the ﬂuid and structural solvers.

The L2 norms in Fig. 5.10 are similar to the case without subcycling in Fig. 5.8.

The BI scheme follows a ﬁrst order trend while the PI scheme is second order. The

most signiﬁcant difference from the solution without subcycling is in temperature.

The PI and PCI schemes follow the second order trendline precisely here, while in

Figs. 5.8(c) the schemes deviate from the line at the largest time steps.

5.2.3 Fluid Subcycling

L2 norms with ﬂuid subcycling in Fig. 5.10 have equal thermal and structural time

steps, and a fluid time step that is four times smaller (∆tT = ∆tS = 4∆tF ) In general the BI scheme is first order and the PI and PCI schemes trend toward second order as before. The BI scheme always has larger errors than the PI and PCI schemes. Note that the velocity, pressure, and heat flux L2 norms converge at a sub-second order rate at the larger time steps for the PI and PCI schemes. This

103 -5 -3.5

-6 -4

-4.5 )

-7 ) 2 2

(L -5 (L 10 10 PI PI log -8 -5.5 PCI log PCI BI -6 BI SI -9 SI 1st Order -6.5 1st Order 2nd Order 2nd Order -10 -7 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Displacement. (b) Velocity.

0 -5 -1 -6 ) )

2 2 -2 (L -7 (L 10 10 PI PI

log log -3 PCI PCI -8 BI BI SI -4 SI 1st Order 1st Order -9 2nd Order 2nd Order -5 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Temperature. (d) Pressure.

0 )

2 -1 (L 10 PI

log -2 PCI BI -3 SI 1st Order 2nd Order -4 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) 10 (e) Heat Flux.

Figure 5.8: Order of Accuracy of panel displacement, velocity, temperature, pressure, and heat ﬂux with ∆tT = ∆tS = ∆tF (Time-accurate aerothermodynamics).

104 -3 -5 -3.5

-6 -4 ) )

2 -4.5 2

(L -7 (L

10 -5 10 log -8 PI log -5.5 PI PCI PCI BI -6 BI -9 1st Order 1st Order -6.5 2nd Order 2nd Order -10 -7 -5 -4.5 -4 -3.5 -3 -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (a) Displacement. (b) Velocity.

-4.5 0

-5 -1 -5.5

) -6 )

2 2 -2 (L -6.5 (L 10 10

log log -3 -7 PI PI PCI PCI -7.5 BI -4 BI 1st Order 1st Order -8 2nd Order 2nd Order -8.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Temperature. (d) Pressure.

0 )

2 -1 (L 10

log -2 PI PCI -3 BI 1st Order 2nd Order -4 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) 10 (e) Heat Flux.

Figure 5.9: Order of Accuracy of panel displacement, velocity, temperature, pressure, and heat ﬂux with ∆tT = 4∆tS = 4∆tF (Time-accurate aerothermodynamics).

105 behavior is most pronounced with the velocity in Fig. 5.10(b). However, the L2 norms converge towards second order at the smaller time steps. The PCI scheme shows slightly lower L2 norms at the higher time steps compared to the PI schemes for these values. Displacement and temperature follow the second order trendline for the entire range of time steps.

5.2.4 Fluid and Structural Subcycling

Finally, the order of accuracy is analyzed for both ﬂuid and structural time stepping, with 4 structural steps per thermal step, and 4 ﬂuid steps per structural step

(∆tT = 4∆tS = 16∆tF ). Consistent with the other three cases, the PI and PCI schemes trend towards second order while the BI has larger errors and trends towards first order. Like the case with fluid subcycling in Fig. 5.10, the PI and PCI schemes deviate from the second order trendline for velocity, displacement, and pressure at the largest time steps, and converges towards second order accuracy at the smaller time steps. The displacement follows the second order line for all time steps as before. The three schemes have similar L2 norms for temperature at larger time steps; however at smaller time steps they diverge to first (BI) and second (PI/PCI) order accuracy.

106 -4 -2.5

-5 -3

) -6 -3.5 2 ) 2 (L (L 10 -4 10

log -7

PI log PI PCI -4.5 PCI BI -8 BI 1st Order -5 1st Order 2nd Order 2nd Order -9 -5.5 -4.5 -4 -3.5 -3 -2.5 -2 -4.5 -4 -3.5 -3 -2.5 -2 log (∆ t) log (∆ t) 10 10 (a) Displacement. (b) Velocity.

-4 0.5

-4.5 0 -5 -0.5 -5.5 ) ) -1 2 -6 2 (L (L

10 -1.5 -6.5 10 log PI log -2 -7 PI PCI PCI -7.5 BI -2.5 BI 1st Order 1st Order -8 -3 2nd Order 2nd Order -8.5 -3.5 -4.5 -4 -3.5 -3 -2.5 -2 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Temperature. (d) Pressure.

1.5

0.5

) 0 2 (L -0.5 10

log -1 PI PCI -1.5 BI 1st Order -2 2nd Order -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) 10 (e) Heat Flux.

Figure 5.10: Order of Accuracy of panel displacement, velocity, temperature, pressure, and heat ﬂux with ∆tT = ∆tS = 4∆tF (Time-accurate aerothermodynamics).

107 -4 -2.5 -4.5

-5 -3

-5.5 -3.5 ) ) 2

-6 2 (L (L

10 -4 -6.5 10 log PI log PI -7 -4.5 PCI PCI -7.5 BI BI 1st Order -5 1st Order -8 2nd Order 2nd Order -8.5 -5.5 -4.5 -4 -3.5 -3 -2.5 -2 -4.5 -4 -3.5 -3 -2.5 -2 log (∆ t) log (∆ t) 10 10 (a) Displacement. (b) Velocity.

0.5 -3 0

-4 -0.5 ) )

2 -1 2 (L -5 (L 10 -1.5 10 log PI log -2 PI -6 PCI PCI BI -2.5 BI 1st Order 1st Order -3 -7 2nd Order 2nd Order -3.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) log (∆ t) 10 10 (c) Temperature. (d) Pressure.

1.5

0.5

) 0 2 (L -0.5 10

log -1 PI PCI -1.5 BI 1st Order -2 2nd Order -2.5 -5 -4.5 -4 -3.5 -3 -2.5 log (∆ t) 10 (e) Heat Flux.

Figure 5.11: Order of Accuracy of panel displacement, velocity, temperature, pressure, and heat ﬂux with ∆tT = 4∆tS = 16∆tF (Time-accurate aerothermodynamics).

108 5.2.5 Summary of Accuracy Analysis

The summary of the computed order of accuracy for displacement, velocity, pres-

sure, temperature, and heat ﬂux is listed in Table 5.3. The four different subcycling

cases are included, and labeled with ”M − N”, where M is the ratio of the thermal to fluid time step size, and N is the ratio of the structural to fluid time step size. The orders are accuracy are computed using the two finest solutions using

Eq. (5.3).

For 1-1 time stepping, the orders of accuracy of the PI, PCI and SI schemes are all at or very close to 2. Interestingly ,the PCI and SI schemes have an accuracy of 1.83 and 1.96 for temperature, respectively. However, they closely follow the second order trendline for all time steps in Fig. 5.8(c). The BI scheme has an order of accuracy of 1 for all variables. The 4-1 time stepping results are similar, with an accuracy of 2 for all the PI and PCI solutions and near 1 for all the BI solutions.

In the 4-4 results the PI and PCI schemes have a reduction in the computed order of accuracy primarily in velocity, and to a lesser extent in pressure and heat ﬂux.

This is consistent with Fig. 5.10, in which the L2 norms for those variables are sub- second order at the larger time steps. However from Fig. 5.10 it is clear that the solutions are trending toward the second order line. The BI scheme is still roughly

ﬁrst order accurate, though the numerical values are slightly above 1. The 16-4 results are very similar to that of the 4-4 results.

109 Table 5.3: Computed orders of accuracies with time-accurate aerothermodynamics.

Case Displacement Velocity Temperature Pressure Heat Flux PI 1-1 2.00 2.00 2.01 2.00 1.99 PCI 1-1 2.00 2.00 1.87 2.00 1.99 SI 1-1 2.00 2.00 1.96 2.00 2.00 BI 1-1 1.00 1.01 1.00 1.00 1.00 PI 4-1 2.00 2.00 2.00 2.00 2.00 PCI 4-1 2.00 2.00 2.00 2.00 1.99 BI 4-1 1.00 0.99 0.98 1.00 1.00 PI 4-4 1.96 1.77 2.02 1.89 1.89 PCI 4-4 1.93 1.77 1.99 1.88 1.88 BI 4-4 1.08 1.22 1.08 1.07 1.06 PI 16-4 1.96 1.78 2.02 1.89 1.89 PCI 16-4 1.94 1.77 2.02 1.89 1.88 BI 16-4 1.05 1.12 1.08 1.05 1.04

110 Chapter 6

Fluid-Thermal-Structural Response:

Quasi-Steady Aerothermodynamics

The impact of the different time marching procedures on the panel response using

quasi-steady aerothermodynamics is investigated for both conﬁgurations. Conﬁg-

uration 1 has a stable response until the material temperature limits are reached,

and Conﬁguration 2 has an in initially stable response which undergoes a snap-

through that initiates ﬂutter.

6.1 Response Study: Conﬁguration 1

Examination on the impact of the different time marching procedures on the panel

response is carried out over a time record of 10 seconds. Time steps between

∆t = 4000µs and ∆t = 62.5µs are considered, based on the convergence of the solutions. The SI scheme solution at the smallest time step considered is used as

111 a benchmark solution. The mid-panel transverse displacement, velocity, and average temperature are shown in Fig. 6.1. The panel heats up and buckles into the

flow, and continues to deform into the flow from the rising temperature. The initial transient vibrations apparent in the velocity damp out to a slow and nearly steady growth of the panel. After roughly 10 seconds, the temperature at the surface of the panel rises above material limits, restricting analysis to the stable region of response. The profiles of displacement, temperature, pressure, and heat flux are in

Fig. 6.2. The pressure and heat flux have similar profiles, with maximum pressure and heating at about 15% of the panel length. The temperature roughly follows the heat flux profile, as expected. The maximum displacement is at about 60% of the panel chord, due to the higher pressure on the front half of the panel.

An important ﬁnding from this study is the dependence of the predicted panel response on the couple scheme used, time step size, and the use of subcycling for the structural time marching. While the panel displacement in Fig. 6.1-6.2 is largely a static response to the rising temperature, ﬁve qualitatively distinct dynamic responses of the higher structural modes are observed. These types of responses are illustrated in the second structural mode solution in Fig. 6.3-6.4, where the converged response from the benchmark solution is shown in red, and the high- lighted behavior is in blue. In the benchmark solution the second mode quickly becomes negative during the initial panel buckling. The initial buckling initiates vibrations in the second mode that eventually damp out, and the second mode response slowly grows negatively.

112 10 550

8 500

6 450 ave w/h T 4 400 Displacement Temperature 2 350

0 300 0 2 4 6 8 10 Time (sec) (a) Average temperature (red) and normalized mid-panel displacement (blue). 20

5 wh(1/sec) ˙w/h 0

-5

-10 0 2 4 6 8 10 Time (sec) (b) Normalized mid-panel velocity.

Figure 6.1: Average temperature, mid-panel displacement, and mid-panel velocity of Conﬁguration 1 over 10 seconds.

113

10 650

600 8 550 t=0s 6 t=10s 500 T (K) w/h 4 450 t=0s t=10s 400 2 350 0 300

0 0.5 1 1.5 0 0.5 1 1.5 x (m) x (m) (a) Normalized displacement. (b) Thickness-averaged temperature.

6000 500

4000 400

t=0s ) 2000 t=10s 2 300 (Pa) b q (kW/m P−P 0 200

−2000 100 t=0s t=10s

−4000 0 0 0.5 1 1.5 0 0.5 1 1.5 x (m) x (m) (c) Net pressure. (d) Heat Flux.

Figure 6.2: Profiles of temperature,displacement, pressure, and heat flux over panel at t = 0 and t = 10s for Configuration 1.

114 The first type of response, in Fig. 6.3(a), is a stable solution that matches with the benchmark. The second type, in Fig. 6.3(b), is a stable solution that overpredicts the transient oscillations. The third is a response with small sustained oscillations that do not damp out, shown in Fig. 6.4(a). The fourth, shown in Fig. 6.4(a), is an unstable response with large, but finitely bounded oscillations. The final type, not shown here, is an immediate divergence of the solution to infinity that arises from inherent numerical instability.

A summary of results is provided by Table 6.1. As expected, the stable response is generally predicted for smaller time steps. The SI scheme solutions are stable for all time steps since the implicit time integrators combined with the strong coupling are inherently numerically stable. However, at the largest time step the SI scheme overpredicts the transient oscillations. The PCI scheme predicts stable solutions at all but the largest time step, if subcycling is not considered. The PI scheme solutions exhibit large unstable oscillations at time steps of 2000 µsec and greater, but are stable for smaller step sizes. Structural subcycling does not change the stability of the response predictions for the PI and PCI schemes, however the schemes overpredict the transient oscillations at structural time steps of 1 msec and above.

The CE scheme solutions diverge to inﬁnity at the largest time step due to the explicit nature of the time integrators used. If subcycling is not used, the response is stable for ∆tS ≤ 2ms, with some improvement over the PI scheme. However, with 25-1 subcycling, the CE scheme exhibits sustained oscillations for any time step larger than 125 µs. The BI scheme always overpredicts the transient oscilla-

115 0.05 PI 1, ∆t=5 ×10 -4 0 SI 1, ∆t=6.25 ×10 -5

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

Modal Displacement per unit thickness -0.35 0 1 2 3 4 5 time (sec) (a) Stable response with correct damping. 0.05 BI 1, ∆t=2.5 ×10 -4 0 SI 1, ∆t=6.25 ×10 -5

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

Modal Displacement per unit thickness -0.35 0 1 2 3 4 5 time (sec) (b) Stable response with underprediction of damping.

Figure 6.3: Examples of stable responses of mode 2 compared to benchmark solution (red dashed line) for Conﬁguration 1.

116 0.1 BI 1, ∆t=5 ×10 -4 SI 1, ∆t=6.25 ×10 -5 0

-0.1

-0.2

-0.3

Modal Displacement per unit thickness -0.4 0 1 2 3 4 5 time (sec) (a) Neutrally stable response with no damping. 6

BI 1, ∆t=2 ×10 -3 2 SI 1, ∆t=6.25 ×10 -5

-2

-4

Modal Displacement per unit thickness -6 0 1 2 3 4 5 time (sec) (b) Unstable response with large but bounded oscillations.

Figure 6.4: Examples of stable responses of mode 2 compared to benchmark solution (red dashed line) for Conﬁguration 1.

117 tions over the range time steps considered, and at larger time steps dynamically unstable responses are observed. Similar to the CE scheme, the use of subcycling in the BI scheme introduces sustained oscillations at smaller time steps.

Table 6.1: Characteristic behavior of responses for Conﬁguration 1.

SI PCI PI BI CE ∆tS (µs) 1-1 1-1 25-1 1-1 25-1 1-1 25-1 1-1 25-1 8000 S* U U U U U U D D 4000 S S S* U U U U D D 2000 S S S* U U U U S O 1000 S S S* S S* U U S O 500 S S S S S O O S O 250 S S S S S S* O S O 125 S S S S S S* S* S S 62.5 S S S S S S* S* S S S: Stable response (Fig. 6.3(a)). S*: Stable response with underdamped vibrations (Fig. 6.3(b)). O: Sustained oscillations (Fig. 6.4(a)). U: Large unstable oscillations (Fig. 6.4(b)). D: Diverges at start of solution.

6.2 Response Study: Conﬁguration 2

The impact of the different time marching procedures on the panel response over a time record of 35 seconds is assessed for this study. Time steps between ∆tS =

1000µs at the coarsest and ∆tS = 7.8125µs at the finest are considered. The SI scheme at the smallest time step is used as the benchmark solution, and the mid- chord transverse displacement and average temperature is shown in Fig. 6.5. Sim- ilar to Configuration 1, the panel heats up and buckles into the flow. At approxi-

118 15 450 Displacement 10 Temperature 412.5 5

0 375 ave w/h T

-5 337.5 -10

-15 300 0 10 20 30 Time (sec)

Figure 6.5: Average temperature (red) and normalized mid-panel displacement (blue) of Configuration 2 over 35 seconds. mately 24.5 seconds, the panel snaps out of the flow due to the increasing pressure loading, consistent with the results from [96]. Profiles of displacement, temperature, pressure, and heat flux immediately before the snap-through event are shown in Fig. 6.6. Note the maximum displacement of the panel in Fig. 6.6(a) occurs at about 65% of the length; the high pressure distribution at the front half of the panel in Fig. 6.6(c) pushes this point further back until it snaps out of the flow. Cen- ter displacement during the initial snap-through, and at the end of the analysis is shown in Fig. 6.7. This snap-through event causes the panel to enter a dynamic

ﬂutter instability in Fig. 6.7(a), and eventually settles into a stable limit cycle oscillation (LCO), as shown in Fig. 6.7(b). Note the Fig. 6.7(a) is over 1 second, while

Fig. 6.7(b) has a smaller window of .05 seconds.

119 14 420 t=0s 12 t=24.5s 400

10 380 8 360 w/h 6 T (K) t=0s 340 t=24.5s 4 320 2 300

0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (a) Normalized displacement. (b) Thickness-averaged temperature.

30 1000

500 25 ) 0 2

(Pa) 20 b

t=0s q (kW/m

P−P −500 t=24.5s 15 −1000 t=0s t=24.5s

−1500 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (c) Net pressure. (d) Heat Flux.

Figure 6.6: Profiles of temperature,displacement, pressure, and heat flux over panel at t = 0 and t = 24.5s (immediately prior to snap-through) for Configu- ration 2.

120 15

0 w/h

-5

-10

-15 24.5 25 25.5 Time (sec) (a) Initial snap-through (24.5 sec).

0 w/h

-5

-10

-15 34.95 34.96 34.97 34.98 34.99 35 Time (sec) (b) Flutter limit cycle (35 sec).

Figure 6.7: Center displacement during initial snap-through and final flutter response for Configuration 2.

121 The coupling schemes are compared through analysis of the post-instability limit cycle response of the panel. The displacement at the 3/4 chord position is shown in Fig. 6.8(a) over the final 0.02 seconds of response for all five coupling schemes. Between the five solutions, there exist two distinct classes of response predictions: a limit cycle predicted by the PI, PCI, and SI (benchmark) solutions dominated by the second and third structural modes (denoted as “LCO 2-3”); and a limit cycle of relatively smaller magnitude predicted by the CE and BI solutions dominated by the third and fourth structural modes (“LCO 3-4”). The oscillation amplitude of the PI and PCI solutions are within 0.02% of the benchmark SI solution, while the amplitude of the LCO predicted by the BI and CE schemes is about

60% smaller. The power spectral density of the response, in Fig. 6.8(b), further illustrates the differences in the two types of response. The dominant frequency of the PI solution is within 1% of the benchmark solution, while the BI and CE are about 16% and 18% higher, respectively. Finally, the spatio-temporal content of the LCOs are compared using the x-t diagrams in Fig. 6.9. The peak magnitude of the oscillations occurs near the 1/4 and 3/4 chord locations for both LCOs, with higher peaks for LCO 2-3. The peak displacements are also almost completely out of phase for LCO 2-3, while they are almost completely in phase for LCO 3-4.

The post-ﬂutter behavior over a range of time steps of each coupling scheme at both 1-1 time-stepping and 25-1 subcycling is summarized in Table 6.2. Four general classes of behavior are observed: LCO 2-3 from Fig. 6.9(a) (1); LCO 3-4 from Fig. 6.9(b) (2); other types of limit cycles and non-periodic transient motion

122 10

SI−1, ∆t=7.8125×10−6 −5 PI−1, ∆t=1.25×10−4 −4 Displacement per unit thickness BI−1, ∆t=1.25×10 CE−1, ∆t=1.25×10−4

−10 34.98 34.985 34.99 34.995 time (sec) (a) 3/4 chord displacement

2 10 SI−1, ∆t=7.8125×10−6 ∆ × −4 0 PI−1, t=1.25 10 10 BI−1, ∆t=1.25×10−4 CE−1, ∆t=1.25×10−4 −2 10

−4 10

PSD (1/Hz) −6 10

−8 10

−10 10 0 200 400 600 800 1000 Frequency (Hz) (b) PSD of 3/4 chord displacement

Figure 6.8: Displacement and power spectral density at 3/4 chord length.

123 (a) Limit Cycle Oscillation 2-3

(b) Limit Cycle Oscillation 3-4

Figure 6.9: x-t diagram of post-ﬂutter limit cycle oscillations.

124 (O); or divergence towards inﬁnity at the start of the solution (D). At time steps

250 µs and larger, neither LCO 2-3 nor LCO 3-4 is observed for any of the coupling schemes. At smaller time steps, the PI, PCI, and SI schemes always predict LCO

2-3, either with or without subcycling. The CE scheme also predicts LCO 2-3 at the smallest time steps, but either limit cycle may be present at larger time steps.

Subcycling also results in prediction of LCO 3-4 a smaller time step for the CE scheme. The BI scheme predicts both LCO 2-3 and LCO 3-4 over most of the range of time steps, without a clear trend emerging. In all but the BI scheme, LCO 2-3 always exists at the smallest time steps.

Table 6.2: Post-ﬂutter behavior of Conﬁguration 2. SI PCI PI BI CE ∆tS (µs) 1-1 1-1 25-1 1-1 25-1 1-1 25-1 1-1 25-1 1000 O O O O O O O D D 500 O O O O O O O D D 250 O O O O O O O O D 125 2-3 2-3 2-3 2-3 2-3 3-4 2-3 3-4 2-3 62.5 2-3 2-3 2-3 2-3 2-3 2-3 3-4 2-3 2-3 30.25 2-3 2-3 2-3 2-3 2-3 3-4 3-4 2-3 3-4 15.625 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 2-3 7.8125 2-3 2-3 2-3 2-3 2-3 3-4 3-4 2-3 2-3 2-3: Limit cycle oscillation dominated by structural modes 2 and 3 3-4: Limit cycle oscillation dominated by structural modes 3 and 4 O: Other limit cycle or non-periodic motion D: Diverges at start of solution

125 6.3 Computational Expense

The computational expense of the analysis is strongly dependent on the coupling

scheme and time steps used. The loosely coupled schemes are cheaper than the

strongly coupled scheme for a given time step size due to the lack of inter-solver

subiterations. A larger time step combined with subcycling can also substantially

improve the computational expense of the response simulation, allowing for in-

creased simulation times or more complex models. Computational cost is pro-

portional to the number of calls to both the structural and thermal solvers, and is

computed as:

−1 ∆tT tsim = t0 + CSN + CT N × (6.1) ∆tS

where tsim is the total simulation time, t0 is overhead time, CS is cost of the structural solver (including pressure model) per time step, CT is the cost of the thermal solver (including heat ﬂux model) per time step, and N is the total number of time steps. Note that any time spent transferring information between the thermal and structural models is considered part of CT , since the transfers only occur at the thermal time steps.

The values for t0, CS, and CT are listed in Table 6.3. Note these values are estimated using a least-squares ﬁt of the computational expenses of the solutions from the previous sections. Conﬁguration 1 is more computationally expensive

126 than Configuration 2 because of the use of CFD surrogates, which are more expensive than the analytical aerothermodynamic models [79, 91]. For Configuration 1 the fluid models dominate the solution time compared to the thermal and structural solvers; the converse is true for Configuration 2. Note that for Configuration

1, the thermal solution is about twice as expensive as the structural solution since the heat ﬂux surrogate is about twice as expensive as the pressure surrogate. For

Conﬁguration 2, the thermal solution is about four times as expensive as the structural solution due to the large number of degrees of freedom in the thermal solver compared to the structural solver.

The cost of the PI and BI schemes are nearly identical since the same time integrators are used, and the extra time spent associated with computing the predictors and interpolations for the PI scheme is small. The PCI scheme has the same cost as the PI and BI schemes for Conﬁguration 1 due to the surrogate models dominating the solution time. For Conﬁguration 2, the PCI scheme is more expensive than the

PI and BI schemes since it requires an additional correction step to the thermal and structural solvers. Note that the structural correction step only increases the structural cost by about 6%, while the thermal correction step increases the thermal cost by 75%. While the CE scheme uses explicit time integrators, the structural cost is about four times more expensive compared to the BI and PI schemes since the pressure and acceleration are computed four times per time step for the Runge-

Kutta algorithm. The SI scheme is about three to four times more expensive than the PI and BI schemes due to the additional subiterations needed to strongly cou-

127 ple the solvers. In general, this scheme requires three subiterations per time step to converge the pressure and heat ﬂux, although during signiﬁcant oscillations more subiterations may be required, making the solution more expensive.

Table 6.3: Computational cost parameters (milliseconds)a Conﬁguration 1 Conﬁguration 2 Scheme t0 CS CT t0 CS CT Conventional Explicit 4500 200 100 160 2.7 2.2 Basic Implicit 4500 50 100 160 0.66 2.4 Predictor Implicit 4500 50 100 160 0.68 2.5 Predictor-Corrector Implicit 4500 50 100 160 0.75 4.2 Strong Implicit 4500 150 300 160 2.7 7.0 a Single core, 2.66 GHz Intel Xeon x5650.

The computational times for the solution at the largest acceptable time step for each coupling scheme, with and without subcycling, is summarized in Table 6.4 for Configuration 1 and Table 6.5 for Configuration 2. Note that an acceptable solution is considered to be a stable response for Configuration 1, and a response that predicts LCO 2-3 for all smaller time steps for Configuration 2. For Config- uration 1, the PCI scheme without subcycling is the most efficient, however the

PCI and PI schemes with subcycling are still more efficient than any of the other schemes. The PCI scheme is 180% more efficient than the best PI scheme solution time, and 280% more efficient than the next fastest solution, from the CE scheme without subcycling. The PI scheme with subcycling is 40% more efficient than the

CE scheme. For Configuration 2, the PI and PCI schemes with 25-1 subcycling are the most efficient. Without subcycling, the PI scheme is 50% more efficient than the

128 PCI scheme, and 200% more efﬁcient than the CE or SI schemes. With subcycling, the PI and PCI schemes have similar computational costs, and are about 1000% more efﬁcient than the CE or SI schemes. Note that the BI scheme does not produce acceptable solutions at the range of time steps tested, so the computational time is greater than the values listed in Tables 6.4 and 6.5.

Table 6.4: Computation time for 10 seconds of response for Conﬁguration 1a 1-1 25-1 Scheme ∆tS (µs) Time (min) ∆tS (µs) Time (min) Conventional Explicit 2000 25 125 270 Basic Implicit b <62.5 >500 <62.5 >144 Predictor Implicit 1000 26 500 18 Predictor-Corrector Implicit 4000 6.5 500 18 Strong Implicit 4000 30 N/A N/A a Single core, 2.66 GHz Intel Xeon x5650. b BI scheme solutions overpredict oscillations at smallest time step.

Table 6.5: Computation time for 35 seconds of response for Conﬁguration 2a 1-1 25-1 Scheme ∆tS (µs) Time (min) ∆tS (µs) Time (min) Conventional Explicit 62.5 44 15.625 104 Basic Implicitb <7.1825 >240 <7.1825 >60 Predictor Implicit 125 15 125 4.0 Predictor-Corrector Implicit 125 23 125 4.5 Strong Implicit 125 45 N/A N/A a Single core, 2.66 GHz Intel Xeon x5650. b BI scheme solutions predict incorrect limit cycle at smallest time step.

129 Chapter 7

Fluid-Thermal-Structural Response: Time

Accurate Aerothermodynamics

The impact of the coupling schemes on the panel response using time-accurate aerothermodynamics is investigated for Configuration 2. First, the numerical stability and relative accuracy of the coupling schemes on the initially dynamically stable panel response is characterized. In a second study, the panel is prescribed with an elevated temperature distribution, and the resulting flutter response is analyzed. The work balance between the fluid and structural solvers is also compared from these responses. A long time record analysis of the panel starting from an undeformed configuration to the post-flutter response is performed. Finally, the computational costs between the different coupling schemes are compared. Note that in these analyses, a single time step is considered for all solvers because the

RANS-based ﬂow solver is stable for large time steps and the ﬂow is nearly quasi- steady.

130 7.1 Initial Stable Response

The numerical stability and relative accuracy of the Predictor Implicit (PI), Predictor-

Corrector Implicit (PCI), Basic Implicit (BI), and Strong Implicit (SI) schemes are

compared for large time steps on the physically stable initial response of the panel

over one second. Eight time step sizes are considered, ranging from 0.01 seconds to 7.8125 × 10−4 seconds at the smallest:

1i−1 ∆ti = 10−1 × (7.1) S 2 where i is the time step level between 1 and 8.

The SI scheme at the smallest time step is used as the benchmark “truth” solution. The mid-panel displacement, velocity, and average temperature are shown in

Fig. 7.1 for one second. The panel immediately buckles into the ﬂow and continues to rise with time due to the increasing temperature. The response is similar to the simulation with quasi-steady analytical aerothermodynamics in Fig. 5.2, although the temperature is about .25 Kelvin less at one second, and the initial vibrations of the panel are more damped in Fig. 7.1.

The error for the mid-panel displacement relative to the benchmark solution is deﬁned as:

131 2.5 304

2 303

1.5

302 ave w/h T 1 Displacement Temperature 301 0.5

0 300 0 0.2 0.4 0.6 0.8 1 Time (sec) (a) Average temperature (red) and normalized mid-panel displacement (blue). 7

wh(1/sec) ˙w/h 2

-1 0 0.2 0.4 0.6 0.8 1 Time (sec) (b) Normalized mid-panel velocity.

Figure 7.1: Average temperature, mid-panel displacement, and mid-panel velocity of panel with time-accurate aerothermodynamics from the benchmark solution.

132 n n n w − wt w(%) = s × 100 (7.2) 1 N P (wn2 ) N n=1 where n is the time step counter, w is the mid-panel displacement, and wt is the benchmark solution (SI scheme with ∆t = 7.8125 × 10−4 sec) mid-panel displace-

ment. The error is normalized by the root mean square of the benchmark displace-

ment over the time period.

The relative errors over one second for select time step levels are shown in

Fig. 7.2 for the SI, BI, PI, and PCI schemes. The ﬁrst through ﬁfth time step level

errors are included for the SI scheme in Fig. 7.2(a). Only the largest time step, ∆t =

.1 sec (blue), is unstable; the structural solution jumps between the two statically-

stable equilibriums from the buckling bifurcation. At all smaller time steps the SI

scheme is stable and has relatively small errors. The BI scheme, with time step

levels 5 through 8 illustrated in Fig. 7.2(b), is unstable for a much wider range of

time steps. Only the smallest two time steps have a stable response. Even at the

smallest time step level, ∆t = 7.8125×−4 (orange), has a relatively large oscillations in the error. For the PI scheme in Fig. 7.2(c), time step levels 4 through 8 are shown.

The solution at ∆t = 1.25×10−2 (black) is unstable, while smaller time steps remain stable over the solution time. Time step levels three to seven are given for the PCI scheme in Fig. 7.2(d). At ∆t = 1.25 × 10−2 sec (black) and smaller, the solution remains stable with relatively small errors. The stability of the PI and PCI schemes are signiﬁcantly better than that of the BI scheme in Fig. 7.2(b), and the PCI scheme

133 is stable at double the time step compared to the PI scheme.

5 5

(%) 0 (%) 0 w w ǫ ǫ ∆t =1 ×10 -1 ∆t =5 ×10 -2 ∆t =6.25 ×10 -3 ∆t =2.5 ×10 -2 ∆t =3.125 ×10 -3 ∆t =1.25 ×10 -2 ∆t =1.5625 ×10 -3 ∆t =6.25 ×10 -3 ∆t =7.8125 ×10 -4 -5 -5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (sec) time (sec) (a) Strong Implicit, time step levels 1 to 5. (b) Basic Implicit, time step levels 5 to 8.

5 5

(%) 0 (%) 0 w w ǫ ǫ ∆t =1.25 ×10 -2 ∆t =2.5 ×10 -2 ∆t =6.25 ×10 -3 ∆t =1.25 ×10 -2 ∆t =3.125 ×10 -3 ∆t =6.25 ×10 -3 ∆t =1.5625 ×10 -3 ∆t =3.125 ×10 -3 ∆t =7.8125 ×10 -4 ∆t =1.5625 ×10 -3 -5 -5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (sec) time (sec) (c) Predictor Implicit, time step levels 4 to 8. (d) Predictor-Corrector Implicit, time step levels 3 to 7.

Figure 7.2: Relative errors of the mid-panel displacement compared to benchmark solution for SI, BI, PI, and PCI schemes over select time step sizes.

The total error of each scheme at each time step is given in Table 7.1, which is computed using Eq. (7.3). Errors greater than 1% are colored in red, corresponding with solutions that are unstable; errors smaller than 0.1% are colored in blue. Note that several of the unstable solutions were halted by the CFD solver after the ﬂuid

134 solution diverged. As expected, the BI scheme performs the worst, and the SI performs the best. The BI scheme is only stable for the smallest two time step levels, and the errors are still above 0.1% at the ﬁnest resolution. The PI scheme shows signiﬁcant improvement over the BI scheme, with 4 stable levels and 2 below 0.1% error. The PCI scheme is a further improvement with 5 stable levels, with 3 below

0.1% error. In general, the errors of the PCI scheme are comparable to the errors of the PI scheme at the next time step level. Finally, the SI scheme is the most stable and accurate, with all but the coarsest time step remaining stable. However, at smaller time steps the strong coupling does not show improvement over the

PCI scheme. In fact, to achieve 0.1% error the PCI scheme and SI scheme required the same time step. While the SI scheme provided enhanced stability, as expected, the coarser time step solutions required more subiterations for convergence. The computational costs are discussed in detail in Section 7.4.

v u N u 1 X n 2 Ew = t ( ) (7.3) N w n=1

7.2 Flutter Response with Elevated Initial Temperature

The response of the panel undergoing a ﬂutter instability for the various coupling schemes over a range of time steps is investigated next. Based on the response of the panel with quasi-steady analytical aerothermodynamic models in Section 6.2,

135 Table 7.1: Relative errors of mid-panel displacement compared to benchmark. Val- ues in red indicate errors larger than 1%; values in blue indicate errors under 0.1%.

∆tS (ms) BI PI PCI SI 100 126 168 179 160 50 150 150 175 .91 25 161 189 2.08 0.62 12.5 142* 81.4* 0.550 0.47 6.25 83.4* 0.61 0.154 0.16 3.125 2.79 0.13 0.063 0.063 1.5625 0.790 0.045 0.036 0.036 0.78125 0.330 0.0093 0.00150 † * CFD solver diverged during fluid subiterations. † Benchmark solution. the panel is expected to remain stable for an extended period of time before snap- ping out of the flow and undergoing flutter. The structural response during the stable period is predominantly static. To reduce the computational cost of the simulations, the structure is prescribed with an initial elevated temperature such that the panel is near the point of snap-through.

7.2.1 Initialization: Static Aeroelastic Equilibrium

The temperature distribution used as an initial condition for the thermal solver is extracted from the simulation using the quasi-steady analytical aerothermodynamic models from a speciﬁc point in time. With the temperature prescribed, a static aeroelastic simulation is performed to generate an initial condition in equilibrium for the ﬂuid-thermal-structural analysis. The static aeroelastic simulation is performed by a simple algorithm:

136 1. Prescribe a temperature distribution on the panel and use the quasi-steady analytical aerothermodynamics solution as an initial estimate for the structural displacement.

2. Deform the CFD mesh to conform the deformed panel and compute the aerodynamic pressure.

3. Prescribe the updated pressure on the panel and compute the new static deformation.

4. Repeat steps 2-3 until the displacement converges.

If the panel bows out of the ﬂow during any iteration, then the panel loads are too great and the panel has snapped through. In this case, the temperature distribution is chosen from an earlier point in time from the quasi-steady simulation.

Once an acceptable solution is found, a transient aeroelastic simulation with a is performed with a constant-in-time panel temperature distribution, until the resid- uals from the CFD solver drop to machine precision. This solution is then used as a steady-state initial condition for the ﬂutter analysis.

The temperature distribution chosen as the initial condition near snap-through is shown in Fig. 7.3. The average temperature across the panel is 366 K, corresponding to a time of 17.5 seconds in the quasi-steady simulation. Note this is sig- niﬁcantly different than the snap-through time of 24.5 seconds in the quasi-steady simulation. The primary difference is from the pressure loading computed by the

CFD solver compared to the quasi-steady simpler analytical aerothermodynamic model. The displacements, heat ﬂux, and pressure from the aeroelastic equilibrium

137 380

375

370

365

T (K) 360

355

350

345 0 0.2 0.4 0.6 0.8 1 x (m)

Figure 7.3: Prescribed temperature distribution from quasi-steady analytical aerothermodynamics at t = 17.5 sec. solution are shown in Fig. 7.4 for both the CFD-based solution and quasi-steady

fluid solution. The flowfield pressure is illustrated in Fig. 7.5. The displacement is more depressed on the front half of the panel for the CFD solution compared to the quasi-steady analytical solution; the shape is similar to the quasi-steady solution just prior to snap-through in Fig. 6.6(a). The pressure load from the CFD solution has a higher peak pressure near the middle of the panel and higher pressures along the entire back half. This higher pressure near the peak pushes the panel out of the

ﬂow further than the quasi-steady pressure load, which is the cause of the earlier snap-through. Finally, note the heat ﬂux is similar for both solutions except near the front of the panel, due to the differences in the panel deformation.

138 12 1000 Approximate 10 CFD

500 8

6 (Pa) w/h b 0 p−p 4 −500 2 Approximate CFD

0 −1000 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (a) Normalized displacement. (b) Net Pressure.

25 )

2 20

q (kW/m 15

10 Approximate CFD

5 0 0.2 0.4 0.6 0.8 1 x (m) (c) Heat Flux.

Figure 7.4: Initial panel solution with prescribed temperature distribution in aeroelastic equilibrium, compared to quasi-steady analytical aerothermodynamics.

139 Figure 7.5: Flowﬁeld pressure of static aeroelastic solution

140 7.2.2 Flutter Analysis

The panel snap-through and subsequent ﬂutter is analyzed over a range of time steps for the various coupling schemes for three seconds of response. Five time steps are considered: ∆t = 500µs, ∆t = 250µs, ∆t = 125µs, ∆t = 62.5µs, and

∆31.25µs.

The SI scheme at ∆t = 31.25µs is used as the benchmark case. The average temperature, center displacement, and 3/4 chord displacement are shown in Fig. 7.6.

At the start of the analysis, the panel is in stable aeroelastic equilibrium. How- ever, the small increase in displacement and pressure from the rising temperature causes the panel to snap through and enter ﬂutter almost immediately. Like the quasi-steady response in Fig. 6.5, the panel exhibits aperiodic transient behavior for a period of time before settling into a limit cycle. The limit cycle has maximum amplitude near the 1/4 and 3/4 positions on the panel at about ±7.5 times the panel thickness, and has a dominant frequency of about 90 Hz.

The ﬂutter displacement envelope and power spectral density for displacement at the 3/4 chord location is shown for the four coupling schemes at ∆t = 62.5µs in

Fig. 7.7. The flutter envelope represents the bounds of the flutter oscillations, and is computed using a moving min/max filter. In Fig. 7.7(a) the SI, PI, and PCI all have a similar flutter boundary consistent with the benchmark solution in Fig. 7.6(c).

The BI scheme solution, has a signiﬁcantly smaller displacement envelope than the others. The limit cycle predicted by the PI, PCI, and SI schemes is dominated

141 375 15

370 0 w/h T (K)

-5

-10

365 -15 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time (sec) time (sec) (a) Average temperature. (b) Normalized mid-panel displacement.

0 w/h

-5

-10

-15 0 0.5 1 1.5 2 2.5 3 time (sec) (c) Normalized 3/4 panel displacement.

Figure 7.6: Temperature rise and panel displacement during snap-through and subsequent ﬂutter.

142 by the second and third structural modes (“LCO 2-3”), while the one predicted by the BI scheme is dominated by the third and fourth modes (“LCO 3-4”). The PSD of the response between two and three seconds in Fig. 7.7(b) further illustrates the differences in the response. The PI, PCI, and SI schemes all have the same dominant frequency centered at 90 Hz, while the BI scheme has a sharper dominant frequency at 125 Hz. Note that there still exists signiﬁcant noise is the PSD for the

SI, PI, and PCI solutions, which is caused from transients in the initial aperiodic response of the snap-through. If the simulations are carried out for a longer time record the transients are expected to damp out, as shown later in Section 7.3.

Space-time contour plots of the normalized panel displacement over the last

0.05 seconds of response for the solutions are shown in Fig. 7.8. As expected, the SI,

PI, and PCI solutions have the same trend, while the BI solution is fundamentally different. The displacements are smaller across the entire panel for the BI scheme, and the maximum displacement occurs at the 15% and 85%. For the SI, PI, and

PCI schemes, positive displacement on the front half of the panel coincides with negative displacement on the back half at any given point in time. For the BI scheme, positive displacement happens at the same time and the front and back of the panel, while the middle of the panel is negative. Note that these LCOs are qualitatively similar to the LCOs seen in the quasi-steady response in Fig. 6.9.

The results for the time steps considered are summarized in Table 7.2 for each coupling scheme. Three distinct types of response are observed: divergence of the

ﬂuid solver at the subiteration level (DIV); the limit cycle with structural modes 2

143 15

5 SI BI 0

w/h PI PCI -5

-10

-15 0 0.5 1 1.5 2 2.5 3 Time (sec) (a) Normalized Flutter displacement envelope.

10 2 SI BI 10 0 PI PCI

10 -2

10 -4 PSD (1/Hz)

10 -6

10 -8 0 200 400 600 800 1000 ω (Hz) (b) Normalized Power Spectral Density.

Figure 7.7: Flutter displacement envelope and PSD at 3/4 chord location of SI, BI, PI, and PCI schemes at ∆t = 62.5µs.

144 (a) SI scheme. (b) BI scheme.

Figure 7.8: Space-time contours of normalized displacement over .05 seconds for SI (benchmark), BI, PI, and PCI schemes, ∆t = 62.5µs.

145 and 3 (2-3); and the limit cycle with modes 3 and 4 (3-4). At ∆t ≥ 500µs the ﬂuid solver fails to converge during the ﬂutter response for every coupling scheme at around one second. At smaller time steps, the SI, PI, and PCI scheme all predict

LCO 2-3. The BI scheme, however, does not consistently predict the same limit cycle over the range of time steps. At larger time steps, it predicts the LCO 2-3, but the solutions at ∆t = 62.5µs and ∆t = 31.25µs are LCO 3-4. The BI scheme was also simulated at a smaller time step ∆t = 15.625µs, and recovered LCO 2-3. This behavior is in agreement with the trend seen with the quasi-steady ﬂuid models in

Section 6.2, in which the BI scheme did not reliably converge towards a consistent limit cycle with decreasing time step, while the other schemes did.

Table 7.2: Post-ﬂutter behavior for each coupling scheme at different time steps.

∆tS (µs) BI PI PCI SI 500 DIV DIV DIV D IV 250 2-3 2-3 2-3 2-3 125 2-3 2-3 2-3 2-3 62.5 3-4 2-3 2-3 2-3 31.25 3-4 2-3 2-3 2-3 15.625 2-3 - - - DIV: CFD solver diverged during ﬂuid subiterations. 2-3: Limit cycle oscillation dominated by structural modes 2 and 3. 3-4: Limit cycle oscillation dominated by structural modes 3 and 4.

7.2.3 Fluid-Structure Work Balance

Loosely coupled schemes are not capable of perfectly matching the work/energy exchange across the ﬂuid-structure interface. Strongly coupled schemes can per-

146 fectly match the exchange only in the limit of inﬁnite subiterations. The addition

of ﬁctitious energy can destabilize the solution, thus it is important to minimize

the violation of energy between the ﬂuid and structural solvers [40]. The work

performed by each solver over a single time step is estimated by [42]:

L Wn+1 ≈ Wn + wn+1 − wn| pn+1 + pn (7.4a) S S 2 S S S S L Wn+1 ≈ Wn − wn+1 − wn | pn+1 + pn (7.4b) F F 2 F F F F

n+1 where S is the work done for either the ﬂuid (F ) or structural (S) solver, and L is

n n n n+1 n the panel length. The work violation is W = WS +WF . Note that pS is simply pF

n+1 n+1 for the BI scheme, the pressure predictor pp for the PI scheme, and pF for the

PCI scheme. The displacements for the BI, PI and SI schemes are the same between the ﬂuid and structural solvers, while for the PCI scheme the ﬂuid displacement is based on the initial predicted displacement, and the structural displacement is

n+1 n+1 n+1 based on the corrected displacement wc . For the SI scheme pS is pF from the second to last subiteration at each step. For a strongly coupled scheme the work violation converges to zero as the pressure converges between subiterations.

The work violation over one second of response for time steps ranging from

500µs to 62.5µs for the BI, PI, PCI, and SI schemes is shown in Fig. 7.9. Note that the violation is normalized by the work done by the structural solver. In general the violation increases with time; the largest rise occurs as the panel begins to snap through during the ﬁrst 0.2 seconds, and then the growth stalls as the panel

147 reaches a limit cycle oscillation. The BI scheme clearly performs the worst. Even at

the smallest time step, the work violation is at about 10%. The PI scheme has some

improvement over the BI scheme, with a work violation at around 0.015% at the

smallest time step. The PCI scheme has the smallest work violation at about .01%

even at the largest time step, and about 1×10−8 at the smallest. The SI scheme does has a relatively consistent work violation between .0001%-.01%, since the tolerance to terminate the subiterations in Eq. 3.13 requires fewer subiterations at smaller time steps. The convergence rate for each scheme is shown in Fig. 7.10 at one second. The BI scheme has a sub-ﬁrst-order rate of convergence at about 0.9, while the PI and PCI schemes have a rate of about 2.6 and 4.5, respectively. Note that the SI scheme has the lowest work violation at the largest time step, but the PCI scheme has better agreement at the smallest time steps. This is at least partially due to the difference in the source of the work violation. In the SI scheme the violation arises from the mismatch of the ﬂuid loads, while in the PCI scheme the violation is from the mismatch of the structural loads.

7.3 Long Time Record Analysis

Based on the ﬂutter response using an elevated initial temperature to determine an appropriate coupling scheme and time step, a long-time record analysis of the panel response starting at the initial stress-free temperature is carried out. Since the SI, PI and PCI schemes all predicted the same behavior over the range of time

148 10 0 10 0

10 -2 10 -2

10 -4 10 -4 ------

S -6 S -6 W W

W 10 W 10 ------

∆t =5 ×10 -4 ∆t =5 ×10 -4 10 -8 10 -8 ∆t =2.5 ×10 -4 ∆t =2.5 ×10 -4 ∆t =1.25 ×10 -4 ∆t =1.25 ×10 -4 10 -10 10 -10 ∆t =6.25 ×10 -5 ∆t =6.25 ×10 -5

10 -12 10 -12 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (sec) time (sec) (a) BI scheme. (b) PI scheme.

10 0 10 0

∆t =5 ×10 -4 ∆t =5 ×10 -4 10 -2 10 -2 ∆t =2.5 ×10 -4 ∆t =2.5 ×10 -4 ∆t =1.25 ×10 -4 ∆t =1.25 ×10 -4 10 -4 10 -4 ∆t =6.25 ×10 -5 ∆t =6.25 ×10 -5 ------S S -6 -6 W W W W 10 10 ------

10 -8 10 -8

10 -10 10 -10

10 -12 10 -12 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (sec) time (sec) (c) PCI scheme. (d) SI scheme.

Figure 7.9: Work violation over one second, normalized by work done by structural solver, for the BI, PI, PCI, and SI schemes.

149 0

-2 - - -

S -4 W W - - - 10

log -6 BI PI -8 PCI SI -10 -4.4 -4.2 -4 -3.8 -3.6 -3.4 -3.2 log (∆ t) 10

Figure 7.10: Normalized work violation at one second at each time step size for the BI, PI, PCI, and SI schemes. steps in Table 7.2, the simplest scheme, PI, is chosen for the analysis with a time step of 125µs. The simulation is carried out for 30 seconds of response.

The average temperature, mid-panel displacement, and 3/4 chord panel displacement are shown in Fig. 7.11. As before, the panel heats up and buckles into the ﬂow, and at approximately 21 seconds, snaps out of the ﬂow and begins to

ﬂutter and settles into LCO 2-3. Note that average temperature of the panel at snap-through is 372 K, approximately 6 K higher than the average temperature rise used as an initial condition in the previous short time record ﬂutter simulation.

The magnitudes of the oscillations at the mid-panel and 3/4 chord agree with those from the short time record simulation in Fig. 7.6(b-c). The power spectral density of the 3/4 chord panel displacement from the long and short time record simulations are shown in Fig. 7.12. Note that the PSD for the short time record simulation

150 is computed between the 2 and 3 seconds, while the PSD for the long time record simulation is computed between 23 and 24 seconds, which corresponds to 2 to

3 seconds after snap-through. The two solutions have good agreement, with the peak frequency at about 90 Hz.

400 15

10 380

5 360 0 w/h T (K) 340 -5

320 -10

300 -15 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time (sec) time (sec) (a) Average Temperature. (b) Normalized mid-panel displacement.

0 w/h

-5

-10

-15 0 5 10 15 20 25 30 time (sec) (c) Normalized 3/4 chord panel displacement.

Figure 7.11: Average temperature, mid-panel displacement, and 3/4 chord panel displacement of long-time record analysis with time-accurate CFD.

The prediction from the unsteady CFD ﬂow solver is compared to predictions from the quasi-steady analytical aerothermodynamic predictions in Chapter 6 and

151 10 2 Long Time Record (PI) Short Time Record (SI)

10 0

10 -2 PSD (1/Hz)

10 -4

10 -6 0 200 400 600 800 1000 ω (Hz)

Figure 7.12: Power Spectral density of ﬂutter between long time record simulation and short time record simulation (SI scheme, ∆t = 62.5µs). a prediction using quasi-steady surrogate models based on the same CFD grid.

Note that these surrogates are designed similar to the ones described for Con-

ﬁguration 1 in Chapter 4, though for the current geometry and ﬂow conditions.

A comparison of the three responses is shown in Figs. 7.13-7.14. The maximum, minimum, and average temperatures over the 30 second time period is shown in

Fig. 7.13(a). The unsteady CFD and surrogate predictions show excellent agreement, while the temperatures are generally higher for the quasi-steady analytical aerothermodynamics prediction. At 30 seconds, the mean and maximum temperature rise predicted using the analytical aerothermodynamics model, is about 7% higher than that predicted using the CFD and surrogate solutions. Note that when

ﬂutter begins, there is an abrupt change in the minimum and maximum tempera-

152 ture rate of change; though it is not as apparent in the mean temperature.

420 15

400 10

380 5

360 0 w/h T (K)

340 -5

Max 320 -10 CFD Mean Analytic Min Surrogate 300 -15 0 5 10 15 20 25 30 0 5 10 15 20 25 30 time (sec) time (sec) (a) Minimum, maximum, and average tempera- (b) Mid-panel ﬂutter displacement envelope. ture.

15 10 2 CFD 10 Analytic 10 0 Surrogate 5 10 -2 0 w/h 10 -4 -5 PSD (1/Hz)

CFD 10 -6 -10 Analytic Surrogate -15 10 -8 0 5 10 15 20 25 30 0 200 400 600 800 1000 time (sec) ω (Hz) (c) 3/4 chord ﬂutter displacement envelope. (d) PSD of 3/4 chord (29.5-30 sec).

Figure 7.13: Temperature, ﬂutter displacement envelope, and power spectral density of panel with unsteady CFD aerothermodynamics (blue), quasi-steady surrogate aerothermodynamics (red), and quasi-steady analytical aerothermodynamics (black).

From displacement envelope between the three predictions in Fig. 7.13(b-c), the CFD and surrogate models predict a consistent time to ﬂutter at approximately

21 seconds, while the analytical aerothermodynamics model predicts the time to

ﬂutter at 24.5 seconds. Despite the fact that the CFD-based prediction has lower

153 temperatures, it predicts flutter about 3.5 seconds ahead of the analytical aerothermodynamics solution. Profiles of the displacement, temperature, and fluid loads immediately prior to flutter for the three predictions are shown in Fig. 7.14. The unsteady CFD and surrogate predictions have excellent agreement. The quasi- steady analytical aerothermodynamics predictions has a higher heat flux, driv- ing higher temperatures and displacements. The analytical model also predicts a larger change in pressure across the panel, as expected since the deformation is higher. However, the average pressure across the panel is higher for the CFD and surrogate predictions, producing a larger force on the panel and making it more susceptible to snap-through.

Although there are significant differences in the time to flutter, the post-flutter behavior is similar between all three predictions. The flutter displacement en- velopes in Fig. 7.13(b-c) have nearly identical amplitude for the limit cycle at the mid-panel and 3/4 chord locations. At 30 seconds, the limit cycle amplitude predicted from the quasi-steady analytical aerothermodynamics solution is about 4% larger than that of the unsteady CFD solution, while the difference between the

CFD and surrogate solutions are less than 1%. Furthermore, the peaks of the PSD of the solutions as shown in Fig. 7.13(d) at the 3/4 chord location between 29.5 and

30 seconds are in good agreement as well. Note that at the end of the simulation the transient noise present in Fig. 7.12 are small enough so that the lower-powered peaks are clearly identiﬁed. All three solutions predict the same frequency of the dominant peak at about 108 Hz. The unsteady CFD and quasi-steady surrogate

154 solutions have predicted frequency peaks within 0.5% of each other, while the

quasi-steady analytical aerothermodynamics solution predicts peak frequencies

approximately 2% higher. The good agreement between these predictions sug-

gests that the quasi-steady assumption holds for this conﬁguration, and that the

surrogate aerothermodynamics model perform well in capturing the ﬂuid physics

for transient panel deformations, while the analytical aerothermodynamics model

qualitatively captures the response.

7.4 Computational Cost

The computational cost is compared between each coupling to determine which is

the most computationally efﬁcient for the cases considered. The cost for the simu-

lation is proportional to the number of ﬂuid time steps and amount of subcycling:

−1 −1 ∆tS ∆tT tsim = t0 + CF NF + CSNF + CT NF (7.5) ∆tF ∆tF

where t0 is the overhead time, NF is the number of ﬂuid time steps, and CF , CS, and CT are the computational costs per time step of the ﬂuid, structural, and thermal solvers, respectively. Note that this is a measure of walltime, so the computational cost is considered per-processor for parallel solutions. Mesh deformation is included in CF , transfer of pressure and displacements are part of CS, and transfer

of heat ﬂux and temperature are in CT . Finally, note that the ratio of time steps

represent the amount of subcycling. If subcycling is not present, Eq. 7.5 reduces to

155 12 400 CFD Analytic 10 Surrogate 390

8 380

6 370 w/h T (K) CFD Analytic 4 360 Surrogate

2 350

0 340 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (a) Normalized Displacement. (b) Temperature.

1000 30

500 25 ) 0 2 20 (Pa) b CFD 15 p-p -500 Analytic q (kW/m Surrogate -1000 10 CFD Analytic Surrogate -1500 5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (c) Net Pressure. (d) Heat Flux.

Figure 7.14: Displacement, temperature, pressure, and heat ﬂux across panel immediately prior to ﬂutter with unsteady CFD (20.8 seconds), quasi-steady surrogates (20.8 seconds), and quasi-steady analytical aerothermodynamics (24.5 seconds).

156 tsim = t0 + (CF + CS + CT )NF = t0 + CNF (7.6) where C is the cost of the coupled solver over one time step.

Because the thermal and structural solvers used in this study have very minimal computational cost compared to the ﬂuid solver, Eq. 7.6 is used to charac- terize the time-to-solution. The three loosely coupled schemes, BI, PI, and PCI, have equivalent computational costs because the ﬂuid solver is updated once per time step. Note that the PCI scheme’s structural and thermal solver solution times are about twice as expensive due to the corrector step, however this difference is negligible since CFD solver dominate the solution time. The computational cost parameters, t0 and CF , are given in Table 7.3. The cost per time step for the loosely coupled schemes is about 6.6 seconds, with about a 10% standard deviation between any solution. The SI scheme is at least twice as expensive, and up to about

10 times, depending on the number of subiterations of the coupled system. In general larger time steps require more subiterations compared to smaller time steps.

Table 7.3: Computational cost parameters, in seconds.

t0 CF BI/PI/PCI 210 ± 30 6.6 ± 0.6 SI 210 ± 30 13 to 66

The simulation walltime tsim for the stable response analysis from Section 7.1 are given in Table 7.4 for each scheme at each time step considered. Note that the

157 underlined value represents the largest time step solution with a center displacement error of less than 1%, and the bolded and underlined value for errors less than 0.1%. For either criteria, the PCI scheme has the lowest computational cost, followed by the PI scheme. While the SI scheme is more accurate at larger time steps, up to 10 subiterations were required to converge the solution at each time step, reducing its computational efﬁciency. The PCI scheme is 2.5 times cheaper than the SI scheme to achieve 1% error, and about 4.5 times cheaper for 0.1% error.

Compared to the PI scheme, the PCI scheme is about twice as fast. The BI scheme is 6.7 times as expensive as the PCI scheme for 1% error. The error is greater than

0.1% for the smallest time step considered for the BI scheme, making it at least seven times as expensive compared to the PCI scheme.

Table 7.4: Walltime for one second of stable response, in minutes. Underlined values correspond to fastest time with center displacement errors under 1% for each scheme. Bolded and underlined values correspond to fastest time with center displacement errors under 0.1% for each scheme. Time steps are given in ms. Each simulation performed using 36 2.10GHz Intel Xeon E5-2620 cores.

∆tS (ms) BI PI PCI SI 100 4.3 4.6 3.5 13 50 5.5 4.9 4.5 25 25 8.3 6.6 7.3 47 12.5 - - 10 79 6.25 - 19 20 132 3.125 44 36 34 160 1.5625 67 67 71 280 0.78125 155 134 131 400 - : CFD solver diverged during ﬂuid subiterations.

The simulation walltime tsim for 3 seconds of ﬂutter response from Section 7.2

158 are listed in Table 7.5 for each scheme at each time step. The largest time step used in the flutter analysis, ∆t = 500µs, is omitted from the table since the fluid solver diverged for each scheme. The PI, PCI, and SI schemes all predicted LCO 2-3 for the range of time steps shown, however the BI scheme was inconsistent. The PI and PCI schemes have similar walltimes, while the SI scheme is about three times as expensive at the larger time steps, and about twice as expensive at the smaller time steps. Thus, the PI and PCI schemes are the most efficient, and a third of the cost of the SI scheme at the largest acceptable time step.

Table 7.5: Walltime for three seconds of ﬂutter response, in hours. Time steps are given in µs. Each simulation performed using 36 2.10GHz Intel Xeon E5-2620 cores.

∆tS (µs) BI PI PCI SI 250 23 24 25 77 125 44 45 45 135 62.5 84∗ 87 86 215 31.25 162∗ 170 172 356 15.625 335 - - - ∗: The BI scheme predicts LCO 3-4 at these time steps.

Finally, note that these walltimes are for a single time step size across all solvers since the thermal and structural solvers were computationally inexpensive, and the RANS CFD solver remained stable over relatively large time steps. However, if a complex structure is modeled with a comparable number of degrees of freedom to the fluid solver per processor, and the fluid solver requires a smaller time step, subcycling can be beneficial. A fluid solver with an explicit time integrator

159 could be suitable for subcycling since it is significantly cheaper per step yet requires smaller time steps compared to an implicit solver. One such example is [62] for fluid-structural interactions, where a time step ratio of 10 was used with an explicit DNS flow solver coupled to an implicit structural solver.

As an example, consider a system with equal ﬂuid, thermal, and structural costs per processor, e.g. CT = CS = CF . In the limit of inﬁnite subcycling (∆tT ∆tS

∆tF ) , the computational cost would be 1/3 of that without subcycling with an equivalent ﬂuid time step. Even for low amounts of subcycling, e.g. ∆tT = 4∆tS =

16∆tF as used in Section 5.2, the computational cost is still reduced to 44%.

160 Chapter 8

Concluding Remarks

8.1 Principal Conclusions Obtained in This Study

The focus of this research was to develop and apply time integration strategies for coupling of fluid-thermal-structural interactions in high-speed flows. Time integration procedures for both quasi-steady flow and unsteady flow are considered, with specific emphasis on maintaining the second order temporal accuracy of the individual solvers. The schemes are also designed to enable subcycling, in which each solver has a potentially different time step. The studies on the time-accuracy and comparisons to other coupling schemes leads to several useful conclusions.

Time Accuracy

1. The quasi-steady and time-accurate predictor implicit (PI) and predictor-corrector

implicit (PCI) schemes maintain global second order temporal accuracy with

the use of the extrapolating predictors for ﬂuid loads. Conversely, the basic

161 implicit (BI) scheme, which neglects the predictors, reduces the global accu-

racy of all solution variables to ﬁrst order. For the conventional explicit (CE)

scheme, although a fourth order accurate Runge-Kutta scheme was used for

the structural solver, the ﬁrst order errors from the Euler time integrator in

the thermal solver reduced the global accuracy to ﬁrst order.

2. The strong implicit (SI) scheme has a nearly identical errors compared to the

PI and PCI schemes for both quasi-steady and time-accurate ﬂuid modeling

at the smaller time steps. This indicates that the ﬂuid load predictors are as

accurate as the ﬂuid loads computed from the strong coupling at sufﬁciently

small time steps; in contrast the BI scheme is always less accurate.

3. The PI and PCI schemes maintain second order temporal accuracy when

ﬂuid or structural subcycling is used with the interpolations of displacement

and temperature. Structural subcycling does not have an effect on the rate

of convergence for the time steps considered for either the quasi-steady or

time-accurate ﬂuid solver schemes. For time-accurate CFD, ﬂuid subcycling

impacts the rate of convergence at larger time steps, however second order

accuracy is recovered at smaller steps.

162 Panel Response with Quasi-Steady Fluid Models

4. The numerical stability is investigated from the stable response of a titanium

panel in Mach 9.5 flow with surrogate-based fluid models (Configuration 1)

between the PI, PCI, BI, CE, and SI schemes. The SI scheme remains stable for

the range of time steps considered, though it overpredicts the transient oscil-

lations of the second structural mode at the largest time step. The BI scheme

overpredicts the oscillations over the entire range of time steps considered.

The PCI scheme is stable for all time steps except the largest, while the CE

scheme requires a time step half of that, and the PI scheme requires a time

step four times smaller. With subcycling, the PI and PCI schemes require

smaller time step to accurately capture the oscillations, and the CE scheme

requires a signiﬁcantly smaller time step.

5. For Conﬁguration 1, the computational expense is dominated by the ﬂuid

surrogate models. The PCI scheme without subcycling is found to be the

most computationally efﬁcient of the coupling schemes based on the stability

of the structural response. The PCI scheme is 180% more efﬁcient than the

PI scheme, and 280% more efﬁcient than the CE scheme. The CE scheme is

about 20% faster than the SI scheme. Because the BI scheme overpredicted

the transient oscillations for the entire range of time steps considered, only

a lower bound for its computational cost can be made; it is at least 20 times

more computationally expensive than the PCI scheme.

163 6. The ﬂutter response of an aluminum panel in Mach 4 ﬂow using approxi-

mate aerothermodynamic models (Conﬁguration 2) is investigated between

the four coupling schemes. While each solution predicted the time to ﬂutter

consistently, characteristically-distinct ﬂutter behaviors were observed. At

the largest time steps, all coupling schemes are either unstable or predict

limit cycles not seen at smaller time steps. At time steps of 120 µs and less,

the PI, PCI and SI schemes predict a ﬂutter limit cycle dominated by struc-

tural modes 2 and 3. Subcycling does not affect the limit cycle observed. A

second type of limit cycle dominated by modes 3 and 4 at a higher frequency

is predicted by the BI and CE at some time step sizes, though the CE scheme

consistently predicts the ﬁrst type of limit cycle at smaller time steps. The

BI scheme switches between both types of limit cycles over the range of time

steps considered.

7. Unlike Conﬁguration 1, the computational cost for Conﬁguration 2 is primar-

ily due to the thermal and structural solvers. The PI and PCI schemes are the

most computationally efﬁcient, based on consistently predicting the proper

limit cycle. With subcycling, the PI and PCI schemes are over 1000% more

efﬁcient than the CE or SI schemes. Even without subcycling, the PI scheme

is 275% more efﬁcient than the CE or SI schemes.

Panel Response with Time-Accurate CFD

8. Analysis of the aluminum panel in Mach 4 ﬂow using time-accurate CFD was

164 investigated for the PI, PCI, BI, and SI schemes. During the initial stable re-

sponse of the panel, the SI scheme was numerically stable for the largest time

steps of any scheme, as expected. In order to remain stable with displace-

ment errors under 1%, the PCI scheme required a time step 4 times smaller,

the PI scheme 8 times smaller, and the BI scheme 32 time smaller. Although

the SI scheme was the most stable per time step, the PCI and PI schemes are

more computationally efﬁcient since they do not require subiterations. The

PCI scheme is 150% more efﬁcient than the SI scheme, and the PI scheme

about 125% more efﬁcient. For a smaller error criterion of <0.1%, the PCI

and PI are even more efﬁcient than the SI scheme, at 350% and 225% faster.

The BI scheme is the least efﬁcient, requiring 6.7 times more computational

time than the PCI scheme to achieve <1% errors.

9. Using an elevated initial temperature on the panel, the ﬂutter response is

investigated with each coupling scheme. Similar to the quasi-steady ﬂutter

analysis, the SI, PCI, and PI schemes consistently predicted a ﬂutter limit cy-

cle dominated by the second and third structural modes, while the BI scheme

predicted a higher frequency limit cycle dominated by modes 3 and 4 for

some time steps. Both the PI and PCI schemes are 3 times as efﬁcient com-

pared to the SI scheme.

10. Based on an analysis of the work performed across the interface between the

ﬂuid and structural solvers, the BI scheme has the largest violation of the

165 three loosely coupled schemes, while the violation of the PCI scheme is sig-

niﬁcantly smaller than either the BI or PI schemes. The net work violation

decreases by a sub-ﬁrst order rate for the BI scheme, at about 2.5 for the PI

scheme, and about 4.5 for the PCI scheme. The work violation does not de-

crease monotonically for the SI scheme because of the ﬂuid load tolerance

that terminates the subiterations. At the largest time step the SI scheme has

the lowest work violation of any of the coupling schemes.

11. A 30 second analysis of the panel starting from an undeformed shape through

snap-through and ﬂutter is performed using the PI scheme. To the author’s

knowledge, this is the ﬁrst long time record analysis of ﬂuid-thermal-structural

interactions performed using a time accurate CFD solver. The panel ﬂut-

ter limit cycle is the same as seen from the analysis with elevated temper-

atures. The response is compared to predictions using quasi-steady surro-

gate aerothermodynamics and analytical aerothermodynamics. The surro-

gate prediction has excellent agreement with the CFD-based prediction in

both the pre and post-ﬂutter response, while the analytical aerothermody-

namics model predicts 7% higher heating, leading to higher temperatures

and deformation. The CFD and surrogate simulations predict the snap-through

and ﬂutter onset about 3.5 seconds sooner than the analytical aerothermo-

dynamics simulation. However, at 30 seconds the ﬂutter response between

the three simulations is approximately the same. The amplitude and peak

166 frequencies of the limit cycle between the CFD and quasi-steady surrogate

predictions agree to within 1%, while the analytical aerothermodynamics

simulation predicts 4% larger amplitudes and 2% higher frequencies. The

good agreement in the dynamic response suggests that the quasi-steady sur-

rogate models are capable of capturing the transient ﬂow physics, and the

quasi-steady analytical aerothermodynamic models can capture the qualita-

tive trends of the panel response.

8.2 Recommendations for Future Research

In this dissertation, coupling schemes for fluid-thermal-structural interactions for both quasi-steady and time-accurate fluid solvers are developed. They are demonstrated to maintain second order temporal accuracy and compare favorably to other basic loose coupling schemes and a strongly coupled scheme. However, there still exist many challenges of fluid thermal structural interactions in high speed flows that need to be addressed.

A fundamental issue in fluid-structural interactions is the lack of experimental data for validation. While the individual models can be verified and validated against known benchmarks, there is little to no experimental data available that captures all of the coupled interactions. Such experimental benchmarks are needed to gain confidence in the ability for the numerical models to accurately predict the coupled response. Furthermore, such experimental benchmarks could be used

167 to determine the ﬁdelity and amount of coupling required if the experiments are representative of the conditions experienced by actual ﬂight vehicles.

The coupling procedures here all considered a constant time step throughout the simulation. However, the disparate time step requirements between the dynamically stable and ﬂutter portions of the panel response suggest that an adaptive time step could be used. An adaptive time step could provide signiﬁcant computational savings by using large time steps for the stable response, and switching to a smaller time step when dynamic instabilities occur. The couplings schemes here are amenable to changing time step sizes, with the only requirement that the ratio of time steps remain integer values for subcycling.

While these coupling schemes have been demonstrated to be effective, other predictor-based schemes could be constructed as well. Structural predictors using extrapolations have been successful in fluid-structure interactions [42], and could be used for FTSI as well. Similarly, a temperature-based predictor could potentially be used for the fluid-thermal coupling. The authors in [42] also recommended off- setting the structural and fluid solvers by half a time step to improve the accuracy and stability even further. Such an asynchronous coupling scheme may also be beneficial for FTSI, however two of the three solvers would need to remain syn- chronous.

The procedures developed here were tested on a simple panel in two dimensional ﬂow. Further study is recommended on more complex structures and ﬂow-

ﬁelds more representative of a hypersonic cruise vehicle. More complex inter-

168 actions with the flow should be investigated as well, such as shock interactions and transient turbulent loading. Such loads, along with engine noise excitation in the flowfield, can have a significant impact on the structural response [111]. The current CFD-based analysis was performed using a Reynolds-Averaged Navier-

Stokes CFD solver, which averages out the transient turbulent loads. Therefore, it is recommended that such loads be included in the coupling framework, either from semi-empirical models or higher ﬁdelity CFD solvers that model the transient turbulence.

While subcycling was shown to maintain second order temporal accuracy, the computational benefits of subcycling with a time-accurate flow solver could not be assessed. The computational cost of the simple structural and thermal solvers were trivial compared to the CFD solver, and the RANS CFD solver remained stable for large time steps. To assess subcycling, a more complex structure encom- passing hundreds of thousands (or more) degrees of freedom is required, ideally with a similar number of degrees of freedom per processor as the flow solver. This study also used a one-to-one mapping between the solvers at the fluid-structure interface, however for more complex geometries and meshes data interpolation algorithms are required to transfer the loads, which can contribute to the relative cost of the thermal and structural solvers. Finally, more complex configurations that model shock interactions, turbulence, or engine noise excitation will add flow unsteadiness, requiring smaller fluid time steps.

Even with effective subcycling, a fundamental bottleneck in the computational

169 cost is the large number of time steps required for the fine time-scale solver (ie. the structural solver for quasi-steady fluid modeling, or the fluid solver for time- accurate fluid modeling). Time marching is generally a serial procedure, since the solution at each time step is dependent on the solution at the previous step. The cost of each time step can be reduced through parallelization of the spatial domain, though the scaling is limited by the spatial partitioning and inter-process communication. Promising approaches to tackling this issue are parallel-in-time algorithms, in which a solution with a coarse time step is used as a starting point to compute fine time-scale responses in a parallel manner [112]. If such approaches can be successfully applied to fluid-thermal-structural interactions, they may provide a means for long time record analyses of large scale systems in a reasonable amount of time.

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181 Appendix A

Least Squares Based Linear Fitting

Least-squares based ﬁtting is used for averaging of the velocity and displacement

n+1 (˜un+1 and ˜˙u ) for the quasi-steady PI scheme, and the extrapolation of the pres-

m+1 n+1/2 sure and heat ﬂux (pP and qP ) for the time-accurate PI scheme.

Using an arbitrary function f(τ), the ﬁt is described as:

f(τ) ≈ a + bτ (A.1)

where a and b are the fit coefficients. The function is defined over the domain

t ∈ [0,T ]. The local error e(t) is:

e(τ) ≡ f(τ) − a − bτ (A.2)

The global error can be deﬁned as a generalized linear operator L (e(τ)2). For

example this operator could be a continuous integral (A.3a), or a discrete summa-

tion (A.3b):

182 Z τ0 L(f(τ)) ≡ f(τ)dτ (A.3a) 0 N X L(f(τ)) ≡ fn for fn = f(τn) (A.3b) n=0

Equation (A.3b) is used since the values from the solvers are already discretized.

The global error E = Le(τ 2) is minimized by setting the derivatives of E to be zero with respect to a and b:

∂E ∂e2 ∂e = L = L(−2e) = L − 2(fτ) − a − bτ = 0 (A.4a) ∂a ∂2 ∂a ∂E ∂e2 ∂e = L = L(−2eτ) = L − 2τ(fτ) − a − bτ = 0 (A.4b) ∂a ∂2 ∂b

Note that because L is a linear operator, the partial derivative may be passed

inside to its arguments. These equations reduce to:

aL(1) + bL(τ) = L(f(τ)) (A.5a)

aL(τ) + bL(τ 2) = L(f(τ)τ) (A.5b)

Solving for a and b yields:

183 L(τ 2)L(f(τ)) − L(τ)L(f(τ)τ) α = (A.6a) L(1)L(τ 2) − L(τ)2 L(1)L(f(τ)τ) − L(τ)L(f(τ)) β = (A.6b) L(1)L(τ 2) − L(τ)2

Using the discrete summation from Eq. (A.3b), and assuming τn = n∆τ, the

equations reduce to:

N N P P (4N + 2) fn − 6 nfn a = n=0 n=1 (A.7a) (N + 1)(N + 2)

N N 2 P nf − P f 6 n n b = × n=1 n=0 (A.7b) N∆τ (N + 1)(N + 2)

When operating on vectors, a and b are computed for each element in the vector independently.

184 Appendix B

Temperature-Dependent Material Properties

The structural and thermal properties for titanium (Ti-6Al-2Sn-4Zr-2Mo) are given in Table B.3 and B.4, respectively. The structural and thermal properties for Alu- minum (Al-7075) are given in Table B.1 and B.2, respectively. These values are extracted from [113].

185 Table B.1: Coefﬁcient of thermal expansion and Young’s modulus for Al-7075

µm T (K) E (GP a) α m·K 270.0 74.08 22.14 280.0 73.67 22.21 290.0 73.27 22.29 300.0 72.86 22.36 310.0 72.46 22.44 320.0 72.05 22.51 330.0 71.65 22.59 340.0 71.24 22.66 350.0 70.84 22.74 360.0 70.43 22.81 370.0 69.90 22.90 380.0 69.11 23.00 390.0 68.32 23.11 400.0 67.53 23.21 410.0 66.74 23.32 420.0 65.95 23.42 430.0 64.86 23.53 440.0 63.67 23.63 450.0 62.49 23.74 460.0 61.31 23.84 470.0 60.12 23.95 480.0 58.82 24.05 490.0 57.11 24.15 500.0 55.40 24.25

186 Table B.2: Thermal capacitance and thermal conductivity for Al-7075 J J T (K) c kg·K κ s·K 275.0 830.26 128.55 295.0 846.84 131.38 315.0 863.42 134.02 335.0 880.00 136.47 355.0 896.58 138.74 375.0 913.16 140.82 395.0 929.74 142.72 415.0 946.32 144.44 435.0 962.90 145.97 455.0 979.48 147.31 475.0 996.06 148.47 495.0 1012.6 149.45

Table B.3: Coefﬁcient of thermal expansion and Young’s modulus for Ti-6Al-2Sn- 4Zr-2Mo µm T (K) E (GP a) α m·K 294.4 113.8 7.04 311.1 112.9 7.20 366.7 109.2 7.74 422.2 106.4 8.28 477.8 103.5 8.64 533.3 100.7 9.00 588.9 98.07 9.18 644.4 95.56 9.36 700.0 92.15 9.54 755.6 87.83 9.63 811.1 83.05 9.72

187 Table B.4: Thermal capacitance and thermal conductivity for Ti-6Al-2Sn-4Zr-2Mo J J T (K) c kg·K κ s·K 275.00 459.31 6.6167 311.11 464.76 7.0105 366.66 473.13 7.6164 422.22 498.25 8.3088 477.77 523.37 9.0012 533.33 552.68 9.6936 588.88 586.18 10.386 644.44 623.86 11.078 700.00 661.55 11.771 755.55 703.42 12.550 811.11 749.47 13.329 850.00 781.71 13.874

188 Appendix C

CFD Mesh Convergence Study

The mesh used for the time-accurate CFD study was chosen based on its agreement

with solutions from a ﬁner mesh. The coarse mesh described in Section 4.3 consists

of 612 cells in the streamwise direction, including 500 cells over the panel, and 140

cells in the normal direction. The ﬁne mesh doubles the number of cells in each direction, with 1224 cells in the streamwise direction (1000 cells over the panel), and 280 cells in the normal direction. The coarse mesh has a total of 85,680 cells, and the ﬁner mesh has 342,720 cells.

The pressure and heat flux distributions are compared between the two meshes using two deformation profiles, shown in Fig. C.1. The freestream properties are from Configuration 2 in Table 4.1, and the panel wall temperature is set to 300 K.

The first deformation profile in Fig. C.1(a) is based on the panel deformation immediately prior to snap-through in Section 7.2.1; the deformation is constructed primarily by the first and second structural modes. The second deformation pro-

ﬁle in Fig. C.1(b) is based on the maximum amplitude of the limit cycle seen in Sec-

189 tion 7.3; the deformation is dominated by the second and third structural modes.

10 10

5 5

0 0 w/h w/h

−5 −5

−10 −10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (a) Deformation proﬁle 1. (b) Deformation proﬁle 2.

Figure C.1: Deformation proﬁles used to compare mesh convergence.

The pressure and heat ﬂux for both deformation proﬁles are given in Fig. C.2.

The pressure distributions in Figs. C.2(a,c) are indistinguishable between the two meshes; the L2 norm based difference between the two meshes is 0.14% for proﬁle

1, and 0.25% for proﬁle 2. The heat ﬂux distributions in Figs. C.2(b,d) also have good agreement, though small differences are observed near the heating peaks.

The difference of the heat flux is 0.69% for profile 1, and 0.86% for profile 2.

A ﬁnal transient analysis is performed based on an oscillating panel. The panel is prescribed to oscillate in the second structural mode with a maximum amplitude of 2 cm (8 panel thicknesses) at a frequency of 100 Hz. The simulation is carried out to one and a quarter cycles (.0125 sec) with a time step ∆t = 10−4 sec. The generalized aerodynamic force (GAF), deﬁned in Eq. (C.1), is shown in Fig. C.3 for both meshes. The GAF represents the pressure loading that acts on the second structural mode. Both meshes have excellent agreement of the GAF over the time

190 35 3000

30 2500 25 ) 2

2000 20 p (Pa) Coarse

Coarse q (kW/m Fine 15 Fine 1500 10

1000 5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (a) Pressure (Profile 1). (b) Heat flux (Profile 1).

45 3500 Coarse Coarse 40 Fine Fine 3000 35 ) 2500 2 30

p (Pa) 25 2000 q (kW/m 20 1500 15

1000 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x (m) x (m) (c) Pressure (Profile 2). (d) Heat flux (Profile 2).

Figure C.2: Pressure and heat ﬂux comparisons for both deformation proﬁles.

191 period, with a difference of 0.411% computed by the L2 norm. Furthermore, the difference of the pressure distribution over the time period is 0.112%. The difference of the heat ﬂux is 0.93%.

Z L GAF (t) = φ2(x)p(x, t)dx (C.1) 0

150

100

0 GAF (N/m) −50

−100 Coarse Fine

−150 0 0.002 0.004 0.006 0.008 0.01 0.012 Time (sec)

Figure C.3: Generalized aerodynamic force (GAF) over 1.25 cycles of forced oscillation.

Between all three analyses, the pressure distribution between the two mesh agree within 0.25%, and the heat ﬂux within 1%. Thus, the coarser mesh is considered sufﬁciently converged and is used in this dissertation.

192 Appendix D

Thermal Solver Veriﬁcation

The thermal solver is verified against Abaqus R , a commercial nonlinear finite element solver. The model is based on the aluminum panel from Configuration 2 in Chapter 4, including the temperature-dependent thermal properties. For the verification study, both models use a consistent mesh with 100 elements along the length of the panel, and 5 elements through the thickness. A constant in time heat

ﬂux proﬁle, shown in Fig. D.1 is applied to the upper surface of the panel. The simulations are run for 20 seconds of response with a time step of 0.01 seconds.

The mid-panel temperature over time and temperature on the upper surface at

20 seconds are shown in Fig. D.2 for both the current model and Abaqus. Both solutions show excellent agreement. At 20 seconds, the mid-panel temperature of the current solver is within 0.0125% of the Abaqus solution, and the maximum difference at any node in the panel is 0.032%.

193 6

4 ) 2 3 q (W/m 2

0 0 0.2 0.4 0.6 0.8 1 x (m)

Figure D.1: Heat ﬂux proﬁle prescribed to panel upper surface.

500 500

450 450

400 400 T (K) T (K)

350 350

Solver Solver Abaqus Abaqus 300 300 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Time (sec) x (m) (a) Mid-Panel temperature over time. (b) Upper surface temperature at 20 seconds.

Figure D.2: Comparison of current thermal model to Abaqus using a constant heat ﬂux proﬁle.

194 Appendix E

Structural Solver Veriﬁcation

The structural model is veriﬁed using Abaqus for two separate studies. In the ﬁrst study a rising uniform temperature is applied to the panel to verify the nonlinear response to large thermal loads. In the second study a random in time, uniform in space pressure loading is applied to the panel to verify the nonlinear dynamic response of the panel. The models are based on the aluminum panel from Con-

ﬁguration 2 in Chapter 4, including the temperature-dependent material properties and Rayleigh damping. In order to match the properties of the Abaqus beam model to the cylindrical bending panel model, the modulus of elasticity and coef-

ﬁcient of thermal expansion of the beam are computed using Eqs. (E.1) and (E.2).

The Abaqus model consists of 100 cubic-interpolation beam (B23) elements, and the current solver uses 15 mode shapes integrated over 501 points along the panel.

Epanel Ebeam ≡ (E.1) 1 − ν2

195 450

400 T (K)

350

300 0 5 10 15 Time (sec)

Figure E.1: Uniform in space temperature loading over time.

αbeam ≡ αpanel (1 + ν) (E.2)

For the ﬁrst veriﬁcation study, a uniform in space temperature rise is prescribed to the panel for 15 seconds, as shown in Fig. E.1. In addition to this temperature load a relatively small uniform load (1 Pa) is applied. This load is required to bias the panel to buckle upwards; without any loading the panel would remain undeformed in an unstable equilibrium.

The center displacement over the ﬁrst second, and over the full 15 seconds are shown in Fig. E.2. The panel quickly buckles upwards, and after initial vibrations damp out the panel displacement slowly grows with the rising temperature. The current solver and Abaqus have excellent agreement both with the initial buckling and vibrations in Fig. E.2(a), and the long term response in Fig. E.2(b) up to 17.5 panel thicknesses. At 15 seconds with a center displacement of 17.5 panel thick-

196 nesses, the two solvers vary by only 0.03%. 5 20

4 15

w/h 10 w/h 2

5 1 Solver Solver Abaqus Abaqus 0 0 0 0.2 0.4 0.6 0.8 1 0 5 10 15 Time (sec) Time (sec) (a) Normalized mid-panel displacement over (b) Normalized mid-panel displacement over 15 ﬁrst second. seconds.

Figure E.2: Comparison of current structural solver to Abaqus with a rising temperature loading.

For the second veriﬁcation study, a uniform in space but random in time pressure load is applied to the panel for 20 seconds, shown in Fig. E.3(a). This random loading is deﬁned to have uniform energy content up to 500 Hz, and zero energy in higher frequencies, as shown in Fig. E.3(b). The root mean square (RMS) of the loading is set at 32 Pa, or about 124 dB SPL.

The center displacement over the ﬁrst two seconds, and the power spectral density of the center displacement of the 20 seconds of response are shown in Fig. E.4.

In addition to the current solver and Abaqus, a linear response prediction using

Abaqus is included to demonstrate that the response is signiﬁcantly nonlinear.

The solutions from the current solver and Abaqus again show excellent agreement overall, while the linear response predicts larger amplitudes at lower frequencies.

From the PSD in Fig. E.4(b), the dominant frequency is centered at 12 Hz for both

197 10 150 10

100 0 10

50 −10 10 /Hz) 2 0 p (Pa) −20 10

−50 PSD (Pa

−30 −100 10

−40 −150 10 0 5 10 15 20 200 400 600 800 1000 time (sec) ω (Hz) (a) Random Loading over time. (b) PSD of random loading.

Figure E.3: Random in time, uniform in space, pressure loading applied to panel for transient response veriﬁcation. nonlinear solutions, and 6 Hz for the linear solutions. Less dominant peaks occur at about 58 Hz, 157 Hz, and 305 Hz with good agreement between the nonlinear solutions.

3 0 10

−2 10 1

Solver 0 −4 w/h 10 Abaqus Abaqus (Linear) −1 PSD (1/Hz) −6 Solver 10 −2 Abaqus Abaqus (Linear) −8 −3 10 0 0.5 1 1.5 2 0 100 200 300 400 500 Time (sec) ω (Hz) (a) Normalized mid-panel displacement over (b) PSD of normalized mid-panel displacement ﬁrst 2 seconds. over 20 seconds.

Figure E.4: Comparison of current structural solver to Abaqus (linear and nonlinear) with a random pressure loading.

198