University of Southern California Dissertations and Theses

Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations

(USC Thesis Other)

Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations

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Content Numerical study of sho c k-w a v e/turbulen t b oundary la y er in teractions on flexible and rigid panels with w all-mo deled large-eddy sim ulations b y Jonathan Ho y A Dissertation Presen ted to the F A CUL TY OF THE USC GRADUA TE SCHOOL In P artial F ulfillmen t of the Requiremen ts for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) Decem b er 2022 Cop yrigh t 2022 Jonathan Ho y Dedication I dedicate this Thesis to m y father Dennis Ho y and to m y late mother Karen Ho y who passed a w a y in Octob er 2016 early during m y PhD at USC. Without their lo v e and supp ort, I w ould nev er ha v e b een able to reac h m y full p oten tial. I w ould also lik e to thank m y girlfriend Natasha who has giv en me unconditional lo v e and supp ort ev er since w e met four y ears ago in April 2018. I w ould lik e to also ac kno wledge t w o of m y closest men tors at Cal P oly P omona, Dr. Kevin Anderson and the late Dr. Angela Shih who b eliev ed in m y abilities and c hallenged me to pursue academic rigor. I w ould lik e to thank m y former labmates Zaki Hasnain and Xiangyu Gao whose friendship and camaraderie ga v e me the strength to con tin ue w orking hard and long nigh t s. I w ould lik e to thank m y advisor, Dr. Iv an Bermejo-Moreno. I appreciate that I w as able to b e one of the founding mem b ers of the computational aerospace lab along with Jonas Buc hmeier and Xiangyu Gao. My in terest in computational science and rigor op ened do ors in m y career suc h as in ternships at Sandia National Lab and Math w orks. I am thankful to ha v e b een giv en opp ortunities to attend conferences and trainings in Seattle, Chicago, Pho enix, W ashington DC, San F rancisco, San ta Barbara, and San Diego to name a few. Finally , I w ould lik e to thank m y committee mem b ers Dr. Aiic hiro Nakano and Dr. Carlos P a n tano-R ubino for agreeing to serv e on m y PhD defense committee and taking the time to review m y w ork and pro vide feedbac k. ii A c kno wledgmen ts This w ork has b een partially supp orted b y NSF a w ard 2143014, NASA gran t 80NSSC18M0148, and DOE/NNSA a w ard DENA0003993. Computational resource s w ere pro vided b y an INCITE a w ard allo cation on Theta sup ercomputer at Argonne Leadership Computing F acilit y (ALCF), Argonne National Lab oratory , an allo cation on Quartz sup ercomputer at La wrence Liv ermore National Lab oratory under the Predictiv e Science A cademic Alliance Program (PSAAP), and b y the Cen ter for A dv anced R esearc h Computing at the Univ ersit y of Southern California. iii Con ten ts Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii A c kno wledgmen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of T ables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Chapter 1: In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 R otating w edge reac hing a constan t setp oin t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 R otating w edge oscillating b et w een t w o setp oin ts . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3: Computational Metho dology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 C oupled solv er arc hitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Flo w Solv er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Computational Metho dology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Mesh and computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.3 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.3.1 T op Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.3.2 Bottom W all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.3.3 Inlet, outlet, and span wise b oundary conditions . . . . . . . . . . . . . . . . 23 3.2.3.4 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.4 Numerical Instrumen tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Solid Solv er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 iv 3.3.1 Computational Metho dology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Mesh and computational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.3 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.4 Numerical Instrumen tation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4 Me sh Deformation Solv er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.1 Computational Metho dology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Solv er Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.1 Flo w domain to solid domain coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5.2 Solid domain to flo w domain coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter 4: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1 Sim ulations with the rotating w edge reac hing a constan t setp oin t . . . . . . . . . . . . . . . . 38 4.1.1 Assessmen t of the fully dev elop ed turbulen t b oundary la y er . . . . . . . . . . . . . . . 39 4.1.2 Assessmen t of span wise effects using a full-span sim ulation Z d = 50δ 0 . . . . . . . . . 40 4.1.3 STBLI sensitivit y to rear w edge length and incoming TBL thic kness . . . . . . . . . . 42 4.1.4 T emp oral ev olution of panel displacemen t . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.4.1 F ully-coupled FSI sim ulations . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.4.2 Assessmen t of FSI coupling effects . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.5 Flo w c haracterization on span wise-normal cen ter plane . . . . . . . . . . . . . . . . . . 55 4.1.5.1 Instan taneous quan tities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.5.2 Time a v eraged quan tities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.6 Influence of panel elasticit y on the temp oral ev olution of w all quan tities . . . . . . . . 71 4.1.7 Time-a v eraged w all quan tities in the quasi-steady regime . . . . . . . . . . . . . . . . 74 4.1.7.1 Stream wise profiles of time- and span wise-a v eraged w all pressure . . . . . . . 74 4.1.7.2 Stream wise profiles of time- and span wise-a v eraged w all shear stress . . . . . 76 4.1.8 Influence of panel elasticit y and w edge deflection angle on spatial mean panel statistics 79 4.1.9 Sp ectral analysis of w all pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1.10 Analysis of the flo w separation bubble dynamics . . . . . . . . . . . . . . . . . . . . . 90 4.1.10.1 Probabilit y distributions of flo w rev ersal . . . . . . . . . . . . . . . . . . . . . 96 4.2 Sim ulations with the rotating w edge oscillating b et w een t w o setp oin ts . . . . . . . . . . . . . 99 4.2.1 Mean panel pressure in resp onse to forced w edge oscillation . . . . . . . . . . . . . . . 104 4.2.2 P anel vibrations in resp onse to forced w edge oscillation . . . . . . . . . . . . . . . . . 104 v 4.2.3 V olume of separation bubble in resp onse to forced w edge oscillation . . . . . . . . . . 105 Chapter 5: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 App endix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 A Dam ping ratio ζ estimate from measured signal data . . . . . . . . . . . . . . . . . . . . . . . 110 B I n viscid top b oundary condition calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 C F lo w c haracterization on span wise-normal cen ter plane . . . . . . . . . . . . . . . . . . . . . . 115 Bibliograph y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 vi List of T ables 2.1 Flo w prop erties for exp erimen tal configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Prop erties of the elastic panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Solv er time in tegration and coupling frequency v alues . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Flo w mesh spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Flo w solv er n umerical instrumen tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4 Solid mesh spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Solid solv er n umerical instrumen tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.6 Me sh deformation solv er constrain ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Prob e displacemen t comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 P eak w all pressure statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Me an separation and reattac hmen t lo cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Me an pressure statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 R egion lo cations for stream wise PSD a v eraging . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Se paration bubble cen troid statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7 Se paration bubble v olume statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.8 W edge oscillation frequencies f w used for rigid and flexible panel sim ulation . . . . . . . . . . 100 4.9 Ste ady state parameters used for rigid and flexible panel gain calculation . . . . . . . . . . . 100 B.1 S ummary of top b oundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 vii List of Figures 1.1 ST BLI Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Sc hematic of the problem setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 W edge angle v ersus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 F ull span riv eted plate in xz and yz views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 W edge angle v ersus time for con tin uous w edge oscillation case . . . . . . . . . . . . . . . . . . 12 3.1 FSI solv er high lev el arc hitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Flo w mesh v ertical spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 I llustration of inciden t oblique sho c k and rear Prandtl-Mey er expansion generated b y the mo ving w edge and its in teraction with the top b oundary of the computational flo w domain . 20 3.4 T op b oundary pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 I nlet w all-normal profiles of flo w quan tities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 Spati al do wnsampling of flo w pressure field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Spati al upsampling of the solid displacemen t field . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 W all-normal profiles of the incoming TBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Fle xible full span stream wise and span wise v ariation of w all quan tities . . . . . . . . . . . . . 41 4.3 P arametric studies of w all pressure v ersus b oundary la y er thic kness and w edge length . . . . 43 4.4 F ull span panel displacemen t prob e measuremen ts v ersus time . . . . . . . . . . . . . . . . . . 44 4.5 F ull span w all displacemen t v ersus time and stream wise co ordinate . . . . . . . . . . . . . . . 45 4.6 P anel displacemen t prob e measuremen ts v ersus time . . . . . . . . . . . . . . . . . . . . . . . 46 4.7 W all dis placemen t v ersus time and stream wise co ordinate . . . . . . . . . . . . . . . . . . . . 47 4.8 Static flexible w all displacemen t v ersus stream wise co ordinate . . . . . . . . . . . . . . . . . . 49 4.9 Static prob e error v ersus prob e stream wise lo cation offset . . . . . . . . . . . . . . . . . . . . 51 4.10 P anel displacemen t prob e measuremen ts v ersus time20 mm prob e offset . . . . . . . . . . . 52 4.11 FSI one w a y coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.12 One -w a y coupling FSI panel displacemen t prob e comparison . . . . . . . . . . . . . . . . . . . 56 4.13 C en terplane comparison of instan taneous stream wise x v elo cit y . . . . . . . . . . . . . . . . . 58 viii 4.14 C en terplane comparison of instan taneous w all normal v elo cit y . . . . . . . . . . . . . . . . . . 59 4.15 C en terplane comparison of instan taneous pressure . . . . . . . . . . . . . . . . . . . . . . . . 60 4.16 C en terplane comparison of instan taneous temp erature . . . . . . . . . . . . . . . . . . . . . . 62 4.17 C en terplane comparison of densit y gradien t magnitude . . . . . . . . . . . . . . . . . . . . . . 63 4.18 C en terplane comparison of v orticit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.19 C en terplane comparison of mean stream wise x v elo cit y . . . . . . . . . . . . . . . . . . . . . . 65 4.20 C en terplane comparison of mean stream wise x v elo cit y . . . . . . . . . . . . . . . . . . . . . . 67 4.21 C en terplane comparison of mean pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.22 C en terplane comparison of stream wise x v elo cit y v ariance . . . . . . . . . . . . . . . . . . . . 69 4.23 C en terplane comparison of pressure v ariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.24 W all pre ssure v ersus time and stream wise co ordinate . . . . . . . . . . . . . . . . . . . . . . . 72 4.25 Skin friction co efficien t v ersus time and stream wise co ordinate . . . . . . . . . . . . . . . . . . 73 4.26 Me an of w all pressure v ersus stream wise co ordinate . . . . . . . . . . . . . . . . . . . . . . . . 75 4.27 Stand ard deviation of w all pressure v ersus stream wise co ordinate . . . . . . . . . . . . . . . . 77 4.28 Me an of skin friction co efficien t v ersus stream wise co ordinate . . . . . . . . . . . . . . . . . . 78 4.29 Ne t panel pressure o v er time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.30 Ne t panel cen troid of pressure o v er time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.31 W all pre ssure PSD v ersus stream wise co ordinate . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.32 W all pre ssure PSD o v erla y ed on to w all displacemen t PSD . . . . . . . . . . . . . . . . . . . . 86 4.33 W all pre ssure PSD feature comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.34 F requency band p o w er con tribution v ersus stream wise co ordinate . . . . . . . . . . . . . . . . 89 4.35 B ubble cen troid lo cation v ersus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.36 Join t prob abilit y distribution of instan taneous flo w separation . . . . . . . . . . . . . . . . . . 93 4.37 PSD o f bubble v olume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.38 B ubble v olume v ersus time and probabilit y distribution of bubble v olume . . . . . . . . . . . 95 4.39 Prob abilit y of instan taneous flo w rev ersal v ersus w all normal and stream wise co ordinate . . . 97 4.40 Dis turbance time lag illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.41 Pre ssure and separation bubble time lags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.42 P anel mean pressure v ersus w edge cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.43 P anel pressure gain and phase angle v ersus w edge forcing frequency . . . . . . . . . . . . . . 105 4.44 W all dis placemen t v ersus w edge cycle and panel gain v ersus w edge frequency . . . . . . . . . 106 4.45 Se paration bubble v olume v ersus w edge cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.46 Se paration bubble v olume gain and phase dela y v ersus w edge frequency . . . . . . . . . . . . 107 ix A.1 I llustration damping ratio computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.1 W edge extension geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 C.1 Cen terplane comparison of mean temp erature . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 C.2 Cen terplane comparison of temp erature v ariance . . . . . . . . . . . . . . . . . . . . . . . . . 116 C.3 Cen terplane comparison v ertical v elo cit y v ariance . . . . . . . . . . . . . . . . . . . . . . . . . 117 C.4 Cen terplane comparison of mean resolv ed Reynolds shear stress . . . . . . . . . . . . . . . . . 118 x Nomenclature A cron yms CFD Computational Fluid Dynamics CFL Couran t–F riedric hs–Lewy Condition DES Detac hed Eddy Sim ulation DFT Discrete F ourier T ransform DNS Direct Numerical Sim ulation FEM Finite Elemen t Metho d FFT F ast F ourier T ransform FSI Fluid Structure In teraction FTT Flo w-Through Time FVM Finite V olume Metho d HPC High P erformance Computing ISI Inciden t Sho c k In tersection LES Large Eddy Sim ulation ODE Ordinary Differen tial Equation PDE P artial Differen tial Equation PDF Probabilit y Densit y F unction PIV P article Image V elo cimetry PME Prandtl-Mey er Expansion PSD P o w er Sp ectral Densit y QOI Quan tit y of In terest RANS Reynolds A v eraged Na vier Stok es xi RMS Ro ot mean squared SGS Subgrid Scale SST Shear Stress T ransp ort STBLI Sho c k T urbulen t Boundary La y er In teraction STD Standard Deviation WMLES W all-Mo delled Large Eddy Sim ulation Sym b ols β Inciden t sho c k angle ∆ Cell mesh size δ Boundary la y er thic kness δ 0 Boundary la y er thic kness at the reference stream wise lo cation µ Dynamic viscosit y of the fluid ν Kinematic viscosit y of the fluid ν s P oisson’s ratio of the flexible panel ω k ω k Enstroph y of the fluid ρ Densit y ρ ∞ F reestream densit y θ Compression w edge deflection angle, m easured relativ e to the horizon tal ζ Solid solv er viscous damping ratio a Solid solv er mass prop ortional damping co effi cien t c Sound sp eed of the fluid C f W all shear stress co efficien t E Mo dulus of elasticit y E s Y oung’s mo dulus of the flexible panel e T T otal energy of the fluid p er unit mass f n P anel primary natural frequency f w F orcing frequency of oscillatory w edge rotation h wm W all mo del exc hange heigh t xii l v Viscous lengthscale of the fluid flo w L sep Flo w separation length M Mac h n um b er M ∞ F reestream Mac h n um b er p ∞ F reestream pressure p w W all pressure p cav Ca vit y pressure Pr Prandtl n um b er of the fluid Pr t T urbulen t (eddy) Prandtl n um b er r n Relativ e fluid flo w v elo cit y normal to the me sh face Re δ Reynolds n um b er based on b oundary la y er thic kness Re τ Reynolds n um b er based on w all shear s tress St Strouhal n um b er t Time t ′ Dimensionless time t 0 Time when panel reac hes maxim um deflection for the first cycle starting from rest T ∞ F reestream temp erature T w W all temp erature u Stream wise (x ) fluid flo w v elo cit y comp onen t u m i Fluid flo w mesh v elo cit y u i Fluid flo w v elo cit y v W all normal (y ) fluid flo w v elo cit y comp onen t V s Separation bubble v olume w Span wise (z ) fluid flo w v elo cit y comp onen t x,y,z Stream wise, w all normal, and span wise direc tions, resp ectiv ely x ′ ,y ′ ,z ′ Dimensionless stream wise, w all normal, and span wise directions, resp ectiv ely x I In viscid oblique sho c k impingemen t p oin t x r Stream wise co ordinate of flo w reattac hmen t xiii x s Stream wise co ordinate of flo w separation x CP Cen ter of pressure x p,0 Stream wise co ordinate of the start of the flexible panel x p,L Stream wise co ordinate of the end of the flexible panel Y s W all normal displacemen t of the flexible panel Z d Span wise width of the computational domain xiv Abstract Coupled fluid structure in teraction (FSI) sim ulations that in tegrate a finite-v olume w all-mo deled LES (WMLES) flo w solv er and a finite-elemen t (FEM) solid mec hanics solv er are used to capture the in teraction of oblique sho c ks impinging on turbulen t b oundary la y ers dev elop ed along rigid and flexible panels. The sim ulations replicate prior exp erimen ts conducted at the trisonic wind tunnel (TMK) of the Sup ersonic and Hyp e rsonic T ec hnologies Departmen t at the German A erospace Cen ter (DLR), Cologne. In the exp erimen ts, an incoming sup ersonic turbulen t b oundary la y er (TBL) w as impinged b y an oblique sho c k generated b y a rotating w edge that reac hes a maxim um w edge angle θ max within appro ximately 15 ms . Three flo w configurations are studied, corresp onding to the follo wing com binations of freestream Mac h n um b er, M ∞ , and maxim um w edge deflection angle, θ max : (1) M ∞ = 3.0 , θ max = 17.5 ◦ , (2) M ∞ = 4.0 , θ max = 20.0 ◦ , and (3)M ∞ = 4.0 , θ max = 15.0 ◦ . Eac h configuration results in differen t degrees of FSI and STBLI strength: case (1) is the strongest in teraction, follo w ed b y case (2) and, finally , case (3). Displacemen t signals o v er time and static pressure at differen t panel lo cations are compared to the exp erimen ts and found to ha v e go o d o v erall agreemen t. The effect of panel flexibilit y on the w all pressure as a function of stream wise lo cation and time is assessed, sho wing a mo dulation b y the panel vibration that affects the full panel stream wise exten t. W all pressure p o w er sp ectral densities sho w an elongation of the frequency band asso ciated with the flo w separation region for the flexible panel case o v er the nominal rigid-w all configuration, and excitation asso ciated with panel vibration. A dditionally , for the flexible panel case, the TBL is found to tak e a longer distance to reco v er do wnstream of the in teraction with the oblique sho c k. The region along the panel where lo w-frequency motions of the STBLI are dominan t is broadened for the flexible case but the p eak pressure fluctuations in that region are atten uated. F or the t w o cases with mean flo w separation, the study of separation bubble dynamics rev eals that panel flexibilit y increases the separation length and v olume, but lea v es the stream wise lo cation of the separation bubble cen troid nearly unaffected. T o assess the imp ortance of three-dimensional effects, sim ulations for the M ∞ = 3.0 , θ max = 17.5 case are conducted considering b oth the full span of the panel used in the exp erimen t as w ell as a ten-fold reduced span wise (p erio dic) domain equal to fiv e times the incoming TBL thic kness prior to the in teraction, δ 0 . Small span wise v ariations of panel deflection w ere observ ed in the full-span sim ulation results, with a lesser influence on the w all pressure and shear stress. These results indicate that span wise-p erio dic sim ulations with a reduced span of 5δ 0 can accurately predict the coupled in teraction with significan t reduction in computational cost. In the final part of this computational study , sim ulations are p erformed for a M ∞ = 3 TBL in teracting with a dynamic sho c k-expansion system generated b y a w edge p erio dically rotating b et w een t w o deflection angles, θ = 15.5 ◦ and 17.5 ◦ , at differen t oscillating frequencies, f w , ranging from 50 Hz to 800 Hz , o v er rigid xv and elastic panels. The deflection of the flexible panel is found to b e excited b y w edge oscillation frequencies, f w , close to the panels primary natural frequency , f n = 225 Hz . Increasing the oscillation frequency o v er rigid and flexible w alls results in an atten uated resp onse of the flo w separation bubble. xvi Chapter 1 In tro duction In teractions b et w een sho c k w a v es and turbulen t b oundary la y ers (STBLI) are critical to the design of su- p e rsonic and h yp ersonic flying v ehicles. Researc hed for decades ( Dolling , 2001 ), most STBLI studies ha v e fo c used on in teractions with b oundary la y ers dev elop ed o v er rigid w alls ( Délery & Dussauge , 2009 ). The fluid-structural coupling of STBLIs has not b een studied nearly as extensiv ely despite its relev ance in in- ternal (e.g., scramjet engines) and external (e.g., aero dynamic con trol surfaces) flo ws o v er ligh t w eigh t, thin panels ( McNamara & F riedmann , 2011 ). Of particular concern is the p oten tial coupling of vibrational mo des of the panel (e.g., engine w alls, con trol surfaces) with the c haracteristic lo w frequency motions of the sho c k system and separated flo w region that arise for sufficien tly strong in teractions ( Clemens & Nara y anasw am y , 2014 ), whic h can lead to flo w-induced cyclic loading, structural fatigue, and failure ( Sp ottsw o o d et al. , 2019 ). W all deformation can, in turn, alter the flo w c haracteristics of the STBLI, including the sho c k system, tur- bulence amplification, b oundary la y er dev elopmen t, and separation. Sufficien t ly strong STBLIs pro ducing mean flo w separation are c haracterized b y lo w-frequency motions of the sho c k system and (breathing motions of ) the separation bubble ( Clemens & Nara y anasw am y , 2014 ). The mec hanism resp onsible for causing these motions is a sub ject of debate within the comm unit y but it is widely b eliev ed that for stronger in teractions, where the separation length is m uc h greater than 2δ , where δ is the incoming b oundary la y er thic kness, the mec hanism is most lik ely dominated b y fluid en trainmen t and disc harge in the shear la y er do wnstream of the in teraction ( T oub er & Sandham , 2009 a ). F or w eak er in t eractions, with a separation length less than or equal to 2δ , the mec hanism resp onsible for the lo w frequency motion is b eliev ed to b e dominated b y upstream pressure fluctuations in the turbulen t b oundary la y er ( Clemens & Nara y anasw am y , 2014 ; Pip onniau et al. , 2009 ; Pirozzoli & Grasso , 2006 ; W u & Martin , 2008 ). Orders of magnitude lo w er than the c haracteristic frequencies of sup ersonic turbulen t b oundary la y ers (TBLs), these lo w-frequency motions asso ciated with separated STBLIs can p oten tially couple with (or b e enhanced b y) the natural vibrational frequencies of the structural panels of sup ersonic v ehicles, particularly 1 Figure 1.1: Sc hematic illustration of an impinging STBLI (adapted from Bermejo-Moreno et al. , 2014 ) as the panel flexibilit y increases with thinner and ligh ter structures. Suc h ev en ts could lead to resonance and catastrophic failure of the structure. Consequen tly , it is imp ortan t to gain a deep er understanding of the dynamics of this in teraction and its implications on the design of sup ersonic and h yp ersonic systems. A sc hematic illustration of a strong STBLI is sho wn in figure 1.1 . As noted b y T oub er & Sandham ( 2009 a ) and Willems ( 2017 ), STBLI con tains sev eral k ey comp onen ts. Upstream of the inciden t sho c k, the incoming flo w is a turbulen t b oundary la y er (usually fully dev elop ed). The inciden t oblique sho c k is at an angle β from the horizon tal, stream wise direction. A reflected sho c k forms opp osite of the inciden t sho c k, with b oth sho c ks in tersecting a certain distance from the w all, due to viscous effects. A separation sho c k forms upstream of the inciden t-reflected sho c k in tersection and spans v ertically from the in tersection with the inciden t sho c k do wn to the sonic line (where the Mac h n um b er is M = 1 ), whic h remains in close pro ximit y to the w all prior to flo w separation. A transmitted fourth sho c k forms at the common p oin t of in tersection with the inciden t, reflected, and separated sho c ks. Do wnstream of the separation sho c k, a turbulen t shear la y er is formed, whic h is resp onsible for v ortex shedding and flapping fluid motions do wnstream. The shear la y er increases its distance from the w all starting from the separation sho c k. This distance con tin ues to increase as the b oundary la y er thic k ens due to the adv erse pressure gradien t imp osed b y the sho c k system, un t il in tersecting with the transmitte d oblique sho c k. Behind this shear la y er, the flo w is sub ject to flo w rev ersal where there is a c hange in direction of the flo w (with the stream wise v elo cit y b ecoming negativ e, u < 0 ). This region of flo w rev ersal is called the separation bubble. When flo w separation o ccurs, the w all shear stress c hanges sign from p ositiv e to negativ e. Con v ersely , when flo w reattac hmen t o ccurs, the w all shear stress c hanges sign from negativ e to p ositiv e. The stream wise distance b et w een the p oin ts where the 2 flo w separates, x s , and reattac hes, x r , is called the separation length L sep . Behind the reflected sho c k, an expansion fan forms starting at the curv ed region of the transmitted sho c k. Upstream of this expansion fan but do wnstream of the reflected and inciden t sho c k in tersection, the pressure in the flo w field is highest in the STBLI. F arther do wnstream of the expansion fan, the distance b et w een the shear la y er and the w all b egins to decrease as the sonic line mo v es closer to w ards the w all. The b oundary la y er exp eriences turbulence amplification and thic k ening across the in teraction with the sho c k system. As the flo w is turned to b ecome w all-parallel again do wnstream of the expansion fan, compression w a v es form and coalesce ab o v e the sonic line to form the reattac hmen t sho c k that follo ws b ehind the reflected sho c k. The inciden t oblique sho c k could b e generated b y a compression w edge a w a y from the panel. This can lead to additional flo w features not sho wn in figure 1.1 suc h as Prandtl-Mey er expansions (PME), formed at the rear corner of the w edge of finite stream wise exten t, that ma y in teract with the STBLI. F or flexible panel STBLIs, there is usually a ca vit y with quiescen t flo w conditions b elo w the panel whic h is h eld at an appro ximately constan t pressure p cav . The panel (usually rectangular in shap e) ma y b e fixed on t w o sides and free on the other t w o or fixed on all four sides. Recen tly , an increasing n um b er of exp erimen ts ha v e in v estigated the coupling of STBLIs with flexible panels. In a sequence of exp erimen ts targeting statistically quasi t w o-dimensional configurations of oblique STBLIs impinging on a rectangular thin flexible panel, Willems et al. ( 2013 ), Daub et al. ( 2016 ), and Willems ( 2017 ) in v estigated the effects of the freestream Mac h n um b er, M ∞ , and the inciden t oblique sho c k angle, β . By carefully con trolling the pressure differen tial across the flexible panel, V arigonda & Nara y anasw am y ( 2019 ) in v estigated in teractions resulting in conca v e and con v ex panel curv ature. T ripathi et al. ( 2020 ) assessed the effects of the Reynolds n um b er, sho c k impingemen t lo cation and ca vit y pressure on the panel dynamics and separation bubble c haracteristics in M ∞ = 2 oblique STBLIs. T esting M ∞ = 2 oblique and M ∞ = 1.4 normal STBLI-panel coupling, Gramola et al. ( 2020 ) found a strong influence of the ca vit y pressure on the aerostructural dynamics, suggesting strategies for passiv e con trol of w a v e drag through adaptiv e sho c k con trol bumps. Exp erimen ts of M ∞ = 4 STBLIs impinging on a thin steel panel b y Neet & A ustin ( 2020 ) observ ed a flattened and elongated separation region and a reduction of static pressure in the flexible configuration relativ e to a rigid panel. Sp ottsw o o d et al. ( 2012 ) p erformed oblique sho c k impingemen t exp erimen ts on a complian t panel that w as fixed on all four sides. In that study , a coupled fluid structure in teraction (FSI) sim ulation is p erformed b y solving the Reynolds-A v eraged Na vier Stok es (RANS) equations and using the Men ter k -ω shear stress transp ort (SST) turbulence mo del. A dditionally , they dev elop ed a corresp onding reduced order surrogate mo del using quasi solutions coupled FSI sim ulation and lo cal piston theory . Whalen et al. ( 2020 ) lo ok ed at the STBLI of a flexible panel on a ramp exp osed to a Mac h 6 flo w with transitional and incoming turbulen t b oundary la y ers. The effect of aerothermal heating 3 w as eviden t in the non-linear panel resp onse obtained exp erimen tally , and equiv alen t rigid ramp test rev ealed evidence of feedbac k in to the do wnstream p ortion of the flo w field. Zhou et al. ( 2019 ) studied the mean separation length on rigid panel STBLI as a function of parameters suc h as the Reynolds n um b er, Mac h n um b er, and inciden t sho c k strength. As a result of this analysis, they prop osed t w o mo dels of separation length at lo w and high Mac h n um b ers. Comparing results from the rigid and flexible panel STBLI sim ulations, it is desirable to pro vide reduced-order mo dels that accoun t for panel compliance. This task presen ts c hallenges sinc e the flo w field of the STBLI is affected b y the panel deformation shap e whic h is dep enden t up on additional factors b esides mec hanical and geometric prop erties of the panel, suc h as the in viscid impingemen t p oin t and ca vit y pressure. The net pressure on the upp er surface of the panel and the cen troid of the pressure v ary dep ending on the impingemen t p oin t lo cation. These v ariations can result in panel deflection shap es that are asymmetric ab out the cen ter. Sev eral measuremen t c hallenges remain that prev en t a complete c haracterization of these coupled in- teractions, esp ecially of the near-w all flo w ph ysics, from exp erimen ts alone ( Riley et al. , 2019 ). Numerical sim ulations th us pla y a crucial role to complemen t exp erimen ts and pro vide missing fundamen tal insigh t. T o predict aero dynamic forces acting on the deforming structure, simplified form ulations based on piston, V an Dyk e, and sho c k-expansion theories are common for quasi-steady in teractions, due to the minimal computa- tional cost ( Brou w er & McNamara , 2019 ; McNamara & F riedmann , 2007 ; Sulliv an et al. , 2020 ). F or higher ph y sical fidelit y , prior studies resort to solving the in viscid Euler flo w equations ( Visbal , 2012 ) or the Na vier- Stok es equations via RANS ( Visbal , 2014 ; Gogulapati et al. , 2014 ; Shahriar et al. , 2018 ; Y ao et al. , 2017 ), detac hed-eddy sim ulations (DES) ( Gan & Zha , 2016 ), large-eddy sim ulation (LES) ( P asquariello et al. , 2015 ; Borazjani & Akbarzadeh , 2020 ; Shinde et al. , 2022 ), or DNS ( Shinde et al. , 2018 ), coupled with structural solv ers ( Sc hemmel et al. , 2020 ). Although attractiv e from a computational cost standp oin t, RANS cannot ac- curately predict strong flo w separation and asso ciated lo w-frequency dynamics in STBLIs ( Sadagopan et al. , 2021 ). The stringen t grid resolution requiremen ts of DNS and w all-resolv ed LES render these sim ulation metho ds still prohibitiv ely exp ensiv e b ey ond mo derate Reynolds n um b ers, particularly for the long in tegra- tion times required to capture lo w-frequency motions and under in teractions with s pan wise inhomogeneit y (brough t, for example, b y the panel deflection). W all-mo dele d large-eddy sim ulation (WMLES) ( Larsson et al. , 2016 ; Bose & P ark , 2018 ) greatly reduces the computational cost of sim ulating w all-b ounded turbulence b y mo deling, instead of resolving, the inner region of the turbulen t b oundary la y er up to appro ximately 10% of the b oundary la y er thic kness (including the viscous, buffer, and part of the logarithmic subla y ers). Prior w ork has pro v en WMLES capable of cap- turing the flo w ph ysics of non-equilibrium, separated STBLIs o v er adiabatic rigid w alls, enabling sim ulations 4 with long in tegration times, critical to the analysis of lo w-frequency unsteadiness, and of in teractions with 3D effects ( Bermejo-Moreno et al. , 2014 ). When solving FSI problems through n umerical sim ulations, t w o main strategies exist to couple the fluid and solid domain: monolithic and partitioned/staggered solv ers ( Bazilevs et al. , 2013 ). With the monolithic approac h, the fluid and solid are strongly coupled together b y a set of equations (mesh, solid mec hanics, and fluid mec hanics) that m ust b e satisfied at ev ery time step ( Heil et al. , 2008 ). Within this approac h, there are three t yp es of tec hniques: blo c k-iterativ e, quasi-direct, and direct coupling ( Bazilevs et al. , 2013 ). In the blo c k-iterativ e coupling approac h, the fluid, solid, and mesh are treated as three separate blo c ks and non-linear iterations are p erformed one blo c k at a time in cyclic fashion. With the quasi-direct coupling approac h, the fluid-structure and mesh system are treated as t w o separate blo c ks. Finally , with the direct coupling approac h, the fluid, structure and mesh system is treated as a single blo c k and m ust b e solv ed sim ultaneously . In all three cases, iterations are p erformed within a single time step un til the solution of all the coupled equations con v erge. The partitioned approac h couples the domains together using sp ecialized fluid flo w and solid structure solv ers, not form ulated sp ecifically for FSI. Unlik e the monolithic approac h, the partitioned approac h do es not necessarily require con v ergence at eac h time step. As a result, ph ysical quan t ities at the in terface are exc hanged less frequen tly . The main adv an tage of monolithic solv ers is that they are generally more stable in situations where there is an incompressible fluid surrounding a solid domain or when the solid structure is ligh t and flexible and the fluid is dense. The first disadv an tage for monolithic solv ers includes the requiremen t to write a complete FSI solv er that is not easily able to lev erage existing solv ers. The second disadv an tage of monolithic solv ers is that this metho d is not computationally efficien t for problems where strong coupling is not necessary . Using the staggered, or lo osely-coupled, FSI approac h, the degree of coupling can b e adjusted accordingly to the problem, b y c hanging the exc hange frequency , and existing, sp ecialized solv ers can b e utilized for eac h computational domain (fluid, structure, mesh). This also allo ws for a more robust v erification and v a lidation pro cess of eac h indep enden t solv er. F or example, in circumstances where the turbulen t flo w frequencies are m uc h greater than the natural frequencies of solid structure motion, the frequency of solv er comm unication and domain remeshing can b e reduced, resulting in lo w er computational o v erhead. As a consequence, the use of lo osely-coupled tec hniques for compressible flo w FSI has b e en adopted with success ( Pip erno et al. , 1995 ; Ostoic h et al. , 2013 ; Ostoic h , 2013 ). Within the category of lo ose FSI coupling, there exist differen t strategies to impro v e the accuracy of coupling b y predicting the fluid-solid in terface p osition through extrap olation ( Mic hler et al. , 2004 ; Miller & McNamara , 2015 ). As outlined in c hapter 3 , the FSI solv er used in the presen t study utilizes a simple lo osely-coupled approac h that tak es the instan taneous pressure field from the finer flo w mesh and do wnsamples it on to the coarser solid mesh at a set frequency . 5 The c hange in displacemen t field from the last exc hange p erio d is then upsampled from the solid mesh and transferred bac k to the flo w solv er mesh at set frequency (whic h need not b e the same as that of the exc hange from the flo w field to the solid domain). Due to the explicit R unge Kutta metho ds used b y the solid solv er, a small time step size is required to main tain stabilit y for the solid solv er. The use of small time step sizes in this case reduces the need to impro v e coupling accuracy using the established approac hes outlined ab o v e. The presen t w ork fo cuses on the n umerical sim ulation replicating the exp erimen ts of Daub et al. ( 2016 ) of impinging STBLIs o v er rectangular rigid and flexible panels at t w o differen t freestream Mac h n um b ers (M ∞ = 3 and M ∞ = 4 ) and for rotations of the compression w edge leading to three differen t end deflection angles (θ = 17.5 ◦ , for the lo w est Mac h n um b er; 20 ◦ and 15 ◦ for the highest). The main no v elt y regarding the n umerical metho dology resides in the coupling of WMLES (emplo ying an equilibrium-based w all stress mo del Ka w ai & Larsson , 2012 ) and a finite-elemen t metho d (FEM) solid mec hanics solv er that incorp orates structural damping. The use of WMLES enables the sim ulation to b e p erformed at the same Reynolds n u m b e r (Re ∞ = 49.410 6 /m ), spanning the full panel width and for the same duration as rep orted in the exp erimen ts, allo wing for a more complete c haracterization of STBLI lo w-frequency motions and coupling with the flexible panel dynamics. Previous n umerical studies of the same exp erimen t for the M ∞ = 3 and θ = 17.5 configuration w ere conducted b y P asquariello et al. ( 2015 ) and Zop e et al. ( 2021 ). The former used w all-resolv ed LES and a h yp erelastic Sain t-V enan t-Kirc hoff FEM solid solv er that neglected structural damping, with a cut-cell immersed b oundary metho d treatmen t of the fluid-solid in terface. The sim ulation ran for a shorter duration, not reac hing a quasi-steady state of panel deflection. Zop e et al. ( 2021 ) utilized b oth Dynamic Hybrid RANS/LES (DHRL) and RANS approac hes for the flo w solv er, also neglecting damping in the solid mec hanics solv er. P asquariello et al. ( 2017 ) p erformed an adiabatic rigid-w all LES of the M ∞ = 3 and θ = 20 configuration b y Daub et al. ( 2016 ). In the sim ulation, the w edge angle remained constan t and the top b oundary condition used the in viscid flo w assum ption instead of including the ph y sical geometry of the mo ving w edge in the sim ulation flo w domain. The LES grid used for the analysis of lo w-frequency motions had 363 million cells and w as run for a total of 25.6 ms . Findings of that study supp ort the theory that Görtler-lik e v ortices serv e as a mec hanism for lo w-frequency unsteady motions of the separation sho c k. Moreo v er, they illustrate the fact that the Görtler-lik e v ortices ha v e sufficien t span wise z length suc h that the sim ulation span wise domain width needs to b e large enough (L z > 5δ 0 in this instance) to prop erly capture these structures. Recen tly , there has b een an in terest in STBLI of mo ving and oscillating sho c ks where c hanges in su- p e rsonic fligh t v ehicle sp eed and pitc h angle cause the oblique sho c k prop erties to dynamically c hange in time. The ph ysics of this dynamic in teraction particularly with flexible surfaces is not w ell understo o d. Exp erimen ts b y Currao et al. ( 2019 ) used a can tilev ered flexible panel with a computed primary natural 6 frequency f n = 87 Hz that w as impinged up on b y an oblique sho c k that w as generated b y an upstream w edge whic h deflected the M = 5.8 h yp ersonic flo w do wn w ard b y θ = 10 ◦ . The do wn w ard deflection of the panel caused expansion of the flo w w eak ening the impinging sho c k and c hanging the impingemen t p oin t. Pressure sensitiv e pain t (PSP) measuremen ts along with RANS solutions indicated that p eak w all pressure w as reduced b y increasing panel deflection. Exp erimen ts b y Currao et al. ( 2021 ) lo ok ed at the effect of w edge oscillation on a M = 5.8 h yp ersonic flo w with a freestream Reynolds n um b er of Re ∞ = 710 6 m −1 o v er a rigid plate. The w edge pitc h angle α v aried from α max = 10 ◦ to α min = 0 ◦ with an oscillation frequency f w = 42 Hz . It w as found that the transien t effects induced b y the sho c k motion result in a maxim um bubble length v ariation of 30 p ercen t. A dditionally , pressure and heat flux mo des consisten t with the w edge oscillation frequency w ere found ev erywhere along the plate. This thesis is organized as follo ws. Chapter 1 (this c hapter) in tro duces the problem to b e studied and pro v ides a literature o v erview of the previous w ork done in this field of study . Chapter 2 describ es the setup of the problem to b e studied. Chapter 3 describ es the computational metho dology of the FSI solv er. Chapter 4 presen ts the results from the n umerical sim ulations p erformed and is comp osed in to the follo wing sections. Chapter 5 pro vides conclusions from the results presen ted and discusses future researc h directions that could add on to the findings of this w ork. 7 Chapter 2 Problem Setup The sim ulations conducted in the presen t study are based up on the exp erimen ts of Daub et al. ( 2016 ) conducted at the trisonic wind tunnel (TMK) of the Sup ersonic and Hyp ersonic T ec hnologies Departmen t at the German A erospace Researc h Cen ter (DLR). In the exp erimen ts, oblique sho c k w a v es in teract with sup ersonic turbulen t b oundary la y ers o v er rigid and elastic panels. T w o sim ulation efforts will b e presen ted. In the first set, a sho c k-generating w edge with its b ottom side initially parallel to the freestream is rotated in time un til reac hing a constan t setp oin t, follo wing the exp erimen tal configuration for differen t freestream conditions and rotation angles. In the second set of sim ulations, the w edge rotation oscillates b et w een t w o setp oin ts with differen t rotation frequencies, departing from the exp erimen ts to in v estigate additional fluid- structural coupling leading to limit-cycle oscillations and resonance. This c hapter describ es the problem setup for eac h one of these sim ulation efforts. 2.1 Rotating w edge reac hing a constan t setp oin t The first set of sim ulations in the presen t study seeks to replicate the exp erimen ts of Daub et al. ( 2016 ). Figure 2.1 sho ws a cross section of the test setup in the xy Cartesian plane. The horizon tal (stream wise) co ordinate, x , is aligned with the incoming freestream and has its origin (x = 0 ) at the leading edge of the rigid test apparatus. The v ertical co ordinate, y , is normal to the upp er rigid w all surface, along whic h the turbulen t b oundary la y er dev elops. The span wise co ordinate, z (normal to the page in figure 2.1 ), has its origin at the cen ter of the panel. The inciden t oblique sho c k is generated b y a w edge that is actuated b y a serv o-con trolled motor. The w edge rotates ab out the cen ter p oin t (x rot ,y rot ) = (112,182) mm . In the exp erimen t, the w edge has an angle α = 30 ◦ and is L front−rear = 84 mm . The distance from the rotation cen ter to the rear cen ter is L rot−rear = 39 mm long. F or non-zero deflection angles, at the rear end of the w edge, the flo w undergo es 8 = Figure 2.1: Sc hematic of the problem setup with the flo w computational domain in gra y (the solid domain includes the flexible panel). The in viscid inciden t sho c k impingemen t lo cation, x I , and w edge deflection corresp ond to the case with M ∞ = 3.0 and θ max = 17.5 ◦ . a Prandtl-Mey er t yp e expansion whic h in teracts with the inciden t sho c k causing it to curv e to w ards the stream wise direction. The w edge is initially set with its b ottom side parallel to the incoming flo w direction and parallel to the rigid w all, resulting in a flo w deflection angle θ 0 = 0 (i.e., no inciden t sho c k w a v e). Prior to w edge rotation, the flo w o v er the panel (b oth flexible and rigid) is c haracterized as that of a fully dev elop ed compressible turbulen t b oundary la y er with an incoming b oundary la y er thic kness δ 0 (measured at x = 200 mm ) and a freestream Mac h n um b er M ∞ (measured ab o v e the b oundary la y er). Due to the ph ysical limitations of the serv o motors used and the in ternal con trol system used, the w edge angle do es not reac h its setp oin t instan tly and exhibits some notable degree of o v ersho ot and undersho ot. A ccording to Daub et al. ( 2016 ), these rotary oscillations are on the order of 0.1 ◦ . Digitized data from Daub et al. ( 2016 ) is used to mo del the w edge angle rotation as a function of time. Up o n reac hing the maxim um time plotted b y Daub et al. ( 2016 ), the w edge angle is set to a constan t v alue. Figure 2.2 (a) sho ws the w edge angle v ersus time for the three cases discussed. F or case (1), with an incoming Mac h n um b er M ∞ = 3 , the angular setp oin t θ max = 17.5 ◦ is reac hed within appro ximately 15 ms , but is first o v ershot, reac hing a maxim um of 18 ◦ , and then readjusted to w ards the setp oin t. F or the cases in v olving the incoming M ∞ = 4 turbulen t b oundary la y er, case (3) with θ max = 15 ◦ , and case (2) with θ max = 20 ◦ , the setp oin t is undershot and nev er reac hed, with the angular p osition not reac hing a quasi-steady state but instead approac hing it in time. The implication of the w edge not oscillating around the setp oin t is that a 9 (a) (b) Figure 2.2: (a) W edge angle v ersus time for the three sim ulated cases (1, 2, and 3) digitized from Daub et al. ( 2016 ). (b) Zo omed-in normalized w edge angle v ersus time for the three sim ulated cases Mac h n um b er Pressure T emp erature V elo cit y Max deflection angle In viscid impingemen t p oin t M ∞ p ∞ [kPa] T ∞ [K] u ∞ [m/s] θ max [ ◦ ] x I [mm] 3.0 15.6 97.2 595 17.5 328 4.0 8.7 64.9 650 20.0 426 4.0 8.7 64.9 650 15.0 363 T able 2.1: Flo w prop erties for exp erimen tal configurations. Subscript 1 refers to the freestream conditions. sligh t drift will b e presen t in time-a v eraged statistics o v er time, whic h will b e discussed in more detail in the results c hapter (§ 4 ). This b eha vior is sho wn more clearly in figure 2.2 (b) where the time ev olution of the w edge angle is normalized b y the setp oin t, zo oming in to the region near the quasi-steady state. The incoming flo w prop erties for the three test cases are sho wn in table 2.1 . The ca vit y b elo w the panel is set to ha v e a constan t pressure of p ∞ suc h that the panel is in static equilibrium when there is no oblique sho c k impingemen t. As summarized in table 2.2 , the elastic panel is 1.47 mm thic k and is made of spring steel (CK 75). It has an elastic mo dulus of E = 206 GPa , a densit y of ρ = 7800 kg/m 3 , and a P oisson’s ratio ofν = 0.33 . Based on measuremen ts of panel oscillation rate of deca y , the damping ratioζ is estimated to b e ζ 0.04 . The panel is defined in the exp erimen t as b eing 300 mm long, whic h measures the distance from the fron t to rear edge of the rigid frame holding the elastic panel in place. T w o ro ws of riv ets are staggered from eac h other and coun tersunk on to the fron t and rear of the panel to hold the panel to the frame, as sho wn in figure 2.3 (a). The inner ro ws of riv ets are spaced 308 mm apart and the outer ro ws of riv ets are Y oung’s Mo dulus Densit y P oisson’s ratio P anel thic kness P anel length Damping ratio E [GPa] ρ [kg/m 3 ] ν h [mm] L [mm] ζ 206 7800 0.33 1.47 300 0.04 T able 2.2: Prop erties of the elastic panel 10 spaced 320 mm apart. Lo oking at the test setup in the yz plane as sho wn in figure 2.3 (b), the panel spans 200 mm in the span wise direction. There is foam sealing along the edges to prev en t leakage from the b ottom ca vit y . The presence of this foam sealing pro vides additional system damping and stiffness along the edges reducing the amoun t of do wn w ard deflection. (a) (b) Figure 2.3: Rectangular elastic plate: (a) xz plane view, sho wing the t w o ro ws of riv ets near eac h transv erse edge, mark ed b y gra y circles, and the lo cation of the exp erimen tal fron t, cen ter, and rear prob es of panel displacemen t are mark ed b y red, blue, and green squares, resp ectiv ely . (b) yz plane view, with the ca vit y underneath the panel. In the exp erimen ts b y Daub et al. ( 2016 ), the rigid panel w as instrumen ted with 30 pressure prob es placed along the cen terline at z = 0 , spaced 10 mm apart to measure the time-a v eraged pressure in the stream wise direction. A dditionally , 10 high-sp eed pressure sensors w ere placed at v arious lo cations along the cen terline for p o w er sp ectral densit y data. The instrumen tation for the flexible panel did not include pressure prob es, but three capacitiv e distance sensors w ere used to measure panel deflection o v er time. The three prob es, lab eled fr ont , c enter , and r e ar , w ere placed 75 , 155 , and 225 mm , resp ectiv ely , do wnstream of the edge of the rigid frame in con tact with the elastic plate. This corresp onds to stream wise lo cations x = 295 mm , x = 375 mm , and x = 445 mm , resp ectiv ely , in the global co ordinate system. 11 Figure 2.4: W edge angle v ersus time for select con tin uous w edge oscillation cases f w = (50,200,800) Hz o v er a 20 ms timespan 2.2 Rotating w edge oscillating b et w een t w o setp oin ts In the second part of this w ork, the effect of induced w edge oscillation on STBLI for flexible and rigid panels is in v estigated. T o do this, b oth rigid and flexible sim ulations are initialized using the quasi-steady state from the previously sim ulated case (1), with M = 3 and θ max = 17.5 ◦ . The w edge is then oscillated at a set frequency , f w , b et w een θ min = 15.6 ◦ and θ max = 17.6 ◦ . The v alues of θ max and θ min are c hosen suc h that the flo w exp eriences con tin ual separation and that exp erimen tal data is a v ailable for the steady θ max and θ min cases. T o b etter illustrate this, the w edge angle θ v ersus time t for sev eral differen t oscillation frequencies f w = [50,200,800] Hz is sho wn in figure 2.4 . The minim um w edge frequency of f w = 50 Hz w as c hosen suc h that a sufficien t n um b er of cycles w ould b e able to b e run without prohibitiv e computational cost. Ideally , the frequency should also b e lo w enough suc h that quan tities (suc h as panel displacemen t and w all pressure) can b e appro ximated b y fixed-angle quasi-static solutions of the w edge angle set at θ min and θ max , resp ectiv ely . Con v ersely , the maxim um w edge frequency of f w = 800 Hz w as c hosen suc h that the quan tities obtained w ould b e w ell-appro ximated b y the quasi-static mean w edge angle θ mean = 1 2 (θ min +θ max ) . As the w edge oscillation frequency approac hes the resonan t frequency f n of the panel, the goal is to induce resonance and/or limit cycle oscillation (LCO) on the panel, and analyze these effects on the flo w field. 12 Chapter 3 Computational Metho dology This c hapter is organized in to the follo wing sections: Section 3.1 describ es the high-lev el o v erview of the FSI solv er arc hitecture. Section 3.2 describ es the flo w solv er implemen tation whic h details the w all-mo deled large eddy sim ulation and b oundary conditions. Section 3.3 describ es the solid solv er finite elemen t discretization. Section 3.4 describ es the mesh deformation solv er. Section 3.5 describ es the metho ds used to couple together the solid, flo w, and mesh deformation solv ers. 3.1 Coupled solv er arc hitecture The presen t sim ulations are p erformed using a partitioned FSI solv er (uPDE, dev elop ed in-house) with lo o sely coupled domain-sp ecific solv ers for the fluid flo w and solid structure, complemen ted with a mesh deformation solv er for a sp ecified region of in terest of the flo w d omain. The flo w and solid domains are discretized using unstructured, b o dy-fitted meshes. A t the in terface b et w een the solid and fluid domains, the t w o meshes need not b e conformal, allo wing differen t grid resolutions to b e considered dep ending on the ph y sics that need to b e resolv ed in eac h domain. F or the presen t sim ulations, the fluid domain mesh is m uc h finer than the solid domain mesh at the fluid-solid in terface, with eac h fluid mesh face b eing connected to a single solid face, whereas eac h solid mesh face is connected to (and fully encloses) m ultiple fluid mesh faces. The three solv ers comm unicate to eac h other in the follo wing manner. The flo w solv er sends pressure field information at the fluid-solid in terface to the solid solv er. The solid solv er sends displacemen t information at the fluid-solid in terface to the flo w mesh deformation solv er. The flo w mesh deformation solv er then up dates the c hange in flo w mesh geometry along with the corresp onding cell v elo cit y to the flo w solv er, whic h then up d ates its differen tial op erators. The rate at whic h information is exc hanged b et w een the three solv ers need not b e the same and is targeted to balance computational efficiency with accuracy . A blo c k diagram of this pro c ess is sho wn in figure 3.1 . 13 Flo w Solv er Solid Solv er Mesh Deformation Pro ject w all pressure pw Pro ject w all displacemen t Ys Up date mesh geometry and discrete op erators Figure 3.1: FSI solv er high lev el arc hitecture Solv er time step and coupling frequency ∆t Solid solv er time step 10 −7 s Flo w solv er time step 10 −7 s Mesh deformation frequency 10 −5 s Flo w-to-solid coupling (do wnsample pressure) 10 −6 s Solid-to-flo w coupling (upsample displacemen t) 10 −5 s T able 3.1: Solv er time in tegration and coupling frequency v alues Eac h comp onen t of the FSI solv er is resp onsible for its o wn time in tegration. In the case of the mesh deformation s olv er, the presen t form ulation used do es not consider time effects but is instead used to con v ert displacemen t data at the surface fluid-solid in terface to the finite v olume mesh of the flo w domain. This means that t w o coupling frequencies are presen t, the solid-to-flo w and the flo w-to-solid coupling frequencies. Eac h of these coupling frequencies is c hosen to balance computational efficiency with accuracy . T able 3.1 sho ws the differen t in tegration times and coupling frequencies used in the coupled FSI solv er. A constan t time step size w as c hosen in order to conduct p o w er-sp ectral densit y analyses from the resulting time signals of w all pressure and panel deflection output from the sim ulations. In order to main tain n umerical stabilit y , the time step size w as selected to fall within the Couran t–F riedric hs–Lewy Condition (CFL) limits imp osed b y the fourth-order R unge-Kutta (RK4) time in tegration sc heme used b y eac h domain-sp ecific solv er. 3.2 Flo w Solv er Subsection 3.2.1 details the n umerical metho ds of the flo w solv er. Subsection 3.2.2 describ es the compu- tational domain and mesh. Subsection 3.2.3 describ es the b oundary conditions used b y the flo w solv er (including the w all mo del). Subsection 3.2.4 describ es the n umerical instrumen tation used on the flo w solv er to record data for analysis. 14 3.2.1 Computational Metho dology The flo w solv er in tegrates the spatially-filtered compressible Na vier-Stok es equations using a finite-v olume, cell-cen tered form ulation on unstructured, b o dy-fitted, hexahedral meshes. The starting p oin t is the differ- en tial form of conserv ation la ws of mass (equation 3.1 ), linear momen tum (equation 3.2 ), and total energy (equation 3.3 ) for a compressible flo w without external b o dy forces: ∂ρ ∂t + ∂ ∂x j (ρu j ) = 0, (3.1) ∂ρu i ∂t + ∂ ∂x j (ρu i u j ) = ∂σ ij ∂x j , (3.2) ∂ρe T ∂t + ∂ ∂x j (ρe T u j ) = ∂u i σ ij ∂x j ∂q j ∂x j , (3.3) where Einstein summation notation for rep eated subindices is assumed, ρ is the densit y , u i is the v elo cit y , e T =e+ 1 2 u i u i is the total energy p er unit mass, q j is the heat flux giv en b y F ourier’s la w as q j =k∂T/∂x j (k is the thermal conductivit y and T is the temp e rature), and σ ij is the stress tensor, defined as σ ij =pδ ij +τ ij , (3.4) where p is the thermo dynamic pressure, µ v is the bulk (or v olume) viscosit y (assumed zero in the presen t study), and τ ij is the viscous stress tensor. F or a Newtonian fluid, the viscous stress tensor, τ ij , is defined as follo ws: τ ij = 2µS ij + µ v 2 3 µ ∂u k ∂x k δ ij (3.5) Where S ij is kno wn as the strain-rate tensor (symmetric part of the v elo cit y gradien t tensor) defined b elo w. S ij = 1 2 ∂u i ∂x j + ∂u j ∂x i ! (3.6) The dynamic viscosit y , µ , is mo deled using Sutherland’s la w in the presen t sim ulations: µ =µ ref T T ref 3/2 T ref +T s T +T s (3.7) The reference p oin t used for the curren t sim ulations is T ref = 293.15 K , µ ref = 1.82110 −5 Ns/m 2 , and T s = 122 K . The w orking gas is air, assumed calorically p erfect with constan t v alues of the heat capacities at constan t pressure and v olume, c p and c v , resp ectiv ely , and th us follo wing an ideal gas equation of state, p = ρRT . 15 The gas constan t R = 287 J/(kgK) is related to the sp ecific heat capacities at constan t pressure, c p , and at constan t v olume, c v , b y R =c p c v . The ratio of sp ecific heat capacities, γc p /c v , is 1.4 . The compressible flo w LES equations can b e formally obtained b y in tro ducing a spatiotemp oral filter op e rator,F , acting on a function f with a k ernel filter G o v er a filtering domain, D , in ⃗ ξ =fx;tg b y f F[f] Z D G( ⃗ ξ ⃗ ξ ′ )f( ⃗ ξ ′ )d ⃗ ξ ′ , (3.8) and densit y-a v eraged quan tities ( F a vre , 1965 ), e f ρf/ρ . Assuming that the filter op erator comm utes with the spatiotemp oral deriv ativ es, the filtered go v erning equations can b e written as ( Bermejo-Moreno , 2009 ) ∂ρ ∂t + ∂ρe u j ∂x j = 0 (3.9) ∂ρe u i ∂t + ∂ρe u i e u j ∂x j = ∂p ∂x j + ∂ˇ τ ij ∂x j " ∂ρg u i u j ∂x j ∂ρe u i e u j ∂x j # + " ∂τ ij ∂x j ∂ˇ τ ij ∂x j # (3.10) ∂ρe e T ∂t + ∂ρe e T e u j ∂x j = ∂pe u j ∂x j + ∂ˇ τ ij e u i ∂x j ∂ˇ q j ∂x j " ∂ρ] e T u j ∂x j ∂ρe e T e u j ∂x j # " ∂pu j ∂x j ∂pe u j ∂x j # + " ∂τ ij u i ∂x j ∂ˇ τ ij e u i ∂x j # " ∂q j ∂x j ∂ˇ q j ∂x j # (3.11) where ˇ τ ij τ ij ( e T,e u) =µ( e T) ∂e u i ∂x j + ∂e u j ∂x i ! + µ v ( e T) 2 3 µ( e T) ∂e u k ∂x k δ ij 6=e τ ij , (3.12) ˇ q j ˇ k ∂ e T ∂x j =k( e T) ∂ e T ∂x j =q j ( e T)6=e q j . (3.13) p =RρT =Rρ e T = (γ1)ρe e (3.14) p γ1 =c v ρ e T =ρe =ρe T 1 2 ρu i u i =ρf e T 1 2 ρe u i e u i 1 2 ρ(g u i u i e u i e u i ). (3.15) The terms in square brac k ets in equations ( 3.9 - 3.11 ) require mo deling to accoun t for motions at subfilter scales. In the presen t implemen tation of the LES equations, the filtering is done implicitly b y the mesh discretization, and no additional explicit filter is applied. Therefore, the subfilter scales corresp ond to the subgrid scales b elo w the mesh resolution. Only the subgrid-scale stress tensor, τ S ij =ρ(g u i u j e u i e u j ) and the subgrid-scale heat flux, q S j =ρ( g Tu j e Te u j ) , will b e mo deled in the presen t w ork, neglecting the con tribution from the other subgrid-scale terms. The subgrid-scale stress tensor used follo ws the eddy-viscosit y mo del 16 of V reman ( 2004 ), with a mo del constan t of 0.07. The subgrid-scale heat flux is mo deled follo wing F ourier’s la w with a thermal diffusivit y calculated dividing the subgrid-scale kinematic eddy viscosit y b y the turbulen t Prandtl n um b er, tak en as 0.9. T o correct for FSI effects, the motion and deformation of the flo w mesh is accoun ted for b y an Arbitrary Langrangian Eulerian (ALE) form ulation that mo difies the con v ectiv e terms with the relativ e flo w v elo cit y , accoun ting for the v elo cit y of the mesh. The starting p oin t of the ALE in tegral form of the conserv ation equations is the Reynolds T ransp ort Theorem applied to an arbitrary v olume δV c whose b oundary mo v es with the mesh v elo cit yu m i , as noted b y Donea et al. ( 2004 ). Th us, the filtered conserv ation la ws of equations 3.9 , 3.10 , and 3.11 in in tegral form at eac h mesh cell c with v olume δV c and b ounding surface area δA c are defined as follo ws: d dt Z δVc qdV + Z δAc FdA = Z δAc GdA (3.16) where q = [ρ,ρe u i ,ρe e T ] T is the state v ector of conserv ed v ariables, F = ρr n ,ρr n e u i +pn i ,r n (ρe e T +p) T is the con v ectiv e (Euler) flux v ector and G = h 0,(ˇ τ ij +τ S ij )n j ,ˇ τ ij e u i n j +(ˇ q j +q S j )n j i T is the diffusiv e flux v ector. The comp onen t of the relativ e flo w v elo cit y normal to the surface of the (cell) con trol v olume is r n = (u i u m i )n i , where u m i is the mesh v elo cit y and n i the unitary surface normal. In semi-discrete form, equation ( 3.16 ) can b e expressed for eac h cell c as: d dt q c = 1 δV c X f∈δAc (F f c +G f c )A f c (3.17) whereq c is the state v ector at the cell cen troid, whereasF f c andG f c are the con v ectiv e and viscous fluxes at fac e f of the cell c . Resolving turbulence requires lo w-dissipativ e, cen tered n umerical sc hemes, whereas up wind sc hemes are needed to capture sho c k w a v es, necessarily in tro ducing undesired n umerical dissipation. These t w o opp osing requiremen ts are com bined in the flo w solv er b y means of a solution-adaptiv e approac h. A sho c k sensor distinguishes, at eac h time step, sho c k-free regions, where a second-order cen tered sc heme is emplo y ed to accurately resolv e turbulen t features, from near-sho c k regions, where a second-order essen tially non- oscillatory (ENO) sc heme is used with a Harten–Lax–v an Leer con tact (HLLC) appro ximate Riemann solv er. The sho c k sensor iden tifies near-sho c k regions in cells where ∂u k /∂x k > max 1.2 p ω k ω k ,0.05c/∆ , where ∂u k /∂x k is the dilatation, ω k ω k is the enstroph y , c is the sound sp eed, and ∆ is the mesh cell size. 17 3.2.2 Mesh and computational domain As sho wn in figure 2.1 , the computational flo w domain in the stream wise direction starts at x = 130 mm and ends at x = 570 mm . In the w all normal direction, the flo w domain ranges from the solid w all up to a heigh t y =L y = 100 mm . Finally , the flo w domain in the span wise dimension ranges from z =100 mm to z = 100 mm in the full-span sim ulations (replicating the full panel used in the exp erimen ts b y Daub et al. , 2016 ) and from z = 10 mm to z = 10 mm for the reduced-span p erio dic mesh. It is imp ortan t to note that the ph ysical geometry of the rotating w edge is not included within the computational domain, as the w edge is lo cated ab o v e the top b oundary . Instead, the effects of the w edge (impinging sho c k and Prandtl- Mey er expansion) are accoun ted for b y means of a time-v arying in viscid flo w mo del (based on instan taneous application of steady sho c k-expansion theory) along the top b oundary (y =L y ). In terms of the reference b oundary la y er thic kness, δ 0 , this computational flo w domain corresp onds to an initial size (b efore deformation) of 110δ 0 25δ 0 5δ 0 . The flo w mesh consists of t w elv e zones in total, three in the stream wise direction m ultiplied b y four in the w all normal direction. The stream wise zones are laid out suc h that the first and third zones are ab o v e rigid w alls, while the second zone is ab o v e the flexible panel. A summary of the flo w mes h spacing is pro vided in table 3.2 , whic h follo ws general guidelines for WMLES found in Larsson et al. ( 2016 ). Uniform mesh spacing is used in the stream wise and span wise directions, ∆x 0.08δ 0 and ∆z = 0.05δ 0 , resp ectiv ely . In the w all-normal direction, the first mesh zone extends from the w all to the w all-mo del exc hange lo cation y = h wm = 0.1δ 0 , with a uniform spacing of ∆y = 0.025δ 0 . Therefore, four cells are included b elo w the lo cation where information is passed from the LES solution to the w all mo del, to reduce n umerical errors and a v oid the log-la y er mismatc h ( Ka w ai & Larsson , 2012 ). The second mesh zone in y spans from y = h wm to y = δ 0 , with a w all-normal spacing increasing up to ∆y = 0.05δ 0 follo wing a h yp erb olic tangen t stretc hing. The third zone, whic h spans from y = δ 0 to y = 4δ 0 , follo ws a uniform w all-normal grid spacing, ∆y = 0.05δ 0 , whereas the fourth and final zone spans from y = 4δ 0 to the top b oundary at y = 25δ 0 with another h yp erb olic tangen t stretc hing la w. T able 3.2 , summariz es the exact spacing equations used in the y direction along with the n um b er of ce lls used in eac h direction of the mesh. F or reference, a plot of the mesh size as a function of the w all-normal y co ordinate (in linear and log form) is sho wn in figure 3.2 . Com bining the zone spacings together from table 3.2 , the resulting n um b er of cells for the flo w mesh in eac h co ordinate direction is [N x ,N y ,N z ] = [1408,140,100/1000] , with a total n um b er of cells of nearly 20 million and 200 million in total for the reduced span and full span sim ulations resp ectiv ely . Whereas most of the results presen ted in this w ork corresp ond to the flo w mesh with the nominal grid resolution 18 Stream wise distance, x Spacing equation, 0<ξ < 1 , Num b er of cells, N x x2 [130,210] mm ξ 256 x2 [210,530] mm ξ 1024 x2 [530,570] mm ξ 128 W all-normal distance, y Spacing equation, 0<η < 1 , Num b er of cells, N y y2 [0.0,0.1]δ 0 η 4 y2 [0.1,1.0]δ 0 1.0+1.394tanh[(η1)arctanh(0.717)] 22 y2 [1.0,4.0]δ 0 η 58 y2 [4.0,25]δ 0 1.0+1.166tanh[(η1)arctanh(0.857)] 162 Span wise distance, z Spacing equation, 0<ζ < 1 , Num b er of cells, N z Span wise p erio dic (Reduce d Span): z2 [10,10] mm ζ 100 Span wise symmetric (F ull Span): z2 [100,100] mm ζ 1000 T able 3.2: Flo w mesh spacing. ξ , η , and ζ are natural co ordinates discretized b y uniformly distributed no d es, suc h that the linear maps ξ 7! (xx min )/(x max x min ) , η 7! (yy min )/(y max y min ) , and ζ 7! (zz min )/(z max z min ) ) corresp ond to uniform spacing in the ph ysical co ordinates x , y , and z , resp ectiv ely (where min and max corresp ond to the minim um and maxim um v alues of the co ordinate direction in eac h in terv a l c onsidered). Non-linear maps result in grid stretc hing in the corresp onding co ordinate direction. describ ed ab o v e, additional sim ulations of the rigid-w all configuration w ere conducted on a finer reduced- span (z2 [10,10]mm ) mesh of appro ximately70 million cells to ev aluate grid con v ergence of the results, as sho wn in figure 4.1 . Time in tegration in the flo w solv er emplo y ed a four-stage explicit R unge-Kutta metho d with a constan t time step ∆t = 10 −7 s . This time step size fell w ell within CFL stabilit y limits throughout the duration of all sim ulations that w ere run. (a) (b) Figure 3.2: Flo w mesh cell heigh t ∆y/δ 0 v ersus y/δ 0 in linear (a) and logarithmic (b) scales. 19 3.2.3 Boundary and initial conditions 3.2.3.1 T op Boundary Along the top b oundary of the sim ulation, the effect of the w edge angle on the flo w field is mo deled an- alytically using a time-v arying in viscid flo w assumption that resorts to steady sho c k-expansion theory to calculate instan taneous flo w quan tities at an y giv en time. Under this assumption, the inciden t oblique sho c k generated at the w edge fron t p oin t and the Prandtl-Mey er expansion at the rear p oin t of the w edge in tersects with the top b oundary of the flo w domain at lo cations that dep end on time. This effect is sho wn in figure Figure 3.3: Illustration of inciden t oblique sho c k and rear Prandtl-Mey er expansion generated b y the mo ving w edge and its in teraction with the top b oundary of the computational flo w domain 3.3 where there are three distinct regions of the in viscid flo w field on the top b oundary . The first, most upstream region extends from the domain inlet to the in tersection of the top plane, y =L y , with the oblique sho c k formed at the w edge fron t (x inlet x x ISI ). In this region, the c haracteristic b oundary condition sets the flo w v ariables to the incoming freestream v alues. The second region spans from the in tersection with the inciden t oblique sho c k to the in tersection with the first Mac h w a v e of the Prandtl-Mey er expansion generated at the rear corner of the w edge (x ISI xx PME ). The third region corresp onds to the isen tropic expansion of the flo w whic h spans from the in tersection of the first (head) Mac h w a v e of the Prandtl-Mey er expansion with the top b oundary to the domain outlet (x PME x x outlet ). As the rotation angle of the w edge c hanges in time, the prescrib ed v alues of flo w quan tities and the lo cations of [x ISI ,x PME ] at the top of the b oundary c hange as w ell. App endix B summarizes the sho c k-expansion theory equations used to calculate the fundamen tal flo w quan tities along the top b oundary . T o accoun t for the time ev olution of the w edge angle, the time co ordinate is discretized and m ultiple snapshots corresp onding to differen t rotation angles are generated. As the w edge angle c hanges, linear in t erp o lation of the calculated flo w quan tities is applied b et w een the in viscid flo w solution obtained for t w o 20 discrete w edge angles. As describ ed in c hapter 2 , after the rotation of the w edge reac hes its last digitized v a lue as pro vided b y Daub et al. ( 2016 ), the w edge angle is held constan t and equal to the last recorded v a lue. Figure 3.4 sho ws con tours of the flo w quan tities imp osed at the top b oundary condition as a function of the w edge angle and the stream wise co ordinate for cases (1) and (2). Similar plots (not sho wn) can b e obtained for case (3). 3.2.3.2 Bottom W all An equilibrium, w all-stress w all mo del based on Ka w ai & Larsson ( 2012 ) is emplo y ed on the b ottom w all of the domain to further reduce the dep endence of the sim ulation cost on the Reynolds n um b er. Equilibrium conditions in the inner la y er are assumed in the w all-mo del form ulation, implying that pressure gradien t and con v ectiv e terms in the stream wise direction are neglected. Th us, the momen tum and energy equations in the w all mo del reduce to the follo wing ordinary differen tial equation (ODE) system (see Larsson et al. , 2016 ): d dy (µ+µ t,wm ) du ∥ dy = 0 (3.18) d dy 2 4 c p µ Pr + µ t,wm Pr t,wm ! dT dy +(µ+µ t,wm )u ∥ du ∥ dy 3 5 = 0 (3.19) where T and u ∥ are the temp erature and the w all-parallel v elo cit y resp ectiv ely . c p is the sp ecific heat capacit y at constan t pressure. Pr = c p µ/k is the fluid Prandtl n um b er, and Pr t,wm its turbulen t (or eddy) coun t erpart. µ is the fluid dynamic viscosit y , also mo deled using Sutherland’s la w as in the LES solv er, and µ t is the turbulen t (or eddy) viscosit y , expressed using a mixing-length mo del as: µ t,wm =κρ r τ w ρ y 1e −y + /A + 2 (3.20) The sup erscript + denotes inner units (y + y/l v ), scaled b y the viscous length, l v ν w / p τ w /ρ w , where the subscript w denotes flo w quan tities at the w all surface, the kinematic viscosit y is ν w µ w /ρ w , and τ w is the w all shear stress. κ is the v on Kármán constan t and A + is the dimensionless effectiv e subla y er thic kness. The presen t sim ulations use the follo wing standard v alues for the w all-mo del constan ts: κ = 0.41 , A + = 17 and Pr t,wm = 0.9 . The w all mo del equations are solv ed in the region b et w een the solid w all and the w all mo del exc hange lo cation (y h wm ), mean t to include the viscous and buffer subla y ers and part of the logarithmic la y er for equilibrium b oundary la y ers. The w all mo del tak es the v elo cit y u i , pressure p , and temp erature T from the 21 (1a) (2a) (1b) (2b) (1c) (2c) (1d) (1d) Figure 3.4: Con tour plots of flo w quan tities imp osed at the top b oundary as a function of stream wise co ordinate, x , and w edge angle, θ , for case (1), with M ∞ = 3 , and case (2), with M ∞ = 4 : (a) pressure p/p ∞ , (b) temp erature T/T ∞ , (c) stream wise v elo cit y u/u ∞ , and (d) w all-normal v elo cit y v/u ∞ . Dashed lines represen t the x lo cations of the in tersections with the top b oundary of the inciden t sho c k (ISI) and of the first Mac h w a v e of the Prandtl-Mey er expansion (PME) as a function of θ . 22 LES grid at y = h wm at the exc hange lo cation as its top (i.e., a w a y from the w all) b oundary condition, along with no-slip and either adiabatic or isothermal conditions at the w all (b ottom b oundary condition). After solving the ODE system (equations 3.18 and 3.19 ) in the w all-mo del grid for eac h b oundary face, the w all mo del returns the w all shear stress, τ w , and (for non-adiabatic w alls) the w all heat flux, q w , as the w all b oun dary condition to the LES grid. The presen t sim ulations consider adiabatic w alls q w = 0 . 3.2.3.3 Inlet, outlet, and span wise b oundary conditions Syn t hetic turbulence is generated at the inlet of the flo w domain using a digital filtering tec hnique based on Klein et al. ( 2003 ); Xie & Castro ( 2008 ); T oub er & Sandham ( 2009 a ). Using this tec hnique, the mean profiles of stream wise v elo cit y u , temp erature T , pressure p , and the Reynolds stresses u ′ i u ′ j are sp ecified as a function of the w all-normal distance y . The normalized w all-normal profiles of mean flo w quan tities and Reynolds stresses used as input b y the syn thetic turbulence generator at the inlet for the M = 3 and M = 4 TBLs as function of outer units (y/δ 0 ) are sho wn in figure 3.5 . The thic kness of the inlet TBL is adjusted suc h that the nominal thic kness at the reference lo cation is initially matc hed, after a sufficien tly long distance from the inflo w allo ws the b oundary la y er to b ecome fully dev elop ed. The stream wise turbulen t lengthscale is c hosen equal to the inlet b oundary la y er thic kness, whereas half that v alue is used for the w all-normal and span wise turbulen t lengthscales. A t the outlet of the computational flo w domain, a c haracteristic outflo w b oundary condition is imp osed, that extrap olates flo w quan tities in sup ersonic regions. In the subsonic region within the b oundary la y er, the pressure is set equal to the spatial a v erage of the sup ersonic p ortion of the outlet plane. F or sim ulations including the full span wise width of the panel (z2 [100,100] mm ) in the computational domain, a c haracteristic symmetry b oundary condition is emplo y ed that sets the flux of flo w quan tities across the b oundary to zero, akin to a slip w all. Under this condition, an ti-symmetric mo des induced from the solid solv er along the span wise cen ter plane z = 0 are p ermitted. F or sim ulations using the span-reduced domain (z 2 [10,10] mm ), p erio dic b oundary conditions are applied in the span wise direction instead. Using p erio dicit y , the flo w is recycled along the t w o span wise planes where eac h face along the planes is matc h ed to another face with the same cen troid in x and y co ordinates corresp onding to the other plane. This p erio dicit y effectiv ely imp oses a t w o-dimensional homogeneit y of the flo w, whic h will b e justified from the results presen ted in section 4.1.2 , as long as the span wise length is large enough to minimize span wise coherence of flo w quan tities. F or p erio dicit y to b e used, the matc hed p erio dic faces need to b e geometrically iden tical to eac h other. F or the rigid case, this is a trivial requiremen t since there is no mesh deformation. 23 (a) (b) (c) (d) Figure 3.5: W all-normal profiles of flo w quan tities imp osed at the inflo w b oundary for sim ulation cases with Mac h 3 (a,c) and Mac h 4 (b,d) TBLs. (a,b) stream wise v elo cit y u/u ∞ , temp erature T/T ∞ , and densit y ρ/ρ ∞ . (c,d) Reynolds stresses (u ′ u ′ /u 2 ∞ , v ′ v ′ /u 2 ∞ , w ′ w ′ /u 2 ∞ , u ′ v ′ /u 2 ∞ ). 24 F or the FSI case, as the flo w mesh deforms, care m ust b e tak en to ensure that symmetric deformations of the flo w mesh are enforced. 3.2.3.4 Initialization T o initialize b oth the rigid and the flexible panel sim ulations, a rigid flo w domain with a fully-dev elop ed TBL is first generated. The flo w is first set to the freestream conditions of v elo cit y u ∞ , temp eratureT ∞ , and pressure p ∞ throughout the en tire domain. The flo w solv er is then run for t w o flo w-through times (FTT), based on the stream wise length of the computational domain, to flush out an y transien t start-up effects. F or eac h M ∞ case, this fully-dev elop ed TBL flo w field is then sa v ed to disk as a c hec kp oin t (also kno wn as restart) file to b e used as the starting p oin t for subsequen t sim ulations on rigid and flexible panels. 3.2.4 Numerical Instrumen tation T o record the ev olution of the flo w field o v er time, selected p ortions of the flo w field are sa v ed to disk throughout the duration of the sim ulations. A summary of the instrumen tation used b y the flo w solv er is sho wn in table 3.3 . Instan taneous snapshots of the flo w field in the full computational domain are recorded ev ery 10 4 steps as c hec kp oin ts to restart the sim ulation. T o visualize the time ev olution of the flo w fields, flo w quan tities in all cells on the z = 0 cen terplane are sa v ed to disk ev ery 100 time steps. Using these files, visualization and qualitativ e analysis of the flo w field in the STBLI system is p erformed. T o study the flo w separation bubble, all of the cells in whic h the flo w exp eriences instan taneous rev ersal in the stream wise direction (u < 0 ) are sa v ed to disk ev ery 100 time steps. Using the data obtained from this recording, the dynamics of flo w separation can b e accurately measured. The output data of the w all mo del (w all temp erature T w , w all pressure p w , w all shear stress τ w , and w all displacemen t Y s ) is sa v ed to disk ev ery 100 time steps. This data is used to quan tify span wise effects, measure w all-related quan tities, and allo w for span wise a v eraging. T o measure the sp ectral c haracteristics of the flo w in the stream wise direction, high-sp e ed prob es are placed along the cen terline z = 0 at the w all (y =Y s (x) ) suc h that the flo w quan tities in ev ery cell along that line are recorded ev ery 5 time steps. Finally , a v ertical line prob e is placed on the span wise cen terplane (z = 0 ) at the reference stream wise lo cation (x = 200 mm ) extending from the w all, y =Y s (x) , up toy = 4δ 0 , to measure instan taneous and time-a v eraged w all-normal profiles of flo w quan tities. 25 Name Selection c riteria Recording frequency Flo w field c hec kp oin t file All cells Ev ery 10 4 time steps Flo w field cen terplane Cells where z = 0 (cen terplane) Ev ery 10 2 time steps Separation bubble Cells w here u< 0 (flo w rev ersal) Ev ery 10 2 time steps W all quan tities F aces whe re y =Y s (x) (w all) Ev ery 10 2 time steps Flo w w all line prob e Cells w here y =Y s (x) and z = 0 Ev ery 5 time steps Flo w TBL line prob e Cells where x = 200 mm , 0<y < 4δ 0 , and z = 0 Ev ery 10 2 time steps T able 3.3: Flo w solv er n umerical instrumen tation. 3.3 Solid Solv er Subsection 3.3.1 outlines the finite elemen t metho dology of the solid solv er. Subsection 3.3.2 details the mesh and computational domain. Subsection 3.3.3 describ es the b oundary and initial conditions. Subsection 3.3.4 discusses the n umerical instrumen tation used to record solid field data to disk. 3.3.1 Computational Metho dology The n umerical form ulation of the solid mec hanics solv er starts with the go v erning Cauc h y momen tum equa- tion: σ ij,j +f i =ρ¨ x i (3.21) where σ ij,j is the div ergence of the stress tensor σ ij , f i is the sum external b o dy forces p er unit v olume, and ρ¨ x i is the densit y times the lo cal acceleration. In this section, a comma in the subscript of a field follo w ed b y a subindex is used to denote a spatial deriv ativ e along the corresp onding co ordinate direction index, whereas dot(s) ab o v e a field denote temp oral deriv ativ e(s). F or linear isotropic elasticit y , the stress tensor, σ ij , is related to the strain tens or, ϵ ij b y: σ ij =λσ ij ϵ kk +2µϵ ij (3.22) where µ and λ are the Lamé constan ts. In v ectorial form, the stress (σ ) and strain (ϵ ) tensors: σ T = σ xx σ yy σ zz σ xy σ yz σ xz (3.23) ϵ T = ϵ xx ϵ yy ϵ zz ϵ xy ϵ yz ϵ xz (3.24) are related b y σ = [D]ϵ (3.25) 26 through the stiffness tensor matrix [D] , whic h is expressed in terms of the Y oung’s mo dulus,E , and P oisson’s ratio, ν , of the material as [D] = E (1+ν)(12ν) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1ν ν ν 0 0 0 ν 1ν ν 0 0 0 ν ν 1ν 0 0 0 0 0 0 1−2ν 2 0 0 0 0 0 0 1−2ν 2 0 0 0 0 0 0 1−2ν 2 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (3.26) F or small infinitesimal strains, the strain tensor ϵ ij is relate d to the gradien t of displacemen t v ector u i,j . ϵ i,j = 1 2 u i,j +u j,i (3.27) Using a finite elemen t approac h (see, for example Rao , 2005 ), fields are appro ximated with a set of p o in ts in eac h elemen t that use in terp olation shap e functions. Since natural co ordinates are used for the in t erp o lating approac h, the ph ysical co ordinates (xyz) are related to the natural co ordinates (rst) using the elemen t shap e functions defined later in equation 3.33 . x =x(r,s) = [N]x e (3.28) Eac h elemen t, denoted with the subscript e , con tains a collection of these p oin ts (also referred to as no des). Through the finite elemen t metho d, the go v erning equations ( 3.21 ), when com bined with a mo del relating stress and strain (see equation 3.26 ), and a mo del relating strain to displacemen t (see equation 3.27 ), result in a system of second order initial v alue ordinary differen tial equations (ODEs) that can b e in tegrated in time. [M]¨ x =f ext f int (3.29) The in ternal forcef int is obtained b y adding the damping force with the stiffness force. T o find the stiffness force, the stiffness matrix [K] is in tegrated o v er the displacemen t path of the mesh f int = [C]˙ x+ Z t 0 [K(x)]˙ xdt. (3.30) By up dating the stiffness matrix as the mesh deforms, non-linear effects suc h as geometric hardening can b e accoun ted for. 27 The form ulations used to obtain the stiffness matrix [K] , mass matrix [M] , and damping matrix [C] are sho wn starting with a description of the shap e functions used. The shap e function v ector N used for an order n hexahedral finite elemen t with n 3 no des i s a function of the natural co ordinates (r,s,t) N T = N 1,1,1 (r,s,t) N 1,1,2 (r,s,t) N n,n,n (r,s,t) (3.31) F or problems with v ector quan tities lik e displacemen t, v elo cit y , acceleration, etc., the shap e function matrix [N] , whic h uses en tries from the shap e function v ectorN , is defined in equation ( 3.32 ). Let k = n 3 , where n is the n um b er of no des p er dimension in the elemen t. Then, the shap e function matrix [N] has 3k columns and 3 ro ws. [N] = 2 6 6 6 6 6 4 N 1 0 0 N 2 N k 0 0 0 N 1 0 0 0 N k 0 0 0 N 1 0 0 0 N k 3 7 7 7 7 7 5 (3.32) where N m N i,j,k with m = k +n(j +ni) . The particular p olynomial shap e functions used b y the solid solv er are the pro duct of Lagrange in terp olating p olynomials L(ξ) in three dimensions N i,j,k (r,s,t) =L i (r)L j (s)L k (t) (3.33) These n1 order p olynomials are defined b y n p oin ts as L j (ξ) = Y 1≤m≤n ξξ m ξ j ξ m , m6=j. (3.34) Letting n b e the n um b er of no des in the ξ dimension, whic h spans ξ2 [1,1] , the co ordinate of i -th no de ξ i used in t he construction of the Lagrange p olynomials: ξ i =1+ 2i2 n1 (3.35) T o dev elop finite elemen t form ulations for elasticit y , the elemen t strain matrix, [B] e , is constructed from the gradien t of shap e functions [B] e = [[B 1 ],[B 2 ],[B k ]] (3.36) 28 where [B i ] = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ∂N i ∂x 0 0 0 ∂N i ∂y 0 0 0 ∂N i ∂z ∂N i ∂y ∂N i ∂x 0 0 ∂N i ∂z ∂N i ∂y ∂N i ∂z 0 ∂N i ∂x 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (3.37) T o find the gradien t of shap e functions for the en tries of B w e use the in v erse of the Jacobian matrix [J] 2 6 6 6 6 6 6 6 6 6 4 ∂N i ∂x ∂N i ∂y ∂N i ∂z 3 7 7 7 7 7 7 7 7 7 5 = [J] −1 2 6 6 6 6 6 6 6 6 6 4 ∂N i ∂r ∂N i ∂s ∂N i ∂t 3 7 7 7 7 7 7 7 7 7 5 (3.38) where [J] = 2 6 6 6 6 6 6 6 6 6 4 ∂x ∂r ∂y ∂r ∂z ∂r ∂x ∂s ∂y ∂s ∂z ∂s ∂x ∂t ∂y ∂t ∂z ∂t 3 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 X ∂N i ∂r x i X ∂N i ∂r y i X ∂N i ∂r z i X ∂N i ∂s x i X ∂N i ∂s y i X ∂N i ∂s z i X ∂N i ∂t x i X ∂N i ∂t y i X ∂N i ∂t z i 3 7 7 7 7 7 7 7 7 7 7 5 (3.39) The elemen t stiffness matrix, [K] e , is obtained b y in tegrating the strain matrix along with the stiffness tensor matrix [D] : [K] e = ZZZ [B] T e [D][B] e dV (3.40) Using the same displacemen t mo del that is used in the deriv ation of the elemen t stiffness matrix, the consisten t elemen t mass matrix [M] e , is defined form ally as [M] e = ZZZ ρ[N] T [N]dV (3.41) 29 F or computational efficiency , a form of the mass matrix is sough t that can b e easily in v ertible. T o in v ert the mass matrix [M] , it is cast in to a diagonal form using a lump ed no de appro ximation where the no des in the mesh distribute the mass of the structure. This diagonalized elemen t mass matrix [M] e results [M] e ZZZ ρ[I]NdV (3.42) where [I] is the iden tit y matrix. Ra y leigh’s prop ortional damping mo del is used to express the damping matrix, [C] , as a linear com bina- tion of the mass and stiffness matrices: [C] =a[M]+b[K] (3.43) In the presen t study , only the mass prop ortional damping co efficien t a is considered, setting b = 0 . The elemen t external load v ector,f e,ext , is determined b y in tegrating the face pressure o v er the face area and dis tributing that force to no des of that face f e,ext = ZZ p[N]ndA (3.44) All of the lo cal elemen t matrices are assem bled together to form a global matrix. Presen tly , time in tegra- tion is done using an explicit fourth-order R unge-Kutta metho d with a constan t time step. As discussed in c hapter 9.5.1 of Bathe ( 1996 ), the maxim um allo w able time step size ∆t cr m ust b e less than T n /π where T n is the in v erse of the highest natural frequency in the assem bled finite elemen t system. Since T n is in v ersely prop ortional to the panel stiffness, large deflections in the panel can require v ery restrictiv e time steps to main tain stabilit y . Unlik e with linear stabilit y , the n umerical solution of the non-linear system do es not alw a ys visually app ear to go unstable but can b e quite inaccurate when the critical time step is exceeded. T o ensure elemen t accuracy and a v oid a phenomenon kno wn as shear lo c king, elemen ts with at least quadratic in terp olation (three no des p er dimension) should b e used. As discussed in detail in Bathe ( 1996 ), shear lo c king o ccurs when linearly in terp olated elemen ts are sub jected to a b ending load that generates shearing deformation instead of b ending deformation, yielding in v alid results. The finite elemen ts used in this study are isoparametric 27-no de hexahedra with 8 corner no des, 6 face no des, 12 edge no des, and 1 in t erior cell no de. 30 W e appro ximate v olumetric in tegrals on finite elemen ts using Gaussian quadrature in eac h dimension of the natural co ordinate system (rst ) and the determinan t of the Jacobian [J] as ZZZ fdV = Z 1 −1 Z 1 −1 Z 1 −1 fdet[J]drdsdt n=5 X i=1 n=5 X j=1 n=5 X k=1 w i w j w k f(r i ,s j ,t k )det[J]. (3.45) Curren tly , the n um b er of quadrature p oin ts, n , is fiv e p oin ts p er dimension, resulting in 5 3 ev aluations of the in tegrand. T o accoun t for large deformations with small strains, the stiffness matrix, [K] , and the load v ector,f int , are up dated with c hanges in the elemen t geometry from deformation during the time adv ancemen t. Equation 3.30 is implemen ted using the follo wing step-b y-step pro cedure: 1. Compute the new effectiv e load v ector, f new =f +[K]δx n . 2. A dd the lo cal deformation δx n to the no de co ordinates in the finite elemen t mesh, x n,new =x n +δx n . 3. Recalculate the new stiffness matrix, [K] = [K(x n,new )] . 4. A dd the lo cal deformation to the total deformation,u n,new =u n +δx n . 5. Reset the lo cal deformation to zero, δx n = 0 . 3.3.2 Mesh and computational domain T able 3.4 pro vides a summary of the exten t of the domain and the mesh size for the solid solv er. The computational domain of the panel in the stream wise direction b egins and ends at the outer most ro w of riv ets spanning a length of L = 320 mm . The decision to use 64 elemen ts in the stream wise direction w as made suc h that the elemen ts w ould b e 5 mm wide and the corner no des w ould land directly up on the prob e lo cations found in the exp erimen ts. In the span wise direction of the mesh, eac h elemen t is 10 mm long. F or the reduced and full span v ersion of the mesh that results in 2 and 20 elemen ts resp ectiv ely . This resolution in x and z also allo ws for eac h solid face to b e directly mapp ed to its o wn set of flo w faces without ha ving to share flo w faces b et w een solid faces. In the w all-normal y direction only a single finite elemen t, measuring 1.47 mm long is used. 31 Stream wise distance, x Spacing equation, 0<ξ < 1 , Num b er of cells, N x x2 [210,530] mm ξ 64 W all-normal distance, y Spacing equation, 0<η < 1 , Num b er of cells, N y y2 [1.47,0.0] mm η 1 Span wise distance, z Spacing equation, 0<ζ < 1 , Num b er of cells, N z Span wise p erio dic (reduced span): z2 [10,10] mm ζ 2 Span wise symmetric (full span): z2 [100,100] mm ζ 20 T able 3.4: Solid mesh spacing. ξ , η , and ζ are natural co ordinates discretized b y uniformly distributed no d es, suc h that the linear maps ξ 7! (xx min )/(x max x min ) , η 7! (yy min )/(y max y min ) , and ζ 7! (zz min )/(z max z min ) corresp ond to uniform spacing in the ph ysical co ordinates x , y , and z , resp ectiv ely (where min and max corresp ond to the minim um and maxim um v alues of the co ordinate direction in eac h in terv al cons idered). 3.3.3 Boundary and initial conditions T o initialize the solid solv er, the acceleration, v elo cit y , and displacemen t are all set to zero. The solid solv er uses simple fixed-fixed b oundary conditions along the upstream and do wnstream transv erse edges of the panel. Under the fixed-fixed condition, no des in con tact with those edges are prescrib ed zero displacemen t, v elo cit y , and acceleration in eac h Cartesian direction. Along the b ottom of the panel, the pressure at eac h face is held constan t and equal to the freestream v aluep ∞ , to accoun t for the ca vit y placed b elo w the panel in the exp erimen ts b y Daub et al. ( 2016 ). Along the top of the panel, the instan taneous do wnsampled pressure field from the flo w solv er is applied to eac h face, whic h is sho wn visually in figure 3.6 . F or the reduced-span sim ulations, symmetry along the span wise left (z =z min ) and righ t (z =z max ) planes is enforced b y the solid solv er to conform to the p erio dicit y conditions of the flo w solv er. The steps used to enforce that condition on to the solid domain are: 1. Eac h no de (L ) along the left face of the panel is paired to a no de (R ) whic h has the same stream wise (x ) and w all-normal (y ) co ordinates along the righ t face of the panel. 2. A t eac h time step, the displacemen t, v elo cit y , and acceleration of the t w o no des in the pair is a v eraged: ⃗ u avg = 1 2 (⃗ u L +⃗ u R ) (3.46) 3. Eac h no de in the pair is assigned the a v erage of the pair ⃗ u L =⃗ u avg (3.47) ⃗ u R =⃗ u avg (3.48) 32 Name Selection criteria Recording frequency Solid field c hec kp oin t file All finite elemen ts (cells) Ev ery 10 4 time steps Solid fron t prob e x = 295 mm , y =Y s (x) , and z = 0 Ev ery 10 2 time steps Solid cen ter prob e x = 375 mm , y =Y s (x) , and z = 0 Ev ery 10 2 time steps Solid rear prob e x = 445 mm , y = 0 , and z = 0 Ev ery 10 2 time steps Solid field solution All finite elemen ts (cells) Ev ery 10 2 time s teps Solid w all line prob es y =Y s (x) and z = 0 Ev ery 10 2 time steps T able 3.5: Solid solv er n umerical instrumen tation. 3.3.4 Numerical Instrumen tation T o trac k the w all displacemen t at the exp erimen tal prob e lo cations, three prob es are placed along the cen terline at the b ottom of the panel whic h corresp ond to the stream wise lo cations of the fron t, rear, and cen ter prob es. In addition to the three p oin t prob es, a line prob e 65 p oin ts long (corresp onding to the stream wise lo cation of the corner no des) is inserted along the cen terline extending along the full length of the panel. A summary of this instrumen tation is sho wn in table 3.5 . 3.4 Mesh Deformation Solv er Subsection 3.4.1 describ es the n umerical metho ds used in the mesh deformation solv er. Subsection 3.4.2 describ es the b oundary conditions and imp osed constrain ts in the solv er. 3.4.1 Computational Metho dology A t the fluid-solid in terface, the mesh displacemen t information is transferred from the solid solv er to the flo w solv er. In the fluid mesh, eac h face that is shared b y a pair of adjacen t cells is abstracted as a spring whose spring constan t, k , is in v ersely prop ortional to the square of the length, ℓ , b et w een the cen troids of the cells sharing the face: k / 1/ℓ 2 . After considering all the faces in the fluid mesh, a system of in terconnected springs is obtained. Constrain ts that restrict the motion of one or m ultiple degrees of freedom are applied at selected faces, whic h can include b oundary as w ell as in terior faces. Cons train ts on in terior mesh faces allo w for the restriction of mesh deformation to a region of in terest within the flo w domain, for example, near the solid- fluid in terface where the b oundary displacemen t is imp osed. Using a finite elemen t approac h, eac h spring is treated as a three-dimensional truss elemen t with its stiffness matrix defined as: [K] e,mesh = [λ] T [k e,mesh ][λ] (3.49) 33 where: [k] e,mesh =k 2 6 4 1 1 1 1 3 7 5 (3.50) [λ] = 2 6 4 l ij m ij n ij 0 0 0 0 0 0 l ij m ij n ij 3 7 5 (3.51) l ij = x j x i ℓ , m ij = y j y i ℓ , n ij = z j z i ℓ (3.52) ℓ = q (x j x i ) 2 +(y j y i ) 2 +(z j z i ) 2 (3.53) where (x,y,z) represen t the co ordinates of the cen troid for cells i and j sharing a face. This global assem bly of lo cal matrices results in a linear system of equations whose solution is the new displacemen ts for eac h unconstrained degree of freedom of eac h face of the mesh. [K] = X [K] e,mesh (3.54) The resulting linear system of equation to b e solv ed is: [K]Q =F (3.55) whereQ is the displacemen t v ector of the v ertices (corresp onding to flo w mesh cell cen troids) and F is the v ector of forces applied on the spring system. The assem bled stiffness matrix is rearranged to a blo c k form to solv e for the unconstrained v ertex lo cations 2 6 4 K aa K ab K ba K bb 3 7 5 2 6 4 Q a Q b 3 7 5 = 2 6 4 F a F b 3 7 5 (3.56) where subscript a corresp onds to the unkno wn degrees of freedom (i.e., those without an y imp osed con- strain t s), and subscript b corresp onds to the constrained ones. Q a is the solution v ector and Q b is the constrain ts v ector. Solving for the unkno wn degrees of freedom: [K aa ]Q a =F a [K ab ]Q b (3.57) 34 Cell selection expression Constrain t x< 0.22 δu i = 0 x> 0.52 δu i = 0 y > 4δ 0 δu i = 0 (Span wise p erio dic mesh) z =L z /2 δu z = 0 (Span wise p erio dic mesh) z =L z /2 δu z = 0 T able 3.6: Mesh deformation solv er constrain ts The resulting distributed linear system is solv ed in an iterativ e fashion using the generalized minimal residual (GMRES) metho d through the T rilinos external library . 3.4.2 Boundary Conditions A summary of the constrain ts used in the mesh deformation solv er are sho wn in table 3.6 . T o impro v e the p erformance of the mesh deformation solv er, certain cell regions in the flo w domain are constrained to ensure a zero deformation in one or m ultiple co ordinate directions (δu i = 0 ). First, cells that are not directly ab o v e the flexible panel in the w all-normal direction are constrained in all directions. Next, cells ab o v e the flexible panel whose v ertical ( y ) co ordinate exceeds 4δ 0 are also constrained in all directions. Finally , in the case of the reduced span p erio dic mesh, all cells along the span wise left (z = 10 mm ) and righ t (z = 10 mm ) planes are constrained suc h that deformation in the span wise direction is set to zero (δu z = 0 ). Moreo v er, p erio dicit y is enforced suc h that matc hed cells in the span wise p erio dic planes m ust ha v e iden tical deformation in the stream wise x and w all-normal y directions. F or additional p erformance enhancemen ts, al l cells can b e prev en ted from deforming in the stream wise x and w all-normal z directions, lea ving y as the only unc onstrained direction. 3.5 Solv er Coupling 3.5.1 Flo w domain to solid domain coupling When transferring the flo w pressure field to the solid solv er, a force balance is used to ensure that Newton’s la ws of motion are not violated. F s = Z As p s (x,y,z)n s dA = Z As p f (x,y,z)n f dA =F f (3.58) whereF is the force that results from in tegrating the pressure p o v er the in terface surface area A of normal n , with subscripts s and f denoting solid and fluid, resp ectiv ely . F or the curren t sim ulations, the hexahedral meshes that discretize the fluid and solid domains are suc h that, at the in terface, eac h solid mesh face/edge 35 Figure 3.6: Spatial do wnsampling of facial pressure field from the fine mesh of the flo w solv er to the coarse mesh of the solid solv er fully encloses m ultiple fluid mesh faces/edges without in tersections (i.e., ev ery no de of the solid mesh at the in terface has a corresp ondence with another no de in the fluid mesh, but not the con v erse). Th us, equation ( 3.58 ) can b e expressed discretely for eac h solid mesh face, i , at the in terface, in the direction normal to the face as p s,i = ni X j=1 p f,j A f,j A s,i (3.59) where p s,i is the resulting pressure acting on the i -th solid face, p f,j is the pressure of the j -th flo w face inside the i -th solid face, and n i is the n um b er of fluid flo w mesh faces enclosed b y the i -th solid face under consideration. The instan taneous flo w pressure field at the fluid-solid in terface is transferred ev ery 10 time steps, corresp onding to a coupling frequency of 1.25 MHz . An example of this do wnsampling is sho wn in figure 3.6 , where the high frequency v ariations are visibly smo othed out. 3.5.2 Solid domain to flo w domain coupling T o transfer no dal data from the solid domain surface to the flo w domain, the no des in the flo w domain are first matc hed to a face in the solid domain mesh. The matc hing at eac h face is p erformed using Gauss- Newton’s metho d ( Björc k , 1996 ) to find the natural co ordinates (r,s) that minimize the Euclidean distance b e t w een the target p oin t of the flo w domain and the source face of the solid domain: d 2 = [x(r,s)x t ] 2 +[y(r,s)y t ] 2 +[z(r,s)z t ] 2 (3.60) 36 Figure 3.7: Spatial upsamping of the coarse mesh solid solv er displacemen t (sho wn righ t) to the finer mesh of the flo w solv er (sho wn left) Using the shap e function as w eigh ting v alues, for ann th-order face withn 2 no des, the in terp olated target no dal displacemen t,u target , is found from the no dal displacemen ts of the matc hed source f ace,u i, source , as: u target = n 2 X i=1 N i (r,s)u i, source (3.61) where N i (r,s) are the shap e functions of the finite-elemen t form ulation. The displacemen t field at the fluid- solid in terface is transferred from the solid solv er to the mesh deformation solv er ev ery 100 time steps. This results in a solid to flo w coupling frequency of 100 kHz . An example of upsampling of a no dal displacemen t field using the in terp olation functions in equation ( 3.61 ) is sho wn in figure 3.7 . The no dal v elo cit y field is upsampled in the same manner as the no dal displacemen t field. 37 Chapter 4 Results The results are organized in to t w o sections. The first (§ 4.1 ), analyzes results obtained from STBLI sim ula- tions with rigid and flexible panels, replicating exp erimen ts b y Daub et al. ( 2016 ) with the w edge rotation un t il a constan t setp oin t is reac hed. Within this section, sev eral subsections are presen t. Subsection 4.1.1 assesses the c haracteristics of the fully-dev elop ed turbulen t b oundary la y er upstream of the inciden t oblique sho c k at a reference lo cation. Subsection 4.1.2 analyzes results obtained from the full-span (Z d = 50δ 0 ), fully-coupled FSI sim ulations and assesses the v alidit y of using a m uc h more computationally affordable p e- rio dic reduced span sim ulation (Z d = 5δ 0 ) instead. Subsection 4.1.3 assesses the sensitivit y of w all pressure profiles to the incoming b oundary la y er thic kness and to the length b et w een the fron t w edge tip (whic h pro- duces the inciden t sho c k) and the rear w edge tip (whic h pro duces PME w a v es curving the inciden t sho c k), in comparison with exp erimen tal data. Subsection 4.1.4 describ es static and dynamic measuremen ts of the flexible panel displacemen t and also pro vides comparison to exp erimen ts, along with error sensitivit y analysis to prob e lo cation. Subsequen t subsections quan tify the effect of panel elasticit y on the STBLI, b y comparing results obtained from sim ulations with rigid and flexible panels on cen terplane snapshots of the flo w field (§ 4.1.5 ), w all quan tities (§ 4.1.6 - 4.1.7 ), net w all pressure force and cen ter of pressure (§ 4.1.8 ), sp ectral anal- yses of w all pressure (§ 4.1.9 ), and the c haracteristics of the separation bubble (§ 4.1.10 ). The second and last section in this c hapter (§ 4.2 ) in v estigates results obtained oscillating the mo ving w edge b et w een t w o setp oin ts, θ2 [θ min ,θ max ] , for differen t frequencies, f w , o v er b oth rigid and flexible panels. 4.1 Sim ulations with the rotating w edge reac hing a constan t setp oin t Results from the three test cases in tro duced in c hapter 2 are referred to as cases (1), (2), and (3). Case (1), M ∞ = 3.0 and θ max = 17.5 ◦ , is the strongest STBLI studied, has the largest amoun t of panel deflection, and the largest mean separation bubble. Case (2), M ∞ = 4.0 and θ max = 20.0 ◦ , is in termediate in terms 38 of panel deflection, STBLI strength, and separation length. Case (3), M ∞ = 4.0 and θ max = 15.0 ◦ , is the w eak est STBLI sim ulated and has the least amoun t of panel deflection. Eac h case includes sim ulations for a rigid panel configuration and a flexible panel configuration in order to study the effects of panel flexibilit y in the STBLI. Unless otherwise noted, red and blue colors used in line plots in this c hapter denote results from the rigid and flexible panel configurations, resp ectiv ely . In cases (1) and (2), the panel stream wise length is 320 mm , corresp onding to the distance b et w een the t w o outer ro ws of riv ets that attac h the flexible panel to the rigid frame. In case (3), the panel stream wise length is 300 mm long corresp onding to the length of the ca vit y in the rigid frame under the flexible panel. This discrepancy in mo deling b et w een the cases is due in part to the deflection in case (3) b eing is less than in cases (1) and (2). Static deflection tests rev ealed that using a shorter length of 300 mm instead of 320 mm for case (3), yielded closer agreemen t with the exp erimen t. This suggests that under smaller loads, the riv eted panel b eha v es closer to an ideal fixed-fixed or clamp ed b oundary condition. As the panel load increases, the stiffness of the panel increases as w ell, but not at the same rate predicted b y geometric hardening of a non-riv eted panel. This is consisten t with the findings of Daub et al. ( 2016 ), where ANSYS 3D sim ulations using fixed-fixed b oundary conditions without mo deling riv ets underestimated the deflection of the cen ter panel for a giv en pressure differen tial, ∆p . 4.1.1 Assessmen t of the fully dev elop ed turbulen t b oundary la y er Do wnstream of the inlet of the computational domain, at a reference lo cation upstream of the in teraction with x = 200 mm , the incoming turbulen t b oundary la y er dev elop ed o v er the rigid panel is c haracterized for the case with M ∞ = 3.0 . Using the Pitot rak e measuremen ts of Daub et al. ( 2016 ), the prior n umerical w all-resolv ed LES of P asquariello et al. ( 2017 ) with a rigid-panel configuration estimated a b oundary la y er thic kness based on 99% of the freestream v elo cit y equal to δ 0 = 4 mm , and a Reynolds n um b er based on the momen tum thic kness, Re θ = 1410 3 . The presen t WMLES used for this assessmen t attain v alues of δ 0 = 3.88 mm and Re θ = 13,836 at that stream wise lo cation. W all-normal profiles of mean stream wise v elo cit y and Reynolds stresses obtained from the WMLES at that reference lo cation are compared with w all-resolv ed LES data of a zero-pressure gradien t, flat-plate incompressible turbulen t b oundary la y er at a comparable incompressible R eynolds n um b er (Re θ,inc 5,650 in the reference data, and 5,746 in the presen t WMLES, once corrected for compressibilit y b y using the dynamic viscosit y at the w all rather than in the freestream). The v an Driest transformation is used to compare mean stream wise v elo cit y data from the compressible and incompressible turbulen t b oundary la y ers. These w all-normal profiles are presen ted in figure 4.1 for t w o WMLES sim ulations: one corresp onding to the nominal grid resolution describ ed in section 3.2.2 , and another finer-grid WMLES that refines the grid b y 39 (a) (b) Figure 4.1: W all-normal profiles of the incoming b oundary la y er at the reference lo cation ( x = 2 mm , Re θ,inc 5,650 ). (a) v an Driest transformed mean stream wise v elo cit y , u + VD , as a function of the w all- normal distance in inner scaling, y + . (b) Ro ot-mean-square v elo cit y fluctuation in the stream wise ( u rms ), w a ll-normal (v rms ), and span wise (w rms ) directions, and Reynolds shear stress (u ′ v ′ ), normalized b y the freestream v elo cit y , as a function of the w all-normal distance in outer scaling (y/δ 0 ). Dashed lines corresp ond to the w all-resolv ed LES data of a flat plate incompressible turbulen t b oundary la y er at Re θ = 5,746 b y Eitel-Amor et al. ( 2014 ). The gra y areas indicate the exten t of the w all-mo deled region in the presen t WMLES. Dotted lines in the v an Driest transformed v elo cit y plot corresp ond to the linear (viscous) and logarithmic (with constan ts κ = 0.41 and B = 5.5 ) la ws. a factor of 1.6 in x , and 1.5 in y and z , resulting in a mesh with appro ximately 70 , rather than 20 , million cells. The exc hange heigh t of the w all mo del is k ept constan t b et w een the t w o sim ulations. In the plots sho wn in figure 4.1 , the w all-mo deled region is indicated with a gra y shade. The mean stream wise v elo cit y plot blends the w all-mo del and LES solutions (figure 4.1 a). In con trast, the profiles of Reynolds stresses (figure 4.1 b) include only the solution obtained from the LES grid, extending all the w a y t o the w all (y = 0 ), without an y blending with the w all-mo deled solution (whic h do es not pro vide Reynolds stress data). Despite a small o v erprediction of the mean stream wise v elo cit y in the w ak e region and underprediction of the transv erse and span wise Reynolds stresses relativ e to the reference data, the o v erall agreemen t is deemed satisfactory and the results sufficien tly grid con v erged with resp ect to the turbulen t b oundary la y er statistics. 4.1.2 Assessmen t of span wise effects using a full-span sim ulation Z d =50δ 0 W e assess in figure 4.2 the three-dimensionalit y of the flo w through con tour plots of the panel deflection, skin friction co efficien t, and w all pressure, mapp ed on the flexible panel and pro jected on to the horizon tal ( xz ) plane, time-a v eraged after the initial transien t. Three-dimensional effects are most noticeable in the panel deflection (figure 4.2 a) near the longitudinal edges (jz ′ j> 15 ), where the deflection exceeds that of the panel cen terline, esp ecially for stream wise lo cations of maxim um w all pressure (0≲ x ′ ≲ 20 ). The foam sealing (not mo deled in the presen t sim ulations) used in the exp erimen ts along the longitudinal edges ( Daub et al. , 40 (a) (b) (c) (d) Figure 4.2: Case (1) M ∞ = 3.0 , θ max = 17 ◦ time a v eraged stream wise x and span wise z v ariation of (a) w all displacemen t Y s /δ 0 , (b) skin friction co efficien t C f 10 3 , (c) w all pressure p w /p ∞ , and (d) w all temp erature T w /T ∞ . 41 2016 ) ma y add a sligh t stiffness along the edges, reducing the deflection difference b et w een the cen terline and the edges of the panel. Consisten t with our fixed-fixed (along transv erse edges) panel mo del, detailed static finite-elemen t sim ulations p erformed b y Willems et al. ( 2013 ), whic h accoun ted for panel riv eting, also found the panel deflection to b e larger along the edges of the panel. Con tours of the time-a v eraged skin friction co efficien t (figure 4.2 b) sho w a predominan tly t w o-dimensional c haracter, except v ery near the longitudinal edges (jz ′ j > 20 ) do wnstream of the in viscid impingemen t lo c ation (x ′ > 0 ), whic h ma y b e attributed to the symmetry (slip-w all) b oundary conditions imp osed b y the flo w solv er on the side b oundaries of the computational domain for this full-span sim ulation. Do wnstream of the b oundary-la y er reattac hmen t (15≲x ′ ≲ 40 ), the time-a v eraged skin friction exhibits stream wise streaks leading to ragged span wise con tours. These are attributed to larger-scale turbulen t flo w structures pro duced b y the shear la y er resulting from the STBLI (see figures 2.1 and 4.13 ) and Görtler-lik e v ortices, requiring m uc h longer a v eraging time p erio ds to homogenize. Görtler-lik e v ortices w ere previously found in w all- resolv ed LES of the rigid-w all STBLI p erformed b y P asquariello et al. ( 2017 ) with a span wise w a v elength of appro ximately 2δ 0 . The time-a v eraged w all pressure con tours sho wn in figure 4.2 (c) exhibit minimal three- dimensionalit y . The w all temp erature (figure 4.2 d), sho ws minimal three-dimensionalit y except less than 2δ 0 from the longitudinal edges of the panel in similar fashion to the w all shear stress. The presen ted assessmen t of three-dimensional effects b eing mostly confined to the vicinit y of the side edges of the panel, with the rest of the span wise domain a w a y from the side edges b eing c haracterized b y a (statistically) nearly t w o-dimensional in teraction, pro vides a partial justification for the subsequen t use of less computationally exp ensiv e, span wise-p erio dic sim ulations on a reduced span of 5δ 0 , c hosen based on prior n umerical studies of STBLIs. Still, the use of reduced-span domains with p erio dicit y is kno wn to result often in an underprediction of separation bubble length for prior high-fidelit y sim ulations (as discussed, for example, in Pirozzoli & Bernardini , 2011 ; Morgan et al. , 2013 ; T oub er & Sandham , 2009 b ; Bermejo-Moreno et al. , 2014 ) and care m ust b e tak en when in terpreting the results in comparisons with full-span sim ulations and exp erimen ts. 4.1.3 STBLI sensitivit y to rear w edge length and incoming TBL thic kness T w o main sources of uncertain t y when comparing the presen t sim ulations with exp erimen ts stem from vis- cous effects of the flo w around the compression w edge and the dev elopmen t of the incoming b oundary la y er. The compression w edge that generates the oblique sho c k impinging on the panel is not included in the com- putational domain, but its effects are mo deled through the top b oundary of the flo w computational domain, where time-v arying Rankine-Hugoniot and Prandtl-Mey er in viscid flo w conditions are prescrib ed, follo wing 42 (a) (b) Figure 4.3: Sensitivit y of stream wise profiles of time- and span wise-a v eraged w all pressure o v er a rigid panel to v ariations in: (a) w edge length (ξ ) for an incoming b oundary la y er thic kness with η = 1.2 ; and (b) in- coming b oundary la y er thic kness (η ) for a w edge extension with ξ = 1.069 . Sym b ols represen t exp erimen tal measuremen ts b y Daub et al. ( 2016 ) P a squariello et al. ( 2017 ). While effectiv e at reducing the computational cost, suc h top b oundary condition neglects the w edge b oundary la y ers and w ak e, whic h can alter the Prandtl-Mey er expansion generated at the rear corner of the w edge and its in teraction with the inciden t oblique sho c k, th us affecting the strength of the STBLI on the panel (see figure 2.1 ). A second source of uncertain t y arises from an incomplete c haracter- ization of the incoming b oundary la y er in the exp erimen ts, whic h relied on Pitot-rak e and global turbulen t in t ensit y (longitudinal and transv erse) measuremen ts at one lo cation (x = 150 mm ) upstream of the STBLI. The strength of the in teraction and, in turn, the exten t of flo w separation dep end on the incoming b oundary la y er thic kness ( Zhou et al. , 2019 ). W e assess in figure 4.3 the effects of these uncertain ties on the w all-pressure profiles c haracterizing the STBLI from t w o parametric studies conducted on span wise-p erio dic rigid-w all sim ulations b y: 1) extending the w edge length b y a factor ξ (in 2 -mm incremen ts), for the same w edge angle of 30 ◦ , whic h dela ys the in teraction of the expansion fan and the sho c k; 2) mo difying the incoming b oundary la y er thic kness b y a factor η relativ e to the reference v alue inferred exp erimen tally of δ 0 = 4 mm . A subset of the sim ulation results is presen ted in figure 4.3 , for clarit y . As seen in figure 4.3 (a), w edge length extensions strengthen the STBLI, whic h is initiated farther upstream and reac hes a larger pressure p eak for increasing ξ , while preserving a similar pressure profile shap e. In con trast, a thic k er incoming b oundary la y er alters the shap e of the pressure profile b y also bringing the initial pressure rise (corresp onding to the separation sho c k) farther upstream while decreasing the maxim um w all pressure reac hed throughout the STBLI (figure 4.3 b). Comparison with exp erimen tal w all-pressure profiles pro vided the b est agreemen t for a 6mm w edge extension (ξ = 1.069 ) and an incoming b oundary la y er 20% thic k er than the 4 mm reference v alue (η = 1.2 ). These 43 −2 0 2 4 6 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 Y s /δ 0 0 5 10 15 20 25 30 35 40 t [ms] −5 −4 −3 −2 −1 0 Y s [mm] Figure 4.4: Case (1)M ∞ = 3.0 ,θ max = 17 ◦ panel displacemen t comparison b et w een exp erimen ts Daub et al. ( 2016 ) (dashed lines), full-span (Z d = 50δ 0 ) FSI-WMLES sim ulation (solid lines), previous w all-resolv ed LES sim ulation b y P asquariello et al. ( 2015 ) (blue dotted line) at the fron t prob e (red lines), cen ter prob e (blue lines), and rear prob e (green lines). v a lues are used in the span wise p erio dic, rigid- and flexible-w all sim ulations presen ted in the remaining sections to ev aluate the effect of panel flexibilit y on the STBLI. 4.1.4 T emp oral ev olution of panel displacemen t 4.1.4.1 F ully-coupled FSI sim ulations In the exp erimen ts b y Daub et al. ( 2016 ), three capacitiv e distance prob es (denoted fron t, rear, and cen ter) w ere placed at v arious lo cations along the panel (see figure 2.3 in § 2.1 ). F or case (1), P asquariello et al. ( 2015 ) p erformed a prior FSI sim ulation using w all-resolv ed LES. In that w ork, structural damping w as neglected and only the cen ter prob e w as plotted for 20 ms . The panel reac hed its maxim um deflection at around 20 ms , with subsequen t cycles increasing in amplitude. A comparison of those prior (exp erimen tal and n umerical) results and the presen t full-span (Z d = 50δ 0 ) FSI sim ulation (using WMLES and structural damping) is sho wn in figure 4.4 . Ov erall, there is go o d agreemen t ac hiev ed for all three prob es (fron t, rear, and cen ter) in b oth the transien t phase of the sim ulation t < 20 ms and in the quasi-steady phase t > 20 ms . The fron t prob e displacemen t app ears to o v ersho ot the exp erimen t at the start of the first panel cycle at t = t 0 . Con v ersely , the rear prob e app ears to undersho ot the exp erimen t at the same time at around (t = t 0 ). As time progresses, the static defle ction error of the prob es in comparison to the exp erimen t sho ws go o d agreemen t for the fron t and 44 Figure 4.5: Case (1): M ∞ = 3.0 , θ max = 17 ◦ . F ull span (Z d = 50δ 0 ) WMLES FSI sim ulation. Span wise a v eraged w all displacemen tY s /δ 0 v ersus timef n (tt 0 ) for the flexible panel. Dashed v ertical lines corresp ond to the nominal x lo cations of the fron t, cen ter, and rear prob es (resp ectiv ely). cen ter prob es. Here, static defle ction refers to the deflection obtained b y a v eraging in time during the quasi- steady regime (c haracterized b y smaller oscillations), after the initial transien t during whic h the w edge angle rapidly rotates. The rear prob e in the n umerical sim ulation sligh tly o v erestimates the quasi-static deflection. In addition, the rear prob e oscillates with a longer sustained p eak-to-p eak amplitude than in the exp erimen t. Curiously , this is not the case for the other t w o prob es, whic h app ear to deca y in oscillation magnitude at a similar rate to the exp erimen t. This indicates that non-linear oscillations are presen t in b oth the exp erimen t and sim ulation suc h that there is a large sensitivit y of the temp oral ev olution of prob e displacemen tY s to the stream wise lo cation of the prob es. T o illustrate this sensitivit y , anxt diagram of v ertical panel displacemen t, Y s , for the full-span sim ulation is sho wn in figure 4.5 . F or reference, the stream wise lo cations of the prob es are o v erla y ed on to the diagram with dashed v ertical lines. Shifting atten tion to the reduced-span (Z d = 5δ 0 ), span wise-p erio dic FSI-WMLES sim ulations, a com- parison with the exp erimen tal measuremen ts of Daub et al. ( 2016 ) is sho wn in figure 4.6 . A dditionally , as done for the full-span sim ulation, individual prob e data for those three lo cations are complemen ted with the con tours of v ertical displacemen t giv en for a line prob e along the stream wise direction on the cen terplane (z = 0 ), with uniformly spaced p oin ts co v ering the full flexible panel length, as sho wn in figure 4.7 . These span wise-p erio dic sim ulations, whic h ha v e a reduced computational cost b y a factor of 10 compared to the full-span coun terpart, also sho w a reasonable agreemen t with the exp erimen ts. The reduced computational cost enables a more efficien t prediction of the longer-term ev olution. 45 −2 0 2 4 6 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 Y s /δ 0 0 5 10 15 20 25 30 35 40 t [ms] −5 −4 −3 −2 −1 0 Y s [mm] (1) −2 0 2 4 6 8 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 Y s /δ 0 0 10 20 30 40 50 60 t [ms] −5 −4 −3 −2 −1 0 1 Y s [mm] (2) 0 2 4 6 f n (t−t 0 ) −0.4 −0.3 −0.2 −0.1 0.0 Y s /δ 0 0 10 20 30 40 50 60 70 80 t [ms] −1.5 −1.0 −0.5 0.0 Y s [mm] (3) Figure 4.6: P anel displacemen t comparison b et w een exp erimen ts b y Daub et al. ( 2016 ) (dashed lines), presen t FSI-WMLES sim ulation (solid lines), and previous w all-resolv ed LES sim ulation P asquariello et al. ( 2015 ) (blue dotted line) at the fron t prob e (red line), cen ter prob e (blue line), and rear prob e (green line). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 46 (1) (2) (3) Figure 4.7: Span wise-a v eraged w all displacemen t Y s /δ 0 v ersus time f n (t t 0 ) for the flexible panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . Dashed v ertical lines corresp ond to the nominal x lo c ations of the fron t, cen ter, and rear prob es (resp ectiv ely). 47 F or eac h case, the oblique sho c k caused b y the w edge rotation impinges at the rear edge of the panel and mo v es farther upstream as the w edge reac hes it angular setp oin t. This causes the pane l to first deform near the do wnstream edge and then the deformation propagates upstream. This can b e easily seen in figure 4.6 , where the rear prob e is the first to resp ond, follo w ed b y the cen ter prob e, and, finally , the fron t prob e. This initial transien t causes a small p ositiv e panel deflection to w ards the fron t of the panel (sho wn in a gra yscale colormap in figure 4.7 ) that is quite short in time duration. As time progresses, the panel transitions from a transien t state c haracterized b y large oscillations in panel deflection to w ards a quasi-steady state in whic h the smaller panel oscillations are caused b y unsteady effects of the STBLI and the w edge serv o-con trol system. F or all cases, the maxim um panel deflection is reac hed within the first 20 ms of the sim ulation. As time progresses, the panel oscillates ab out a quasi-steady state in an underdamp ed fashion. F or all three cases there is considerable sensitivit y of the recorded temp oral prob e displacemen t to the stream wise lo cation of the prob es. F or eac h configuration, the first negativ e p eak of panel deflection is tak en as a time datum ( t 0 ) for the non-dimensionalization of time, t ′ = f n (t t 0 ) , used in subsequen t plots. T o analyze the temp oral ev olution of the coupled in teraction, it is helpful to distinguish b et w een the t w o phases/regimes men tioned earlier. The first phase (denoted as the transien t phase), for time f n (tt 0 ) < 3 , is c haracterized b y large c hanges to the w all displacemen t and flo w quan tities, and is not suitable for time a v eraging. The second phase (denoted as the quasi-steady phase), for f n (t t 0 ) > 3 , is defined b y smaller, damp ed oscillation ab out a static e quilibrium , and is suitable for time a v eraging. The transien t phase, corresp onding to the fast w edge rotation used to create the oblique sho c k and establish the STBLI, is w ell captured b y the presen t FSI sim u lations for the cen ter prob e in all configurations (cases 1, 2, and 3), as seen in figure 4.6 . After the w edge reac hes its final p osition, the exp erimen tal prob es oscillate ab out a static equilibrium p oin t with initially deca ying amplitude. In the cases with mean flo w separation, and asso ciated lo w-frequency unsteady motions, the panel still oscillates without reac hing a true static equilibrium p oin t b y the end of the sim ulations. T o estimate the quasi-static equilibrium deflection, the panel displacemen t w as time-a v eraged o v er the p erio d t2 [25,70] ms for case (1), t2 [30,75] ms for case (2), and t2 [30,75] ms for case (3). Lo oking at the quasi-static deflection curv e v ersus stream wise co ordinate x in figure 4.8 , it app ears that agreemen t with the exp erimen ts could impro v e if the prob e lo cations w ere shifted upstream. This trend is consisten t for all three exp erimen tal prob es and for all three cases, with the exception of the rear prob e for case (3) and the cen ter prob e for case (1). Once shifted, the magnitude of deflection and the shap e of the deflection curv e agree quite w ell with the exp erimen tal results of Daub et al. ( 2016 ). This result is notew orth y considering that only 128 HEX-27 finite elemen ts w ere used to construct the solid panel (instead of plate elemen ts) and that the riv ets in the panel w ere not directly mo deled or indirectly mo deled using 48 −20 −10 0 10 20 30 40 50 (x−x I )/δ 0 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 Y s /δ 0 (1) −30 −20 −10 0 10 20 30 40 (x−x I )/δ 0 −0.8 −0.6 −0.4 −0.2 0.0 Y s /δ 0 (2) −50 −40 −30 −20 −10 0 10 20 (x−x I )/δ 0 −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0.00 Y s /δ 0 (3) Figure 4.8: Time and span wise a v eraged w all displacemen t profiles Y s /δ 0 for flexible WMLES cases. Quasi- steady a v erages of the fron t (red), cen ter (blue), and rear (green) prob es at their nominal stream wise x lo c ation in exp erimen ts b y Daub et al. ( 2016 ) are plotted alongside. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 49 springs (as w as done b y P asquariello et al. ( 2015 )). It is imp ortan t to note that the deflection curv e is not symmetric ab out the geometric yz cen terplane of the panel. As the sho c k impingemen t p oin t, x I , mo v es farther do wnstream, the cen ter of pressure mo v es farther do wnstream as w ell, and the asymmetric features b ecome more pronounced. As exp ected, the magnitude of the maxim um panel deflection increases as the net panel pressure also increases. Due to geometric hardening of the panel, the relation b et w een the rate of panel deflection and the w all pressure is non-linear. Under geometric hardening, the panel increases its stiffness with an increase in deflection. As a consequence of the previous observ ations, a sensitivit y study is conducted to find the mo dified lo cations of the prob es in the n umerical sim ulations that minimize the discrepancy (error) with exp erimen ts of the time-a v eraged prob e signals in the quasi-steady regime. The quasi-static error for eac h prob e (fron t, rear, and cen ter) and the mean error of all three, e mean = 1 3 (e front +e center +e rear ) , is plotted in figure 4.9 as a function of the offset in the stream wise lo cation of the prob es, ∆x . F or case (1), the o v erall error for the quasi-static deflection is found to b e a minim um at ∆x =21 mm , with the error remaining lo w b et w een 15 and25mm . F or case (2), the prob e error is at its minim um for an offset of 10 mm , with the o v erall error remaining relativ ely lo w for an offset ranging from 5 to 20 mm . F or case (3), the optimal prob e shift is found to b e 15 mm with the error remaining lo w b et w een 10 and 30 mm . Due to the o v erall go o d agreemen t for the t w o cases with w eak er STBLIs and a lo w o v erall prob e error for all of the runs with this offset, a global prob e shift of 20 mm in the stream wise direction is applied to all prob e lo cations (fron t, rear, cen ter) for all cases (1, 2, and 3). The results of this prob e shift are sho wn in figure 4.10 . Lo oking at case (1), the effect of applying the ∆x =20 mm prob e lo cation offset is clearly visible b y comparing figures 4.6 and 4.10 , resulting in an impro v ed agreemen t with the quasi-steady state deflection of the exp erimen t. W e see the effect of this prob e lo cation shift in the initial motion of the prob es: the prob es tak e longer to resp ond to the oblique sho c k impacting the rear of the panel. Lo oking at the transien t stage of the sim ulation for this particular case, w e see that non-linear oscillation is presen t and some of those dynamics are not captured prop erly b y the sim ulation. F or case (2), the cen ter and fron t prob e sho ws go o d agreemen t in b oth the transien t and quasi-steady phases. The rear prob e in the sim ulation has go o d quasi-steady state agreemen t with the exp erimen t but app ears to con tain more than one dominan t mo de of vibration, in con trast to the exp erimen t. F or case (3), the sim ulation accurately captures the primary and secondary mo des presen t in the exp erimen ts. Unlik e cases (1) and (2), the rear prob e sho ws a larger displacemen t than the cen ter prob e in the quasi-steady phase. The reason for this app ears to b e the impingemen t p oin t of the inciden t sho c k b eing mo v ed closer to w ards the rear of the panel. The rear prob e has b etter agreemen t with the exp erimen t without the prob e shift applied. This sensitivit y analysis illustrates the c hallenges 50 −40 −30 −20 −10 0 10 20 30 40 Δx [mm] 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 e [mm] e mean e front e center e rear (1) −40 −30 −20 −10 0 10 20 30 40 Δx [mm] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 e [mm] e mean e front e center e rear (2) −40 −30 −20 −10 0 10 20 30 40 Δx [mm] 0.0 0.1 0.2 0.3 0.4 0.5 e [mm] e mean e front e center e rear (3) Figure 4.9: Static prob e error, e v ersus prob e lo cation offset, ∆x , for the fron t (red), cen ter (blue), and rear (green) prob es. Mean prob e error e mean = 1 3 (e front +e center +e rear ) is sho wn in blac k. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 51 −2 0 2 4 6 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 Y s /δ 0 0 5 10 15 20 25 30 35 40 t [ms] −5 −4 −3 −2 −1 0 Y s [mm] (1) −2 0 2 4 6 8 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 Y s /δ 0 0 10 20 30 40 50 60 t [ms] −5 −4 −3 −2 −1 0 1 Y s [mm] (2) 0 2 4 6 f n (t−t 0 ) −0.4 −0.3 −0.2 −0.1 0.0 Y s /δ 0 0 10 20 30 40 50 60 70 80 t [ms] −1.5 −1.0 −0.5 0.0 Y s [mm] (3) Figure 4.10: P anel displacemen t comparison b et w een exp erimen ts Daub et al. ( 2016 ) (dashed line), WMLES sim ulation (solid line), previous w all-resolv ed LES sim ulation P asquariello et al. ( 2015 ) (blue dotted line) at the fron t prob e (red line), cen ter prob e (blue line), and rear prob e (green line). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . Prob e lo cations ha v e b een shifted 20 mm upstream for case (1), (2), and (3) in the presen t sim ulations. 52 Case Primary F ron t Cen ter Rear Damping frequency prob e prob e prob e ratio f n [Hz] Y s,front [mm] Y s,center [mm] Y s,rear [mm] ζ (1) Exp erimen t 230.3 2.13 3.95 2.62 0.038 SP-FSI-n 237.5 2.95 3.81 2.02 0.04 SP-FSI-s 237.5 2.17 3.99 2.61 0.04 FS-FSI 233.0 2.11 3.91 2.75 0.046 1C-FSI 143.0 3.19 3.38 1.32 0.088 (2) Exp erimen t 180.0 1.40 3.29 2.62 0.045 SP-FSI-n 185.0 1.84 3.24 2.37 0.04 SP-FSI-s 185.0 1.57 3.18 2.63 0.04 1C-FSI 191.7 2.04 3.27 2.26 0.043 (3) Exp erimen t 111.1 0.33 0.98 1.08 0.041 SP-FSI-n 112.5 0.48 1.21 1.16 0.044 SP-FSI-s 112.5 0.38 1.14 1.25 0.044 1C-FSI 114.0 0.52 1.25 1.16 0.037 T able 4.1: Comparison of flexible panel primary natural frequency , quasi-static prob ed displacemen t, and damping ratio obtained from exp erimen ts ( Daub et al. , 2016 ) and p erio dic reduced-span sim ulations (SP-FSI) considering nominal prob e lo cations (-n) and a 20 mm upstream shift (-s), full span sim ulation (FS-FSI) for cases (1), and one w a y FSI coupling using rigid w all pressure (1C-FSI)M ∞ = 3.0 ,θ max = 17 ◦ , (2)M ∞ = 4.0 , θ max = 20 ◦ , and (3) M ∞ = 4.0 , θ max = 15 ◦ . of p erforming exact p oin t-to-p oin t comparisons b et w een exp erimen ts and sim ulations for highly non-linear panel deflection dynamics. In addition to estimating the quasi-static panel deflection error, the time signals of panel deflection at the probing lo cations obtained from the sim ulations are used to estimate the primary natural frequency , f n , and the damping ratio, ζ , of the flexible panel. The results are sho wn in table 4.1 , comparing v alues retriev ed from sim ulations and exp erimen ts. The primary natural frequencies of panel vibration are w ell predicted b y the sim ulations in all studied cases. F urther details on the metho d used to estimate the damping ratio (whic h w as not rep orted in the exp erimen ts b y Daub et al. , 2016 ) are presen ted in App endix A . Giv en their influence in these coupled in teractions for b oth transien t and quasi-steady phases, a c haracterization of the damping prop erties of the flexible panels will b e imp ortan t in future join t n umerical-exp erimen tal studies, to reduce uncertain ties in the comparisons. 4.1.4.2 Assessmen t of FSI coupling effects F or problems in v olving a w eak flo w field dep endence on the deformation induced b y the solid domain, the follo wing first-order appro ximation can b e made. An FSI problem can b e appro ximately decomp osed in to separate fluid and solid mec hanics problems. The fluid mec hanics solv er is run first with the rigid domain assumption for the required in tegration time. The recorded w all pressure field from the flo w solv er is then imp osed as time-v arying b oundary condition for a subsequen t sim ulation of the solid mec hanics solv er. Under 53 this assumption, the flo w field is unaffected b y the motion of the solid domain and th us the system can b e treated as the op en-lo op system sho wn in figure 4.11 . As the flo w solv er b ecomes more insensitiv e to the outputs of the solid solv er, the accuracy of this appro ximation impro v es. Flo w Solv er Solid Solv er W all pressure pw Figure 4.11: FSI one w a y coupled system Computationally , for parametric studies in v olving the panel resp onse, this decoupled appro ximation can b e cost-effectiv e compared to a fully-coupled FSI solv er, since only the solid solv er needs to b e run m ultiple times for estimates of panel deformation, after the flo w field solution has b een calculated once from a single (more computationally exp ensiv e) CFD sim ulation. Th us, in the design pro cess for the fligh t structure, if the flo w field is mostly decoupled from the solid structure resp onse, lo w-cost parametric studies can b e run whic h attempt to optimize for a desired quan tit y of in terest (QOI) of the structure. As an example, in this w ork, c hanges to the panel length, L , and mass damping co efficien t, a , w ould induce a differen t panel resp onse. Rather than run the fully-coupled FSI sim ulation to obtain an estimate of the solid panel resp onse, w e can attempt to apply the time-dep enden t pressure field obtained from a previous sim ulation (rigid or flexible panel). This one-w a y coupling sim ulation approac h op erates on the assumption that small c hanges to the panel displacemen t cause ev en smaller c hanges to the flo w pressure field. This approac h is useful when used iterativ ely in tandem with the m uc h more exp ensiv e fully-coupled sim ulations, as the accuracy and o v erall v alidit y of the w eak-coupling assumption m ust b e assessed. The follo wing steps summarize the design pro cess of this one-w a y coupling metho d to guide more exp ensiv e sim ulations: 1. R un rigid-w all STBLI fluid flo w sim ulation and record w all pressure as a function of time t , stream wise x , and span wise z co ordinate. 2. R un parametric studies for selected input c haracteristics of the panel using the recorded w all pressure from the previous rigid-w all STBLI flo w sim ulation. 3. Select the solid solv er sim ulation whic h b est matc hes the exp erime n tal data using a sp ecified metric. 4. R un a fully-coupled FSI STBLI sim ulation with selected input panel parameters from the b est matc hed solid solv er sim ulation and record w all pressure as a function of time t , stream wise x , and span wise z co ordinate. 5. Ev aluate whether the stopping criteria for the iterativ e metho d are satisfied. Otherwise, pro ceed bac k to step 2 using the flexible-w all STBLI w all pressure. 54 As an example of the application of this one-w a y coupling approac h, the recorded w all pressure from the rigid w all sim ulation of cases (1), (2), and (3) w as used to generate a loading input file to the solid solv er, whic h run subsequen tly . The panel resp onse of the solid solv er w as recorded and compared to the prob es exp erimen tally measured b y Daub et al. ( 2016 ), as sho wn in figure 4.12 . The results sho wn in figure 4.12 for one-w a y coupled sim ulations can b e directly compared to those presen ted earlier in figure 4.6 for fully-coupled FSI sim ulations. F or case (1), whic h has the strongest FSI, the prediction accuracy is not satisfactory for neither the magnitude, frequency , nor phase of the prob ed panel deflection signal, indicating that fully- coupled sim ulations are required to accurately capture the underlying ph ysics. F or cases (2) and (3), whic h ha v e w eak er FSI effects, the prediction accuracy is b etter in regard to the quasi-static deflection. The fully- coupled sim ulations agree b etter with the exp erimen ts also in the frequencies of vibration of the flexible panel, as seen b y the phase of deflection prob e time signals. Case (2), whic h has an in termediate STBLI strength, has decen t initial agreemen t using the one-w a y coupling appro ximation, but b ecomes more out of phase with the exp erimen t as time progresses. Finally , for case (3), the one-w a y coupling appro ximation pro v ides a go o d agreemen t with the exp erimen t. This indicates that the influence of the panel deflection on the flo w field is minimal for that case (3) and w ould serv e as an ideal candidate for the aforemen tioned FSI optimization metho d. 4.1.5 Flo w c haracterization on span wise-normal cen ter plane T o illustrate the effects of panel elasticit y on the flo w ph ysics of STBLI, span wise-normal cen terplane slices for v arious flo w quan tities (instan taneous and time-a v eraged statistics) are compared b et w een rigid- and flexible-panel sim ulations for eac h case (1-3). F or impro v ed clarit y , the region of in terest sho wn in the follo wing figures is confined to directly ab o v e the panel up to a distance of y = 4δ 0 . In this region, viscous effects are imp ortan t and, in the case of the flexible panel, the mesh deformation m ust b e accoun ted for. Also note that, in these figures, the scale of the x and y axes differs to highligh t the w all deflection and the differences of the flo w fields near the w all. Eac h of the instan taneous flo w quan tities sho wn corresp ond to a sim ulation time t = 50 ms , in the quasi-steady phase. Time a v eraging starts when the panel is in a quasi- steady state, b oth for meaningful, con v erged statistics and, for the flexible case sim ulations, to minimize Lagrangian effects that results when a v eraging a mo ving mesh. A dditional cen terplane comparison of flo w fields whic h are not discussed in this section are included in App endix C , for reference. 55 −2 0 2 4 6 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 Y s /δ 0 0 5 10 15 20 25 30 35 40 t [ms] −5 −4 −3 −2 −1 0 Y s [mm] (1) −2 0 2 4 6 8 f n (t−t 0 ) −1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 Y s /δ 0 0 10 20 30 40 50 60 t [ms] −5 −4 −3 −2 −1 0 1 Y s [mm] (2) 0 2 4 6 f n (t−t 0 ) −0.4 −0.3 −0.2 −0.1 0.0 Y s /δ 0 0 10 20 30 40 50 60 70 80 t [ms] −1.5 −1.0 −0.5 0.0 Y s [mm] (3) Figure 4.12: Comparison of one-w a y coupling FSI using recorded pressure snapshots from previous rigid WMLES. P anel displacemen t comparison b et w een exp erimen ts Daub et al. ( 2016 ) (dashed), previous w all- resolv ed LES sim ulation P asquariello et al. ( 2015 ) ( dotted), one-w a y coupling FSI using recorded pressure snapshots from previous rigid WMLES sim ulation (solid), at the fron t prob e (red line), cen ter prob e (blue), and rear prob e (green). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 56 4.1.5.1 Instan taneous quan tities Figure 4.13 sho ws the stream wise v elo cit y comp onen t, u , normalized b y the freestream v elo cit y u ∞ . Up- stream of the oblique sho c k, a turbulen t b oundary la y er is visible. The instan taneous region of flo w rev ersal (mark ed b y the white con tour line in the figure) is visibly larger at that time instan t for the flexible- than for the rigid-panel sim ulations. F or c ases (1) and (2), the instan taneous regions of flo w rev ersal extend farther upstream and do wnstream in the flexible-panel sim ulations than in the rigid-w all coun terparts. It is imp ortan t to note that this observ ation for the instan taneous flo w is not an anomaly . It will b e sho wn later in the time-a v eraged analysis, that this widening of the separated flo w region in comparison the the rigid panel is a consisten t effect of the panel flexibilit y . In case (3), the amoun t of flo w separation is smaller than the first t w o cases and the o v erall differences b et w een the stream wise v elo cit y fields of rigid and flexible panel configurations is quite small. Con tours of instan taneous normalized v ertical flo w v elo cit y , v/u ∞ , are sho wn in figure 4.14 . The color map sho wn is cen tered at zero with increasingly up w ard v elo cit y correlating with red in tensit y and increasingly do wn w ard v elo cit y correlating with blue in tensit y . This map is useful in that it helps to sho w v ertical flo w direction and in tensit y . Lo oking at case (1) and case (2), sev eral flo w features of the STBLI are easily visible. In the flexible configuration, upstream of the STBLI and ab o v e the panel, the v ertical v elo cit y of the flo w has a small do wn w ard comp onen t whic h is a result of the panel deflection directing the flo w sligh tly do wn w ard. By con trast, for the rigid case, the flo w has no v ertical comp onen t upstream of the STBLI (with the exception of turbulen t eddies in the TBL). The separation and inciden t sho c ks are easily visible in this plot b ecause of the abrupt c hange of flo w direction along the stream wise direction when encoun tering the inciden t sho c k and the separation sho c k from the upstream direction. The distance b et w een the inciden t-reflected sho c k in t ersection and the do wnstream compression w a v es is longer for the flexible panel configuration than for the rigid panel configuration. The instan taneous pressure along the cen terplane is sho wn in figure 4.15 . As the air flo ws o v er the flexible panel, the pressure drops b elo w the freestream pressure, p ∞ , as it undergo es an expansion induced b y the do wn w ard deflection. With a rigid panel, the flo w pressure remains nearly constan t un til encoun tering the oblique sho c k. Up on b ecoming separated, the flo w pressure in all cases and panel configurations exp eriences a sharp rise follo w ed b y gradual decline after reattac hing with the w all. F or cases (1), (2) and (3), a large amoun t of pressure can b e seen do wnstream of the in tersection of the impinging main oblique sho c k and the resulting reflected sho c k, whic h is exp ected o wing to the in teraction of the inciden t and separation sho c ks giving rise to the reflected and transmitted sho c ks. In cases (1) and (2), the pressure buildup is larger in this region for the rigid panel than the flexible panel. 57 (1) (2) (3) Figure 4.13: Cen terplane comparison of instan taneous stream wise x v elo cit y u/u ∞ , zo omed in to the region directly ab o v e the panel. The white con tour line represen ts u = 0 and marks the region of flo w rev ersal (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 58 (1) (2) (3) Figure 4.14: Cen terplane comparison of instan taneous w all-normal y v elo cit y v/u ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 59 (1) (2) (3) Figure 4.15: Cen terplane comparison of instan taneous pressure p/p ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 60 The instan taneous temp erature along the cen terplane is sho wn in figure 4.16 . As a result of the increased region of flo w separation, the flexible panel in cases (1) and (2) also exp eriences a wider region of exp osure to elev ated w all temp erature. In case (3), t he thermal impact of panel elasticit y is not visually apparen t, with the region of elev ated w all temp erature also b eing narro w er than in cases (1) and (2). The magnitude of the densit y gradien t v ectorial field, jrρj , is sho wn in figure 4.17 as a n umerical analog to sc hlieren imaging sho wn in compressible flo w exp erimen ts. As men tioned in c hapter 1 (see figure 1.1 ) and also illustrated b y T oub er & Sandham ( 2009 a ), notable flo w features of the STBLI include the inciden t, separation, transmitted, and reflected sho c ks, the separation bubble (for sufficien tly strong STBLIs), the shear la y er, the expansion fan (stemming from the curv ed transmitted sho c k ab o v e the sonic line), and the compression w a v es coalescing in to a reattac hmen t sho c k. As it will b e later evidenced b y the time-a v eraged plots of m ultiple flo w quan tities, all sim ulated cases sho w a separation sho c k ab o v e the sonic line, but only cases (1) and (2) presen t mean flo w separation. The distance b et w een the shear la y er and the separation sho c k is larger for the flexible panel than the rigid panel. The distance b et w een the reflected sho c k and the compression w a v es is m uc h larger for the flexible case than for the rigid case. F or case (2), a small separation sho c k is pro duced for the elastic panel. Similar to case (1), in case (2), the distance b et w een the reflected sho c k and the compression w a v es is greater for the flexible configuration than for the rigid configuration. F or case (3), the differences b et w een the rigid and flexible panel configurations app ear to b e m uc h smaller and subtle, consisten t with the m uc h smaller panel deflection induced b y a w eak er STBLI. The distance b e t w een the reflected sho c k and the reattac hmen t compression w a v es is m uc h smaller than in cases (1) and (2). F or case (3), the STBLI is not strong enough to pro duce a visually noticeable separation sho c k in the instan taneous n umerical sc hlieren image, but a small amoun t of flo w separation is presen t. The instan taneous span wise v orticit y , ω z , is sho wn in figure 4.18 . F or all cases, the span wise v orticit y is confined to the flo w regions dominated b y viscous shear and is amplified b y the in teraction with the sho c k system. F or the strongest in teraction (case 1), v orticit y amplification increases and extends o v er a larger region in the presence of panel flexibilit y . Baro clinic v orticit y generation resulting from misalignmen t of pressure and densit y gradien ts along the curv ed sho c ks is also visible, predominan tly for the strongest STBLI of case (1). 4.1.5.2 Time a v eraged quan tities Con tours of normalized time-a v eraged mean stream wise v elo cit y ,u/u ∞ , are sho wn in figure 4.19 , with regions of mean flo w rev ersal iden tified b y the white iso con tour lines (corresp onding to u = 0 ). The expansion undergone b y the flo w (resulting in an increased stream wise v elo cit y , u > u ∞ ) ab o v e the flexible panel is 61 (1) (2) (3) Figure 4.16: Cen terplane comparison of instan taneous temp erature T/T ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 62 (1) (2) (3) Figure 4.17: Cen terplane comparison of n umerical sc hlieren (gradien t of densit y magnitudejjrρjj ), zo omed in to the region directly ab o v e the panel (1)M ∞ = 3.0 ,θ max = 17 ◦ , (2)M ∞ = 4.0 ,θ max = 20 ◦ , (3)M ∞ = 4.0 , θ max = 15 ◦ 63 (1) (2) (3) Figure 4.18: Cen terplane comparison of span wise v orticit y ω z , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 64 (1) (2) (3) Figure 4.19: Cen terplane comparison of time-a v eraged stream wise x v elo cit y u/u ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 65 visible for all three cases. In case (1), the region of mean flo w separation ( u < 0 ) is m uc h larger in the presence of panel flexibilit y . The w eak er STBLIs for cases (2) and (3) translate in to smaller regions of mean flo w separation, in comparison to case (1). Con tours of the time a v eraged v ertical v elo cit y v is sho wn in figure 4.20 . The expansion undergone b y the flo w ab o v e the flexible panel upstream of the STBLI is clearly also manifested in the do wn w ard v ertical v elo cit y observ ed upstream of the STBLI (not presen t for the rigid-w all sim ulations). F or case (1), the region of up w ard motion imp osed b y the separation bubble (whic h originates the separation sho c k) is significan tly enlarged in the presence of panel flexibilit y , with the corresp onding lengthening of the transmitted sho c k Do wnstream of the STBLI, in the reco v ery region, the flo w main tains a gradual increase in v elo cit y when mo ving a w a y from the w all, due in part to the in teraction with the Prandtl-Mey er expansion generated at the rear corner of the w edge, whic h causes the flo w to expand. Con tours of normalized mean pressure, p/p ∞ , are sho wn in figure 4.21 . F or the stronger STBLIs of cases (1) and (2), the mean pressure is highest b ehind the i n tersection of the inciden t and separation sho c k, as exp ected from in viscid sho c k theory . In case (1), whic h has the strongest STBLI, the pressure buildup b e hind this in tersection p oin t is m uc h higher for the rigid w all than the flexible w all. The lo w er pressure buildup for the flexible w all is attributed to the region b ounding the inciden t sho c k, reflected sho c k, and expansion w a v es b eing larger for the flexible configuration, and presen ting higher unsteadiness. F or case (2), the pressure buildup b ehind the sho c k in tersection region is nearly iden tical for b oth rigid- and flexible-w all cases. F or the w eak STBLI of case (3), the pressure buildup is higher for the rigid configuration than the flexible configuration. The pressure v ariance p ′ p ′ /p 2 ∞ is sho wn in figure 4.23 . F or the strongest STBLI of case (1), the p eak of pressure fluctuations o ccurs at the in tersection of the inciden t sho c k and the start of the do wnstream expansion w a v es b ehind the reflected sho c k. The o v erall p eak fluctuations are similar b et w een the rigid and flexible panel configurations, but the flexible panel causes the surrounding region adjacen t to the p eak fluctuations to ha v e a higher v ariance in comparison. The small motions of the flexible panel app ear to cause greater stream wise motions of the reflected sho c k than the rigid-w all baseline. F or case (2), the pressure v a riance is qualitativ ely similar for the rigid and flexible cases. Con tours of the normalized stream wise v elo cit y v ariance, u ′ u ′ /u 2 ∞ (stream wise Reynolds stress), are sho wn in figure 4.22 . F or case (1), the p eak of these fluctuations o ccurs at the lo cation of the separation sho c k. This result is though t to b e due to the strong bac k and forth lo w-frequency motions of the separation sho c k. The presence of the flexible panel in tensifies and enlarges the region of this separation sho c k fluctuations. The effects of panel flexibilit y in in tensifying stream wise Reynolds stresses is clearly seen for all three cases, regardless of STBLI strength. 66 (1) (2) (3) Figure 4.20: Cen terplane comparison of time-a v eraged w all-normal y v elo cit y v/u ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 67 (1) (2) (3) Figure 4.21: Cen terplane comparison of time-a v eraged pressure p/p ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 68 (1) (2) (3) Figure 4.22: Cen terplane comparison of time-a v eraged stream wise x v elo cit y v ariance u ′ u ′ /u 2 ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 69 (1) (2) (3) Figure 4.23: Cen terplane comparison of pressure v ariance p ′ p ′ /p 2 ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 70 4.1.6 Influence of panel elasticit y on the temp oral ev olution of w all quan tities Figure 4.24 compares the span wise-a v eraged w all pressure as a function of the stream wise direction and time for the rigid and flexible panel sim ulations. The effect of the w edge rotation is seen in all the sim ulations as a con tin uous pressure increase un til reac hing the final w edge p osition, θ max , with the oblique sho c k imping- ing first on the rear of the panel and propagating fast upstream un til reac hing the statistically stationary impingemen t lo cation. F rom this comparison, it is inferred that the deflection of the panel in the flexible case th us has a clear effect on the w all pressure. This effect extends upstream a significan t distance, as seen b y the lo w pressure traces starting at time t ′ f n (t t 0 ) = 0 extending from the leading edge of the panel (sho wn with a dashed v ertical line) to a stream wise lo cation just upstream of the main STBLI. This is in con trast with the rigid case upstream of the in teraction, for whic h the normalized w all pressure, p w /p ∞ , remains constan t at unit y . Mo dulations of panel pressure upstream of the sho c k are presen t in all flexible-w all sim ulation cases and are consisten t with panel motion, with dips in pressure corresp onding to deflections of the panel. In terestingly , for case (3), despite ha ving a m uc h smaller amoun t of deflection than cases (1) and (2), the effect of panel vibration is clearly sho wn with s econdary and primary panel mo des. A t the rear of the panel, for case (1), the unsteady motion of the panel results in noticeable mo dulations of w all pressure, not presen t in the rigid case. A dditionally , the p eak of w all pressure is higher and lo cated farther upstream for the rigid case. Mo dulations of the p eak w all pressure are presen t in cases (1) and (2) for the flexible case un til a quasi-steady state is reac hed for the panel displacemen t. In case (3), the p eak w all pressure magnitude is notably higher in the flexible panel configuration than for the rigid panel case, whic h is opp osite to the first t w o cases. This could b e a result of the fact that the STBLI is quite w eak in comparison to cases (1) and (2) and that the panel slop e is p ositiv e at the impingemen t p oin t, dY s /dxj x I > 0 , in c on trast to cases (1) and (2). The span wise-a v eraged stream wise skin friction co efficien t, C f , is sho wn in figure 4.25 as a function of the stream wise co ordinate and time. T o highligh t the region of the w all where the flo w exp eriences separation (C f < 0 ), the colorbar go es from white in color when the skin friction is zero C f = 0 , to full blac k when the skin friction is at its minim um v alue. F or case (1), the flexible panel has a larger degree of flo w separation along the w all. As the flo w follo ws along the path of the deformed flexible panel, a drop in the stream wise w all shear stress o ccurs. F or the rigid panel, no suc h drop o ccurs, and the w all shear stress sho ws a mild increase un til it exp eriences flo w separation from the in teraction with the inciden t sho c k. In case (2), the maxim um p eak w all shear stress is increased for the flexible panel and the flo w separates further upstream and reattac hes further do wnstream. The effect of panel oscillations are sligh tly visible, with mo dulations of the w all stress co efficien t upstream of the STBLI b eing presen t. In case (3), the t w o skin friction profiles are 71 (1a) (1b) (2a) (2b) (3a) (3b) Figure 4.24: Span wise-a v eraged w all pressure p w /p ∞ v ersus time f n (tt 0 ) for (a) rigid and (b) flexible panel configurations (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . Dashed v ertical lines represen t the upstream and do wnstream lo cations of the panel edges. 72 (1a) (1b) (2a) (2b) (3a) (3b) Figure 4.25: Span wise a v eraged w all skin friction co efficien t C f v ersus time f n (tt 0 ) for (a) rigid and (b) flexible panel configurations. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . Dashed v ertical lines represen t the upstream and do wnstream lo cations of the panel edges. 73 Case P anel Maxim um pressure Maxim um pressure lo cation p max /p ∞ x ′ max (1) Rigid ( Daub et al. , 2016 ) 4.09 10.25 Rigid (sim ulation) 4.06 7.88 Flexible (sim ulation) 3.91 11.71 (2) Rigid ( Daub et al. , 2016 ) 6.07 14.36 Rigid (sim ulation) 5.91 11.57 Flexible (sim ulation) 5.86 14.30 (3) Rigid ( Daub et al. , 2016 ) 4.56 10.92 Rigid (sim ulation) 4.66 11.14 Flexible (sim ulation) 4.97 11.84 T able 4.2: P eak mean w all pressure magnitude and lo cation of p eak mean w all pressure (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . Comparison b et w een rigid and flexible panel c ases for the presen t sim ulations, and exp erimen ts b y Daub et al. ( 2016 ) for the rigid panel case. nearly iden tical to eac h other but the p eak skin friction at x ′ 15 is higher for the flexible panel than for the rigid panel. 4.1.7 Time-a v eraged w all quan tities in the quasi-steady regime 4.1.7.1 Stream wise profiles of time- and span wise-a v eraged w all pressure Figure 4.26 sho ws w all pressure stream wise profiles, a v eraged in time and along the span wise direction, p w /p ∞ , for rigid- and flexible-panel configurations in all three cases. Comparison to exp erimen tal data b y Daub et al. ( 2016 ) for the rigid panel is sho wn as w ell. F or simplicit y in the notation, the bar/o v erline op e rator (e.g., p ) in this discussion denotes not only time-a v eraging, as in the con tour plots on cen ter planes presen ted earlier, but also span wise a v eraging. T able 4.2 pro vides statistics ab out the maxim um w all pressure of the STBLI. F or case (1) with the flexible panel configuration, the w all pressure b egins to decrease once the flo w is ab o v e the panel. This b eha vior is consisten t with the previous discussion on the expansion induced b y the do wn w ard panel deflection and agrees with results from previous n umerical sim ulations b y P asquariello et al. ( 2015 ) and Zop e et al. ( 2021 ) (not sho wn). In addition, the initial linear slop e of the pressure gradien t up to the p oin t of separation closely matc hes b et w een exp erimen ts and the rigid-w all sim ulation. This slop e is indep enden t of the mo de of inducing separation, according to the free in teraction theory of Chapman et al. ( 1957 ). F or cases (1) and (2), whic h ha v e stronger STBLIs, the flo w separates farther upstream for the flexible configuration than the rigid baseline. F or case (3), whic h has the w eak est STBLI, the difference in the p oin t of separation is negligible b et w een b oth rigid- and flexible-w all sim ulations. An indication of flo w separation strength is the large reduction of the pressure gradien t ( d 2 p w /dx 2 < 0 ), near the in viscid inciden t sho c k impingemen t p oin t along the w all, x = x I , after separation, follo w ed b y an inflection p oin t in the w all pressure (d 2 p w /dx 2 = 0 ), where the pressure gradien t b egins to increase again. The p eak of the 74 −30 −20 −10 0 10 20 30 40 50 (x−x I )/δ 0 1 2 3 4 p w /p ∞ (1) −40 −30 −20 −10 0 10 20 30 40 (x−x I )/δ 0 1 2 3 4 5 6 p w /p ∞ (2) −50 −40 −30 −20 −10 0 10 20 (x−x I )/δ 0 1 2 3 4 5 p w /p ∞ (3) Figure 4.26: Time and span wise a v eraged stream wise mean profiles of w all-pressure, p w /p ∞ . Rigid (red) and flexible (blue) WMLES sim ulations (solid lines), compared with the rigid-w all exp erimen tal data of Daub et al. ( 2016 ) (dashed with sym b ols) (1, 2, 3) and w all-resolv ed FSI-LES data of P asquariello et al. ( 2015 ) (dotted). F ron t and rear edges of the panel are represen ted b y dashed v ertical lines, and unitary normalized w all pressure (p w = p ∞ ) is represen ted b y a solid blac k horizon tal line. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 75 w all pressure is lo w er in magnitude and o ccurs farther do wnstream for the flexible-panel configuration of cases (1) and (2). This is a result of the flo w reattac hing farther do wnstream for the flexible panel. In case (3), for the flexible panel, the p eak w all pressure do es o ccur farther do wnstream, but it is also higher than for the rigid panel, unlik e the previous t w o cases. The cause of this higher p eak pressure is attributed to flo w compression from the panel slop e b eing p ositiv e, dY/dx > 0 , at the lo cation of p eak pressure. F arther do wnstream, the w all pressure decreases as a consequence of the Prandtl-Mey er expansion w a v es generated at the rear tip of the rotating w edge. This expansion along the top b oundary is mo deled through in viscid theory assumptions, neglecting viscous effects suc h as v ortex shedding, and could result in discrepancies with the exp erimen tal results. Figure 4.27 sho ws the standard deviation of w all pressure along the stream wise direction for the three cases. F or cases (1) and (2), three distinct p eaks in w all pressure v ariance are presen t. F or case (3) only t w o of those p eaks are clearly visible, with the middle p eak b eing virtually non-existen t. The three p eaks corresp ond to the follo wing effects mo ving do wnstream: the lo w-frequency motions, the separation bubble, and the p eak of mean w all pressure. The incipien t flo w separation bubble of case (3) translates in to a m uc h reduced spik e in w all pressure fluctuation stemming in that region, compared to the other t w o cases (with significan t mean separation). Lo oking at case (1) in more detail, w e see that the magnitude of w all pressure fluctuations drops ab o v e the flexible panel but remains relativ ely uniform for the rigid panel. As the flo w b e gins to separate, an lo cal p eak of pressure fluctuations caused b y the lo w-frequency motions of the STBLI can b e seen. In terestingly , the pressure fluctuations of the lo w-frequency motions are higher for the rigid panel than for the flexible panel, but note that the affected region of the lo w-frequency motions is wider for the flexible panel than the rigid panel. The second p eak app ears to corresp ond to the lo cation of minim um stream wise skin friction C f where flo w rev ersal is strongest (as sho wn in figure 4.28 ). The third and final p e ak app ears to corresp ond to the lo cation of maxim um pressure for the STBLI. W e see that the flexible panel has a higher pressure v ariation at its p eak than the rigid panel despite ha ving a lo w er o v erall p eak pressure. 4.1.7.2 Stream wise profiles of time- and span wise-a v eraged w all shear stress Figure 4.28 sho ws the time and span wise-a v eraged skin friction co efficien t, C f , along the stream wise direction as a function ofx for cases (1), (2), and (3) with rigid and flexible panel configurations (sho wn in red and blue line colors resp ectiv ely). When C f c hanges from p ositiv e to negativ e, the flo w exp eriences mean separation. Con v ersely , when C f c hanges from negativ e to p ositiv e, the flo w exp eriences mean reattac hmen t. Upstream of the panel insert, the rigid- and flexible-panel configurations ha v e iden tical mean w all shear, as exp ected. 76 −30 −20 −10 0 10 20 30 40 50 (x−x I )/δ 0 0.1 0.2 0.3 0.4 q p ′ w p ′ w /p ∞ p w /p ∞ (1) −40 −30 −20 −10 0 10 20 30 40 (x−x I )/δ 0 0.2 0.4 0.6 q p ′ w p ′ w /p ∞ p w /p ∞ (2) −50 −40 −30 −20 −10 0 10 20 (x−x I )/δ 0 0.1 0.2 0.3 0.4 0.5 q p ′ w p ′ w /p ∞ p w /p ∞ (3) Figure 4.27: Time and span wise a v eraged stream wise profiles of w all pressure standard deviation, q p ′ w p ′ w /p ∞ . Rigid (red) and flexible (blue) WMLES sim ulations (solid lines), fron t and rear edges of the panel are represen ted b y dashed v ertical lines. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 77 −30 −20 −10 0 10 20 30 40 50 (x−x I )/δ 0 0.0 0.5 1.0 1.5 2.0 2.5 C f · 10 3 (1) −40 −30 −20 −10 0 10 20 30 40 (x−x I )/δ 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 C f · 10 3 (2) −50 −40 −30 −20 −10 0 10 20 (x−x I )/δ 0 0.0 0.5 1.0 1.5 2.0 2.5 C f · 10 3 (3) Figure 4.28: Time and span wise-a v eraged stream wise mean profiles of skin friction co efficien t C f 10 3 for the rigid (sho wn in red) and flexible (sho wn in blue) WMLES cases. F ron t and rear edges of the panel are represen ted b y dashed v ertical lines, and a skin friction co efficien t of zero (denoting flo w separation or reattac hmen t) is represen ted b y a solid blac k horizon tal line. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 78 Case P anel Separation lo cation Reattac hmen t lo cation Separation Length x ′ s x ′ r L sep /δ 0 (1) Rigid 6.41 1.37 7.78 Flexible 9.45 3.14 12.59 (2) Rigid 0.12 5.10 5.22 Flexible 0.49 6.43 6.91 (3) Rigid 1.54 2.47 0.93 Flexible 1.77 2.94 1.17 T able 4.3: Mean separation (x ′ s ) and reattac hmen t (x ′ r ) lo cations, and mean separation length (L sep /δ 0 ) (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . F arther do wnstream, the w all shear stress for the rigid and flexible cases b egin to div erge from eac h other. In the rigid panel, the shear stress con tin ues to increase sligh tly as the turbulen t b oundary la y er thic k ens. Ov er the flexible panel, the w all shear stress b egins to decrease along the stream wise direction. F or cases (1) and (2), the flexible panel configuration results in the flo w separating further upstream than its corresp onding rigid panel configuration. F or case (3) the opp osite is true, the flo w b egins to separate further upstream for the rigid configuration than the flexible one. F or all three cases (1), (2), and (3) the flo w reattac hes further do wnstream for the flexible configuration. F or case (3), near the rear edge of the flexible panel, the C f exhibits anomalous do wnshifts and upshifts whose exact cause is unkno wn. It is h yp othesized that the non-smo oth pressure drop along the rear and fron t edges of the panel (on discrete elemen ts m uc h larger than the flo w cells) ma y cause the equilibrium w all-mo del to pro duce these sharp rises and drops. The cause of the non-smo oth pressure drop is sp eculated to b e the use of linear in terp olation instead of the quadratic in terp o lation shap e functions in the solid-to-flo w coupling, whic h result in a discon tin uous slop e dY s /dx . T able 4.3 summarizes the mean separation lo cation, x ′ s , the mean reattac hmen t lo cation, x ′ r , and the mean separation length, L sep = x ′ r x ′ s , of all three cases for the rigid and flexible panel configurations. The observ ed increase of separation length for flexible panels has b een previously rep orted b y Neet & A ustin ( 2020 ) for comparable non-dimensional panel deflections. A dditionally , Zop e et al. ( 2021 ) found an increase in the separation length from the rigid baseline configuration using RANS-based metho ds. 4.1.8 Influence of panel elasticit y and w edge deflection angle on spatial mean panel statistics The flo w pressure field imp osed on to the panel (rigid and flexible) v aries in stream wise ( x ), span wise (z ), and temp oral (t ) dimensions. The instan taneous angle of deflection imp osed b y the mo ving w edge, θ(t) , pla ys a direct role in the pressure field. In the case of the flexible panel, it is also eviden t that the w all pressure is altered b y the deflection of the panel. It is useful to reduce the w all pressure field, p w (x,z,t) , subtracted b y 79 the pressure b eneath the b ottom ca vit y of of the panel, p cav , in to a single time-v arying net p oin t load, p(t) , that is applied at a single time-v arying stream wise lo cation (the cen ter of pressure), x CP (t) , calculated as p(t) = RR (p w (x,z,t)p cav )dxdz RR dxdz , x CP (t) = RR x(p w (x,z,t)p cav )dxdz RR (p w (x,z,t)p cav )dxdz . (4.1) The ca vit y pressure b eneath the panel, p cav , is set to the freestream pressure, p ∞ , for all sim ulated cases. The results of this reduction exercise help to elucidate the relation b et w een the input w edge angle and panel deflection on the w all pressure. Since the panel deflection, Y s , v aries in the stream wise direction it is helpful to use the geometric cen ter of the panel as a single reference p oin t for correlation. Figure 4.29 plots the time ev olution of the spatial mean pressure on the panel. Both the rigid and flexible mean panel pressure sho w oscillations that are correlated to the instan taneous w edge angle sho wn in figure 2.2 . On the righ t-hand side, the difference b et w een the t w o ( p flexible p rigid ) is plotted in blac k. F or all three cases, there is a small net pressure increase for the fle xible configuration o v er the rigid configuration. F or case (1), there is a small difference b et w een the instan taneous net pressure on the panel, and the lev el of noise in the differen tial is high enough suc h that the correlation of panel oscillation on the differen tial is not strongly discernible. F or cases (2) and (3) the instan taneous panel deflection is clearly correlated to the instan taneous pressure differen tial b et w een the flexible and rigid panels, with the signal-to-noise ratio b eing m uc h less. F or all cases, there is a clear influence of the w edge angle (including the small scale oscillations as sho wn in figure 2.2 (b)) on the net pressure. F rom these observ ations, w e see that the w edge angle is the primary v ariable in determining the net panel pressure and that the mo dest amoun t of panel deflection in the sim ulations pro vides a minor (but noticeable) role in altering the pressure field. Figure 4.30 plots the time ev olution of the x cen troid of pressure x CP on the panel for all three cases. On the left hand side, the rigid and flexible panel pressure cen troids, x CP , are plotted. On the righ t hand side, the difference ( x CP,flexible x CP,rigid ) b et w een the t w o is plotted in blac k. F or all three of the cases, w e see that panel deformation causes the cen ter of pressure to b e shifted do wnstream in relation to its rigid coun terpart, increasing in magnitude in tandem with increasing panel deflection. The magnitude of differences b et w een the cen ter of pressures is correlated strongly to panel cen ter motion. The influence of the serv o-con trolled w edge angular p osition, θ(t) , pla ys a primary role in the cen ter of pressure: small oscillations of the w edge angle (see figure 2.2 b) result in corresp onding oscillations of the cen ter of pressure when accoun ting for time dela y effects in the flo w field. F or instance, case (1) has an o v ersho ot in w edge angle initially and oscillates to w ards the setp oin t. The cen ter of pressure for this case (particularly the rigid case whic h is not influenced b y panel elasticit y) b eha v es in an iden tical fashion, o v ersho oting at first and then correcting in an oscillating 80 0 1 2 3 4 5 6 7 f n (t− t 0 ) 1.26 1.28 1.30 1.32 1.34 p/p ∞ (1a) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 (p flexible − p rigid )/p ∞ (1b) 0 2 4 6 8 10 f n (t− t 0 ) 1.60 1.65 1.70 1.75 1.80 1.85 1.90 p/p ∞ (2a) 0 2 4 6 8 10 f n (t− t 0 ) 0.00 0.02 0.04 0.06 0.08 0.10 (p flexible − p rigid )/p ∞ (2b) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 p/p ∞ (3a) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 0.00 0.01 0.02 0.03 0.04 0.05 (p flexible − p rigid )/p ∞ (3b) Figure 4.29: (a) Net panel pressure o v er time p for flexible (sho wn in blue) and rigid (sho wn in red) panel configurations. (b) Differen tial b et w een flexible and rigid cases p flexible p rigid (sho wn in blac k). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 81 0 1 2 3 4 5 6 7 f n (t− t 0 ) 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 (x CP − x I )/δ 0 (1a) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (x CP,flexible − x CP,rigid )/δ 0 (1b) 0 2 4 6 8 10 f n (t−t 0 ) 18 19 20 21 22 23 (x CP −x I )/δ 0 (2a) 0 2 4 6 8 10 f n (t− t 0 ) 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 (x CP,flexible − x CP,rigid )/δ 0 (2b) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 (x CP − x I )/δ 0 (3a) 0 1 2 3 4 5 6 7 f n (t− t 0 ) 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 (x CP,flexible − x CP,rigid )/δ 0 (3b) Figure 4.30: (a) Net panel pressure x -cen troid o v er time, x CP (t) , for flexible (sho wn in blue) and rigid (sho wn in red) panel configurations. (b) Differen tial b et w een flexible and rigid cases, x CP,flexible x CP,rigid (sho wn in blac k). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 82 Case P anel Mean net pressure Mean cen ter of pressure p/p ∞ (x CP x I )/δ 0 (1) Rigid 1.29 13.5 Flexible 1.32 14.8 (2) Rigid 1.75 12.8 Flexible 1.81 14.3 (3) Rigid 0.92 -0.97 Flexible 0.95 -1.76 T able 4.4: Time a v eraged net pressure magnitude p/p ∞ and cen ter of pressure lo cation (x CP x I )/δ 0 (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . manner. The time-a v eraged summary of these mean statistics in the quasi-steady state is sho wn in table 4.4 . 4.1.9 Sp ectral analysis of w all pressure The motiv ation for the sp ectral analyses presen ted in this section stems from the study of lo w-frequency dynamics that c haracterize STBLIs ( Clemens & Nara y anasw am y , 2014 ), and ho w those ma y b e affected b y panel flexibilit y . T o analyze the frequency con tributions of the w all pressure, algorithms whic h accurately estimate the p o w er sp ectral densit y (PSD) are desired. F or this w ork, the metho d of W elc h ( 1967 ) is em- plo y ed. As sho wn b y Prieb e & Martin ( 2012 ), this metho d is based on a v eraging m ultiple segmen ts and applying a Hann windo w function on eac h segmen t whic h o v erlap for FFT computation. Since the accuracy of the estimated PSD using W elc h’s metho d increases with a longer input sequence, a long in tegration time of the solv ers is desired. While the FSI sim ulations conducted in the presen t study are still computationally exp ensiv e, the use of WMLES w ithin them enables m uc h longer in tegration times and, th us, a b etter c har- acterization of lo w-frequency dynamics than previous studies. The t w o parameters to b e adjusted in W elc h’s metho d are the FFT length and the p ercen t o v erlap b et w een segmen ts. A longer FFT length allo ws for a wider sp ectrum of frequencies to b e calculated, but less segmen ts can b e used for a v eraging. An o v erlap fraction of zero represen ts no o v erlap b et w een segmen ts and an o v erlap fraction of one represen ts complete o v erlap b et w een the segmen ts. As the o v erlap fraction approac hes one , the computational cost approac hes O(N 2 ) and can b e prohibitiv ely exp ensiv e with diminishing returns in terms of a v eraging accuracy . The follo wing sim ulation time ranges w ere selected for eac h of these three cases: t 2 [20,110] ms for case (1), t2 [15,90] ms for case (2), and t2 [20,100] ms for case (3). In figure 4.31 , a comparison b et w een the rigid and flexible panel PSD of the w all pressure signals obtained along the stream wise direction is presen ted. The sampling in terv al for all stream wise PSD prob es is ∆t = 5.010 −7 s . The horizon tal axis represen ts the stream wise co ordinate of the high-sp eed prob e. The v ertical axis in figure 4.31 represen ts the frequencies of the PSD, non-dimensionalized in to a Strouhal n um b er, 83 (1a) (1b) (2a) (2b) (3a) (3b) Figure 4.31: W all pressure p o w er sp ectral densit y v ersus stream wise co ordinate for the (a) rigid and (b) flexible panel configuration. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 84 St =fδ 0 /u ∞ , based on the reference b oundary la y er thic kness δ 0 . The con tour plot sho ws the magnitude of the PSD prem ultiplied b y frequency , and normalized at eac h stream wise lo cation indep enden tly . In all cases, the incoming turbulence is c haracterized b y high frequencies cen tered around St 1 . F arther do wnstream, the flo w exp eriences separation and is c haracterized b y lo w-frequency motions ( St 10 −3 –10 −2 ) stemming from the unsteadiness of the separation sho c k ( Clemens & Nara y anasw am y , 2014 ). In the separated flo w region, the w all pressure is c haracterized b y an in termediate band of frequencies (St 0.1 –1 ) commensurate with the eddies inside flo w separation bubble. As the flo w reattac hes farther do wnstream, the high frequencies c haracteristic of the incoming TBL reco v er and b ecome dominan t again. As a res ult of panel flexibilit y , the lo c ation, size, and sp ectral distribution of these regions are affected. Figure 4.32 sho ws an o v erla y of the rigid w all pressure PSD (in red) with the flexible w all pressure PSD (blue). Regions in purple are indicativ e of similarit y b et w een rigid and flexible, while regions in red sho w a dominance of the rigid panel p o w er sp ectrum, and regions in blue sho w a dominance of the flexible panel p o w er sp ectrum. Finally , the flexible w all displacemen t PSD, that is w eigh ted b y o v erall magnitude and not normalized at eac h station, is sho wn alongside in gra yscale. F or cases (1) and (2), whic h ha v e m uc h larger panel deflection than case (3), the flo w separates further upstream for the flexible case, causing the lo w-frequency motions to o ccur further upstream as w ell. As a result of the flexible-panel configuration exp eriencing more flo w separation in comparison to the rigid w all baseline, a wider region that is dominated b y these medium frequency motions is visible in cases (1) and (2), with the high-frequency TBL sp ectral con ten t requiring a longer stream wise distance to reco v er. In case (3), the t w o PSD plots lo ok nearly iden tical, but the rigid case app ears to separate sligh tly upstream. T o gain a b etter understanding of the effects of panel flexibilit y on the stream wise ev olution of w all pressure p o w er sp ectral densit y , t w o regions of the STBLI are analyzed and compared to eac h other. The first region corresp onds to the lo w-frequency motions and is lo cated near the separation sho c k fo ot. The second region corresp onds to the separation bubble, where medium-range frequencies dominate. T o obtain more accurate and represen tativ e results of the sp ectral con ten t in these regions, the p o w er sp ectra are a v eraged in the stream wise direction o v er a sp ecified range. T able 4.5 sho ws the start and end p oin t of PSD a v eraging along these dimensions. The results of this a v eraging are sho wn in figure 4.33 . When comparing the flexible panel results (sho wn in blue) with the rigid panel results (sho wn in red), the follo wing is observ ed. In case (1), there is a distinct do wn w ard shift in dominan t frequencies in the w all pressure PSD for the separation bubble region and a sligh t do wn w ard shift for the lo w-frequency motions as w ell. There is also evidence of a do wn w ard shift in the flo w separation bubble frequencies mo ving to w ards the vibrational mo des of the panel. F or case (2), there is also a sizable do wn w ard shift in dominan t frequencies in the w all pressure PSD for the separation bubble region. Ho w ev er, the p o w er sp ectrum of the lo w-frequency 85 (1) (2) (3) Figure 4.32: W all pressure p o w er sp ectral densit y for rigid distribution (shaded in red), and flexible distribution (shaded in blue) o v erla y ed on to panel displacemen t (shaded in gra yscale) p o w er sp ectral densit y . (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 86 10 −3 10 −2 10 −1 10 0 10 1 St = fδ 0 /U ∞ 0.000 0.002 0.004 0.006 0.008 0.010 f·PSD(p w )/ R PSD(p w )df (1) 10 −3 10 −2 10 −1 10 0 10 1 St = fδ 0 /U ∞ 0.000 0.005 0.010 0.015 0.020 0.025 f·PSD(p w )/ R PSD(p w )df (2) 10 −3 10 −2 10 −1 10 0 St = fδ 0 /U ∞ 0.000 0.002 0.004 0.006 0.008 0.010 f·PSD(p w )/ R PSD(p w )df (3) Figure 4.33: Prem ultiplied PSDs of w all-pressure a v eraged along the stream wise co ordinate (see table 4.5 ) in regions corresp onding to the lo w-frequency motions of the separation sho c k (solid lines) and the separation bubble (dashed lines), comparing the rigid (red) and flexible (blue) panel cases. Angle brac k ets denote a v eraging o v er a stream wise range. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 87 Case P anel Band n ame Start (xx I )/δ 0 End (xx I )/δ 0 (1) Rigid Lo w-frequency motions -9.5 -8.6 Rigid Separation bubble -5.0 10.0 Flexible Lo w-frequency motions -12.7 -11.6 Flexible Separation bubble -7.5 12.0 (2) Rigid Lo w-frequency motions -2.5 -2.0 Rigid Separation bubble 3.0 5.0 Flexible Lo w-frequency motions -3.5 -3.0 Flexible Separation bubble 3.0 5.0 (3) Rigid Lo w-frequency motions -1.3 0.8 Rigid Separation bubble 1.5 4.0 Flexible Lo w-frequency motion -1.0 0.5 Flexible Separation bubble 2.0 4.5 T able 4.5: Stream wise ranges used for a v eraging the w all-pressure PSD of the rigid and flexible panel sim ulations in the regions of lo w-frequency motions and of the separation bubble. (1)M ∞ = 3.0 ,θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . motions app ears to b e relativ ely unaffected. Finally for case (3), no notable differences in the lo w-frequency motions are presen t. Cases (2) and (3) sho w evidence of the w all displacemen t mo des influencing the w all pressure mo des b oth upstream and do wnstream of the STBLI. It is imp ortan t to note that the p o w er sp ectra obtained from the flexible panel sim ulations w ere a v eraged o v er the quasi-steady p erio d, where oscillations w ere relativ ely small in magnitude and the main differences in the flexible panel and rigid panel results w ere due to quasi-static and not dynamic deflection. T o b etter elucidate these differences, it is helpful to quan tify the con tributions to the PSD o v er three frequency ranges. The first range, for St > 0.5 , considers the high-frequency con tribution to the p o w er sp ectrum and corresp onds to the fluctuations c haracteristic of the incoming turbulen t b oundary la y er. The second range of 0.05 St 0.5 represen ts the medium-range frequencies consisten t with the separation bubble. The final range of St < 0.05 represen ts the c haracteristic lo w-frequency motions of the separation sho c k. The individual band con tribution η(x) as a function of stream wise co ordinate, x , is calculate d as η(x) = R f b fa PSD(x,f)df R ∞ 0 PSD(x,f)df (4.2) Figure 4.34 sho ws the band con tributions for these three ranges. The turbulen t b oundary la y er clearly separates sligh tly farther upstream for the flexible panel than for the rigid panel for cases (1) and (2), resulting in an immediate drop in the high-frequency band con tribution. This drop is accompanied b y a subsequen t spik e in lo w-frequency band con tribution. In case (1), the p eak con tribution of the lo w-frequency motions is only sligh tly higher for the flexible panel, but the lo w-frequency motions band is m uc h wider for the flexible case. F arther do wnstream, starting at xx I for case (1) and xx I 5δ 0 for case (2), for b oth 88 −15 −10 −5 0 5 10 15 (x− x I )/δ 0 0.0 0.2 0.4 0.6 0.8 η(x) (1) −15 −10 −5 0 5 10 15 (x− x I )/δ 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 η(x) (2) −4 −2 0 2 4 (x− x I )/δ 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 η(x) (3) Figure 4.34: F requency band p o w er con tribution v ersus stream wise co ordinate x , for the flexible (sho wn in blue) and rigid (sho wn in red) panel configuration. Lo w-frequency band (solid line: 0<St< 0.05 ) , medium frequency band (dashed line: 0.05<St< 0.5 ), high frequency band (dotted line: 0.05<St<1 ). 89 the rigid and flexible panel cases, the high frequency con tribution starts increasing again, with the turbulen t b o undary la y er reco v ering in a shorter distance for the rigid case. This is esp ecially visible for case (2), in whic h the rate of TBL reco v ery app ears to b e more rapid than for case (1). This slo w er TBL reco v ery is an indication of STBLI strength. F or case (2), the stream wise exten t of the lo w-frequency bands in the flexible- and rigid-panel sim ulations are roughly equiv alen t. The p eak con tribution of the lo w-frequency band is actually sligh tly higher for the rigid case than the flexible case. F or case (3), the rigid panel configuration exp eriences flo w separation further upstream than the flexible case but the lo w-frequency con tributions are nearly iden tical in shap e and size with eac h other. In conclusion, the exten t and p eak of lo w-frequency motions dep end on ho w m uc h the separation sho c k mo v es along the stream wise direction, whic h can result from b oth flexible panel unsteady deflections as w ell as STBLI strength, in turn a consequence of the incoming TBL c haracteristics and the inciden t sho c k lo c ation. F urthermore, the latter also dep ends on the panel curv ature at the impingemen t lo cation, whic h v a ries among the differen t cases (1-3). 4.1.10 Analysis of the flo w separation bubble dynamics In order to quan tify the effects of panel elasticit y on STBLI, an analysis on the geometric structure of the flo w separation bubble where flo w rev ersal o ccurs ( u < 0 ) is p erformed. T o conduct this analysis, the flo w field cells undergoing separation w ere recorded to disk ev ery 10 2 time steps. Key statistics w ere extracted from eac h snapshot of the flo w separation bubble: the v olume, V/(Z d δ 2 0 ) , non-dimensionalized b y the span wise domain length and the reference b oundary la y er thic knessδ 2 0 , and the dimensionless co ordinates of its cen troid in the stream wise, (xx I )/δ 0 , and v ertical y/δ 0 ) directions. F rom these statistics, insigh t in t o ho w panel elasticit y affects the dynamics of flo w separation is obtained. The separation bubble cen troid is sho wn in figure 4.35 as a function of time for eac h of the sim ulated cases. Once fully established, the v ertical cen troid of the separation bubble remains relativ ely constan t o v er time for the rigid sim ulations. In con trast, for the flexible panel sim ulations and for all thre e cases considered, fluctuations in the v ertical cen troid strongly correlate with the v ertical p osition of the flexible panel b elo w the separation bubble. F or case (3), despite ha ving the smallest panel displacemen t, the higher-order mo des of the panel deflection are also clearly visible. This increased sensitivit y is a direct consequence of the separation bubble b eing smaller in v olume than for cases (1) and (2), and th us closer to the w all p osition. F or cases (2) and (3), the bubble stream wise cen troid is slo w er to con v erge in time relativ e to case (1). This slo w er con v ergence is attributed to the w edge rotation not completely con v erging to a steady p osition and also b eing undershot b y the serv o-con trol system for cases (2) and (3). This is in con trast to case (1), whic h 90 2.5 5.0 7.5 10.0 12.5 15.0 f n (t−t 0 ) −3.0 −2.5 −2.0 −1.5 (¯ x− x I )/δ 0 (1a) 2.5 5.0 7.5 10.0 12.5 15.0 f n (t− t 0 ) −0.75 −0.50 −0.25 0.00 0.25 0.50 ¯ y/δ 0 (1b) 2 4 6 8 f n (t−t 0 ) 2 3 4 5 (¯ x−x I )/δ 0 (2a) 2 4 6 8 f n (t− t 0 ) −0.6 −0.4 −0.2 0.0 0.2 0.4 ¯ y/δ 0 (2b) 1 2 3 4 5 6 7 f n (t−t 0 ) 0 1 2 3 4 (¯ x−x I )/δ 0 (3a) 2 4 6 f n (t− t 0 ) −0.2 −0.1 0.0 0.1 0.2 ¯ y/δ 0 (3b) Figure 4.35: Separation bubble cen troid lo cation v ersus time for the flexible (sho wn in blue) and rigid (sho wn in red) panel configurations. The stream wise cen troid, x , is sho wn in (a) and the v ertical cen troid,y , is sho wn in (b). (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 91 Case P anel Mean x -cen troid Std dev x -cen troid Mean y -cen troid Std dev y -cen troid (xx I )/δ 0 (xx I )/δ 0 y/δ 0 y/δ 0 (1) Rigid -2.09 0.176 0.379 0.022 Flexible -2.24 0.22 -0.403 0.028 (2) Rigid 2.94 0.33 0.29 0.02 Flexible 3.50 0.35 -0.45 0.038 (3) Rigid 1.68 0.17 0.14 0.012 Flexible 2.08 0.16 -0.15 0.023 T able 4.6: Quasi-steady state statistics of the separation bubble cen troid (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . exp eriences sligh t o v ersho ot but is quic k er to reco v er to the targeted setp oin t. After 40 ms , the w edge is also artificially k ept constan t in the sim ulation due to a lac k of exp erimen tal data b ey ond that time in Daub et al. ( 2016 ). The w edge angle v ersus time can b e view ed in more detail in figure 2.2 . While there is a sligh t absolute difference in the bubble stream wise cen troid b et w een the rigid and flexible panel configurations for cases (1), (2), and (3), there is a remarkable similarit y in the temp oral ev olution of x for b oth the rigid and flexible configurations despite ha ving m uc h larger o v erall v olumes, esp ecially in case (1) and case (2). Suc h a finding suggests that for a mo dest amoun t of elasticit y , the separation bubble lo cation, x , is largely unaffected, but the v olume of total separated flo w is quite sensitiv e. F rom the probabilit y densit y functions sho wn in figure 4.36 , a sligh t negativ e correlation is observ ed b et w een the stream wise lo cation of the separation bubble cen troid and its v olume (higher bubble v olume correlating with a more upstream bubble cen troid), with the exception of the rigid-panel case (1). The absence of a strong correlation suggests that there are no significan t stream wise con traction mo des of the bubble where stream wise motion of the bubble results in shrinkage or expansion. In fact, despite ha ving large v ariations in v o lume, the bubble lo cation remains relativ ely consisten t and app ears to b e more dep enden t up on the small scale v ariations of the instan taneous w edge angle. As exp ected, in case (1), the join t probabilit y densities of the rigid and flexible panel are farthest apart from eac h other and gradually merge to w ards eac h other as the o v erall panel deflection decreases (cases 2 and 3). F or reference, tables 4.6 and 4.7 summarize k ey statistics of the separation bubble in the quasi-steady regime. One imp ortan t observ ation is that the fluctuations in v o lume are roughly prop ortional to mean. The bubble v olume p o w er sp ectral densit y sho wn in figure 4.37 rev eals that a do wn w ard shift in dominan t frequencies of separation bubble v olume fluctuations is presen t, particularly for cases (1) and (2). F or case (3), the sp ectral distribution is nearly iden tical b et w een the rigid and flexible cases. The time ev olution of the separation bubble v olume is sho wn in figure 4.38 (1-3a). There is a sligh t drift in bubble v olume for cases (2) and (3) due to the w edge not reac hing equilibrium during the time frame plotted. In figure 4.38 (1-3a), the probabilit y distribution of the bubble v olume is calculated for e ac h case 92 (1) (2) (3) Figure 4.36: Join t probabilit y distribution of bubble x cen troid, and v olume for the flexible (sho wn in blue) and rigid (sho wn in red) panel configurations. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 93 10 −3 10 −2 10 −1 St = fδ 0 /U ∞ 0 20 40 60 80 100 120 PSD(V b /V b ) (1) 10 −3 10 −2 10 −1 St = fδ 0 /U ∞ 0 20 40 60 80 PSD(V b /V b ) (2) 10 −3 10 −2 10 −1 St = fδ 0 /U ∞ 0 20 40 60 80 100 PSD(V b /V b ) (3) Figure 4.37: P o w er sp ectral densit y of mean normalized bubble v olume, V b /V b , for flexible (sho wn in blue) and rigid (sho wn in red) panel configurations. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 94 2.5 5.0 7.5 10.0 12.5 15.0 f n (t−t 0 ) 0 2 4 6 8 10 V b /(Z d δ 2 0 ) (1a) 4 6 8 10 V b /(Z d δ 2 0 ) 0.00 0.25 0.50 0.75 1.00 1.25 probability density (1b) 2 4 6 8 f n (t−t 0 ) 1 2 3 4 5 V b /(Z d δ 2 0 ) (2a) 2 3 4 5 V b /(Z d δ 2 0 ) 0.0 0.5 1.0 1.5 probability density (2b) 2 4 6 f n (t−t 0 ) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 V b /(Z d δ 2 0 ) (3a) 0.3 0.4 0.5 0.6 0.7 0.8 V b /(Z d δ 2 0 ) 0 2 4 6 8 probability density (3b) Figure 4.38: (a) Separation bubble v olume for rigid (sho wn in red) and flexible panels (sho wn in blue) v ersus time. (b) PDF of separation bubble v olume. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . 95 Case P anel Mean v olume Std dev v olume Ratio µ[V/(Z d δ 2 0 )] σ[V/(Z d δ 2 0 )] σ/µ (1) Rigid 4.30 0.37 0.087 Flexible 9.08 0.72 0.079 (2) Rigid 2.25 0.23 0.104 Flexible 3.58 0.38 0.105 (3) Rigid 0.50 0.056 0.111 Flexible 0.57 0.063 0.112 T able 4.7: Quasi-steady state statistics of the separation bubble v olume (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . using discrete bins and a Gaussian fit is applied alongside. In all three cases, the v olume PDF is appro ximated quite w ell with a Gaussian fit. Case (1) whic h has the largest amoun t of fluid structure in teraction sho ws the greatest disparit y b et w een the flexible and rigid panel distributions. Case (2) has lesser disparit y than case (1), and, finally , case(3) sho ws the t w o distributions almost o v erlapping. Lo oking at table 4.7 , the standard deviation of v olume is found to b e roughly prop ortional to the mean v olume regardless of panel flexibilit y . 4.1.10.1 Probabilit y distributions of flo w rev ersal A commonly used tec hnique for determining the shap e of the mean flo w separation bubble is to a v erage the v elo cit y field o v er a sp ecified in terv al. In this analysis, an alternate approac h is tak en where the probabilit y of instan taneous flo w rev ersal is assessed as a function of the stream wise, x , and v ertical,y , co ordinates. During the sim ulation, cells whic h exp erience flo w rev ersal ( u < 0 ) are recorded to disk ev ery 100 time steps. The individual cell v olumes, along with their corresp onding cen troids (x,y ) are then gathered and used as input to a Gaussian k ernel densit y estimate (KDE) metho d that allo ws for the estimation of a m ultidimensional PDF in a non-parametric w a y from scattered input data ( Scott , 2015 ). The output from KDE algorithm can b e plotted on a structured grid, sho wing the probabilit y of instan taneous flo w rev ersal as a function of x and y . Plots sho wing b oth the rigid and flexible panel separation probabilit y densit y are sho wn in figure 4.39 for eac h case under consideration. The spatially-in tegrated probabilit y of instan taneous flo w separation for flexible cases are larger than for the equiv alen t rigid case (prop ortional to the mean separation bubble v o lumes). In all three cases, the con tour plots of flo w rev ersal probabilit y for the rigid and flexible panel cases sho w qualitativ e agreemen t with eac h other. This also supp orts the observ ation that the do wn w ard panel deflection causes the bubble gro wth to expand out w ard in all directions a w a y from the w all and not drift do wnstream or upstream. In case (1), the p eak of instan taneous flo w rev ersal probabilit y o ccurs near the halfw a y p oin t b et w een the in viscid impingemen t p oin t, x I , and separation bubble cen troid, x . In cases (2) and (3), the p eak of instan taneous flo w separation probabilit y o ccurs close to the mean separation bubble 96 (1a) (1b) (2a) (2b) (3a) (3b) Figure 4.39: Time- and span wise-a v eraged probabilit y distributions of flo w rev ersal for the rigid (top) and flexible (b ottom) panel cases in the quasi-stationary p erio d. (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ . The mean w all p osition, Y s (x) , is sho wn with dashed lines and the separation bubble cen troid mark ed with a cross for reference. 97 cen troid. Case (3) sho ws little difference b et w een the rigid and flexible panel configurations and has a m uc h smaller fo otprin t than the first t w o cases. T o gain a b etter understanding of ho w elasticit y affects the shap e of the flo w separation bubble, the t w o shap es are collapsed on to eac h other using con tour lines of flo w separation probabilit y . Figure 4.39 (b) compares geometrical differences b et w een the t w o flo w separation regions not related to w all curv ature and increased flo w separation. In b oth cases, the region of flo w rev ersal is biased to w ards the region b et w een the separation bubble cen troid and the sho c k impingemen t p oin t at the w all. The flo w rev ersal regions are predominan tly triangular, with panel flexibilit y resulting in a sligh t flattening, shortening its heigh t, when normalized b y the separation length. The separation bubble shap es for rigid (red) and flexible (blue) cases are compared b y sup erimp osing con tour lines of the probabilit y distributions sho wn in co ordinates normalized b y the separation length, L sep , and relativ e to the mean separation bubble stream wise cen troid, x , and to the m ean w all v ertical lo cation, Y s (x) . 98 4.2 Sim ulations with the rotating w edge oscillating b et w een t w o setp oin ts A k ey ob jectiv e of this study is to in v estigate STBLIs (with and without flexible panels) in resp onse to the inciden t sho c k angle, β(t) , and the in viscid impingemen t lo cation, x I (t) , c hanging in time at a constan t frequency of oscillation, f w . This w orks builds up on the previous section whic h lo ok ed at cases with a constan t w edge angle setp oin t attained after a transien t. The test configuration and computational mesh is exactly the same as case (1): M ∞ = 3.0,θ max = 17.5 ◦ . Ho w ev er, the w edge angle starts at θ = θ max , and then rotates do wn to θ = θ min = 15.5 ◦ , oscillating at a constan t frequency , f w , b et w een those t w o angles afterw ards. The decision to c ho ose the particular v alue of θ max = 17.5 ◦ , corresp onding to case (1), is that in order to initialize the sim ulation, the flo w field of a strong STBLI that w as i n quasi-steady state (for b oth flexible and rigid panels w as needed. When c ho osing θ min = 15.5 ◦ , t w o goals w ere presen t. The first goal w as to induce a large resonan t resp onse for the flexible panel, whic h is obtained b y increasing the difference b et w een θ max and θ min . The second goal w as to k eep the flo w separation bubble from b ecoming in termitten t during the sim ulation. In order to accomplish this, θ min m ust b e large enough to main tain separation consisten tly . F or θ < 15 , the STBLI starts b ecoming w eak enough to start ha ving in termitten t separation. Because of this, θ min = 15.5 ◦ w as selected. Rather than only study a single w edge oscillation frequency , f w f n , to induce resonance, a range of differen t frequencies w as c hosen to study the STBLI resp onse as a function differen t frequencies. Using the measured panel natural frequency of case (1) in Daub et al. ( 2016 ) f n 229 Hz as a middle p oin t, the minim um b ound of f w = 50 Hz w as c hosen suc h that there w ould b e a sufficien t n um b er of cycles o v er a time-p erio d of 100 ms . The upp er b ound of f w w as c hosen to b e 800 Hz so that f n w ould reside close to the middle of the frequencies c hosen when using a log scale. The v alues of f w used for eac h individual sim ulation are listed in table 4.8 and range fromf w = 50Hz tof w = 800Hz . The time dimension is non-dimensionalized b y the w edge oscillation frequency f w and the measured time lag t l of the signal to c hanges in the w edge angle t ′ = f w (tt l ) . T able 4.9 states the c haracteristic parameters of the in teractions in the quasi-steady state corresp onding to the t w o limiting angles of oscillation, for rigid and flexible panel configurations. One imp ortan t consideration is the time dela y asso ciated with disturbances to the w edge angle, θ(t) , and ho w long it tak es for these disturbances to propagate throughout the coupled FSI system. An estimate of the minim um time dela y for a disturbance in the w edge angle to reac h the b ottom w all, t d , is obtained b y 99 f w Hz Flexible Rigid 50 ⋆ ⋆ 75 ⋆ 100 ⋆ ⋆ 150 ⋆ 200 ⋆ ⋆ 225 ⋆ 250 ⋆ 300 ⋆ 400 ⋆ ⋆ 500 ⋆ 600 ⋆ 800 ⋆ ⋆ T able 4.8: W edge oscillation frequencies f w used for rigid and fl exible panel sim ulation P arameter Rigid: θ min = 15.5 ◦ θ max = 17.5 ◦ Flexible: θ min = 15.5 ◦ θ max = 17.5 ◦ Net panel pressure, p/p ∞ 1.01 1.21 1.038 1.236 P anel cen ter displacemen t, Y s /δ 0 - - 0.92 1.0 Separation bubble v olume, V/Z d δ 2 0 1.3 2.4 1.8 3.8 T able 4.9: Steady state parameters for the flexible and rigid panel configurations with θ max = 17.5 ◦ and θ min = 15.5 ◦ used in calculation of gain the time it w ould tak e for the comp onen t of the flo w v elo cit y parallel (tangen tial) to the inciden t sho c k, v t (whic h is conserv ed across the sho c k) v t =u ∞ cos(β) = q u 2 2 +v 2 2 cos(βθ) (4.3) to tra v el the length of the inciden t sho c k, L shock , from the w edge fron t to the sho c k impingemen t p oin t on the w all L shock = q (x I x f ) 2 +y 2 f (4.4) In the presen t sim ulations, the computational domain do es not include the ph ysical w edge. Instead, the b oun dary condition on the top b oundary of the flo w solv er domain applies instan taneously the in viscid solution without accoun ting for an y time lag effects from the w edge disturbance to propagate to that top b oun dary . Hence, when estimating the time dela y of disturbances within the computational flo w domain, the sho c k length to b e used should instead b e the distance from the in tersection of the inciden t sho c k at the top of t he domain (x ISI ,L y ) to the sho c k impingemen t p oin t on the w all, (x I ,0) . L ′ shock = q (x I x ISI ) 2 +L 2 y (4.5) 100 Figure 4.40: Illustration of disturbance time lag t d (the do wnstream flo w is parallel to the b ottom surface of the w edge and not the sho c k angle β ) and th us the dela y time results t d = L ′ s,shock v t (4.6) Figure 4.40 aids in illustrating the estimation of the disturbance time lag for the sim ulated STBLI system. This time lag can b e view ed as the theoretical estimate of dela y for disturbances in the w edge angle to propagate to the panel surface using the in viscid flo w assumption. Since the w edge oscillates b et w een θ min and θ max , the disturbance time lag v aries temp orally from t d = 3.5810 −4 s when θ = θ max to t d = 3.6910 −4 s when θ = θ min . F or simplicit y , a disturbance time lag of t d 3.610 −4 s is considered in this case as a reference for comparison w ith other time lags presen t in the STBLI system (e.g., asso ciated to the separation bubble breathing motions). Figure 4.41 sho ws the time lags of the pressure field and the separation bubble v olume in resp onse to c hanges in the w edge angle along the top b oundary of the flo w domain. When studying vibrations problems, it is common to define a gain factor, G , that represen ts the am- plification of oscillation magnitude in relation to the difference b et w een the steady state solutions of the problem at the maxim um and minim um setp oin ts. In STBLI problems o v er flexible panels, the gain factor is dep enden t up on sev eral factors in the system, suc h as the damping ratio of the panel,ζ , the forcing frequency ratio, f w /f n , the minim um θ min and maxim um θ max angles of the sho c k-generating w edge, and the incoming Mac h n um b er, M ∞ . T o calculate the gain factor, the resp ectiv e quasi-steady state solutions for θ min and θ max are used as reference. As the w edge forcing frequency approac hes infinit y ( f w !1 ) the solution should approac h the quasi-static solution of the mean w edge angle, whic h represen ts a zero gain factor (G = 0 ). This 101 (a) (b) Figure 4.41: Calculated time lags normalized b y the theoretical minim um time panel time lag t l /t d for (a) w all pressure and (b) separation bubble v olume v ersus non-dimensional w edge forcing frequency f w /f n . Rigid and fl exible panel results are plotted in red and blue mark ers resp ectiv ely . is due in part to the fact that there is a time dela y b et w een disturbances in the incoming inciden t oblique sho c k and the time it tak es for disturbances to propagate across the flo w field of the STBLI. Similarly , as the forcing frequenc y approac hes zero (f w ! 0 ), the solution should approac h the quasi-static solution of the curren t (non-mean) w edge angle whic h represen ts a unitary gain factor (G = 1 ). Finally , the panel should reac h maxim um gain G (in terms of panel cen ter displacemen t) as the w edge forcing frequency approac hes its primary natural frequency (f w ! f n ). F or a completely undamp ed linear elastic panel (ζ = 0 ), this w ould result in an unstable system with a theoretical gain factor tending to infinit y ( G!1 ). T o calculate the gain, G , sev eral p oten tial metrics w ere considered. The first metric is akin to a cross- correlation b et w een a time-v arying input signal, ζ(t) , and a complex exp onen tial e −2πfwt with the w edge forcing frequency f w , based on the discrete F ourier transform of the signal, defined as ˆ ζ k N X n=0 ζ i exp −2πfwt /N. (4.7) The phase angle,ϕ , is found from the real (< ) and imaginary (= ) comp onen ts of the previous cross-correlation as ϕ = arctan =( ˆ ζ k ),<( ˆ ζ k ) . (4.8) Under this metho d, the time dela yt l is found b y dividing the calculated phase angleϕ b y the w edge oscillating frequency f w t l = ϕ 2πf w . (4.9) 102 Careful consideration m ust b e tak en when the calculated phase angle is outside the b ounds of 0 < ϕ < π/2 since the time lag is defined in this instance as a ph ysical dela y in resp onse to a disturbance to the w edge angle. If, for instance, a sine w a v e output comes immediately (without dela y) from a cosine w a v e input, the phase angle ϕ w ould b e π/2 but the ph ysical time lag could b e in terpreted to b e zero. The gain is defined as G = 4 q <( ˆ ζ k ) 2 +=( ˆ ζ k ) 2 ζ qss (θ =θ max )ζ qss (θ =θ min ) , (4.10) where ζ qss refers to the quasi-steady state v alue of the v ariable ζ . An alternate approac h to finding time dela y , t l , in v olv es directly computing the cross -correlation of the t w o signals as (ζ ⋆θ)[j] = N−1 X i=0 θ i−j ζ i . (4.11) The lag time t l =j∆t whic h maximizes the cross-correlation is tak en to b e the computed time dela y . The second gain metric (used for panel displacemen t) simply measures the difference b et w een the input signal extrema. While simple, this can b e effectiv e at measuring the p eak signal v ariation. This metho d has the dra wbac k of not considering time-a v eraged effects and can b e sensitiv e to instan taneous signal spik es. F or input signals not sub ject to instan taneous spik es suc h as the panel displacemen t, Y s , this simple metric is quite effectiv e. G = max(ζ)min(ζ) ζ qss (θ =θ max )ζ qss (θ =θ min ) (4.12) The final gain metric (used for separation bubble v olume and mean pressure) tak es the time lag t l found using equation 4.11 and correlates the w edge angle with l ag correction θ(tt l ) to the signal ζ(t) . G = ζ(θ >θ max ∆θ)ζ(θ <θ min +∆θ) ζ qss (θ =θ max )ζ qss (θ =θ min ) (4.13) The n umerical a v erage of the signal ζ(t) when the w edge angle θ is at maxim um and minim um is calculated and then the absolute difference b et w een the t w o is tak en. Since the w edge angle is at maxim um and minim um for an infinitely small amoun t of time, a tolerance ∆θ is added when p erforming the n umerical a v eraging for the gain calculation. An adv an tage of this gain metric calculation is that the noise in the signal is accoun ted for b y a v eraging the signal at the relativ e maxima and minima resp ectiv ely . 103 (a) Rigid (b) Flexible Figure 4.42: P anel mean pressure p/p ∞ of (a) rigid panel and (b) flexible panel configurations v ersus non- dimensional cycle time f w (tt l ) . 4.2.1 Mean panel pressure in resp onse to forced w edge oscillation T o quan tify the effects of the distributed pressure loading on the panel o v er time, the pressure field is a v eraged in t o a single p oin t load with magnitude, p , acting at a cen troid lo cation, x CP . These t w o quan tities pro vide a simplified mo del of the panel loading. Figure 4.42 sho ws the temp oral ev olution of the instan taneous net mean panel pressure (for b oth flexible and rigid panels) o v er a subset of the differen t w edge frequencies sim ulated. It is inferred that, for b oth the rigid and flexible panels, the pressure oscillations main tain a nearly consisten t v alue o v er the frequency ranges studied and that the algorithm used to calculate the time lags of the signals is relativ ely successful, giv en the collapse of temp oral signals. Figure 4.43 sho ws the calculation of the gainG and phase angleϕ of the net pressure in resp onse to the w edge oscillation frequency f w . In the case of the rigid panel, as the w edge oscillation frequency increases, there is a decrease in the p e ak-to-p eak instan taneous w all pressure. The flexible panel exhibits a similar b eha vior to the rigid panel, but exp eriences a nearly 10% rise in the gain for oscillating frequency near the primary natural frequency of the panel, f w /f n 1 . The gain obtained at this frequency is h yp othesized to result from the amplification of panel oscillations stemming from vibrational resonance. In con trast, for the rigid panel, there is no amplification of the w all pressure oscillations near the natural frequency , but rather a con tin ued atten uation with increasing w edge frequency . Analyzing the phase angle, the theoretical estimate of the phase dela y is nearly iden tical to the those calculated b y the time dela y algorithms, as sho wn in figure 4.43 (b). 4.2.2 P anel vibrations in resp onse to forced w edge oscillation Figure 4.44 (a) sho ws the temp oral deflection of the flexible panel cen ter in resp onse to the w edge oscillation. Since the panel pressure pro duces a sin usoidal output in resp onse to the w edge oscillation, this also causes 104 (a) (b) Figure 4.43: (a) Normalized gain,jGj , and (b) normalized phase angle,ϕ/2π , for panel pressure, as a function of the normalized w edge forcing frequency f w /f n . The estimated phase angle represen ting the time lag in the propagation of sho c k disturbances in tro duced b y the w edge oscillation, ϕ = 2πt d f w , is sho wn with a dashed blac k line. Rigid and flexible panel results are plotted in red and blue mark ers, resp ectiv ely . corresp onding oscillation in the panel. F or the frequencies sho wn in the figure, the largest amplitude of panel vibration corresp onds to a frequency of oscillation closest to panel natural frequency , f w f n . Figure 4.44 (b) sho ws clear evidence of panel excitation caused b y resonance, with an increasing gain obtained for f w !f n . 4.2.3 V olume of separation bubble in resp onse to forced w edge oscillation T o study the resp onse of the separation bubble to w edge disturbances, the v olume of the separation bubble w as recorded in a similar fashion to the studies with a fixed angular setp oin t. The time lag, t l , b et w een the input w edge angle disturbances and the output disturbances of the separation bubble v olume w as computed for all of the sim ulated cases. As sho wn in figure 4.41 (b), the time lag w as found to b e on the order of three times the theoretical w all pressure disturbance lag, t d , with the flexible panel ha ving a sligh tly higher lag time on a v erage that the rigid panel. This higher lag time of the separation bubble is attributed to the larger subsonic zone of the TBL dev elop ed ab o v e the flexible panel taking longer for disturbances to propagate in t o the separation bubble in comparison to the rigid panel. It is observ ed in figure 4.45 , that the time lag algorithm in use can align the bubble v olume signals on to eac h other alb eit the Gaussian noise that is inheren t to the temp oral signal of separation bubble. Analyzing the rigid panel results, an atten uation of the bubble v olume v ariation can b e seen for the higher frequencies plotted. The gain and phase angle of the separation bubble v olume as a function of the w edge oscillation frequency are sho wn in figure 4.46 . The gain sho ws a clear atten uation in the abilit y of the separation bubbles to resp ond to fluctuations in the w edge (for b oth rigid and flexible panels) at higher frequencies. In terestingly , the gain 105 (a) (b) Figure 4.44: (a) T emp oral ev olution of flexible panel v ertical displacemen t at the panel cen ter, Y s /δ 0 . (b) Magnitude of the gain of panel displacemen t, jGj , calculated from panel deflection v ersus non-dimensional w edge oscillation frequency f w /f n in the quasi-steady state. (a) Rigid (b) Flexible Figure 4.45: Non dimensional Flo w separation bubble v olume V/Z d δ 2 0 of (a) rigid and (b) flexible panel configurations v ersus non-dimensional cycle time f w (tt l ) for a subset of sim ulated w edge oscillation fre- quencies. 106 (a) (b) Figure 4.46: Flo w separation bubble v olumeV/Z d δ 2 0 (a) gainjGj and (b) normalized phase angleϕ/2π v ersus w edge forcing frequency f w /f n . The estimated phase angle represen ting the time lag in the propagation of sho c k disturbances in tro duced b y the w edge oscillation, ϕ = 2πt d f w , is sho wn with a dashed blac k line. Rigid and flexible panel results are plotted in red and blue mark ers resp ectiv ely . for the flexible panel exp eriences a sharp drop of appro ximately 50% in the p eak-to-p eak v ariations of the separation bubble v olume mo ving from f w = 200 Hz to f w = 225 Hz . This result is surprising at first glance in that the larger up w ard-do wn w ard panel motions are somewhat analogous to cycling b et w een a deformed panel in quasi-steady state to an almost flat (pseudo-rigid) panel. F rom the fixed-w edge angle setp oin t studies presen ted in the previous section, large v ariations b et w een the separation bubbles of rigid and flexible panel configurations w ere observ ed. As suc h, it could b e exp ected that cycling b et w een these t w o states migh t cause large v ariations in bubble v olume. Ho w ev er, if w e consider a p oten tial mismatc h b et w een the phase dela y angles of the panel and of the bubble, it is plausible that, b y the time the separation bubble resp onds to the STBLI w edge angle due to phase dela y , the panel deflection is small. On the other end, when the STBLI angle is smaller and pro duces a smaller separation bubble v olume, the panel deflection ma y b e large. These effects could p oten tially com bine together to create smaller v ariations in the separation bubble, th us causing a drop in the gain. Another observ ation from figure 4.46 is that, as the w edge oscillation frequency increases past the natural frequency , f w > f n , only a mo dest reco v ery in the gain factor is observ ed for the next t w o sim ulations tested (for f w = 250 and 300 Hz ). Con tin uing past 300 Hz , the separation bubble v olume sho ws greater atten uation with increasing w edge oscillation frequency for b oth the flexible and rigid panels. 107 Chapter 5 Conclusions A high-fidelit y sim ulation study of STBLIs o v er rigid and flexible panels has b een pres en ted that utilizes a lo osely-coupled, partitioned FSI solv er metho dology consisting of a w all-mo delled LES flo w solv er, an isoparametric high order finite elemen t solid solv er, and a spring-s ystem analogy mesh deformation solv er. The prop osed n umerical metho dology , incorp orating an equilibrium w all mo del in the flo w solv er, signifi- can tly reduces the computational cost, enabling long in tegration times while main taining ph ysical fidelit y , b o th necessary to study the strongly separated STBLIs and their coupling with the flexible panel dynam- ics. The computational metho dology of the unstructured grid FSI solv er used in this study allo ws for the future sim ulation of more practical three-dimensional geometries found in indust rial applications. Previous studies b y P asquariello et al. ( 2015 ) used w all-resolv ed LES form ulations w ere m uc h more computationally exp ensiv e and lac k ed the long in tegration times needed to analyze the imp ortan t lo w-frequency dynamics that c haracterize separated STBLIs. In this first part of this w ork, the effects of panel flexibilit y on a sho c k/turbulen t-b oundary-la y er in ter- action at Mac h 3 with a flo w deflection angle θ = 17.5 ◦ and Mac h 4 with flo w deflection angles θ = 20 ◦ and θ = 15 ◦ for ha v e b een c haracterized, comparing rigid and flexible panel configurations. The in v estigation is conducted b y p erforming high-fidelit y n umerical sim ulations that replicate the conditions tested in previous exp erimen ts b y Daub et al. ( 2016 ). Comparisons of panel deflection with exp erimen tal measuremen ts at three prob e lo cations along the panel cen terline indicate that the incorp oration of structural damping in the solid mec hanics solv er significan tly impro v es the predictions relativ e to prior sim ulation efforts b y P asquariello et al. ( 2015 ). Time signals obtained from the fron t and rear prob es are found to b e sensitiv e to small v aria- tions in the stream wise lo cation particularly for the strongest STBLI case, with M ∞ = 3 and θ max = 17.5 ◦ . After a transien t p erio d dictated b y the initial compression w edge rotation, a quasi-steady state is ac hiev ed in whic h lo w-amplitude panel oscillations induced b y the flo w STBLI dynamics and w edge micro-oscillations are attained. The exp e rimen tal deflection v alues and frequencies are repro duced with reasonable accuracy b y 108 all three sim ulation cases. The effects of three-dimensionalit y induced b y the panel deformation are assessed b y conducting a sim ulation for the case with M ∞ = 3 and θ max = 17.5 ◦ , considering the full span of the panel and comparing the results to those of a sim ulation with span wise-p erio dic reduced domain. The full- span sim ulation rev eals that the three-dimensional effects are noticeable in the panel deflection, esp ecially near the side b oundaries, but are nearly negligible in the flo w w all quan tities (e.g., friction, temp erature, and pressure), making span wise-p erio dic sim ulations suitable to study for this particular configuration. The effects of panel flexibilit y on the STBLI are c haracterized, finding a significan tly larger separation bubble, alterations to the w all pressure and skin friction co efficien t, as w ell as the lo w-frequency motions, particularly for the strongest STBLIs, compared with rigid-panel sim ulations. In the second part of this w ork, the effects of w edge oscillation on (b oth rigid and flexible panel) STBLI are n umerically studied using the same test conditions for the previous M ∞ = 3 incoming TBL. The w edge angle is oscillated b et w een θ min = 15.5 ◦ and θ max = 17.5 ◦ at a constan t frequency f w that ranges from 50 Hz to 800 Hz in a logarithmic fashion. In the flexible panel case, resonance is observ ed as the forcing frequency of the w edge approac hes the primary natural frequency of the panel f n . This resonance causes a sligh t, 10% increase in the amoun t of cyclic loading on the panel, in comparison to the rigid baseline. As w edge the oscillation frequency f w increases, the v olume of fluid inside the flo w separation bubble, V b , struggles to k eep up with the rapid temp oral c hanges to the w edge angle θ . Resonance on the flexible panel resulted in an appro ximately 50% drop in the separation bubble v olume fluctuations when going from f w = 200 Hz to f w = 225 Hz Analysis of time dela ys rev ealed that the separation bubble for b oth the rigid and the flexible panel to ok nearly 3 times longer to resp ond to c hanges in the w edge angle than the w all pressure p w . Giv en the recen t renew ed in terest in scramjet engine design and STBLIs in teracting with w alls inside them, the prop osed FSI sim ulation metho dology could b e used to study h yp ersonic flo ws o v er flexible com- pression ramps and activ e con trol surfaces (sub ject to oscillations) in future researc h. 109 App endix A Damping ratio ζ estimate from measured signal data In order to determine the damping ratio, ζ , and natur al frequency , f n , of the panel displacemen t, Y s , for a particular signal, the first step in v olv es selecting the relativ e minima and maxima p oin ts. T o calculate the natural frequency , f n , the n um b er of maxim um cycles, N maxima 1 , is divided b y the difference in time, ∆T maxima , b et w een the last maxim um p oin t and the first maxim um p oin t: f n,maxima = (N maxima 1)/∆T maxima (A.1) This same logic applies to the the minima p oin ts, suc h that the natural frequency f n is the a v erage b et w een the t w o estimates: f n = 1 2 (f n,minima +f n,maxima ) (A.2) Next, a p olynomial curv e fit is applied to the relativ e minima and to the maxima as sho wn in figure A.1 . The t w o fits Y s,max−fit and Y s,min−fit are then subtracted from eac h other to obtain a p olynomial fit of oscillation amplitude v ersus time. The ev aluated p oin ts along the p olynomial fit are then con v erted to an exp onen tial fit Ae −bt : Ae −bt =Y s,max−fit Y s,min−fit (A.3) Using the approac h in Rao ( 2005 ), the damping ratio ζ is directly prop ortional to the fitted rate of deca y b and the panel natural frequency f n . ζ = b 2πf n (A.4) 110 Figure A.1: Illustration of damping ratio ζ and natural frequency f n calculation using relativ e extrema B In viscid top b oundary condition calculation T o calculate the in tersection of the oblique sho c k with the top b oundary (x ISI ), the lo cation of the w edge fron t p oin t (x front , y front ) m ust b e calculated using x front =x rot L front−rot cos(θα/2) (B.1) y front =y rot +L front−rot sin(θα/2) (B.2) A sc hematic of w edge used in the sim ulation is sho wn in more detail in figure B.1 . In order to calculate the flo w conditions do wnstream of the inciden t oblique sho c k, it is useful to start with the θ -β -M relation (see Anderson , 1990 ) whic h establishes an implicit relationship for the sho c k angle, β , as a fun ction of the deflection angle, θ , and the Mac h n um b er, M , upstream of the sho c k tanθ = 2cotβ M 2 1 sin 2 β1 M 2 1 (γ +cos(2β))+2 . (B.3) 111 Figure B.1: W edge geometry Once β is determined, the upstream Mac h n um b er of the flo w in the direction normal to the oblique sho c k can b e obtained, and the thermo dynamic and kinematic v ariables (p , T , u , andv ) do wnstream of the oblique sho c k are calculated using normal-sho c k relations. p 2 p 1 = 1+ 2γ γ +1 M 2 1 sin 2 β1 (B.4) ρ 2 ρ 1 = (γ +1)M 2 1 sin 2 β (γ1)M 2 1 sin 2 β +2 (B.5) T 2 T 1 = p 2 p 1 ρ 1 ρ 2 (B.6) M 2 = 1 sin(βθ) v u u t 1+ γ−1 2 M 2 1 sin 2 β γM 2 1 sin 2 β γ−1 2 ! (B.7) µ(M) = arcsin 1 M (B.8) u 1 =M 1 p γRT 1 =u ∞ (B.9) v 1 = 0 (B.10) u 2 =M 2 p γRT 2 cosθ (B.11) v 2 =M 2 p γRT 2 sinθ (B.12) 112 Once the w edge fron t p oin t lo cation is kno wn, to calculate the in tersection of the first Mac h w a v e of the PME with the top b oundary (x PME ), the lo cation of the w edge rear tip (x rear , y rear ) is calculated using y d =tan(α/2)L front−rear (B.13) x d =L rot−rear (B.14) θ d = arctan2(y d ,x d ) (B.15) θ ′ =θ d θ+α/2 (B.16) x rear =x rot + q x 2 d +y 2 d cosθ ′ (B.17) y rear =y rot + q x 2 d +y 2 d sinθ ′ (B.18) As the flo w expands mo ving in the do wnstream direction across the Prandtl-Mey er expansion generated at the w edge rear corner, the Mac h n um b er of the flo w increases. The p oin t of in tersection b et w een the inciden t sho c k and the top b oundary is then calculated as x ISI = y front L y tanβ +x front (B.19) Lik ewise, the in tersection of the fron t Mac h w a v e of the Prandtl-Mey er expansion (PME) and the top b oun dary is giv en b y x PME =x rear + y rear L y tan(µ(M 2 )+θ) (B.20) The (in viscid) inciden t sho c k impingemen t p oin t in tersects the w all (b ottom b oundary condition) at a stream- wise lo cation calculated as x I = y front tanβ +x front (B.21) T o find the Mac h n um b er M(x) along the top b oundary as a function of stream wise co ordinate in the Prandtl-Mey er expansion region, equation B.20 is mo dified to accoun t for the turning of the flo w. The Mac h angle function (equation B.8 ) and the Prandtl-Mey er function ν(M) ν(M) = r γ +1 γ1 arctan r γ1 γ +1 (M 2 1)arctan p M 2 1 (B.22) pro v ide for the follo wing implicit relation: µ(M(x))+θν(M(x)) = arctan y rear L y xx rear (B.23) 113 x inlet x<x ISI x ISI x<x PME x PME xx outlet M(x,L y ) M ∞ M 2 (equation B.7 ) M(x) (equation B.23 ) p(x,L y ) p ∞ p 2 (equation B.4 ) p(x) (equation B.26 ) T(x,L y ) T ∞ T 2 (equation B.6 ) T(x) (equation B.27 ) u(x,L y ) u ∞ u 2 (equation B.11 ) u(x) (equation B.28 ) v(x,L y ) 0 v 2 (equation B.12 ) v(x) (equation B.29 ) w(x,L y ) 0 0 0 T able B.1: Summary of top b oundary conditions Once the Mac h n um b er M(x) at the sp ecific x lo cation is found, the pressure p and temp erature T are found using isen tropic relations p 0 =p 2 1+ γ1 2 M 2 2 γ γ−1 (B.24) T 0 =T 2 1+ γ1 2 M 2 2 (B.25) p(x) = p 0 1+ γ−1 2 M(x) 2 γ γ−1 (B.26) T(x) = T 0 1+ γ+1 2 M(x) 2 (B.27) where subscript 0 is used to denote stagnation v ariables, and v ariables without a subscript corresp ond to the static coun terparts. A cross the expansion, the flo w exp eriences a c hange in direction. The stream wise u and w all-normal v comp onen ts are found b y u(x) = cos θν(M(x))+ν(M 2 ) M(x) p γRT(x) (B.28) v(x) =sin θν(M(x))+ν(M 2 ) M(x) p γRT(x) (B.29) where x PME xx outlet . T able B.1 summarizes the equations used to calculate the flo w quan tities in eac h of the three regions in whic h the top b oundary condition is split. F or the t w o Mac h n um b er conditions sim ulated as giv en in table 2.1 , surface plots of the top b oundary conditions of p , T , u , and v as a function of the stream wise co ordinate x and w edge deflection angle θ are giv en in figure 3.4 . 114 C Flo w c haracterization on span wise-normal cen ter plane This app endix includes figures with con tours of additional time-a v eraged flo w quan tities on the cen terplane (z = 0 ) extracted from the span wise-p erio dic reduced-span sim ulations of STBLIs o v er rigid and flexible panel configurations, complemen ting those figures sho wn in section 4.1.5 . (1) (2) (3) Figure C.1: Cen terplane comparison of time-a v eraged temp erature T/T ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 115 (1) (2) (3) Figure C.2: Cen terplane comparison of temp erature v ariance T ′ T ′ /T 2 ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 116 (1) (2) (3) Figure C.3: Cen terplane comparison of time-a v eraged v ertical v elo cit y v ariance (v ertical Reynolds stress), v ′ v ′ /u 2 ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 117 (1) (2) (3) Figure C.4: Cen terplane comparison of time-a v eraged Reynolds shear stress,u ′ v ′ /u 2 ∞ , zo omed in to the region directly ab o v e the panel (1) M ∞ = 3.0 , θ max = 17 ◦ , (2) M ∞ = 4.0 , θ max = 20 ◦ , (3) M ∞ = 4.0 , θ max = 15 ◦ 118 Bibliograph y Anderson, John D a vid 1990 Mo dern c ompr essible flow: with historic al p ersp e ctive , , v ol. 12. 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Creator Hoy, Jonathan (author)

Core Title Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2022-12

Publication Date 08/16/2022

Defense Date 07/29/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag finite element method,fluid structure interaction,OAI-PMH Harvest,shock turbulent boundary layer interaction,wall-modelled large eddy simulation

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Bermejo-Moreno, Ivan (committee chair), Nakano, Aiichiro (committee member), Pantano-Rubino, Carlos (committee member)

Creator Email hoyj@usc.edu,jonhoy34@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111376963

Unique identifier UC111376963

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Document Type Dissertation

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Rights Hoy, Jonathan

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Source 20220819-usctheses-batch-974 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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