Proceedings of the 15th ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM15)

Proceedings of the 15 th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM15) Editors: Stefan Hickel & Maria Vittoria Salvetti doi: 10.5281/zenodo.17280519 22-24 September 2025 Dubrovnik, Croatia ETMM15 PROCEEDINGS C ONTENT S KEYNOTE SPEAKERS 7 1 IMPOSING NEAR -WALL SCALING C ONSTRAINT S IN MACHINE- LEARNED S YMBOLIC SUBGRID- SCALE CL OSURES 12 2 A NEW FORMA LISM FOR IMPROV ING PREDICTIONS WITH R ANS EDD Y VISC OSITY MODEL S 18 4 TURBULENT DRAG REDUC TION IN WA TER -L UBRICA TED CHANNEL FL OW OF HIGHL Y VISC OUS OIL 21 5 NOISE REDUC TION OF A CIRCULA TION-C ONTROLLED WING WITH POROUS INSER T 27 8 TURBULENT FL OW O VER A PERMEABLE WALL UNDER A R ANGE OF REYNOLDS NUMBER S 33 9 RANS MUL TI- SCALE TURBULENCE MODELLING AND IT S APPLICA TION TO HIGH FREES TREAM TURBULENT BOUNDARY LAY ER S 39 10 EFFEC T OF TURBULENT INFL OW C ONDITIONS ON THE D YNAMIC S OF A 3D SUPER SONIC CA VIT Y AS YMMETRIC FL OW 45 12 IMP ACT OF WA LL TEMPERA TURE ON TRANSONIC GAS TURBINES VA NE FL OW D YNA MIC S: A WALL -MODELED LES APPROA CH 51 13 WALL C OOLING EFFECT S ON ENTROPY PRODUC TION IN SHOCK- BOUNDARY LAY ER INTERAC TIONS 57 14 NUMERICAL INVES TIGA TION ON FREE-F ALLING DROPLET S: LIMIT A TIONS AND GAP S FOR IMPRO VEMENT 63 16 CFD MODELLING OF THE HYDROD Y NAMIC S IN AN UNBAFFLED V ORTEX REA CT OR 71 17 MERGING FIL TERING, MODELING AND DISCRETIZA TION TO SIMULA TE LARGE EDDIES IN BURGER S’ TURBULENCE 77 19 IMP ACT OF WIND DIREC TION ON FL OW OVER A REALIS TIC URBAN AREA: A LARGE-EDD Y SIMULA TION S TUD Y 83 20 HIGH-ORDER SY MMETRY -PRESERVING DISCRETIZA TIONS: APPLICA TION T O REPEA TED MA TRIX BL OCK S TRUCTURES 89 21 USING IN- HOUSE FRAMEW ORK UNIC ONES ON TRANSIENT FL OW SIMULA TIONS 95 23 EXPERIMENT AL METHODS ON THE C OLLEC TION OF MONO-C OMPONENT FREEL Y F ALLING DROPLET S 100 24 A C ONSISTENT APPRO ACH T O WALL MODELLING F OR THE P ARTIALL Y -AVERA GED NAVIER -S TOKES METHOD 106 25 T OWARDS EXA- SCALE SIMULA TIONS OF TURBOMACHINERY C OMPONENT S 112 26 C OMBINED EFFEC T S OF PRESSURE GRADIENT A ND ROUGHNESS IN CHANNEL FL OWS 118 27 AGGL OMERA TION D YNAMIC S OF NON-SPHERICAL NANOP ARTICLES IN HOMOGENEOU S ISO TROPIC TURBULENCE 124 28 AC CURA CY AND C ONSISTENCY OF LA TTICE BOL T ZMANN METHODS FOR TURBULENT CHANNEL FL OWS A T Reτ = 180 130 29 IMP ACT OF ASPEC T RA TIO ON THE C OLLISION OF NON- SPHERICAL ELLIPSOID P AR TICLES IN TURBULENT CHANNEL FL OW 136 30 DNS S TUD Y OF C ORIOLIS AND CENTRIFUGAL FORCES IN CANONICA L BOUNDARY LAY ER FL OW 142 32 T OWARDS THE USAGE OF GEOMETRY A GNOS TIC β- VARIA TIONAL -AUT OENC ODERS F OR MODEL ORDER REDUC TION OF TURBULENT FL OWS 148 33 NUMERICAL S TUD Y OF THE SETTLING HY DROD YNAMIC S OF VARIOUS SHA PES OF SOLID P ARTICLES 154 34 AN ENHANCED GENERALISED MUL TIPHASE MODELLING APPROACH F OR SL UG FL OW BOILING 160 35 NUMERICAL S TUDIES OF HYPER SONIC MUL TI-S T A TE FL OW P AST A C ONE-SLICE-R AMP BY TR -IDDES - SPOM 166 36 SCALAR DISPER SION NEAR AN AC TIVE WALL A T HIGH REYN-OLDS NUMBER S USING 2D- PIV AND TRA CER PLIF 172 37 ASSES SING THE IMP AC T OF EXCHANGE L OCA TION HEIGHT ON HIGH-ORDER WALL -MODELED LES 178 38 EXPERIMENT AL INVES TIGA TION OF TURBULENCE EFFECT ON CAMBERED NA CA AIRF OIL S 184 39 PREDIC TION OF HIGH-DENSITY P ARTICLE-LADEN CHA NNEL FL OWS U SING AR TIFICIAL NEURAL NETWORK S 190 40 BOUNDARY LAY ER TRANSITION MECHANISMS IN A HIGH-SPEED L OW-PRES SURE TURBINE BLADE 196 2 41 TURBULENCE CL OSURE FOR THE C OMPRESSIBLE RANS EQU A TIONS ASSIS TED B Y FIELD-INVER SION & MA CHINE- LEARNING 202 42 WALL -MODELED LARGE EDD Y SIMULA TIONS FOR OP TICAL TURRET S 208 44 ENERGY TRA NSFER MECHANISMS IN C OMPRES SIBLE TWO-PHASE TURBULENT FL OWS 214 45 NUMERICAL SIMULA TION OF FREE-OSCILLA TION BL UNT C ONE C OUPLED WITH HYPERSONIC TR ANSITION 218 46 AN SURROGA TE-INFORMED SP AR SE-GRID APPROA CH FOR FLASHBACK PREDIC TION IN H2-FUELED PERF ORA TED BURNERS 223 47 EXPERIMENT AL INVES TIGA TION ON DY NAMIC S T ALL OF SY MMETRIC AIRFOIL S IN HARMONIC MOTION 229 48 C OMP ARISON OF BI–DISPERSE A ND MONO–DISPER SE FL OW WITH HIGH S T OKES NUMBER IN HORIZ ONT AL ISO THERMAL PIPES 235 50 ENSEMBLE-BASED D A T A AS SIMILA TION OF PIV DA T A F OR TURBULENT FL OW P AST A SURF ACE-MOUNTED CUBE 241 51 WALL -RESOL VED LARGE EDD Y SIMULA TIONS OF A PITCHING NA CA0012 IN DEEP D YNA MIC ST ALL 247 52 HYPERREDUC TION IN A METHOD C OUPLING ST OCHASTIC REDUCED MODEL S AND D A T A AS SIMILA TION TO PREDIC T TURBULENT FL OWS 253 53 REYNOLDS NUMBER EFFEC T S ON SHOCK -WA VE/TURBULENT BOUND ARY -LA YER INTERA C TIONS OVER RIDGE- TY PE ROUGHNESS 256 54 DA T A ASSIMILA TION AND UNCERT AINTY QUANTIFICA TION FOR THE R ANS PREDIC TION OF A SEP ARA TED FL OW 262 55 C OMPUT A TIONAL EVIDENCE QUESTIONING THE LA RGE- T O-SMALL BREA KUP CASCADE IN LIQUID JET A TOMIZ A TION: DNS STUD Y A ND IMPLICA TIONS FOR MODELING 268 56 GENERALIZED DEEP LEARNING MODEL F OR PREDIC TING DRAG REDUC TION IN PUL SA TING TURBULENT PIPE FL OW WITH ARBITR ARY A C CELERA TION AND DECELERA TION 274 57 LES-ADM AC C URAC Y AS SESSMENT VIA AN ERROR -LANDSCAPE APPROA CH 280 58 USING PHY SIC S INFORMED NEUR AL NETWORK (PINN) T O IMPRO VE A k − ω TURBULENCE MODEL 286 59 IMP ACT OF FEA THER -INSPIRED SURF A CE STRUC TURES TO AN AIRF OIL BOUNDA RY LA YER 292 60 WALL -PRESSURE BASED S TOCHAS TIC ES TIMA TION OF VEL OCITY FL UCTU A TIONS IN HIGH REYNOLDS NUMBER PIPE FL OW 298 61 C OMP ARA TIVE EXPERIMENT AL STUD Y OF FLAME-WALL INTERA C TION FOR METHANE AND HY DROGEN 304 62 L ONG HEA VY FIBER S IN WALL TURBULENCE - EFFEC T S OF FLEXIBILITY , LENG TH AND FL UID INERTIA 310 63 USE OF GENETIC EXPRES SION PROGRAMMING F OR INFERRING ROUGHNESS C ORRELA TIONS FROM A DNS DA T ABASE 313 64 INTRODUC TION OF A FL OW PENETRA TION METRIC FOR EV AL UA TING THE IMP AC T OF BUILDING C O VERING VEGET A TION ON WIND-DRIVEN NA TURAL VENTILA TION OF INDOOR SP ACES 319 65 GL OBAL DRA G OF HETEROGENEOUS SURF ACES WITH V ARYING REY NOLDS NUMBER 325 66 THREE-DIMENSIONAL V ARIA TIONAL DA T A ASSIMILA TION OF LIMITED EXPERIMENT AL DA T A 331 67 ROLE OF DISCRETIZA TION ERROR DURING TRAINING OF MACHINE- LEARNED TURBULENCE MODEL S 337 68 TURBULENCE CHARA CTERIZ A TION IN COMPRES SOR T ANDEM BLADES AEROD YNA MIC S 342 69 SIMUL T ANEOUS OP TIMIZA TION OF AC TU A TOR PLACEMENT A ND C ONTROL POLICY FOR A CTIV E FL OW C ONTROL 348 70 C OMP ARA TIVE ASSES SMENT OF SCALE ADA P TIVE TURBULENCE MODEL S FOR AEROD YNAMIC UNCER T AINTY QUA NTIFICA TION 351 71 TRANSIENT V ORTEX -INDUCED VIBRA TIONS ON A SQUA RE CYLINDER UNDER A C CELERA TING FL OW C ONDITIONS 357 72 DNS OF TURBULENT NA TURAL C ONVEC TION FL OW S IN A DIFFERENTIALL Y HEA TED CAVIT Y WITH C ONJUGA TE HEA T TRANSFER 363 73 LINEAR OR ANNULAR CASCADE? T OWARDS UNDERS T ANDING THE IMP AC T OF SIMPLIFIED RESEARCH C ONFIGURA TIONS IN TURBO MACHINERY 369 74 EV A L U A TION OF BL O WING A ND SUC TION C ONTROL S T O DEV EL OPING A ND DECA Y ING PROCES SES OF S TREA MWISE V OR TEX 374 76 A N EMBEDDED Z ONA L LES METHOD A PPLIED F OR PREDIC TIONS IN TURBOMA CHINER Y A PPLICA TIONS 380 78 DEV EL OPMENT OF A NO V EL S TR A TEGY F OR EMBEDDED LES BASED ON C ONTINUOU S HY BRID R A NS/LES METHODS 386 79 NEA R - WA LL NUMERICA L C OHERENT S TRUC TURES IN WA LL - MODELED L A RGE- EDD Y SIMUL A TION 392 80 A PPLICA TION OF RIBLET S T O SEP A R A TING TURBULENT BOUND A R Y L A Y ER S 396 TURBULENT F ORCED C ONV EC TION O V ER POROU S L A TTICES 402 81 S T A TIS TICA L C ONV ERGENCE OF SELEC TED QU A NTITIES IN CA NONICA L WA LL TURBULENCE 408 82 REL A TION BETWEEN 3D A ND 2D WRINKLING F A C T OR S IN TURBULENT PREMIXED FL A MES 414 84 A MUL TI- FIDELIT Y D A T A AS SIMIL A TION A L GORITHM ENHA NCED B Y C ONV OL UTIONA L NEUR A L NETW ORK S 420 85 TUNING OF A GGREG A TED TURBULENCE MODEL F OR SEP A R A TED FL O W S FROM DIFFERENT D A T ASET S 426 86 GENER A LIZED A PPRO XIMA TE BA Y ESIA N INFERENCE WITH A N EQUIV A RIA NT NEUR A L OPER A T OR FR A MEW ORK 432 87 T O WA RDS A UNIFIED TURBULENCE MODEL THROUGH MUL TI-OB JEC TIV E LEA RNING 434 88 V ISC OU S DROP LINEA R S T A BILIT Y A NA L Y SIS 437 89 DIREC T NUMERICA L SIMUL A TIONS OF TURBULENT DR A G- REDUC TION V IA PIEZ OELEC TRIC A C TU A TION 440 90 SIMUL A TIONS OF ZERO PRES SURE-GR A DIENT TURBULENT BOUND A R Y L A Y ER S O V ER RIBLET S 446 91 SPEC TR A L LES MODELLING F OR P AS SIV E SCA L A R WITH PHASE REL A X A TION TIME IN ISO TROPIC TURBULENCE 452 92 BI- FIDELIT Y GENE EXPRES SION PROGR A MMING F OR R A NS MODELLING 458 93 A GGREG A TED A ND HY BRID MODEL REDUC TION F OR TURBULENT FL O W S ENHA NCED B Y SCIENTIFIC MA CHINE LEA RNING 464 94 LES OF FL O W A ROUND SMOO TH A ND DIMPLED SPHERES IN THE SUPERCRITICA L REGIME: A C OMP A R A TIV E S TUD Y OF THE NON- RO T A TING A ND RO T A TING CASES 466 95 QU A NTIF Y ING THE SCA LE EFFEC T IN HY DROGEN/ A IR EXPL OSIONS U SING HIGH- FIDELIT Y SIMUL A TIONS WITH DET A ILED CHEMIS TR Y 471 97 TURBULENCE A NISO TROP Y CHA R A C TERIZ A TION IN ROUND JET IMPINGEMENT : A DNS A ND EDD Y - RESOL V ING R A NS - R SM S TUD Y 477 98 HIGH- PERF ORMA NCE C OMPUTING SIMUL A TIONS OF BUBBLE- L A DEN TURBULENCE: DEP A R TURES FROM K OLMOGORO V SCA LING 483 99 BUBBLE BREA K UP IN HOMOGENEOU S ISO TROPIC TURBULENCE 486 101 NUMERICA L INV ES TIG A TION OF DET ONA TION D Y NA MIC S IN A N EXPERIMENT A L PUL SE- WA V E GENER A T OR 488 102 C OMP A RISON OF CA MER A CA LIBR A TION METHODS F OR P A R TICLE TR A CKING V EL OCIMETR Y 494 104 DEV EL OPMENT OF G A NS - BASED WA LL MODEL F OR L A RGE EDD Y SIMUL A TION OF WA LL - BOUNDED FL O W 497 106 THE CA P A CIT Y OF RIBLET S T O A L TER THE SURF A CE PRES SURE OF A N A XIA L C OMPRES SOR BL A DE 501 108 PREDIC TION OF EX TREME EV ENT S IN TURBULENT SIGNA L S OF WA LL - BOUNDED FL O W S 507 110 U SING A S T OCHAS TIC BA CK SCA TTER MODEL F OR WA LL - MODELLED L A RGE EDD Y SIMUL A TION 513 111 PHY SIC S - BASED L OCA LIZ A TION METHODOL OGY F OR D A T A AS SIMIL A TION WITH ENSEMBLE K A LMA N FIL TER 519 113 EXPERIMENT S OF MICROC ONFINED HIGH- PRES SURE TR A NSCRITICA L FL UID TURBULENCE 525 114 CHA R A C TERIZ A TION OF THE TIP LEA K A GE V OR TEX A ND C ORNER SEP A R A TION IN A C OMPRES SOR CASCA DE A T NEA R - S T A LL C ONDITIONS 531 115 DEV EL OPING D A T A- DRIV EN NEA R - WA LL PRES SURE- S TR A IN C ORREL A TIONS F OR ELLIP TIC DIFFERENTIA L REY NOLDS - S TRES S MODEL S 537 116 P A R TICLE- RESOL V ED DNS OF TURBULENT SL URR Y FL O W S IN HORIZ ONT A L PIPES 543 3 74 EVA L U A TION OF BLOWING A ND SUCTION C ONTROL S T O DEVEL OPING AND DECA YING PROCES SES OF S TREAMWISE VOR TEX 374 76 AN EMBEDDED Z ONAL LES METHOD APPLIED F OR PREDICTIONS IN TURBOMA CHINERY A PPLICA TIONS 380 78 DEVEL OPMENT OF A NO VEL S TRA TEGY F OR EMBEDDED LES BASED ON C ONTINUOUS HYBRID RA NS/LES METHODS 386 79 NEAR -WALL NUMERICAL C OHERENT STRUC TURES IN WALL -MODELED LARGE-EDD Y SIMULA TION 392 80 APPLICA TION OF RIBLET S TO SEP ARA TING TURBULENT BOUNDARY LAYER S 396 TURBULENT FORCED C ONVECTION O VER POROUS LA TTICES 402 81 S T A TIS TICAL C ONVERGENCE OF SELEC TED QU ANTITIES IN CANONICAL WALL TURBULENCE 408 82 RELA TION BETWEEN 3D AND 2D WRINKLING F AC TOR S IN TURBULENT PREMIXED FLAMES 414 84 A MUL TI-FIDELIT Y DA T A ASSIMILA TION AL GORITHM ENHANCED BY C ONVOL UTIONAL NEURAL NETWORKS 420 85 TUNING OF AGGREGA TED TURBULENCE MODEL FOR SEP ARA TED FL OWS FROM DIFFERENT D A T ASET S 426 86 GENERALIZED APPRO XIMA TE BAY ESIAN INFERENCE WITH AN EQUIV ARIANT NEURAL OPER A TOR FRA MEWORK 432 87 T OWARDS A UNIFIED TURBULENCE MODEL THROUGH MUL TI-OBJEC TIVE LEARNING 434 88 VISC OUS DROP LINEAR S T ABILITY ANAL YSIS 437 89 DIREC T NUMERICAL SIMULA TIONS OF TURBULENT DRAG-REDUC TION VIA PIEZ OELEC TRIC AC TU A TION 440 90 SIMULA TIONS OF ZERO PRESSURE-GRADIENT TURBULENT BOUND ARY LA YER S OV ER RIBLET S 446 91 SPEC TRAL LES MODELLING FOR P ASSIVE SCA LAR WITH PHASE RELAXA TION TIME IN ISO TROPIC TURBULENCE 452 92 BI-FIDELIT Y GENE EXPRES SION PROGRAMMING F OR RANS MODELLING 458 93 AGGREGA TED AND HY BRID MODEL REDUCTION F OR TURBULENT FL OWS ENHA NCED BY SCIENTIFIC MACHINE LEARNING 464 94 LES OF FL OW AROUND SMOO TH AND DIMPLED SPHERES IN THE SUPERCRITICAL REGIME: A C OMP ARA TIVE STUD Y OF THE NON-RO T A TING AND RO T A TING CASES 466 95 QUA NTIFYING THE SCALE EFFEC T IN HY DROGEN/ AIR EXPL OSIONS USING HIGH-FIDELIT Y SIMULA TIONS WITH DET AILED CHEMISTR Y 471 97 TURBULENCE ANISO TROPY CHARA C TERIZA TION IN ROUND JET IMPINGEMENT: A DNS A ND EDDY -RESOL VING RANS-RSM S TUDY 477 98 HIGH-PERF ORMANCE C OMPUTING SIMULA TIONS OF BUBBLE-LADEN TURBULENCE: DEP ARTURES FROM K OLMOGOROV SCALING 483 99 BUBBLE BREAK UP IN HOMOGENEOUS ISO TROPIC TURBULENCE 486 101 NUMERICAL INVES TIGA TION OF DET ONA TION DY NAMIC S IN AN EXPERIMENT AL PUL SE-WA VE GENERA T OR 488 102 C OMP ARISON OF CAMERA CALIBRA TION METHODS FOR P AR TICLE TRACKING VEL OCIMETRY 494 104 DEVEL OPMENT OF GANS-BASED WALL MODEL F OR LARGE EDD Y SIMULA TION OF WALL -BOUNDED FL OW 497 106 THE CAP ACITY OF RIBLET S TO A L TER THE SURF ACE PRES SURE OF AN AXIAL C OMPRESSOR BLADE 501 108 PREDIC TION OF EXTREME EV ENT S IN TURBULENT SIGNALS OF WA LL -BOUNDED FL OWS 507 110 USING A S TOCHAS TIC BA CKSCA TTER MODEL FOR WALL -MODELLED LARGE EDD Y SIMULA TION 513 111 PHYSIC S -BASED L OCALIZA TION METHODOLOGY FOR D A T A AS SIMILA TION WITH ENSEMBLE KALMAN FIL TER 519 113 EXPERIMENT S OF MICROC ONFINED HIGH-PRES SURE TRANSCRITICAL FL UID TURBULENCE 525 114 CHARA CTERIZ A TION OF THE TIP LEAKAGE V ORTEX AND C ORNER SEP ARA TION IN A C OMPRES SOR CASCADE A T NEAR -S T ALL C ONDITIONS 531 115 DEVEL OPING D A T A-DRIVEN NEAR -WALL PRES SURE- S TRAIN C ORREL A TIONS FOR ELLIP TIC DIFFERENTIAL REYNOLDS-S TRESS MODEL S 537 116 P ARTICLE-RESOL VED DNS OF TURBULENT SL URRY FL OWS IN HORIZ ONT AL PIPES 543 4 117 SHAPE OP TIMIZA TION OF DRAG- REDUCING SURF ACE IN TURBULENT FL OW B Y AD JOINT METHOD 547 118 WALL -RESOL VED LARGE EDD Y SIMULA TION OF P ARTICLE-INDUCED EROSION D AMAGE U SING EULERIAN- LAGRANGIA N MP -PIC METHOD 552 119 AN UNC ONDITIONALL Y S T ABLE, ENERGY PRESERVING METHOD F OR MAGNET OHYDROD YNAMIC S 558 120 FUEL EFFEC T S ON TURBULENT ST A TISTIC S IN REACTING TURBULENT CHANNEL FL OW 564 121 MIXED DA T A-SOURCE TRA NSFER -LEARNING F OR A THREE-DIMENSIONAL TURBULENCE MODEL A UGMENTED PHYSIC S -INFORMED NEUR AL NETWORK 569 123 DISC OV ERING FL OW SEP ARA TION C ONTROL STR A TEGIES IN 3D WINGS VIA DEEP REINFORCEMENT LEARNING 575 124 EFFEC T S OF PERFORA TED PLA TE ROUGHNESS ON SKIN FRIC TION DRA G IN TURBULENT BOUNDARY LAY ERS 581 126 TWO-DIMENSIONA L NUMERICAL SIMULA TION STUD Y OF MUL TICOMPONENT MIXING IN A TRANSCRITICAL SHEAR LAY ER: APPLICA TION AND PERFORMANCE ANA L Y SIS OF A T ABULA TED EQUA TION OF ST A TE APPROA CH 586 128 LES OF MUL TI-ARMED VARIA BLE DENSITY JET S 592 129 T OWARDS REDUCED KINETIC MODEL S FOR BURNER - S T ABILIZED FLAMES WITH FLAME RET ARDANT S 598 130 IMP ACT OF HIGH- PRESSURE TRA NSCRITICAL THERMOD YNAMIC S ON TURBULENCE GENERA TION 604 131 DA T A-DRIVEN ENHANCEMENT S T O TRANSITION AND TURBULENCE MODELING UNDER V ARY ING PRESSURE GRADIENT S AND UNS TEADINES S EFFECT S 610 132 BY P ASS TRANSITION ON A ROUGH- RIBBED SURF ACE: EFFECT S OF FREE-S TREAM TURBULENCE 616 134 NUMERICAL SIMULA TION OF WA TER -IN- FUEL EMUL SION JET 622 135 TURBULENCE DECAY IN P ARTICLE-LADEN FL OWS 628 136 HEA T -AND MOMENTUM- TRANSFER SP A TIO- TEMPORAL FEA TURES IN IMPINGING JET S USING L SE-HPOD 632 137 A HIGH-FIDELIT Y NUMERICA L DA T ABASE FOR FREE- STREA M TRANSITION 638 138 TURBULENCE IN THE NEAR -WAKE REGION OF A FREEL Y OSCILLA TING CYLINDER 644 139 TRACKING A ND STEREO- MA TCHING P ARTICLE CL OUDS IN TURBULENCE FOR DISPER SION STUDIES 647 140 A MIMETIC FINITE VOL UME METHOD ON C OLL OCA TED GRIDS FOR INC OMPRESSIBLE FL OWS 650 141 HOW THE TURBULENT S TRUC TURES BEHIND BUBBLES INFL UENCE THE MAS S TRANSFER T O THE BULK LIQUID 656 142 BAY ESIAN OP TIMIZA TION OF THE S TRUC TURAL L O AD OF MUL TI-S TEP CY LINDERS 659 143 THE EFFEC T OF NUCLEA TION AND C ONDENSA TION MODELING ON THE PREDICTION AC CURA CY IN THE MIXING OF TWO JET S USING AN EULER -LAGRANGE A PPROA CH 665 144 EVA L U A TION OF A NEW EXPLICIT WALL -FUNC TION FORMULA TION FOR LARGE EDD Y SIMULA TION 671 145 DA T A-DRIVEN RANS CL OSURES USING A RELA TIVE IMPORT ANCE TERM ANAL YSIS-BASED Z ONAL APPRO ACH FOR 2D AND 3D SEP ARA TED FLO WS 677 146 A POR T ABLE AL GEBRAIC IMPLEMENT A TION FOR RELIABLE INDUS TRIAL LES 683 147 UNCERT AINTY ANAL YSIS OF URANS SIMULA TIONS C OUPLED WITH AN ANISO TROPIC PRESSURE FL UC TU A TION MODEL 688 148 ENFORCING ENTROP Y C ONSERVA TION IN THE NUMERICAL SIMULA TION OF C OMPRES SIBLE FL OWS F OR THERMALL Y PERFECT G AS MODEL S 694 150 MUL TIFIDELITY APPROA CH USING DA T A ASSIMILA TION FOR A TMOSPHERIC REENTRY C OMPUT A TION 699 151 DIREC T NUMERICAL SIMULA TION OF INERTIAL P ARTICLE TRANSPOR T IN S T ABL Y S TRA TIFIED TURBULENT BOUNDARY LAY ER S 705 152 SIMULA TION OF TURBULENT LIQUID-SOLID PIPE FL OW U SING THE TWO-FL UID MODEL 708 153 THE EFFEC T OF DENSITY RA TIO ON FL OW REGIME AND HEA T TRANSFER IN TURBULENT WALL -BOUNDED GAS- LIQUID FL OW 714 154 EFFEC T OF VARIA BLE FL UID PROPERTIES ON THE SUBGRID- SCALE MODELLING OF TURBULENT HEA T TRANSPOR T 716 155 A HIGH-ORDER MOD A L DISC ONTINUOU S G A LERKIN SOL V ER WITH S TRUC TURE- PRESER V ING PROPER TIES F OR SCA LE- RESOL V ING SIMUL A TIONS 722 156 CHA LLENGING DEC ONV OL UTION METHODS F OR A POS TERIORI V A LID A TION OF SUBGRID SCA LE MODEL S: A PPLICA TION T O HIGH SCHMID T NUMBER FIELDS 727 157 A N A NA L Y SIS OF SPLITTING ERROR S IN PRO JEC TION- BASED IMMER SED BOUND A R Y METHODS 733 158 C OMBINING L A RGE A ND SMA LL - SCA LE SIMUL A TIONS T O DIS SEC T THE ONSET OF CL U S TERING 739 159 AS SIMIL A TING ROUGH FEA TURES 741 160 C OMPLEMENT A RIT Y OF S TEA D Y S T A TE R A NS A ND SCA LE- RESOL V ING HY BRID MODEL S F OR EX TERNA L A EROD Y NA MIC A PPLICA TIONS 747 161 MEA N FL O W SCA LING OF S T A BL Y S TR A TIFIED TURBULENT CHA NNEL FL O W S 753 162 SCA LE- RESOL V ING SIMUL A TIONS OF TR A NSITIONA L FL O W O V ER A NEW L O W - PRES SURE TURBINE BL A DE 756 163 DNS OF GR A V IT Y - DRIV EN BUBBLE SWA RMS ON INCLINED CHA NNEL S 762 164 CRITICA L AS SES SMENT OF R A NS THERMA L TURBULENCE MODEL S F OR L O W PR A ND TL NUMBER FL O W S 768 165 S T OCHAS TIC REDUCED ORDER MODEL OF DIURNA L C Y CLES 774 166 DNS - INF ORMED A D JOINT SHA PE OP TIMIZ A TION F OR HIGH PERF ORMA NCE TURBUL A T OR S 779 167 A S Y S TEMA TIC S TUD Y OF DDES A ND IDDES IN A HIGH-ORDER DISC ONTINUOU S G A LERKIN SOL V ER 783 168 L A RGE EDD Y SIMUL A TION OF FL A ME D Y NA MIC S A ND NEA R - WA LL HEA T TR A NSFER IN SOLID FUEL R A MJET S 789 170 BUBBLE PL UME D Y NA MIC S IN A WA TER C ONT A INMENT : A SENSITIZED R A NS - R SM MODELLING S TUD Y 795 171 TURBULENT HY PER SONIC FL O W O V ER A RECES SING L O W - TEMPER A TURE A BL A T OR 801 172 D A T A- DRIV EN CA LIBR A TION OF TR A NSITION MODEL S F OR NA TUR A L C ONV EC TION FL O W S 807 173 SUPER SONIC TURBULENT BOUND A R Y L A Y ER S O V ER BO A T T A IL SHA PES 813 175 FL O W O V ER S TEP -UP CA NY ONS IMMER SED IN BOUND A R Y L A Y ER S: SIMUL A TION A ND A NA L Y SIS 818 176 HIGH- FIDELIT Y SIMUL A TIONS OF A D V ER SE PRES SURE GR A DIENT FL O W O V ER A ROUNDED S TEP F OR TURBULENCE MODEL IMPRO V EMENT V IA D A T A BASE GENER A TION 824 178 MUL TIV A RIA TE MUL TIFIDELIT Y MODELLING F OR TURBULENT FL O W S 830 179 EXPERIMENT A L INV ES TIG A TION OF SEP A R A TION BUBBLE SUB JEC TED T O A C OU S TIC EX CIT A TION 833 182 U SING DNS OF DROP S IN HIT T O DEV EL OP MODEL S F OR UNRESOL V ED SCA LES IN MUL TIPHASE FL O W S 836 183 A R A TIONA L LENG TH SCA LE F OR L A RGE- EDD Y SIMUL A TIONS ON A NISO TROPIC GRIDS 842 184 A N INTERPRET A BLE D A T A- DRIV EN WA KE MODEL: A NA L OGY WITH THE K - Ε- FP F ORMUL A TION 848 185 E F F E C T O F S M A L L A N G L E O F A T T A CK O N F L O W A R O U N D T W O T A N D E M S Q U A R E B U I L D I N G S O F D I F F E R E N T H E I G H T S 854 187 SCA LE- RESOL V ING SIMUL A TION OF A TURBULENT BOUND A R Y L A Y ER UNDER A D V ER SE PRES SURE GR A DIENT 857 188 CL OSURE OF NONLINEA R FREQUENC Y - DOMA IN REDUCED-ORDER MODEL S F OR UNS TEA D Y FL O W S 859 189 A WA LL - MODELLED LES FR A MEW ORK F OR DROPLET - L A DEN TURBULENT FL O W S U SING PHY SIC S - BASED MODEL S INF ORMED B Y DNS 865 PRES SURE A ND TURBULENCE INTENSIT Y EFFEC T S ON LEA N PREMIXED HY DROGEN FL A MES UNDER HIGH S TR A IN 871 191 FL O W -AWA RE SIMUL A TIONS OF TURBULENCE (F AS T): A MA CHINE LEA RNING- BASED A PPRO A CH T O EFFICIENT TURBULENCE SIMUL A TIONS F OR ENGINEERING FL O W S 877 192 L A RGE- EDD Y SIMUL A TION OF RO T A TIONA L EFFEC T S ON BOUND A R Y L A Y ER S O V ER A N A IRF OIL 880 194 INTER SCA LE MOMENTUM TR A NSFER IN WA LL TURBULENCE 886 197 EX A MINING CHA O TIC A ND DETERMINIS TIC D Y NA MIC S F OR A I- BASED CLIMA TE C ONTROL 891 5 155 A HIGH-ORDER MODA L DISC ONTINUOUS GA LERKIN SOL VER WITH S TRUCTURE-PRESERVING PROPER TIES FOR SCALE-RESOL VING SIMULA TIONS 722 156 CHALLENGING DEC ONVOL UTION METHODS FOR A POS TERIORI VALIDA TION OF SUBGRID SCALE MODEL S: APPLICA TION T O HIGH SCHMID T NUMBER FIELDS 727 157 AN ANAL YSIS OF SPLITTING ERROR S IN PRO JECTION- BASED IMMERSED BOUND ARY METHODS 733 158 C OMBINING LARGE AND SMALL - SCALE SIMULA TIONS TO DIS SEC T THE ONSET OF CL US TERING 739 159 ASSIMILA TING ROUGH FEA TURES 741 160 C OMPLEMENT ARITY OF S TEAD Y S T A TE RANS AND SCALE-RESOL VING HYBRID MODEL S FOR EX TERNAL AEROD YNAMIC A PPLICA TIONS 747 161 MEAN FL OW SCA LING OF ST ABL Y S TRA TIFIED TURBULENT CHANNEL FL OW S 753 162 SCALE-RESOL VING SIMULA TIONS OF TRANSITIONAL FL OW OV ER A NEW L OW -PRES SURE TURBINE BLADE 756 163 DNS OF GRA VITY -DRIVEN BUBBLE SWARMS ON INCLINED CHANNEL S 762 164 CRITICAL AS SESSMENT OF RA NS THERMAL TURBULENCE MODEL S FOR L OW PRAND TL NUMBER FL OW S 768 165 S T OCHAS TIC REDUCED ORDER MODEL OF DIURNAL CY CLES 774 166 DNS-INFORMED AD JOINT SHAPE OP TIMIZA TION FOR HIGH PERF ORMANCE TURBULA TOR S 779 167 A S YS TEMA TIC STUD Y OF DDES AND IDDES IN A HIGH-ORDER DISC ONTINUOUS GALERKIN SOL VER 783 168 LARGE EDD Y SIMULA TION OF FLAME D YNAMIC S AND NEAR -WALL HEA T TRANSFER IN SOLID FUEL R AMJET S 789 170 BUBBLE PL UME D YNA MIC S IN A WA TER C ONT AINMENT: A SENSITIZED RANS-RSM MODELLING S TUD Y 795 171 TURBULENT HYPER SONIC FL OW O VER A RECES SING L OW - TEMPERA TURE ABLA T OR 801 172 DA T A-DRIVEN CALIBRA TION OF TRANSITION MODEL S F OR NA TURAL C ONVEC TION FL OWS 807 173 SUPERSONIC TURBULENT BOUND ARY LAYER S O VER BOA T T AIL SHAPES 813 175 FL OW O VER S TEP -UP CANY ONS IMMER SED IN BOUNDARY LAY ERS: SIMULA TION AND ANAL YSIS 818 176 HIGH-FIDELIT Y SIMULA TIONS OF ADVER SE PRES SURE GRADIENT FL OW O VER A ROUNDED S TEP F OR TURBULENCE MODEL IMPROVEMENT V IA DA T ABASE GENERA TION 824 178 MUL TIVARIA TE MUL TIFIDELITY MODELLING FOR TURBULENT FL OW S 830 179 EXPERIMENT AL INVES TIGA TION OF SEP ARA TION BUBBLE SUBJEC TED T O AC OUS TIC EX CIT A TION 833 182 USING DNS OF DROP S IN HIT T O DEVEL OP MODEL S FOR UNRESOL VED SCALES IN MUL TIPHASE FL OWS 836 183 A RA TIONAL LENG TH SCALE FOR LARGE- EDD Y SIMULA TIONS ON ANISO TROPIC GRIDS 842 184 AN INTERPRET ABLE DA T A-DRIVEN WAKE MODEL: ANAL OGY WITH THE K -Ε-FP FORMULA TION 848 185 EFFEC T OF SMALL A NGLE OF A TT ACK ON FL OW A ROUND TWO T ANDEM SQU ARE BUILDINGS OF DIFFERENT HEIGHT S 854 187 SCALE-RESOL VING SIMULA TION OF A TURBULENT BOUNDARY LA YER UNDER A DVER SE PRESSURE GR ADIENT 857 188 CL OSURE OF NONLINEAR FREQUENCY -DOMAIN REDUCED-ORDER MODEL S FOR UNS TEAD Y FL OWS 859 189 A WALL -MODELLED LES FRAMEW ORK FOR DROPLET -LADEN TURBULENT FL OWS U SING PHYSIC S -BASED MODEL S INFORMED B Y DNS 865 PRESSURE A ND TURBULENCE INTENSITY EFFECT S ON LEAN PREMIXED HYDROGEN FLAMES UNDER HIGH S TRAIN 871 191 FL OW -AWARE SIMULA TIONS OF TURBULENCE (F AS T): A MACHINE LEA RNING-BASED APPROA CH T O EFFICIENT TURBULENCE SIMULA TIONS FOR ENGINEERING FL OWS 877 192 LARGE-EDD Y SIMULA TION OF RO T A TIONAL EFFECT S ON BOUNDARY LAY ERS O VER AN AIRF OIL 880 194 INTERSCA LE MOMENTUM TRANSFER IN WALL TURBULENCE 886 197 EXAMINING CHAO TIC AND DETERMINISTIC D YNAMIC S FOR AI- BASED CLIMA TE C ONTROL 891 6 Cetin Kir is Volc ano Platf orms Inc. “Utilization of Larg e Eddy Simulations in Industrial Research & Development” Comput ational Fluid Dynamics (CFD) using Reynolds Aver aged Navier St ok es (RANS) b ased closure modeling has bec ome a corner st one of aerodynamic analysis, driven by adv ancement s in high-perf ormance computing. While eff icient alg or ithms f or rapid s te ady-s ta t e modeling have matur ed RANS simulations over the pas t two decades, mos t e fforts have f ocused on at tached f low c onditions, w hic h r epresent less th an 25% of t he t ot al aerodynam ics development eff ort. A s CFD exp ands beyond design and optimiz ation tow ards Certif ica tion and Qu alif ication by Analysis, there’ s growing int eres t in h igh- f idelity numer ical modeling for flight envelope regions char act er iz ed by larg e separ a ted f lows. These f lows are signific antly influenc ed by larg e-scale turbulenc e dynamics, and most RANS closures have limited succ ess in such conditions. While this has led some manufac turers to incorp or at e scale-resolving simula tions, widespre ad use of technologies lik e Wall Modeled Larg e Eddy Simula tions (WMLES) rema ins limit ed due to the h igh c omput ationa l c os t and lack of aut oma tion in LES -qu ality gr id gener ation f or c omplex geometr ies. This t alk will demons tra te tha t aut omat ed Cartesian octree gr id g ener ation and an accur ate high-R eynolds numb er visc ous immersed boundar y tre atment for c omplex g eometr ies can effec tively address the gr id gener ation bot tleneck. Furt hermore, carefully designed non-dissip ative discretiz ations, implement ed via a highly optimiz ed softwar e st ack le ver aging modern GPUs, can enable ac cur at e LES at a c omputa tional c os t comp arable to RANS simulations using exis ting c ommerc ial software. The present a tion will showc ase det ai led LES investig a tions of engineering problems involving sep ar at ed f low aerodynamics and aero ac oustics. These simula tions were performe d on modest c ompute resour c es using 2 to 8 ge ner al-purpose GPU car ds like the Nvidia RT X -4090 and Nvidia L40S. This bre ak thr ough has the potential to enable LES utiliza tion for a wide r ange of off-design f light char acter istics without requir ing ac c ess t o DOE-class superc omput ers. KEYNOTE S PEAKERS Bhar a t h Ganap at h isubr aman i University of Sout hampton “Horrible histories: Pr essure-gradient hist ory effects on turbulent boundary layer s” T ur bulent boundar y layers sub ject ed t o s treamwise var ying pr essure gradient s are abundant in eng ineer ing applica tions and environmenta l f lows. The inter action between imposed pressure gr adient and tur bulent boundar y lay er signif icant ly alters the f low stru cture and in turn affec ts ov er all p erformanc e, load di str ibutions, vibr a tions and noise. In this t alk, I will present result s from a ser ies of high-Reynolds number wind tunnel exper iment s on smooth and rough-wa ll tur bulent boundary layers subject ed to pressure gr adient his tor ies of differ ent forms and s trengths. The inf luence of these different pressur e gr adient s on mean f low , flu ctu ations and tur bulent s tructures will b e present ed. I wi ll discuss new insights on these c omplex f lows and s how our eff orts tow ards development of da t a-dr iven models for predic ting su ch non-equilibr ium flows. BIOGRAPHY Bhar at hr am Ganap ath isubr amani is Professor of E xper iment al Fluid Mechanics in the Department of Aeronautics and A s tronautics at the University of Southampton. His researc h and te aching interes t s are aerodynamics, hydrodynamics and propulsion relev ant to transport ation, energy gener ation and autonomous sys tems. Bhara th was born and brought up in Chennai (India). He c omplet ed his undergr adu at e degree in Nav al Arc hitec ture and Oc ean Engineering at the Indian Ins titute of T echnology-Madras (1995-1999). Upon completion, he moved to the University of Minnesot a in the US w here he secured his Mast ers and PhD in Aer osp ac e Engineer ing (1999-2004). He follow ed this with a s tint as a pos tdo ct or al rese arc h fellow at the University of T exas at Aus tin in the Centre for Aeromechan ics resear ch (2004-2006). He moved to the UK as a Lec turer (A ssist ant Professor) in the Department of Aeronautics at Imper ial College London (2007-2010). He has been in Southampton sinc e 2010, f irs t as Senior Lec turer and then as Professor . He currently ser ves as an A ssocia te Edit or for E xper iments in Fluids and Flow (the former focuses on the deve lopment and applic ation of exper imental me thods in f luid flows w hile the lat ter on pr actic al applic ations of f luid mechanics). He is a F ellow of the Alan T ur ing Insitute as well as an Associat e Fellow of AIAA. He is a member of the executive and management bo ards of the Na tional Wind T unnel F acility . He a lso ser ves on the e xecutive and advisor y c ommit tees of v ar ious interna tional c onfer enc es. 7 BIOGRAPHY Aft er a distinguis hed 35-year journey a t NASA, Dr . Cetin Kiris is now the CEO and founder of an e ar ly-s tag e technology s tartup that focuses on providing physics-b ased simulation cap abilities to ac c eler at e digital transf ormation of physica l prot otyping to predictive, fas t , and c os t -effec tive computing. Pr ior to his departur e from NASA, Dr . Kir is ser ved as the br anch chief of the Computa tional Aeroscienc es Br anch at NASA Ames Resear ch Center . He initiat ed and orches tr at ed the development of LAVA, a comput a tional framewor k for Launch, A sc ent, and Veh icle Aerodynamics. He rec eived his mas t er ’ s degr ee and Ph.D. in Aeronautics and A s tronautics from St anfor d University . He has published over 200 techn ical pap ers and co-author ed a book on numeric al simulations of incompressi ble flows. He has r ece ived numerous honors and awards, including NASA Outs t anding Le aders hip Medal; NASA E x ceptiona l Achievement Medal; NASA Softw are of the Y ear Award; NASA Commerc ial Invention of the Y ear Award for c o-developing the NASA-DeBak e y V entr icular Assist Device. Denn ic e Gayme Johns Hopkins Univer sity “ A coher ent structure-based model of wall-bounded shear flows” The prevalenc e of str eamwise c oherent motions and their r ole in ener gy growth, momentum transf er and the self -sust aining proc esses under lying wa ll-bounded tur bulence, has motiv ated the deve lopment of r educ ed order repr esent ations that emphasize s tre amwise coher ent motion. In the res tr ict ed nonline ar (RNL) mo deling p ar adigm, this emphasis is embedded into the decomposition of the dynamics into a s tre amwise c ons t ant mean flow , t hat is viewed as the large-scale c omponent, and small-sc ale pertur b ations about this mean. Order r eduction is achieved through a res tr iction of the nonline ar inter actions t o those c ontr ibuting to t his larg e-scale me an. RNL simula tions have been s hown t o produc e self-sus tain ing tur bulent activity with low-order s t atis tics, s truc tur al fe atur es and ener ge tics consis t ent with modera t e Reynolds num ber canon ical wall- bounded turbulent f lows at vas tly reduc ed c omput ational cos ts. In this talk we highlight rec ent ex tensions of the RNL model to tempor ally developing boundar y layers. Wa ll-resolved RNL large eddy simulations of tempor ally developing boundar y layers over wa lls with cons t ant, increasing and decreasing velocity are performed. The result s demonstr a te that RNL simula tions over an acc eler ating, dec eler ating and a unif orm veloc ity wall produc e comp arable st atis tical behavior to direc t numer ical simulations (DNS) of spa tially de velopi ng flows subjec t to favor able, adverse and zer o pressure gr adients. We then fo cus on the dec eler ating wall conf igura tion, w here signif icant changes to the s tructur al fea tures, tur bulenc e s ta tis tics, and energ y spectr a are obser ved as the r ate of dec eler ation is incr eased. Th is se t ting also repr oduc es t he growing e x tended w ak e r egions in the me an ve locity pr ofi les, and the development of an outer layer pe ak in the sec ond order s ta tis tics seen with increasingly adverse pressure gradient boundar y layers. These results demons tr at e tha t simulations of RNL flow over dec eler ating walls capture the key char act er istics of adverse pressur e gr adient f lows, and highlight t he model’ s potential as a c omput ationa lly effic ient means of explor ing developing f lows that are pr ev alent in a wide r ang e of applica tions. BIOGRAPHY Dennic e F . Gayme is a Professor in Mechanica l Eng ineer ing a t Johns Hopkins University . She rec eived her B. Eng. & Society in Mechanica l Engineer ing fr om McMast er University in 1997, an M.S. in Mechanical Engineering fr om the University of Calif ornia at Berk eley in 1998, and her Ph.D. in Control and Dynamic al Sys tems from the Calif ornia Institut e of T echnology in 2010. Her resear ch interes ts are in modeling, analysis and c ontrol of sp atially dis tr ibut ed and larg e-scale netw or ked sys tems, su ch as wind f arms, wall- bounded she ar f lows, and power syst ems. She was a recipient of a JHU Ca t alys t A ward in 2015, ONR Y oung Inves tig at or and NSF CA REER awar ds in 2017, a Whiting School of Engineering Johns Hopkins Alumni Associa tion E x c ellenc e in T eac hing Awar d in 2020, and the T urbulenc e and Shear Flow Phenomena (T SFP12) Nobuhide Kasagi Awar d in 2022. She is a fellow of the Americ an Physical Socie ty (APS) and ser ves as a Member -A t -Larg e in the E xecutiv e Commit tee of the APS Division of Fluid Dynamics (APS/DFD), the S tanding Chair of the Women in Contr ol Commit tee of the Control Sys tems Soc iety (C S S) of the IEEE, and on the edit or ial bo ar ds of the Annua l Review of Fluid Mechan ics, Physical R eview Fluids and PRX Ener gy . 8 Rene P ecn i k Delft Un iversity of T echnology “T urbulence modeling f or compr essible flows and heat tr ansfer” Unders t anding the imp act of c ompressi bility effects on tur bulent f lows is crucial for a wide rang e of engineer ing applica tions, as they inf luenc e the perfor mance, effic iency and re liability of aer osp ace veh icles, g as tur bines, and hea t exchang ers. T ur bulence in compressi ble f low involves two main mechanisms: heat transf er and intr insic compressi bility effec ts, also termed ‘true ’ compressi bility in tur bulenc e. Hea t tr ansfer mainly causes mean var iations in density and viscosity , and density flu ctu ati ons due t o entropy changes. In c ontr as t, intr insic compr essibility effect s are associat ed with v olume chang es in response t o pressur e f luc tu ations. In this t alk , we will discuss how these mechanisms c an be effectiv ely c har act er ized and modeled. The effec ts of hea t tr ansfer on t he mean f low - captured by chang es in fr iction velo city and viscous length sca les - natur ally lead t o a semi-loc al scaling fr amewor k, whose or igins tr ac e back to the wor k of Mor kovin (1962) and v an Dries t (1951). We present a gener aliz ed version of this fr amewor k, w hich enables the der ivation of sc aling laws and pr ovides guidanc e f or improving tur bulenc e models. In contr as t, intr insic c ompressibi lity effec ts alt er near -wa ll tur bulenc e dynamics through dila ta tional motions that c ounter act the ejection and sweep events driv en by mean s tre amwise vortic es. These effect s, too , can be incorpor at ed into scaling laws and turbulenc e closures, allowing for models tha t are robus t acr oss a wide r ange of c onditions - fr om low t o high Mach numbers and fr om idea l to non-ide al f luids. BIOGRAPHY Rene Pecni k is Professor of Energy T echnology at the Delft University of T echnology in the Nether lands. He earned his PhD in Mechan ical Engineer ing from Gr az Univer sity of T echnolog y , Aus tr ia, in 2007. His doct or al rese arc h specia lized in the numer ica l modeling of aerodynam ic flows in tr ansonic gas tur bines, focusing on unst eady effec ts and laminar - to- turbulent boundar y layer transition. Following his PhD, he joined the Center for T ur bulence Resear ch at St anford University as a pos tdoc tor al fellow , where he wor ked on simulations of re acting tur bulent f lows in hyperson ic propulsion sys t ems. In 2010, he moved to the Ne ther lands to c ontinue his academic c areer a t TU Delft. Pecni k ’ s rese arch br idges fundamenta l s tudies and applied de velopment s. He aims to advanc e know ledge Fi lippo Colet ti ETH Z ür ich “The turbulence along and beneath a fr ee surface ” F ree-surf ace tur bulenc e det ermines transport properties at the g lob al scale, e.g., the uptak e of C O2 and the spreading of f lo ating plastics in oc eans, r ivers, and lakes. Even in the seemingly simple case of a f lat surf ace, t he dynamics are dec eptively c omplex. In the pr esenc e of larg e surf ace deforma tions, the pr oblem is c omplica ted by the mutu al tr ansfer of energy be tween the bulk flow and gr avity-capillary waves. The mechanisms by w hic h wind builds waves and their exc hange of energy with the under lying tur bulenc e are e ven less underst ood. In this talk, I will touch upon fundament al aspec ts of free-surf ac e tur bulence, r anging from cases in w hich the bulk f low b are ly deforms the surfac e; t o cases where it corrug at es it dynamica lly; to cases in w hich tur bulenc e is cre at ed and modulat ed by wind she ar and wind-dr iven waves on the edge of bre aking. Those f lows are inves tigat ed with a suite of exper imental f aci lities and lever ag ing high-resolution imaging, to impr ove our predic tive unders tanding of the two-w ay c oupling between t he surf ac e and t he tur bulenc e underne at h. BIOGRAPHY Filippo Cole tti is Pr ofessor of E xperiment al Fluid Dynamics at ETH Z ur ich, w here he has b een sinc e 2020. Previous ly he was McKnight Land-Gr ant Professor of Aerosp ac e Engineer ing & Mec hanics a t the University of Minnesot a, w hich he joined in 2014. He w as visiting prof essor in IMFT T oulouse and ENS L yon. He rec eived the NSF CAREER award in 2015 and the ERC Consolidat or grant in 2022. He ser ves in t he advisor y c ommitt ee of sever al int ernational conf erenc es (T urbulenc e and She ar Flow Phenomena, Lisbon Symposium on Laser & Imag ing T echniques in Fluid Mechanics, Interna tional Confer enc e of Multiphase Flows) and plays active roles in the Division of Fluid Dynamics of the Amer ican Physica l Society . He founded and co-or gan izes the Fluid Mechanics T our of the Alps, an itiner ant sem inar ser ies fe a tur ing the wor ld’ s top f luid mechan icians. His r esear ch is f ocused on tur bulent multiphase f lows using a r ange of exper imental appr o aches. 9 viously dif ferent to the best constr ained GEP model, resulting from seed 1, τ C o nst r ij = − 2∆ 2 ( − 2 I 1 I 2 3 · T 2 ij + ( I 3 3 + I 3 I 4 ( I 2 + I 3 I 4 − 0 . 4) ) · T 3 ij ) (12) which obe ys the cubic scaling τ C o nst r x ,y ∝ y 3 as in- tended. Although the basis tensors T 1 ij (resulting in an eddy viscosity type contrib ution) and T 4 ij are also sometimes chosen by the genetic optimization, T 2 ij and T 3 ij are by f ar the most frequent basis tensors, and this property is shared by the bes t unconstrained and con- strained models discussed here. It is w orth noting that T 2 ij is, by itself, neutral with respect to the SGS en- er gy transfer b ut still e x erts a SGS force on the flo w field and has therefore an indirect ef fect on the en- er gy dissipation. T ensor T 3 ij is based on the strain rate tensor only and is able to represent forw ard as well as backw ard scatter , i.e., ne g ati v e as well as positi v e SGS ener gy transfer τ ij S ij . Compared to the uncon- strained models, the dimensionless in v ariants I 3 and I 4 are more frequently chosen in the constrained m od- els due to their potential to upscale the w all scaling. W ithout the constraint, the w all scaling of the GEP models is usually lo wer than cubic. 4 Model assessment 0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 0 5 10 15 20 25 K t DNS 256 3 no -m od el 32 3 Smago ri n s ky 3 2 3 S ig m a 32 3 GEP Uncon s t r 32 3 GEP C o nstr 3 2 3 0 0. 002 0. 004 0. 006 0. 008 0. 01 0. 012 0. 014 0 5 10 15 20 25 -d K/ d t t DNS 256 3 no -m od el 32 3 Smago ri n s ky 3 2 3 S ig m a 32 3 GEP Uncon s t r 32 3 GEP C o nstr 3 2 3 DNS B r a ch et e t a l Figure 5: T aylor –Green v orte x at Re 0 = 1600 : V olume- a v eraged kinetic ener gy K (top) and its dissipa- tion rate ε = − ∂K / ∂t (bottom) v ersus non- dimensional time t for the reference DNS and LES using no-model, the Smagorinsk y model, the sin- gular v alues (Sigma) based model as well as the best unconstrained and constrained GEP models. Figure 5 sho ws the temporal e v olution of the v olume-a v eraged kinetic ener gy K = u i u i / 2 and its dissipation rate ε = − ∂K / ∂t which are also used to define the multi-objecti v e cost function as introduced earlier . Apparently , the no-model LES features in- suf ficient dissipation during the high-dissipation phase which peaks roughly at non-dimensional time t = 10 when the flo w is characterized by significant v orte x stretching and breakdo wn, cf. Fig. 2. This is well remedied by the GEP models, whereas both mod- els from the literature are o v erly dissipati v e during the quasi-laminar ini tial stage of the transient, hence, the kinetic ener gy decreases too early . Represent- ing well-established LES models from the literature, the Smagorinsk y model (Smagorinsk y , 1963) and the Sigma model (Nicoud et al., 2011) capture well the peak dissipation, presumably because this state re- sembles homogeneous isotropic turb ulence for which those models ha v e been de v eloped (among other ide- alized states in case of the Sigma model). Ho we v er , also the timing matters in this challenging transient test case and in more comple x technical manifesta- tions of laminar -turb ulent transition. Although the first tar get, the kinetic ener gy , is v ery well reproduced by the GEP models, there is still room for impro v ement re g arding the second tar get, the dissipation of kinetic ener gy , which sho ws minor non-ph ysical oscillations. In this re g ard, the unconstrained and constrained GEP models do not dif fer significantly . The reason for this beha vior is probably the absence of a stabilizing, eddy viscosity type, T 1 ij contrib ution. Ne v ertheless, the best GEP models discussed here are rob ust from a global point of vie w . It remains to be demonstrated that the proper near - w all scaling of the shear stress is reco v er ed for actual w all-bounded flo w simulations. F or this purpose, a w all-refined plane channel flo w LES is conducted us- ing a similar numerical method as for the TGV train- ing. The in v estig ated configuration is a rectangular channel of size L x =2 πH , L y =2 H and L z = πH , where H is the half width and x , y and z denote the stream-wise, the w all-normal and the span-wise direc- tions. The x and z directions are periodic, whereas no-slip w alls are prescribed in the y direction. The flo w is controlled by a constant pressure gradient cor - responding to a friction Re ynolds number of Re τ = (  τ w /ρH ) /ν = 180 . The anisotropic mesh consists of 38 × 54 × 38 cells and is stretched in the w all- normal direction s uch that y + = yu τ /ν ≈ 1 holds for the w all-nearest cell layer . It can be seen in Fig. 6 that the SGS shear stress ⟨ τ sg s x ,y ⟩ for the constrained GEP model obe ys the cubic w all scaling in agreement with the theory . The model contrib ution ⟨ τ sg s x ,y ⟩ peaks at y + ≈ 10 , where the turb ulence intensity is kno wn to reach its maximum in plane channel flo ws. Fur - thermore, it is demonstrated that the prediction of the mean stream-wise v elocity profile ⟨ u ⟩ is impro v ed by using the data -dri v en e xplicit SGS model in compari- Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 16 16 viously dif ferent to the best constr ained GEP model, resulting from seed 1, τ C o nst r ij = − 2∆ 2 ( − 2 I 1 I 2 3 · T 2 ij + ( I 3 3 + I 3 I 4 ( I 2 + I 3 I 4 − 0 . 4) ) · T 3 ij ) (12) which obe ys the cubic scaling τ C o nst r x ,y ∝ y 3 as in- tended. Although the basis tensors T 1 ij (resulting in an eddy viscosity type contrib ution) and T 4 ij are also sometimes chosen by the genetic optimization, T 2 ij and T 3 ij are by f ar the most frequent basis tensors, and this property is shared by the bes t unconstrained and con- strained models discussed here. It is w orth noting that T 2 ij is, by itself, neutral with respect to the SGS en- er gy transfer b ut still e x erts a SGS force on the flo w field and has therefore an indirect ef fect on the en- er gy dissipation. T ensor T 3 ij is based on the strain rate tensor only and is able to represent forw ard as well as backw ard scatter , i.e., ne g ati v e as well as positi v e SGS ener gy transfer τ ij S ij . Compared to the uncon- strained models, the dimensionless in v ariants I 3 and I 4 are more frequently chosen in the constrained m od- els due to their potential to upscale the w all scaling. W ithout the constraint, the w all scaling of the GEP models is usually lo wer than cubic. 4 Model assessment 0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 0 5 10 15 20 25 K t DNS 256 3 no -m od el 32 3 Smago ri n s ky 3 2 3 S ig m a 32 3 GEP Uncon s t r 32 3 GEP C o nstr 3 2 3 0 0. 002 0. 004 0. 006 0. 008 0. 01 0. 012 0. 014 0 5 10 15 20 25 -d K/ d t t DNS 256 3 no -m od el 32 3 Smago ri n s ky 3 2 3 S ig m a 32 3 GEP Uncon s t r 32 3 GEP C o nstr 3 2 3 DNS B r a ch et e t a l Figure 5: T aylor –Green v orte x at Re 0 = 1600 : V olume- a v eraged kinetic ener gy K (top) and its dissipa- tion rate ε = − ∂K / ∂t (bottom) v ersus non- dimensional time t for the reference DNS and LES using no-model, the Smagorinsk y model, the sin- gular v alues (Sigma) based model as well as the best unconstrained and constrained GEP models. Figure 5 sho ws the temporal e v olution of the v olume-a v eraged kinetic ener gy K = u i u i / 2 and its dissipation rate ε = − ∂K / ∂t which are also used to define the multi-objecti v e cost function as introduced earlier . Apparently , the no-model LES features in- suf ficient dissipation during the high-dissipation phase which peaks roughly at non-dimensional time t = 10 when the flo w is characterized by significant v orte x stretching and breakdo wn, cf. Fig. 2. This is well remedied by the GEP models, whereas both mod- els from the literature are o v erly dissipati v e during the quasi-laminar ini tial stage of the transient, hence, the kinetic ener gy decreases too early . Represent- ing well-established LES models from the literature, the Smagorinsk y model (Smagorinsk y , 1963) and the Sigma model (Nicoud et al., 2011) capture well the peak dissipation, presumably because this state re- sembles homogeneous isotropic turb ulence for which those models ha v e been de v eloped (among other ide- alized states in case of the Sigma model). Ho we v er , also the timing matters in this challenging transient test case and in more comple x technical manifesta- tions of laminar -turb ulent transition. Although the first tar get, the kinetic ener gy , is v ery well reproduced by the GEP models, there is still room for impro v ement re g arding the second tar get, the dissipation of kinetic ener gy , which sho ws minor non-ph ysical oscillations. In this re g ard, the unconstrained and constrained GEP models do not dif fer significantly . The reason for this beha vior is probably the absence of a stabilizing, eddy viscosity type, T 1 ij contrib ution. Ne v ertheless, the best GEP models discussed here are rob ust from a global point of vie w . It remains to be demonstrated that the proper near - w all scaling of the shear stress is reco v er ed for actual w all-bounded flo w simulations. F or this purpose, a w all-refined plane channel flo w LES is conducted us- ing a similar numerical method as for the TGV train- ing. The in v estig ated configuration is a rectangular channel of size L x =2 πH , L y =2 H and L z = πH , where H is the half width and x , y and z denote the stream-wise, the w all-normal and the span-wise direc- tions. The x and z directions are periodic, whereas no-slip w alls are prescribed in the y direction. The flo w is controlled by a constant pressure gradient cor - responding to a friction Re ynolds number of Re τ = (  τ w /ρH ) /ν = 180 . The anisotropic mesh consists of 38 × 54 × 38 cells and is stretched in the w all- normal direction s uch that y + = yu τ /ν ≈ 1 holds for the w all-nearest cell layer . It can be seen in Fig. 6 that the SGS shear stress ⟨ τ sg s x ,y ⟩ for the constrained GEP model obe ys the cubic w all scaling in agreement with the theory . The model contrib ution ⟨ τ sg s x ,y ⟩ peaks at y + ≈ 10 , where the turb ulence intensity is kno wn to reach its maximum in plane channel flo ws. Fur - thermore, it is demonstrated that the prediction of the mean stream-wise v elocity profile ⟨ u ⟩ is impro v ed by using the data -dri v en e xplicit SGS model in compari- P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 16 DNS-Moser et al (1999) GEP Constr No-Model 10 0 10 1 10 2 y + 10 − 4 10 − 3 10 − 2 10 − 1 − <τ sg s x,y > /u 2 τ O ( y + ) O ( y + 2 ) O ( y + 3 ) 10 0 10 1 10 2 y + 2 . 5 5 . 0 7 . 5 10 . 0 12 . 5 15 . 0 17 . 5 20 . 0 <u>/ u τ Figure 6: Channel flo w at Re τ = 180 : Normalized and a veraged SGS shear stress −⟨ τ sgs x,y ⟩ /u 2 τ (top) and stream-wise velocity ⟨ u ⟩ /u τ (bottom) depending on the normalized wall distance y + . Note that the marker points indicate the locations of the finite v olume cell centers. son to the no-model LES. Future work will be de voted to the generalizability of the trained models. The GEP models need to be tested for unseen grids and Reynolds numbers – both for the TGV as well as wall-bounded flo ws. Acknowledgments This work is funded by the German Federal Min- istry of Education and Research (BMBF) under grant number 02NUK071 (project iCFD4NS). Refer ences Ferreira, C. (2001). Gene expression programming: a ne w adapti ve algorithm for solving problems. Comple x Systems , 13(2), 87–129. Nicoud, F ., T oda, H. B., Cabrit, O., Bose, S., and Lee, J. (2011). Using singular v alues to build a subgrid-scale model for large eddy simulations. Physics of Fluids , 23(8). Pa wlak, T .P ., W ieloch, B., Krawiec, K. (2015). Semantic backpropagation for designing search operators in genetic programming. IEEE T ransactions on Evolutionary Compu- tation 19, 326–340. Reissmann, M., Hasslberger , J., Sandberg, R.D., and Klein, M. (2021). Application of gene expression programming to a-posteriori LES modeling of a T aylor Green v orte x. J ournal of Computational Physics , 424, 109859. Reissmann, M., Fang, Y ., Ooi, A. S., and Sandber g, R.D. (2025). Constraining genetic symbolic re gression via se- mantic backpropagation. Genetic Pr ogr amming and Evolv- able Machines , 26(1), 12. Rumelhart, D.E., Hinton, G.E., and W illiams, R.J. (1986). Learning representations by back-propagating errors. Na- tur e , 323(6088), 533–536. Smagorinsky , J. (1963). General circulation experiments with the primiti ve equations: I. The basic experiment. Monthly weather r eview , 91(3), 99-164. W aschko wski, F ., Zhao, Y ., Sandberg, R.D., and Klewicki, J. (2022). Multi-objectiv e CFD-dri ven de velopment of cou- pled turb ulence closure models. J ournal of Computational Physics , 452, 110922. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 17 17 A NEW FORMALISM FOR IMPR O VING PREDICTIONS WITH RANS EDD Y VISCOSITY MODELS A. Cimarelli 1 , B. Ni ˇ ceno 2 and Y . T essier Urrecha 1 1 Univ ersit ` a di Modena e Reggio Emilia, DIEF via V ivarelli 10, 41125 Modena, Italy 2 P aul Scherrer Institute (PSI), F orschungsstrasse 111, 5232 V illigen, Switzerland yves.tessier@unimor e.it 2 February 2025 RANS eddy viscosity models are the preferred solution in engineering applications, yet their in- ability to capture turb ulent fluctuations and their dif ficulties with laminar and transitional flo ws represent a hefty trade-of f for their lo w cost, es- pecially for applications in volving e xternal flows. T raditional eddy viscosity models suf fer from stress ov er-prediction in adv erse pressure gradi- ents, which sev erely compromises their ability to predict laminar -to-turb ulent transition. They are also incapable of returning unsteady solu- tions, except in flo ws predisposed to it due to their solutions being characterised by particularly wide ranges of scales — e v en then, only the very lar gest scales can ov ercome eddy viscosity , and the fluctuations the y elicit in the solution are not representati v e of the structures that spawn in the actual realisation of the flo w . This work premiers a ne w formalism for RANS eddy viscosity models based on temporal filtering, leading to an identity for the dynamic definition of the c µ factor that greatly impro v es their performance. The use of the dynamic defi- nition maximises the amount of information for a gi v en grid by allo wing the solution to manifest the turb ulent fluctuations supported by its spatial res- olution, thanks to the now dynamic c µ f actor mod- ulating eddy viscosity . This also allows the sim- ulation to adapt to laminar and transitional flo ws by relaxing the eddy viscosity in those regions, pre v enting the ov er -prediction of stresses. The procedure for calculating c µ requires no empiri- cal factors, correlations, or calibration, and it op- erates in fully automatic fashion. It also allo ws for –as a matter of fact, requires– the elimination of damping functions for eddy viscosity , since the dynamic definition of c µ already addresses the is- sues with near -wall scaling. T o analyse the performance of our solution, we compare results obtained with the k − ε model with and without the dynamic procedure in two flo w cases: The Benchmark on the Aerodynam- ics of a Rectangular Cylinder (B ARC) at Re = 14000 is first used to demonstrate the procedure’ s ability to return a rich unsteady solution. V ari- ants 1, 3, and 4 of the ERCOFT A C T3L bench- mark are then simulated to assess its response to v arious le vels of free stream turb ulence and e v al- uate ho w it compares to the con ventional model and the high-accuracy ILES ref erence by Bassi et al. (2020) in predicting transition and reattach- ment along the round leading edge flat plate. All simulations were ex ecuted in T -Flo ws, a Finite V olume method based code using collocated, un- structured, arbitrary grids. The momentum trans- port scheme is SMAR T , while time integration is performed with the Backward Euler scheme in its first-order form for the stresses and in a second- order v ariant, used in an iterati v e fashion, for the inertial term. The PIMPLE algorithm is applied for Pressure-V elocity decoupling, since time ad- v ancement of the solution is central to all tests performed so far . The tests sho w clear indications that our solution achie v es both set objecti v es and has the potential to bring a lar ge leap forward for indus- trial applications of CFD solutions. The B ARC results obtained with the procedure hold up to the reference DNS by Corsini (2022) at least as well as those of the standard model in terms of integral quantities and mean flo w structures, b ut also re v eal a wide range of turb ulent structures that match those of the DNS. This is clear from the e v olution of the IQs, av erage distrib utions of Reynolds stresses, and instantaneous realisa- tions of the flo w , an example of which can be Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 18 18 A NE W FORMAL I S M F OR IMPR O VING P RED I C TIONS W ITH RANS EDD Y VISCOS ITY M OD ELS A. Cimarelli 1 , B. Ni ˇ ceno 2 and Y . T essier Urrec ha 1 1 Univ ersit ` a di Modena e Reggio Emilia, DIEF via V ivarelli 10, 41125 Modena, Italy 2 P aul Sc herrer Institute (PSI), F orsc hungsstrasse 111, 5232 V illigen, Switzerland yves.tessier@unimor e .it 2 February 2025 RANS eddy viscosity models are the preferred solution in engineering applications, yet their in- ability to capture turb ulent fluctuations and their dif ficulties with laminar and transitional flo ws represent a hefty trade-of f for their lo w cost, es- pecially for applications in v olving e xternal flo ws. T raditional eddy viscosity models suf fer from stress o v er -prediction in adv erse pressure gradi- ents, which se v erely compromises t heir ability to predict laminar -to-turb ulent transition. The y are also incapable of returning unsteady solu- tions, e xcept in flo ws predisposed to it due to their solutions being characterised by particularly wide ranges of scales — e v en then, only the v ery lar gest scales can o v ercome eddy viscosity , and the fluctuations the y elicit in the solution are not representati v e of the structures that spa wn in the actual realisation of the flo w . This w ork premiers a ne w formalism for RANS eddy viscosity models based on temporal filtering, leading to an identity for the dynamic definition of the c µ f actor that greatly impro v es their performance. The use of the dynamic defi- nition maximises the amount of information for a gi v en grid by allo wing the solution to manifest the turb ulent fluc tuations supported by its spatial res- olution, thanks to the no w dynamic c µ f actor mod- ulating eddy viscosity . This also allo ws the sim- ulation to adapt to laminar and transitional flo ws by relaxing the eddy viscosity in those re gions, pre v enting the o v er -prediction of stresses. The procedure for calculating c µ requires no empiri- cal f actors, correlations, or calibration, and it op- erates in fully automatic f ashion. It also allo ws for –as a matter of f act, requires– the elimination of damping functions for eddy viscosity , since the dynamic definition of c µ already addresses the is- sues with near -w all scaling. T o analyse the performance of our solution, we compare results obtained with the k − ε model with and without the dynamic procedure in tw o flo w cases: The Benchmark on the Aerodynam- ics of a Rectangular Cylinder (B ARC) at Re = 14000 is first used to demonstrate the procedure’ s ability to return a rich unsteady solution. V ari- ants 1, 3, and 4 of the ERCOFT A C T3L bench- mark are then simulated to assess its response to v arious le v els of free st ream turb ulence and e v al- uate ho w it compares to the con v entional model and the high-accurac y ILES ref erence by Bassi et al. (2020) in predicting transition and reattach- ment along the round leading edge flat plate. All simulations were e x ecuted in T -Flo ws, a Finite V olume method based code using collocated, un- structured, arbitrary grids. The momentum trans- port scheme is SMAR T , while time inte gration is performed with the Backw ard Euler scheme in its first-order form for the stresses and in a second- order v ariant, used in an iterati v e f ashion, for the inertial term. The PIMPLE algorithm is applied for Pressure-V elocity decoupling, since time ad- v ancement of the solution is central to all tests performed so f ar . The tests sho w clear indications that our solution achie v es both set objecti v es and has the potential to bring a lar ge leap forw ard for indus- trial applications of CFD solutions. The B ARC results obtained with the procedure hold up to the reference DNS by Corsini (2022) at least as well as those of the standard model in terms of inte gral quantities and mean flo w structures, b ut also re v eal a wide range of turb ulent structures that match those of the DNS. This is clear from the e v olution of the IQs, a v erage distrib utions of Re ynolds stresses, and instantaneous realisa- tions of the flo w , an e xample of which can be P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 18 appreciated in Figure 1. The use of the procedure re v eals the turb ulent structures supported by the grid in the T3L study as well, b ut it also allows the solution to recov er the main features of the flo w e videnced in the high-accuracy ILES refer - ence. Meanwhile, the classical dif ficulties of the traditional models are exposed by qualitati vely and quantitati v ely poor solutions (see Figure 2). The enhanced model also responds to changes in free-stream turb ulence as modelled via simple control of inlet boundary conditions. The authors are currently working on a num- ber of simulations in v olving the DES approach and steady-state solutions for the T3L case, in order to compare the performance of the procedure against the former and e v aluate its potential for improving the latter . Refer ences F . Bassi, L. Botti, A. Colombo, A. Criv ellini, M. Franciolini, A. Ghidoni and G. Nov enta (2020), A p- adapti v e Matrix-Free Discontinuous Galerkin Method for the Implicit LES of Incompressible T ransitional Flo ws, Flow , T urbulence and Comb ustion , V ol. 105, pp. 437-470. R. Corsini (2022), Direct numerical simulation of tur- b ulence: the flo w past an irre gular grid and the aerody- namics of a rectangular cylinder , J . Fluid Mech. , V ol. 42, pp. 447-464. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 19 19 Figure 1: Q = 0.15 iso-contours for an instantaneous realisation of the B ARC simulation with the k − ε model in its standard form (left) and enhanced with the dynamic procedure (right). T urbulent structures in the simulation performed with the standard model are strictly spanwise, while the dynamic procedure allo ws the K-H cirri to stretch in x and break into smaller , 3D structures. Hiding areas with positi v e streamwise velocity re veals some structures within the re v erse boundary layer for the simulation with the dynamic procedure. Figure 2: A v erage friction coef ficient for the reference ILES (dashed lines) and the present simulations with the k − ε model with (solid lines) and without (dotted lines) the dynamic procedure, for v arying Tu : 5.6% (gray), 2.3% (red), 0.6% (yellow), 0.2% (green), 0% (blue). The performance of the standard model is conspicuously poor , with stress ov er -prediction being so drastic that transition precedes separation. The main features of the flo w are fully recognizable in the simulations using the dynamic procedure. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 20 20 Figure 1: Q = 0.15 iso-contours for an instantaneous realisation of the B ARC simulation with the k − ε model in its standard form (left) and enhanced with the dynamic procedure (right). T urb ulent structures in the simulation performed with the standard model are strict ly spanwise, while the dynamic procedure allo ws the K-H cirri to stretch in x and break into smaller , 3D structures. Hiding areas with positi v e streamwise v elocity re v eals some structures within the re v erse boundary layer for the simulation with the dynamic procedure. Figure 2: A v erage friction coef ficient for the reference ILES (dashed lines) and the present simulations with the k − ε model with (solid lines) and without (dotted lines) the dynamic procedure, for v arying Tu : 5.6% (gray), 2.3% (red), 0.6% (yello w), 0.2% (green), 0% (blue). The performance of the standard model is conspicuously poor , with stress o v er -prediction being so drastic that transition precedes separation. The main features of the flo w are fully recognizable in the simulations using the dynamic procedure. P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 20 T URBULENT DRA G REDUCTION IN W A TER - LUBRICA TED CHANNEL F LO W OF HIGHL Y VISCOUS OIL A. Roccon 1 , 2 , F . Zonta 3 and A. Soldati 1 , 2 1 P olytechnic Department of Engineering and Arc hitecture, Univ ersity of Udine, Italy 2 Institute of Fluid Mechanics and Heat T ransfer , TU W ien, A ustria 3 School of Engineering, Newcastle Univ ersity , United Kingdom alessio.r [email protected] Abstract W e study the problem of drag reduction (DR) in a lubricated conduit, in which a thin layer of water is injected in the near -wall re gion and facilitates the transport of a core of highly viscous oil. In the present in vestigation, the flo w instance is a channel flo w , and consequently we ha ve one thin layer of lo w-viscosity fluid lubricating each wall. W e run direct numerical simulations (DNS) of this flo w instance. A phase- field method (PFM) is used to describe the dynamics of the liquid-liquid interface. As this technique is tai- lored to ward the transport of v ery viscous fluids like oils, we study the drag reduction performance of the system by keeping fix ed the lubricating fluid proper - ties (water) and by considering tw o dif ferent types of oil characterized by dif ferent viscosities, 10 and 100 times more viscous than water , respecti vely . As in real instances the presence of impurities and surfactants – which act by locally reducing the local v alue of the surface tension – is ine vitable, we consider , for each type of transported oil, a clean and a surfactant-laden interface. For all four tested configurations, we unam- biguously sho w that significant DR can be achie v ed. Reportedly , compared to the single-phase case, we ob- serve a reduction of the mean pressure gradient do wn to p x /p x,sp =0 . 25 for the lar gest viscosity oil. By analyzing the features of turb ulence in the lubricat- ing layer , and the close interaction with the perturba- tions induced by the oil-water interf ace deformation, we elucidate the physical mechanisms leading to DR and we underline the ef fects of viscosity ratios and of surfactants. 1 Intr oduction Pipelines are commonly used to transport hea vy oils. Ho wev er , due the v ery high viscosity , moving hea vy oil is an extremely ener gy-intensi ve process and requires a lar ge pumping po wer . T o reduce the re- quired pumping po wer and the corresponding operat- ing cost, dif ferent DR techniques hav e been de veloped and tested in the past, including: DR by polymers, sur- factants, and fibers, (which act to modify the rheologi- cal properties of the flo w , and thus the apparent viscos- ity (T oms 1949; Forrest & Grierson 1931, V irk 1975; V irk et al 1967); DR by riblets and activ e wall oscilla- tions (which act to modify the turb ulence regeneration cycle (Quadrio et al. 2009; Ricco et al. 2009) or DR by injection of a lo w viscosity fluid – e.g. water – in the near -wall re gion of the pipeline, so that the wall friction, induced by the low-viscosity fluid in contact with the wall, is lo wer (so-called water -lubricated oil transport). Among the dif ferent DR techniques, the water -lubricated oil transport has emer ged as one of the most promising. This technique takes adv antage of the natural tendency of w ater to form a stable layer that remains in contact with the wall of the pipe, and that lubricates the oil flo w (Joseph et al. 1984; Hu et al. 1945; Joseph et al. 1997). In this work, we e xtend our pre vious studies (Roc- con et al. 2019 & 2021) to a more practically rele v ant configuration, by acting on three dif ferent aspects: i) we consider the presence of a lubricating layer on both the top and bottom walls, hence resembling the core- annular flo w configuration obtained in pipes; note in- deed that near -wall turb ulence in pipes and channels (which is the main character in the generation of the flo w resistance) is very similar , in spite of the e x- isting geometrical dif ference; ii) we consider a very lar ge dif ference in viscosity between the two phases, thus mimicking the crude oil/water cases; iii) we also take into account the presence of surf actants, which are very often present in these applications and act by locally reducing the interfacial tension. W e consider the flow of tw o immiscible fluids, hea vy oil and water , inside a rectangular flat channel. The top and bottom parts of the channel are occupied by two thin lubricating layers of w ater having thick- ness h w , density ρ w and viscosity η w , while the core part of the channel is occupied by oil and has thickness h o , density ρ o and viscosity η o . T o mimic a realistic hea vy oil/water system, we assume that the two fluids ha ve the same density ( ρ w = ρ o = ρ ) b ut different vis- cosities so that a viscosity ratio λ = η o /η w can be de- fined. As mentioned abov e, in nearly all cases of prac- tical importance for this kind of flo w , contaminants, impurities, and surfactants are commonly found at the interface between the tw o phases. Therefore, we con- sider both surfactant-free (clean) and surf actant-laden Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 21 21 systems. The dynamic of the system is described by coupling direct numerical simulation of the Na vier - Stokes equations (flo w field) with a two-order param- eter phase-field method – interface and surf actant con- centration – (Soligo et al. 2019). Simulations are performed using a Constant Po wer Input (CPI) frame- work (Hase ga wa et al. 2014; Roccon at al. 2019) which means that the flo w is dri v en by imposing a constant amount of po wer – product between flo w- rate and pressure gradient –, and thus adjusting the imposed pressure gradient to the actual flo w-rate. 2 Methodology W e consider the flow of tw o immiscible fluids in- side a rectangular flat channel. At the top and bottom walls, tw o thin water -lubricating layers are used to fa- v or the transport of a thicker central layer of hea vy oil. T o capture the dynamics of the system, we cou- ple direct numerical simulation of the Na vier -Stokes equations, used to describe the flow field, with a two- order -parameter phase-field method, used to describe the deformation of the oil/water interf ace and the sur- factant concentration. Phase-field modeling of interfacial phenomena In the frame work of the tw o-order -parameter phase-field method, two order parameters are used to describe the interfacial w a ves and the surf actant con- centration. A first-order parameter , the phase field, ϕ , describes the shape and position of the interface. A second order parameter , ψ , describes the surfactant concentration. The behavior of these tw o v ariables is described by two Cahn-Hilliard-lik e equations: ∂ϕ ∂t + u · ∇ ϕ = 1 Pe ϕ ∇ 2 µ ϕ , (1) ∂ψ ∂t + u · ∇ ψ = 1 Pe ψ ∇ · [ ψ (1 − ψ ) ∇ µ ψ ] , (2) where u =( u x ,u y ,u z ) is the velocity v ector , Pe ϕ and Pe ψ are the phase field and the surfactant P ´ eclet numbers and µ ϕ and µ ψ are the corresponding chemi- cal potentials. The two P ´ eclet numbers are defined as follo ws: Pe ϕ = u Π h M ϕ β ; Pe ψ = u Π hα M ψ β 2 , (3) where u Π is the characteristic velocity (equation 11), h is the channel half-height, M ϕ and M ψ the phase field and the surfactant mobilities, while α and β are posi- ti v e constants used in the dimensionless procedure. The chemical potentials µ ϕ and µ ψ are defined as the functional deri v ati v e of a two-order -parameter Ginzb urg-Landau free-ener gy functional that accounts for interfacial motion and surf actant dynamic. The re- sulting expressions of the chemical potentials are: µ ϕ = δ F δϕ = ϕ 3 − ϕ − Ch 2 ∇ 2 ϕ, (4) µ ψ = δ F δψ = Pi log  ψ 1 − ψ  − (1 − ϕ 2 ) 2 2 + ϕ 2 2 E x . (5) The phase field v ariable is uniform in the b ulk of the two phases ( ϕ eq = ± 1 ) and under goes a smooth transition follo wing a hyperbolic tangent pro- file throughout the thin transition layer . Similarly , the concentration of surfactant is uniform in the b ulk of the two phases, ψ b = ψ eq ( ϕ = ± 1) , and reaches its maximum v alue at the interface, ψ 0 = ψ eq ( ϕ = 0) . Hydr odynamics T o describe the hydrodynamics of the multiphase system, the two Cahn-Hilliard-like equations are cou- pled with the Na vier -Stokes equations. Recalling that in the present study we consider two fluids with the same density ( ρ = ρ o = ρ w ) b ut dif ferent viscos- ity ( η o  = η w ), continuity and Na vier -Stokes equations can be written as follo ws: ∇ · u =0 , (6) ∂ u ∂t + u · ∇ u = − ∇ Π − ∇ p + 1 Re Π ∇ · [ η ( ϕ )( ∇ u + ∇ u )] + 3 Ch √ 8 We Π ∇ · (7) where p is the pressure field, ∇ Π=( p x , 0 , 0) is the mean pressure gradient that dri v es the flo w , η ( ϕ ) is the viscosity map accounting for the viscosity contrast between the two phases, f σ ( ψ ) is the surface tension equation of state (EOS) and τ c is the K ortewe g tensor . The latter is defined as follo ws: τ c = | ∇ ϕ | 2 I − ∇ ϕ ⊗ ∇ ϕ, (8) where I is the identity matrix. The interfacial term, composed of the surface tension EOS and the K o- rte we g tensor , accounts for both normal and tangen- tial components of interfacial forces that arise when surface tension is not uniform. The EOS adopted in this work is a modified Langmuir equation in which the minimum surface tension v alue is set to half of the clean interface surf ace tension v alue: f σ ( ψ )= σ ( ψ ) σ 0 = max    1+ β s log (1 − ψ )    Langmuir EOS , 0 . 5    , (9) where σ ( ψ ) is the dimensional surface tension of the surfactant-laden interf ace, σ 0 is the surface tension of a clean interface and β s the elasticity number . This lat- ter parameter quantifies the strength of the surfactant: for a fixed concentration of surfactant, a higher sur - face tension reduction can be obtained with a stronger surfactant (higher elasticity number β s ). The dimensionless groups appearing in equation (7) are the po wer Re ynolds number , Re Π , and the W e- ber number , We Π , which are defined as: Re Π = ρu Π h η w , We Π = ρu 2 Π h σ 0 . (10) Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 22 22 systems. The dynamic of the system is described by coupling direct numerical simulation of the Na vier - Stok es equations (flo w field) with a tw o-order param- eter phase-field method – interf ace and surf actant con- centration – (Soligo et al. 2019). Simulations are performed using a Constant Po wer Input (CPI) frame- w ork (Hase g a w a et al. 2014; Roccon at al. 2019) which means that the flo w is dri v en by imposing a constant amount of po wer – product between flo w- rate and pressure gradient –, and thus adjusting the imposed pressure gradient to the actual flo w-rate. 2 Methodology W e consider the flo w of tw o immiscible fluids in- side a rectangular flat c h a nn e l. At the top and bottom w alls, tw o thin w ater -lubricating layers are used to f a- v or the transport of a thick er central layer of hea vy oil. T o capture the dynamics of the system, we cou- ple direct numerical simulation of the Na vier -Stok es equations, used to describe the flo w field, with a tw o- order -parameter phase-field method, used to describe the deformation of the oil/w ater interf ace and the sur - f actant concentration. Phase-field modeling of interfacial phenomena In the frame w ork of the tw o-order -parameter phase-field method, tw o order parameters are used to describe the interf acial w a v es and the surf actant con- centration. A first-order parameter , the phase field, ϕ , describes the shape and position of the interf ace. A se cond order parameter , ψ , describes the surf actant concentration. The beha vior of these tw o v ariables is described by tw o Cahn-Hilliard-lik e equations: ∂ϕ ∂t + u · ∇ ϕ = 1 Pe ϕ ∇ 2 µ ϕ , (1) ∂ψ ∂t + u · ∇ ψ = 1 Pe ψ ∇ · [ ψ (1 − ψ ) ∇ µ ψ ] , (2) where u =( u x ,u y ,u z ) is the v elocity v ector , Pe ϕ and Pe ψ are the phase field and the surf actant P ´ eclet numbers and µ ϕ and µ ψ are the corresponding chemi- cal potentials. The tw o P ´ eclet numbers are defined as follo ws: Pe ϕ = u Π h M ϕ β ; Pe ψ = u Π hα M ψ β 2 , (3) where u Π is the characteristic v elocity (equation 11), h is the channel half-height, M ϕ and M ψ the phase field and the surf actant mobilities, while α and β are posi- ti v e constants used in the dimensionless procedure. The chemical potentials µ ϕ and µ ψ are defined as the functional deri v ati v e of a tw o-order -parameter Ginzb ur g-Landau free-ener gy functional that accounts for interf acial motion and surf actant dynamic. The re- sulting e xpressions of the chemical potentials are: µ ϕ = δ F δϕ = ϕ 3 − ϕ − Ch 2 ∇ 2 ϕ, (4) µ ψ = δ F δψ = Pi l og  ψ 1 − ψ  − (1 − ϕ 2 ) 2 2 + ϕ 2 2 E x . (5) The phase field v ariable is uniform in the b ulk of the tw o phases ( ϕ eq = ± 1 ) and under goes a smooth transition follo wing a h yperbolic tangent pro- file throughout the thin transition layer . Similarly , the concentration of surf actant is uniform in the b ulk of the tw o phases, ψ b = ψ eq ( ϕ = ± 1) , and reaches its maximum v alue at the interf ace, ψ 0 = ψ eq ( ϕ = 0) . Hydr odynamics T o describe the h ydrodynamics of the multiphase system, the tw o Cahn-Hilliard-lik e equations are cou- pled with the Na vier -Stok es equations. Recalling that in the present study we consider tw o fluids with the same density ( ρ = ρ o = ρ w ) b ut dif ferent viscos- ity ( η o  = η w ), continuity and Na vier -Stok es equations can be written as follo ws: ∇ · u =0 , (6) ∂ u ∂t + u · ∇ u = − ∇ Π − ∇ p + 1 Re Π ∇ · [ η ( ϕ )( ∇ u + ∇ u )] + 3 Ch √ 8 We Π ∇ · (7) where p is the pressure field, ∇ Π=( p x , 0 , 0) is the mean pressure gradient that dri v es the flo w , η ( ϕ ) is the viscosity map accounting for the viscosity contrast between the tw o phases, f σ ( ψ ) is the surf ace tension equation of s tate (EOS) and τ c is the K orte we g tensor . The latter is defined as follo ws: τ c = | ∇ ϕ | 2 I − ∇ ϕ ⊗ ∇ ϕ, (8) where I is the identity matrix. The interf acial term, composed of the surf ace tension EOS and the K o- rte we g tensor , accounts for both normal and tangen- tial components of interf acial forces that arise when surf ace tension is not uniform. The EOS adopted in this w ork is a modified Langmuir equation in which the minimum surf ace tension v alue is set to half of the clean interf ace surf ace tension v alue: f σ ( ψ )= σ ( ψ ) σ 0 = max    1+ β s lo g (1 − ψ )    Langmuir EOS , 0 . 5    , (9) where σ ( ψ ) is the dimensional surf ace tension of the surf actant-laden interf ace, σ 0 is the surf ace tension of a clean interf ace and β s the elasticity number . This lat- ter parameter quant ifies the strength of t he surf actant: for a fix ed concentration of surf actant, a higher sur - f ace tension reduction can be obtained with a stronger surf actant (higher elasticity number β s ). The dimensionless groups appearing in equation (7) are the po wer Re ynolds number , Re Π , and the W e- ber number , We Π , which are defined as: Re Π = ρu Π h η w , We Π = ρu 2 Π h σ 0 . (10) P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 22 The Reynolds number represents the ratio between inertial and viscous forces and is defined based on the viscosity of the two lubricating layers (w ater), η w , while the W eber number is the ratio between inertial and surface tension forces. The W eber number is here defined using the surface tension of a clean interf ace, σ 0 , as a reference. These two dimensionless param- eters, as well as the P ´ eclet numbers, are defined em- ploying u Π as a v elocity scale. As anticipated, in the present work we emplo y the CPI approach (Hasega wa et al. 2014; Roccon et al. 2019) , which is based on dri ving the flo w by a constant pumping po wer , P p . Naturally , to keep the pumping po wer constant o ver time, the mean pressure gradient is dynamically ad- justed according to the ov erall flo w rate, Q t . W ithin the CPI approach, the follo wing characteristic v eloc- ity is introduced: u Π = √ D  P p h 3 η w , (11) where η w is the water viscosity while D is a coef ficient used to account for the presence of a core layer with dif ferent viscosity in the central part of the channel. The gov erning equations (1)-(2)-(6) and (7) are solved using the open-source massi vely parallel spec- tral solver FLO W36 (Roccon et al. 2025) based on transforming the field v ariables into wa v enumber space via a combination of Fourier series (homoge- nous directions) and Chebyshe v polynomials (inho- mogeneous wall-normal direction). The collocations points for all v ariables (velocity and phase field) are equally spaced along the x and y directions while they are stretched along the w all-normal direction. Na vier -Stokes equations are solv ed using a wall- normal velocity-v orticity formulation while the phase field and surfactant concentration transport equations are solved in their original formulation. The computational domain consists of a plane channel with dimensions L x × L y × L z =4 πh × 2 πh × 2 h . W e run fi ve dif ferent simulations: a reference sim- ulation of a single-phase turb ulent channel flo w and four multiphase simulations of water -lubricated chan- nels. F or the water -lubricated cases, two lubricating layers, thickness h w =0 . 15 h , are injected in the near- wall re gion of the channel, so to fa vor the transport of a central layer of oil, thickness h o =1 . 7 h . All mul- tiphase simulations consider water as lubricating fluid and oil – in one case 10 times more viscous than wa- ter (viscosity ratio λ = η o /η w = 10 ) and in the other case 100 times more viscous than water (viscosity ra- tio λ = η o /η w = 100 ) – as transported fluid. For each viscosity ratio, we perform the simulation of a clean system and of a surfactant-laden system. 3 Results W e start by looking at the flow field in the lubri- cating layers, and at its connection with the beha vior of the interfacial w a ves. Figure 1 a sho ws an instanta- neous three dimensional sketch of the oil-w ater inter - face (referring to the case λ = 10 ). Figure 1 b, c sho ws a two dimensional vie w from the top of the interface (for λ = 10 , panel b , and λ = 100 , panel c ) while fig- ure 1 d, e shows the corresponding v olume rendering of the turb ulent kinetic energy , TKE =( u ′ 2 x + u ′ 2 y + u ′ 2 z ) / 2 , in the bottom lubricating layer . First, we examine the ef fect of the viscosity ra- tio. F or λ = 10 (figure 1 b, d ), we notice that the interface deformation is characterized by the stream- wise and spanwise propagation of w av es with dif fer- ent amplitude and wa v elength, which interact and gen- erate an highly irregular interf ace shape. W e define crests as interface locations f arther from the bottom wall and troughs as interf ace locations closer to the bottom wall (see figure 1 a ). As expected, the inter- face shape has a direct influence on the beha vior of the flo w field in the water lubricating layer (panel d ). In particular , where the interface has high crests and rough shape, turbulence seems to be acti ve in the lu- bricating layer; where the interface has deep troughs – and smoother shape –, turb ulence acti vity in the lubri- cating layer appears weak, since there is not enough room for the turb ulence cycle to be sustained and the flo w tends to laminarize. Overall, this results in the coexistence of laminar and turbulent re gions. These observ ations, which highlight the existing strong cor - relation between the interface deformation and turb u- lence acti vity , are consistent with pre vious in vestiga- tions sho wing turb ulence suppression and reacti v ation (Roccon et al. 2019 & 2021) depending on the lo- cal thickness of the lubricating layer . F or λ = 100 (figure 1 c, e ), we notice that the interface is smoother and more regular . Gi ven the relativ ely smaller inter - face deformation, crests and troughs are no w less pro- nounced, indicating a more uniform, and turb ulent, flo w field in the lubricating layer (panel e ). T o quantify the effects of these turb ulence mod- ifications on macroscopic flo w parameters, we com- pute the flo w rates and the pressure drop of the water - lubricated channels. Results are normalized by the corresponding single-phase flo w of oil. In figure 2 a , we plot the flo w-rate as a function of the applied pres- sure gradient, i.e. ( p x , Q t ) pairs. The black solid line indicates the v alue of the po wer (proportional to the product between p x and Q t ), which is kept constant among all simulations. In figure 2 b , we plot the be- ha vior of the pressure gradient, p x , and of the flo w-rate of the oil core and of the entire channel, Q o and Q t , respecti v ely , as a function of the viscosity ratio, λ . Looking at figure 2 a , we observe that the total flo w-rate, Q t , is always lar ger than that of the single- phase flo w . For λ = 10 , the oil flo w rate is 1 . 7 times lar ger than that of the corresponding single-phase flo w , while for λ = 100 , it is 4 times lar ger . As simula- tions are performed keeping constant the po wer input, an increase of the total flo w-rate corresponds to a de- crease of the applied pressure gradient ( p x ≃ 0 . 6 for Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 23 23 Bottom w all Bottom lubricating la y er In te rfacial w a v e s Flo w direc tion C r e s t s T r o u g h s y /h 0 π 2 π y /h x /h 0 π 2 π 0 π 2 π 3 π 4 π x /h 0 π 2 π 3 π 4 π TKE 0246 λ = 10 - clean λ = 100 - clean Interf acial w a v es Interf acial w a v es Rendering of TKE in the lubricating layer Rendering of TKE in the lubricating layer a ) b ) c ) d ) e ) Figure 1: P anel a : three dimensional vie w of the oil-w ater interf ace near the bottom w all for the case λ = 10 . P anels b, c : top vie w of the instantaneous deformation of the oil/w ater interf ace for the clean cases at λ = 10 ( b ) and λ = 100 ( c ), and corresponding v olume rendering (panel d for λ = 10 , panel e for λ = 100 ) of the turb ulence kinetic ener gy , TKE =( u ′ 2 x + u ′ 2 y + u ′ 2 z ) / 2 , in the bottom lubricating layer . P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 24 24 Bottom w all Bottom lubricating la y er In te rfacial w a v e s Flo w direc tion C r e s t s T r o u g h s y /h 0 π 2 π y /h x /h 0 π 2 π 0 π 2 π 3 π 4 π x /h 0 π 2 π 3 π 4 π TKE 0246 λ = 10 - clean λ = 100 - clean Interf acial w a v es Interf acial w a v es Rendering of TKE in the lubricating layer Rendering of TKE in the lubricating layer a ) b ) c ) d ) e ) Figure 1: P anel a : three dimensional vie w of the oil-w ater interf ace near the bottom w all for the case λ = 10 . P anels b, c : top vie w of the instantaneous deformation of the oil/w ater interf ace for the clean cases at λ = 10 ( b ) and λ = 100 ( c ), and corresponding v olume rendering (panel d for λ = 10 , panel e for λ = 100 ) of the turb ulence kinetic ener gy , TKE =( u ′ 2 x + u ′ 2 y + u ′ 2 z ) / 2 , in the bottom lubricating layer . P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 24 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 0 . 00 . 51 . 01 . 52 . 0 Q t Q t ( oil ) p x p x ( oil ) Cons tant pow er Singl e-pha se λ = 10 − clean λ = 10 − surf. λ = 100 − clean λ = 100 − surf. Drag Redu ction Drag In crease 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 . 0 10 100 λ Q t /Q t ( oil ) Q o /Q t ( oil ) p x /p x ( oil ) a ) b ) Figure 2: Steady-state values attained by the oil flo w-rate, Q o , total flo w-rate, Q t , and pressure gradient p x in physical units and normalized by the single-phase v alue of the corresponding oil. Panel a sho ws the achie ved flo w-rates against the required pressure gradient. P anel b sho ws the effect of the viscosity ratios on the flo w-rates and pressure gradient. λ = 10 , and p x ≃ 0 . 25 for λ = 100 ) . The presence of surfactants induces only minor dif ferences to the ov erall flo w-rates/pressure gradients: surfactant-laden cases exhibit slightly smaller flo w rates with respect to the surfactant-free counterparts. This ef fect is a bit more visible for λ = 10 while it v anishes for λ = 100 . Overall, these results indicate that a remarkable DR is observed in w ater -lubricated oil channels, which clearly depends on the considered viscosity ratio be- tween the transported fluid and the fluid used to lubri- cate the flo w . At the same time, the presence of sur- factants and impurities has only little ef fect on the drag reduction performances, and this ef fect tends to v anish when lar ger viscosity ratios are considered ( λ = 100 ). Finally , it is interesting to observe that the mechanisms leading to drag reduction in the lo w and high viscos- ity cases are slightly dif ferent: for λ = 10 , the larger amplitude of the interfacial w av es leads to the gener - ation of laminar patches, which induce lo wer strain rates and, together with the smaller water viscosity , leads to drag reduction. This DR mechanism resem- bles the one observed in our pre vious works (Roccon et al. 2019 & 2021) performed at moderate viscosity ratios with a single lubricating layer where turb ulence- interface interactions are the main f actor leading to DR. For λ = 100 , the former drag reduction mech- anism (generation of laminar patches) tends to v anish, and drag reduction is mainly dri v en by the small vis- cosity of the lubricating layer (water). Hence, com- pared to pre vious in vestigations (Roccon et al. 2019 & 2021), the observed DR mechanism is dif ferent and is mainly linked to the lo w viscosity of the lubricating layer . 4 Conclusions W e performed direct numerical simulations of a water -lubricated channel flo w , a flo w instance in which two near -w all thin lubricating layers of water are used to fa v or the transport of a more viscous fluid (e.g. oil). The simulations rely on DNS of turb ulence, cou- pled with a phase-field method, which is used to de- scribe the dynamics of the oil-water interf ace and the presence of surfactants/contaminants. W e considered a channel geometry , and by keeping the viscosity of the two lubricating layers fix ed (water), we considered two types of oil: an oil 10 times more viscous than water (viscosity ratio λ = η o /η w = 10 ) and an oil 100 times more viscous (viscosity ratio λ = 100 ). For each viscosity ratio, a clean and a surfactant-laden case ha ve been analyzed. By analyzing the drag reduction performance, we observe that oils, 10 and 100 times more viscous than water , can be transported can be transported applying a remarkably lo wer pressure gra- dient, p x . Specifically , compared to the corresponding single phase of oil, we observ ed a reduction do wn to p x /p x,sp ≃ 0 . 25 for λ = 100 . Present results high- light the potential of the water -lubricated technique in ef fecti v ely reducing the drag and thus fa v oring the transport of highly viscous fluids. Acknowledgments W e acknowledge PRA CE for awarding us access to HA WK at GCS@HLRS, Stuttgart, Germany , via the project 2020235507 (R UBIN). Refer ences B. A. T oms (1949), Some observ ations on the flo w of linear polymer solutions through straight tubes at large Re ynolds numbers, in Pr oc. 1st Intl Congr . Rheol. , 2, 135–141. F . F orrest and G. Grierson (1931), Friction losses in cast iron pipe carrying paper stock, P ap. T rade J . 92, 39. P . S. V irk (1975), Drag reduction fundamentals, AIChE J our - nal 21, 625. P . S. V irk, E. Merrill, H. Mickley , K. Smith, and E. Mollo- Christensen (1967), The toms phenomenon: turbulent pipe flo w of dilute polymer solutions, J . Fluid Mech. 30, 305. M. Quadrio, P . Ricco, and C. V iotti (2009), Streamwise- tra velling w av es of spanwise wall v elocity for turb ulent drag reduction, J . Fluid Mech. 627, 161. P . Ricco, M. Skote, and M. A. Leschziner (2021), A re vie w of turb ulent skin-friction drag reduction by near-w all trans- verse forcing, Pr og. Aer osp. Sci. 123, 100713. D. Joseph, M. Renardy , and Y . Renardy (1984), Instability of the flo w of two immiscible liquids with dif ferent viscosities in a pipe, J . Fluid Mech. 141, 309. H. Hu, T . Lundgren, and D. Joseph (1990), Stability of core- Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 25 25 ducted on a Coand ˘ a jet-equipped high-lift flap using perturbed compressible Na vier–Stokes equations, superimposed on a steady RANS background. This study incorporates the influence of small-scale turb u- lence generated by the Coand ˘ a jet. T o mitigate noise, a porous material (P A 80–110) was applied to the aft section of the flap. The application of homogeneous porous material resulted in enhanced flo w through the porous medium, which in-turn thickens the turb ulent boundary layer on the suction side of the flap. The thickened boundary layer f acilitated the de v elopment of larger eddies o v er the porous surface. Analysis of turb ulent sound sources re vealed that the porous treatment increased the po wer spectral density of surface pressure fluctuations across the spectrum, due to strong flo w through the material. Enhanced span-wise coherence was particularly notable between 4–8 kHz, while high-frequenc y coherence abov e 10 kHz persisted ov er lar ge separations, attrib uted to intense small-scale turb ulence at the fluid–porous interface. Streamwise correlations indicated a decel- eration of eddies, fa v oring noise suppression. T ime-av eraged flo w fields sho wed a weakened v ortex at the flap kink compared to the solid config- uration, reducing its acoustic contribution. Far -field analysis demonstrated clear noise reduction in the forward-do wnward direction (radiation directions θ between 180 and 270 degrees), both for trailing-edge noise alone and full-flap noise. In conclusion, the findings underscore the critical role of flo w-through and boundary layer thickening in noise mitigation, while highlighting curvature noise and the noise related to the span-wise v ortices as ke y tar gets for further noise reduction in circulation- controlled wings. Acknowledgments The research presented in this paper was made possible through the generous support of the Ham- b urg Uni versity of Applied Sciences. The authors also gratefully ackno wledge the North-German Supercom- puting Alliance (HLRN) for providing computational resources under project ID nii00132. Refer ences Akkermans, R.A.D., Bernick e, P ., Ewert, R. and Dierke, J. (2018), Zonal Overset-LES with Stochastic V olume Forcing, Int. J. Heat Fluid Flow , V ol. 70, pp. 336–347. Ananthan, V .B., Bernicke, P . and Akkermans, R.A.D. (2020), Aeroacoustic Analysis of a Circulation-Controlled High-Lift Flap by Zonal Overset Lar ge-Eddy Simulation. AIAA J . , V ol. 58, pp. 5294–5305. Bae, Y ., Jeong, Y .E. and Moon, Y . (2009), Effect of porous surface on the flat plate self-noise, AIAA 2009-3311. Bernicke, P ., Akkermans, R.A.D, Ananthan, V .B., Ewert, R., Dierke, J. and Rossian, L. (2019), A Zonal Noise Prediction Method for T railing-Edge Noise with a Porous Model, Int. J . Heat Fluid Flow , V ol. 80, pp. 108469. Brooks, T .F ., Pope, D.S. and Marcolini, M.A. (1989), Air- foil Self-Noise and Prediction N ASA Reference Publication 1218. Document ID: 19890016302. Delfs, J.W ., Faßmann, B., Lippitz, N., et al (2014), SFB 880: Aeroacoustic research for lo w noise take-of f and landing, CEAS Aer onaut. J. , V ol. 26, pp. 403–417. Englar , R.J. (2005), Overvie w of Circulation Control Pneu- matic Aerodynamics:Blo wn Force and Moment Augmenta- tion and Modification as Applied Primarily to Fixed-W ing Aircraft, In: Applications of Cir culation Contr ol T ec hnol- ogy , pp. 23–68. Ewert, R. and Schr ¨ oder , W . (2003), Acoustic perturbation equations based on flo w decomposition via source filtering, J . Comput. Phys. , V ol. 188, pp. 365–398. Ewert, R. (2008), Broadband slat noise prediction based on CAA and stochastic sound sources from a fast random particle-mesh (RPM) method, Comput. Fluids , V ol. 37, pp. 369–387. Francois, D., Radespiel, R. and Seeman, R. (2018), Nu- merical In vestigations of Spanwise-V aried Unsteady Coanda Actuation on High-Lift Configuration, J. Air cr . , V ol. 55, pp. 1720–1730. Fr ¨ ohlich, J. and von T erzi, D. (2008), Hybrid LES/RANS Methods for the Simulation of T urbulent Flo ws, P ro g . Aer osp. Sci. , V ol. 44, pp. 349–377. Geyer , T . and Sarradj, E. (2010), Porous airfoils: Noise re- duction and boundary layer ef fects, Int. J . Aer oacoustics , V ol. 9, pp. 787–820. Hu, F .Q., Hussaini, M.Y . and Manthey , J.L. (1996), Lo w- dissipation and lo w-dispersion runge-kutta schemes for computational acoustics, J. Comput. Phys. , V ol. 124, pp. 177–191. Kato, C., T akano, Y ., Fujitha, H. and Ike gaw a, M. (1993), Numerical prediction of aerodynamic noise radiated from lo w mach number turb ulent wake, AIAA 93-0145. Kumar , P . and Radespiel, R. (2017), Aerodynamic sensiti vi- ties of 2D high lift airfoil coanda flap configured with porous trailing edges, AIAA 2017-3562. Ma, Z. and Zhang, X. (2009), Numerical in v estigation of broadband slat noise attenuation with acoustic liner treat- ment, AIAA J . , V ol. 47, pp. 2812–2820. Nishino, T ., Hahn, S. and Sharif f, K. (2010), Large-eddy simulations of a turb ulent Coanda jet on a circulation con- trol airfoil, Phys. Fluids , V ol. 22. pp. 125105. Oberai, A.A., Roknaldin, F . and Hughes, T .J.R. (2002), Computation of trailing-edge noise due to turb ulent flo w ov er an airfoil, AIAA J . , V ol. 40, pp. 2206–2216. Rossian, L., Suryadi, A., Rossignol, K.S., Ewert, R., Herr , M. and Delfs, J. (2018), Numerical and experimental in- sights into the noise generation of a circulation control air- foil, AIAA 2018-3139. T am, C.K.W . and W ebb, J.C. (1993), Dispersion-relation- preserving finite dif ference schemes for computational acoustics. J . Comput. Phys. , V ol. 107, pp. 262–281. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 32 32 ducted on a Coand ˘ a jet-equipped high-lift flap using perturbed compressible Na vier–Stok es equations, superimposed on a steady RANS background. This study incorporates the influence of small- scale turb u- lence generat ed by the Coand ˘ a j et. T o mitig ate noise, a porous material (P A 80–110) w as applied to the aft section of the flap. The application of homogeneous porous material resulted in enhanced flo w through the porous medium, which in-turn thick ens the turb ulent boundary layer on the suction side of the flap. The thick ened boundary layer f acilitated the de v elopment of lar ger eddies o v er the porous surf ace. Analysis of turb ulent sound sources re v ealed that the porous treatment increased the po wer spectral density of surf ace pressure fluctuations across t he spectrum, due to strong flo w through the material. Enhanced span-wise coherence w as particularly notable between 4–8 kHz, while high-frequenc y coherence abo v e 10 kHz persisted o v er lar ge separations, attrib uted to intense small-scale turb ulence at the fluid–porous interf ace. Streamwise correlations indicated a decel- eration of eddies, f a v oring noise suppression. T ime-a v eraged flo w fields sho wed a weak ened v orte x at the flap kink compared to the solid config- uration, reducing its acoustic contrib ution. F ar -field analysis demonst rated clear noise reduction in the forw ard-do wnw ard dir ection (radiation directions θ between 180 and 270 de grees), both for trailing-edge noise alone and full-flap noise. In conclusion, the findings underscore the critical role of flo w-through and boundary layer thick ening in noise mitig ation, whil e highlighting curv ature noise and the noise related to the span-wise v ortices as k e y tar gets for further noise reduction in circulation- controlled wings. Ackno wledgments The research presented in this paper w as made possible through the generous support of the Ham- b ur g Uni v ersity of Applied Scienc es. The authors also gratefully ackno wledge the North-German Supercom- puting Alliance (HLRN) for pro viding computational resources under project ID nii00132. Refer ences Akk ermans, R.A.D., Bernick e, P ., Ewert, R. and Dierk e, J. (2018), Zonal Ov erset- LES with Stochastic V olume F orcing, Int. J . Heat Fluid Flow , V ol. 70, pp. 336–347. Ananthan, V .B., Bernick e, P . and Akk ermans, R.A.D. (2020), Aeroacoustic Analysis of a Circulation-Controlled High-Lift Flap by Zonal Ov erset Lar ge-Eddy Simulation. AIAA J . , V ol. 58, pp. 5294–5305. Bae, Y ., Jeong, Y .E. and Moon, Y . (2009), Ef fect of porous surf ace on the flat plate self-noise, AIAA 2009-3311. Bernick e, P ., Akk ermans, R.A.D, Ananthan, V .B., Ewert, R., Dierk e, J. and Rossian, L. (2019), A Zonal Noise Prediction Method for T railing-Edge Noise with a Porous Model, Int. J . Heat Fluid Flow , V ol. 80, pp. 108469. Brooks, T .F ., Pope, D.S. and Ma rcolini, M.A. (1989), Air - foil Self-Noise and Prediction N ASA Reference Publication 1218. Document ID: 19890016302. Delfs, J.W ., F aßma nn, B., Lippitz, N., et al (2014), SFB 880: Aeroacoustic research for lo w noise tak e-of f and landing, CEAS Aer onaut. J . , V ol. 26, pp. 403–417. Englar , R.J. (2005), Ov ervie w of Circulation Control Pneu- matic Aerodynamics:Blo wn F orce and Moment Augmenta- tion and Modification as Applied Primarily to Fix ed-W ing Aircraft, In: Applications of Cir culation Contr ol T ec hnol- og y , pp. 23–68. Ewert, R. and Schr ¨ oder , W . (2003), Acoustic perturbation equations based on flo w decomposition via source filtering, J . Comput. Phys. , V ol. 188, pp. 365–398. Ewert, R. (2008), Broadband slat noise prediction based on CAA and stochastic sound sources from a f ast random particle-mesh (RPM) method, Comput. Fluids , V ol. 37, pp. 369–387. Francois, D., Radespiel, R. and Seeman, R. (2018), Nu- merical In v estig ations of Spanwi se-V aried Unsteady Coanda Actuation on High-Lift Configuration, J . Air cr . , V ol. 55, pp. 1720–1730. Fr ¨ ohlich, J. and v on T erzi, D. (2008), Hybrid LES/RANS Methods for the Simulation of T urb ulent Flo ws, P ro g . Aer osp. Sci. , V ol. 44, pp. 349–377. Ge yer , T . and Sarradj, E. (2010), Porous airfoils: Noise re- duction and boundary layer ef fects, Int. J . Aer oacoustics , V ol. 9, pp. 787–820. Hu, F .Q., Huss aini, M.Y . and Manthe y , J.L. (1996), Lo w- dissipation and lo w-dispersion runge-kutta schemes for computational acoustics, J . Comput. Phys. , V ol. 124, pp. 177–191. Kato, C., T akano, Y ., Fujitha, H. and Ik e g a w a, M. (1993), Numerical prediction of aerodynamic noise radiated from lo w mach number turb ulent w ak e, AIAA 93-0145. K umar , P . and Radespiel, R. (2017), Aerodynamic sensiti vi- ties of 2D high lift airfoil coanda flap configured with porous trailing edges, AIAA 2017-3562. Ma, Z. and Zhang, X. (2009), Numerical in v estig ation of broadband slat noise attenuation with acoustic liner treat- ment, AIAA J . , V ol. 47, pp. 2812–2820. Nishino, T ., Hahn, S. and Sharif f, K. (2010), Lar ge-eddy simulations of a turb ulent Coanda jet on a circulation con- trol airfoil, Phys. Fluids , V ol. 22. pp. 125105. Oberai, A.A., Roknaldin, F . and Hughes, T .J.R. (2002), Computation of trailing-edge noise due to turb ulent flo w o v er an airfoil, AIAA J . , V ol. 40, pp. 2206–2216. Rossian, L., Suryadi, A., Rossignol, K.S., Ewert, R., Herr , M. and Delfs, J. (2018), Numerical and e xperimental in- sights into the noise generation of a circulation control air - foil, AIAA 2018-3139. T am, C.K.W . and W ebb, J.C. (1993), Dispersion-relation- preserving finite dif ference s chemes for computational acoustics. J . Comput. Phys. , V ol. 107, pp. 262–281. P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 32 T URBULENT F LO W O VER A PERMEABLE W ALL UNDER A RANGE OF R EYNOLDS NUMBERS W . Sado wski 1 and F . di Mare 1 1 Ruhr-Univ ersit ¨ at Bochum, Boc hum, Germany wojciec h.sadowski@ruhr -uni-bochum.de Abstract The flo ws o ver permeable, porous w alls can be en- countered in se v eral significant areas of research and engineering. The presence of such a wall impacts the flo w properties, shifting the shape of the adja- cent boundary layer , increasing generation of turbu- lence and modifying the skin friction drag. T o in- crease the current understanding of the influence of the permeable walls on the flo w-field and allow for fur - ther improv ements of the upscaled treatment of such flo ws, the current study considers a set of highly- resolved Lar ge Eddy Simulations in a channel half- filled with a porous medium, which is modelled by an array of cubes. Fi ve dif ferent turbulent flo w condi- tions are considered at a range of Reynolds numbers, Re bulk = 1774 – 5485 , where the results at the highest Re are v alidated against a reference Direct Numerical Simulation. Gathered data has been spatially filtered, to represent the a veraged v elocity inside and o ver the porous structure. A detailed study of the mean veloc- ity and turb ulent fluctuations abov e the top (smooth) and bottom (permeable) walls is presented, followed by a determination of the coef ficients for the correla- tions describing the velocity profile o v er the porous, permeable wall. 1 Intr oduction Flo ws o ver solid packings, or structures which can be modelled as such (e.g., trees, urban canopies or ri v er beds) are highly rele v ant for en vironmental re- search, process engineering and other technological applications. If one considers the influence of the packing on the flo w field, the solid phase can be de- scribed as a porous medium. The region between the freeboard (open space ov er the solid phase) and the packing, can be characterized as a permeable and rough wall, and it “couples” the freeboard with the porous region. Hence, it is often referred to as the por ous-fluid interface (PFI). In comparison to smooth non-permeable walls, PFI increases skin friction and permits the fluid to pass in- side the porous structure, thereby allo wing for a pres- ence of non-zero slip velocity on the PFI. Direct Nu- merical Simulation (DNS) studies (e.g., K uwata and Suga, 2016) ha v e indicated that the permeability of the interface results in an increase of turb ulence in- tensity and stresses in comparison to non-permeable walls with equi valent roughness. The presence of the porous medium influences the transport and distrib u- tion of turb ulent kinetic energy near the PFI, poten- tially leading to the transition to turb ulence at lo wer Reynolds numbers (Sug a et al. , 2010). Understanding the physics go verning the flo w around the PFI is of high importance, for example, to optimize the layout of shelter regions, during the de- sign and modelling of packed-bed reactors, for noise reduction in aerodynamics and for the design of heat shielding systems in aerospace applications. In all of the abov e, the difference between macroscopic and microscopic scales (i.e., de vice and pore scales) is typ- ically too lar ge to resolve the flo w field in between the elements of the solid phase. T o reduce the computa- tional ef fort, the flo w in the porous medium is often upscaled using a filtering procedure similar to the one adopted in Lar ge Eddy Simulations (LES), resulting in the equations for spatially-a veraged v elocity and pres- sure (Breugem and Boersma, 2005). In our pre vious work (Sado wski et al., 2023a,b), we ha ve focused on the mathematical formulation of the spatially-filtered equations at the PFI and the influence of the irre gular- ity of the porous matrix on the mean flo w properties. The results ha ve indicated that careful analysis of the upscaled pore-resolved simulations may pa ve the way to wards better understanding of macroscopic proper - ties of fluid flo ws in porous materials. The present study extends our pre vious efforts, by analysing the changes in the flo w structure and topol- ogy with v arying Re ynolds numbers, in a channel half- filled with a porous medium modelled by an array of cubes. Fi v e highly-resolved LESs are realized, char- acterized by the b ulk Reynolds number in the clear channel Re bulk = U bulk H /ν = 1774 , 2468 , 3155 , 4515 and 5485 . The largest Re ynolds number corre- sponds to the DNS study of Breugem and Boersma (2005) in the same geometry (hereafter referred to as BB2005), which is taken as a reference v erifying the adopted computational procedure. 2 Geometry & flow description The geometry of the computational domain is sho wn in Fig. 1 . The axes of the coordinate system, x , y and z , represent the streamwise, w all-normal, and spanwise direction, respectiv ely . The domain is peri- odic in both streamwise and spanwise directions. As indicated in Fig. 1 , H is the height of the chan- nel. The 5400 cubes with the size of H/ 20 , are ar - Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 33 33 x y z 3 H H H flo w direction domain width: 2 H Figure 1: The schematic of the studied geometry . The gray squares indicate positions of the cubes. ranged in 9 layers at the bottom of the channel, and are spaced e v ery H/ 10 in each direction. The centres of the cubes in the lo west layer are located 3 H/ 40 from the bottom wall. The nominal porosity (defined us- ing representati v e volume containing one cube) of the resultant porous matrix is ϕ c =0 . 875 . The nominal permeability can be characterized by a Darcy number Da c = K c /H 2 =3 . 4 × 10 − 4 , where K c is the perme- ability computed according to Breugem and Boersma (2005). The flo w conditions in the clear channel are de- fined by Re bulk = U bulk H /ν , where U bulk is the a v- erage velocity for y> 0 . For the ease of reproducibil- ity of the simulated flo w conditions, a total Reynolds number can also be defined as Re tot =2 U tot H /ν , where U tot is the a verage v elocity in the whole do- main. The non-dimensional numbers for all conducted simulations along with the reference DNS are gathered in T ab . 1 . The cases are referred to with the names starting with letters “Re” and the first two digits of Re tot , as listed in the same table. 3 Numerical method The setup of the simulations mirrors the configu- ration studied in our pre vious work, and will be de- scribed here briefly for completeness. For a detailed description, including the mesh sensiti vity study , the reader is referred to (Sado wski et al. , 2023a). The flo w is dri v en by a pressure gradient, which is automatically adjusted to ensure the correct b ulk ve- locity . Both spatial and temporal deri v ati v es hav e been discretized with second-order schemes. Considering the b ulk Reynolds number , the clear channel could, in principle, be resolved without the need to model sub-grid scales. On the other hand, the complex ge- ometry of the porous medium requires lar ge computa- tional ef fort (around 60% of cells are “placed” in the porous region). T o facilitate timely computations at v arious flo w conditions, a mesh entailing 31M cells a coarser resolution than the grid used for the BB2005 DNS was adopted for each case. Since, we e xpect that adopted mesh resolution might not meat the DNS requirements in certain parts of the domain, the ef fect of the sub-grid scales has been modelled by means of the W ALE model. The model is kept acti ve e v en for lo wer Re ynolds numbers to make sure that e v ery sim- ulation is performed with the same numerical setup, assuming that for lo w Re cases the influence of the added viscosity on results should weaken. All simulations are performed using the OpenF oam-v2312 toolbox. The fluctuating fields ha ve been a veraged in time (a veraging times are reported in T ab . 2 ) and subsequently in space, o ver a periodically repeating sector of the geometry . In the channel part of the domain, the time-av eraged statistics ha ve been a v eraged in the homogeneous directions. T o analyse the distribution of space-time a ver - aged quantities, the time-a veraged results ha ve also been filtered with a one-dimensional Cellular kernel G ( y )=( ℓ −| y | ) /ℓ 2 , with the kernel width set to ℓ = H/ 10 . The filtering operation is defined as ⟨ ψ ⟩ s =  Ω Gγ ψ d x , where γ is an indicator func- tion equal to 1 in the fluid and 0 otherwise. The re- sult, ⟨ ψ ⟩ s , is a superficial av erage of ψ . An alternativ e intrinsic a verage ⟨ ψ ⟩ can be obtained using Dupuit’ s relation, i.e., ⟨ ψ ⟩ s = ϕ ⟨ ψ ⟩ , where ϕ is the porosity distrib ution. Simulation quality assessment The con ver gence of computed statistics was as- sessed by e v aluating the momentum balance of each simulation. T able 2 also reports the near -wall spac- ing at the top wall and at the surf ace of the cubes, which are in line with typical recommendations for wall-resolv ed LES (Sagout, 2006; Chapman, 1979). Moreov er , the resolution of all simulations was as- sessed using two LES quality indices, the fraction of resolved turb ulent kinetic energy (Pope, 2000) IQ Po p e = u ′ i u ′ i 2 k sgs + u ′ i u ′ i , (1) and the resolution index by Celik (2005) based on the estimated ef fecti v e K olmogorov length scale: IQ η =  1+ α η  ∆ ε 1 / 4 eff ν 3 / 4  m  − 1 . (2) In the abov e, the sub-grid turb ulence kinetic ener gy is estimated as k sgs = ν 2 sgs / ( C w ∆) 2 and ef fecti ve dissi- pation as ε eff =( ν + ν sgs ) k sgs / ( C k ∆) 2 , where C w is the constant of the W ALE model, C k =0 . 0376 and α η =0 . 05 are empirical constants, and ∆ is the cell size (gi v en as cube root of the volume). For both indices, a time-a v eraged v alue abov e 0 . 8 suggests an adequate mesh resolution for LES. While the use of eq. ( 2 ) resulted in v alues greater than 0 . 9 in the whole domain, the resolved ener gy criterion gi v en by eq. ( 1 ) was not satisfied in small re gions of lo w velocity which is a kno wn feature of this quality index (for each simulation IQ Po p e < 0 . 8 was reported in less than 0 . 5% of cells). Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 34 34 x y z 3 H H H flo w direction domain width: 2 H Figure 1: The schematic of the studied geometry . The gray squares indicate positions of the cubes. ranged in 9 layers at the bottom of the channel, and are spaced e v ery H/ 10 in each direction. The centres of the cubes in the lo west layer are located 3 H/ 40 from the bottom w all. The nominal porosity (defined us- ing representati v e v olume containing one cube) of the resultant porous matrix is ϕ c =0 . 875 . The nominal permeability can be characterized by a Darc y number Da c = K c /H 2 =3 . 4 × 10 − 4 , where K c is the perme- ability computed according to Breugem and Boersma (2005). The flo w conditions in the clear channel are de- fined by Re bul k = U bul k H /ν , where U bul k is the a v- erage v elocity for y> 0 . F or the ease of reproducibil- ity of the simulated flo w conditions, a total Re ynolds number can also be defined as Re tot =2 U tot H /ν , where U tot is the a v erage v elocity in the whole do- main. The non-dimensional numbers for all conducted simulations along with the reference DNS are g athered in T ab . 1 . The cases are referred to with the names starting with letters “Re” and the first tw o digits of Re tot , as listed in the same table. 3 Numerical method The setup of the simulations mirrors the configu- ration studied in our pre vious w ork, and will be de- scribed here briefly for completeness. F or a detailed description, including the mesh sensiti vity study , the reader is referred to (Sado wski et al. , 2023a). The flo w is dri v en by a pressure gradient, which is automatically adj u s ted to ensure the correct b ulk v e- locity . Both spatial and temporal deri v ati v es ha v e been discretized with second-order schemes. Considering the b ulk Re ynolds number , the clear channel could, in principle, be resolv ed without the need to model sub-grid scales. On the other hand, the comple x ge- ometry of the porous medium requires lar ge computa- tional ef fort (around 60% of cells are “placed” in the porous re gion). T o f acilitate timely computations at v arious flo w conditions, a mesh entailing 31M cells a coarser resolution than the grid used for the BB2005 DNS w as adopted for each case. Since, we e xpect that adopted mesh resolution might not meat the DNS requirements in certain parts of the dom ain, the ef fect of the sub-grid scales has been modelled by m eans of the W ALE model. The model is k ept acti v e e v en for lo wer Re ynolds numbers to mak e sure that e v ery sim- ulation is performed with the same numerical setup, assuming that for lo w Re cases the influence of the added viscosity on results should weak en. All simulations are performed using the OpenF oam-v2312 toolbox. The fluctuat ing fields ha v e been a v eraged in time (a v eraging times are reported in T ab . 2 ) and subsequently in space, o v er a periodically repeating sector of the geometry . In the channel part of the domain, the time-a v eraged statistics ha v e been a v eraged in the homogeneous directions. T o analyse the distrib ution of space-time a v er - aged quantities, the time-a v eraged results ha v e also been filtered with a one-dimensional Cellular k ernel G ( y )= ( ℓ −| y | ) /ℓ 2 , with the k ernel width set to ℓ = H/ 10 . The filtering operation is defined as ⟨ ψ ⟩ s =  Ω Gγ ψ d x , where γ is an indicator func- tion equal to 1 in the fluid and 0 otherwise. The re- sult, ⟨ ψ ⟩ s , is a superficial a v erage of ψ . An alternati v e intrinsic a v erage ⟨ ψ ⟩ can be obtained using Dupuit’ s relation, i.e., ⟨ ψ ⟩ s = ϕ ⟨ ψ ⟩ , where ϕ is the porosity distrib ution. Simulation quality assessment The con v er gence of computed statistics w as as- sessed by e v aluating the momentum balance of each simulation. T able 2 also re po r ts the near -w all spac- ing at the top w all and at the surf ace of the cubes, which are in line with typical recommendations for w all-resolv ed LES (Sagout, 2006; Chapman, 1979). Moreo v er , the resolution of all simulations w as as- sessed using tw o LES quality indices, the fraction of resolv ed turb ulent kinetic ener gy (Pope, 2000) IQ Po p e = u ′ i u ′ i 2 k sg s + u ′ i u ′ i , (1) and the resolution inde x by Celik (2005) based on the estimated ef fecti v e K olmogoro v length scale: IQ η =  1+ α η  ∆ ε 1 / 4 eff ν 3 / 4  m  − 1 . (2) In the abo v e, the sub-grid turb ulence kinetic ener gy is estimated as k sg s = ν 2 sg s / ( C w ∆) 2 and ef fecti v e dissi- pation as ε eff =( ν + ν sg s ) k sg s / ( C k ∆) 2 , where C w is the constant of the W ALE model, C k =0 . 0376 and α η =0 . 05 are empirical constants, and ∆ is the cell size (gi v en as cube root of the v olume). F or both indices, a time-a v eraged v alue abo v e 0 . 8 suggests an adequate mesh resolution for LES. While the use of eq. ( 2 ) resulted in v alues greater than 0 . 9 in the whole domain, the resolv ed ener gy criterion gi v en by eq. ( 1 ) w as not satisfied in small re gions of lo w v elocity which i s a kno wn feature of this quality inde x (for each simulation IQ Po p e < 0 . 8 w as reported in less than 0 . 5% of cells). P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 34 T able 1: The macroscopic v ariables characterizing each conducted simulation and the reference DNS by Breugem and Boersma (2005) labelled as BB2005. The Reynolds numbers are defined as Re bulk = U bulk H /ν , Re tot =2 U tot H /ν , Re p τ = u p τ H /ν , Re t δ = u t τ δ t /ν , where U tot and U bulk are av erage velocities in the whole domain and in the clear channel, respecti vely . The, skin friction coefficients are defined as C p,t f = 2( u p,t τ /U bulk ) 2 . The columns on the right list the parameters of the fitted velocity profile correlations gi ven by eq. ( 4 ) and ( 6 ). Name Re tot Re bulk Re p τ Re t δ Re K C t f × 10 − 3 C p f × 10 − 3 κd p + 0 h p + r α Re63 6360 5485 684 98 12.4 10.280 31.173 0.164 314.86 141.00 0.25 Re52 5200 4515 553 87 10.0 10.761 30.016 0.209 179.77 63.00 0.27 Re36 3600 3155 383 68 6.9 11.655 29.575 0.242 99.76 28.00 0.30 Re28 2800 2468 296 58 5.4 12.282 28.803 0.282 59.23 13.00 0.33 Re20 2000 1774 202 49 3.7 13.313 25.928 0.310 31.33 5.20 0.38 BB2005 - 5500 669 - 12.4 10.264 29.6 - - - - T able 2: The av eraging times for all simulations reported in flo w through times ( T ft = L x /U bulk ), along with the maxima of near wall spacing in w all units at the top wall and the cubes. Name T av g T ft top wall cubes x + y + z + x + y + z + Re63 521 2.5 0.5 2.5 7.5 1.4 6.4 Re52 409 2.1 0.4 2.1 6.4 1.2 5.6 Re36 555 1.5 0.3 1.5 5.0 1.0 4.6 Re28 351 1.2 0.2 1.2 3.9 0.8 3.6 Re20 595 0.9 0.2 0.9 3.0 0.6 2.8 The scatter plot of cell IQ Po p e vs. the correspond- ing time-a veraged contrib ution of sub-grid scale vis- cosity is visible in Fig. 2 . For most “under -resolved” cells, the added viscosity is considerably smaller than the fluid’ s viscosity . There are, ho we ver , a couple of cells in which ν sgs /ν ≈ 0 . 2 . For these cells, the vis- cous stresses are considerably influenced by the pres- ence of the model, ho we ver , as they are not numerous, we assume that they do not impact the ov erall simula- tion fidelity . 4 Results The friction velocity o v er the porous wall can be computed using the global force balance in the clear channel (Kuw ata and Suga, 2016), u p τ =  − ( u t τ ) 2 + ∂P ∂x H, (3) where u t τ =  τ w ( y = H ) is the friction velocity on the top wall and ∂P / ∂x is the mean pressure gradient dri ving the flo w . Both friction velocities can be used to compute the width of the boundary layer o ver the porous wall, δ p /H =( u p τ ) 2 / { ( u p τ ) 2 +( u t τ ) 2 } and top wall as δ t /H =1 − δ t /H . Hereafter , the superscripts ( · ) p + and ( · ) t + represents v alues normalized using u p τ and u t τ , respecti v ely . 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 IQ Pop e 10 − 3 10 − 2 10 − 1 ν sgs /ν Re63 Re52 Re36 Re28 Re20 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 | u | /U bulk Figure 2: The scatter plot of cell IQ Po p e vs. the correspond- ing time-a veraged contrib ution of sub-grid scale viscosity . The points are colored by the magnitude of the time-a veraged v elocity . Comparing the v alues of the friction coef ficients C p,t f = 2( u p,t τ /U bulk ) 2 between simulations Re63 and BB2005 (see T ab . 1 ), allo ws for the initial v alidation of the numerical setup. The computed v alue of C t f is very close to the one reported for BB2005, while the friction coef ficient at the permeable wall is slightly higher . Importantly , Breugem and Boersma (2005) ev alu- ated u p τ directly from the balance of filtered quanti- ties as ( u p τ ) 2 = −⟨ u ′ w ′ ⟩ + ν∂ ⟨ u ⟩ / ∂y . If we fol- lo w the same procedure, we recov er similar v alues of C p f = 29 . 842 and Re p τ = 669 . 8 . In our opinion, the use of eq. ( 3 ) should be preferred for comparison of dif ferent flo w conditions, as the alternativ e is inher - ently influenced by the filter size. This is especially important near PFI as turb ulent and viscous stresses v ary rapidly in this re gion. Double-a veraged v elocity in the channel The profiles of the double-a veraged streamwise velocity and its fluctuations, computed as u rms =  ⟨ u ′ u ′ ⟩ s , are sho wn in Fig. 3 . First and foremost, the results for the highest Reynolds number are in an e x- cellent agreement with the reference DNS, v alidating Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 35 35 0 0.5 1  u  s /U bulk − 1 0 1 y /H Re20 Re28 Re36 Re52 Re63 BB2005 0 1 2 3 u t + rms 0 0.5 1 1.5 2 v t + rms 0 1 2 w t + rms Figure 3: The double-av eraged velocity and v elocity fluctuations. The legend labels refer to the simulations described in T a- ble 1 . The dashed line with open circles represents the DNS data by Breugem and Boersma (2005). 0 . 0 0 . 5 1 . 0 1 . 5 u p + rms Re p τ 202 296 384 553 684 669 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 y /δ p 0 . 0 0 . 5 1 . 0 v p + rms Figure 4: V elocity fluctuations over the preamble layer .The legend labels refer to the simulations described in T able 1 . The dashed line with open circles rep- resents the DNS data by Breugem and Boersma (2005). performed computations. The presence of the permeable wall leads to much lar ger skin friction than at the top, smooth wall re- sulting in an asymmetrical velocity profile in the clear channel. The decrease of the Reynolds number results in a do wnwards shift of the maximum of the v elocity and lo wer normalized v elocity in the porous region. As expected, the velocity fluctuations decrease with the Reynolds number , both in the clear and porous part of the domain. Interestingly , especially for the higher Reynolds numbers, the streamwise fluctuations seem to penetrate deep into the porous medium—a high v alue of u rms can be still observed near the bottom of the channel (a small peak in the pro- file of u rms is a result of filtering procedure). Near the interface, in the porous medium, both spanwise and streamwise components of fluctuations are compara- ble to what is observed near the top w all, indicating presence of strong macroscopic turb ulence. When fluctuations are scaled with u p τ (Fig. 4 ), the wall-normal component decreases as flo w veloc- ity is reduced, ho we ver , this effect is re versed for u p + rms 10 0 10 1 10 2 ( H − y ) t + 0 5 10 15 20 u t + Re t δ = u t τ δ t /ν 49 57 68 87 98 Re τ 64 80 110 180 Figure 5: The velocity profiles ov er the top (smooth) wall in wall units. The dashed lines represent the DNS data gathered by Tsukahara et al. (2014). abov e the porous wall—the normalized fluctuations are stronger for lo wer Re p τ . Moreov er , the profiles seem to ov erlap in the core flo w region. Boundary layer near the top wall Figure 5 presents the normalized velocity profile ov er the top w all, together with the data gathered by Tsukahara et al. (2014) for lo w Re τ channel flows ( Re τ = u τ δ /ν , where δ is the channel half-width or the width of the boundary layer). In their in vestigation, the authors ha ve reported that a weak turb ulent flo w can be sustained at a minimum Re τ = 64 . Abo ve this Reynolds number , the turbulence appears in a form of intermittent “puf f” structures, which can be consis- tently observed for Re τ = 80 . Their presence and interactions with regions of laminar flo w increases the mean velocity profile in the core of the channel in com- parison to higher Re τ . This is in stark contrast to our results, labelled us- ing the Reynolds number Re t δ = u t τ δ t /ν , which is also based on the boundary layer width. Although, the “full-fledged” logarithmic layer is not discernible in the computed results, the profiles remain close to (1 / 0 . 41) ln( y + ) correlation. This might indicate that, although the flo w near the top wall is characterized by a relati v ely small Re t δ it remains fully turb ulent unlike Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 36 36 0 0.5 1  u  s /U bulk − 1 0 1 y /H Re20 Re28 Re36 Re52 Re63 BB2005 0 1 2 3 u t + rms 0 0.5 1 1.5 2 v t + rms 012 w t + rms Figure 3: The double-a v eraged v elocity and v elocity fluctuations. The le gend labels refer to the simulations described in T a- ble 1 . The dashed line with open circles represents the DNS data by Breugem and Boersma (2005). 0 . 0 0 . 5 1 . 0 1 . 5 u p + rms Re p τ 202 296 384 553 684 669 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 y /δ p 0 . 0 0 . 5 1 . 0 v p + rms Figure 4: V elocity fluctuations o v er the preamble layer .The le gend labels refer to the simulations described in T able 1 . The dashed line with open circles rep- resents the DNS data by Breugem and Boersma (2005). performed computations. The presence of the permeable w all leads to much lar ger skin friction than at the top, smooth w all re- sulting in an asymmetrical v elocity profile in the clear channel. The decrease of the Re ynolds number results in a do wnw ards shift of the maximum of the v elocity and lo wer normalized v elocity in the porous re gion. As e xpected, the v elocity fluctuations decrease with the Re ynolds number , both in the clear and porous part of the domain. Interestingly , especially for the higher Re ynolds numbers, the streamwise fluctuations seem to penetrate deep into the porous medium—a high v alue of u rms can be still observ ed near the bottom of the channel (a small peak in the pro- file of u rms is a result of filtering procedure). Near the interf ace, in the porous medium, both spanwise and streamwise components of fluctuations are compara- ble to what is observ ed near the top w all, indicating presence of strong macroscopic turb ulence. When fluctuations are scaled with u p τ (Fig. 4 ), the w all-normal component decreases as flo w v eloc- ity is reduced, ho we v er , this ef fect is re v ersed for u p + rms 10 0 10 1 10 2 ( H − y ) t + 0 5 10 15 20 u t + Re t δ = u t τ δ t /ν 49 57 68 87 98 Re τ 64 80 110 180 Figure 5: The v elocity profiles o v er the top (smooth) w all in w all units. The dashed lines represent the DNS data g athered by Tsukahara et al. (2014). abo v e the porous w all—the normalized fluctuations are stronger for lo wer Re p τ . Moreo v er , the profiles seem to o v erlap in the core flo w re gion. Boundary lay er near the top wall Figure 5 presents the normalized v eloci ty profile o v er the top w all, together with the data g athered by Tsukahara et al. (2014) for lo w Re τ channel flo ws ( Re τ = u τ δ /ν , where δ is the channel half-width or the width of the boundary layer). In their in v estig ation, the authors ha v e reported that a weak turb ulent flo w can be sustained at a minimum Re τ = 64 . Abo v e this Re ynolds number , the turb ulence appears in a form of intermittent “puf f” structures, which can be consis- tently observ ed for Re τ = 80 . Their presence and interactions with re gions of laminar flo w increases the mean v elocity profile in the core of the channel in com- parison to higher Re τ . This is in stark contrast to our results, labelled us- ing the Re ynolds number Re t δ = u t τ δ t /ν , which is also based on the boundary layer width. Although, the “full-fledged” log arithmic layer is not discernible in the computed results, the profiles remain close to (1 / 0 . 41) l n ( y + ) correlation. This might indicate that, although the flo w near the top w al l is characterized by a relati v ely small Re t δ it remains fully turb ulent unlik e P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 36 0 2 4 6 u  u  t + Re t δ 49 57 68 87 98 0 . 00 0 . 25 0 . 50 0 . 75 v  v  t + Re τ 64 80 110 180 20 40 60 80 100 ( H − y ) t + 0 . 00 0 . 25 0 . 50 0 . 75 1 . 00 1 . 25 w  w  t + Figure 6: The profiles of the velocity fluctuations near the top wall, compared to the data of Tsukahara et al. (2014). in the smooth-walled channels. This can be further analysed by comparing the profiles of the velocity fluctuations plotted in Fig. 6 . While the streamwise fluctuations appear to be dis- trib uted similarly to what is observed in smooth- walled channel flo ws, the spanwise component seems to be consistently greater in our results than in corre- sponding Re τ data by Tsukahara et al. (2014). Ho w- e v er , the most important differences relate to the w all- normal fluctuations. The distrib ution of v ′ v ′ t + has a dif ferent shape with the magnitude of the fluctuations decreasing monotonically to wards the w all. Near the wall it- self, the profile of seems to correspond well to what is observed in smooth-w alled channels at higher Re τ , ho we v er , as y + gro ws, the magnitude of fluctuation increases to wards what is observ ed o ver the perme- able wall (Fig. 3 ). Crucially , for the lo west of the computed Reynolds numbers, the fluctuations are still higher than in the Re τ = 64 case. This change can potentially be attrib uted to the K elvin–Helmhotz insta- bility , originating from the porous wall and amplifying the fluctuations in the wall-normal direction. Boundary layer near the permeable wall Follo wing Breugem et al. (2006) and Kuw ata and Suga (2016), the logarithmic velocity profile abo v e the permeable wall reads u p + = 1 κ log y + d 0 h r (4) where, the intercept of the log-law κ is no longer 0 25 50 75 100 125 150 175 y p + 2 3 4 5 6 7 Ξ Log. la y er: y + ∈ (30 , 150) Re63 y + ∈ (20 , 140) Re52 y + ∈ (10 , 120) Re36 y + ∈ (7 . 5 , 100) Re28 y + ∈ (5 , 60) Re20 Figure 7: The profiles of the indicator function Ξ , eq. ( 5 ), for each simulation along with the ranges of y + used for the least-squares fit of the 1 /κ (repre- sented with dashed lines). constant and d 0 and h r are zero-plane displacement and equi v alent roughness height, respecti vely . In both works, authors apply eq. ( 4 ) to area-a veraged data, and they reco v er the coef ficient via a fitting procedure. A so-called indicator function Ξ is used Ξ=( y p + + d p + 0 ) ∂ u p + ∂y p + (5) which for a correct v alue of d 0 results in a region of y p + , for which Ξ is constant and equal to 1 /κ . This range of y p + corresponds to the logarithmic layer . The profiles of Ξ are plotted in Fig. 7 , along with the ranges of y p + which were used for the fitting procedure and can be considered the extent of the log arithmic re- gion in the boundary layer (the fitted parameters are reported in T ab . 1 ). Although, eq. ( 5 ) allows for the determination for both d 0 and κ , resulting values are sensiti ve to a se- lected fitting region. Moreo ver , as pointed out by Monke witz (2024), e ven for turb ulent boundary lay- ers ov er smooth surf aces at high Reynolds numbers (where d 0 =0 ), no truly flat region in Ξ can be ob- served using current DNS data, resulting in estimation of v on K ´ arm ´ an constant between 0 . 435 and 0 . 385 . Al- though the discussion about the uni v ersality of the law of the wall is be yond the scope of the current work, it is important to keep in mind these both limitations of the use of the indicator function to determine the v alues of κ . Furthermore, following Brugem et al. (2006), the velocity profile inside the porous re gion can be ap- proximated as ⟨ u ⟩− U D U i − U D = exp  α y − y i H  ϕ c Da c  , (6) where, α is the fitting constant, y i is the position of the interface (here assumed as the coordinate where ϕ = Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 37 37 10 0 10 1 10 2 10 3 ( y + d ) p + 0 2 4 6 8 10  u  p + Re K = u p τ √ K c /ν 3 . 7 5 . 4 6 . 9 10 . 0 12 . 4 Figure 8: The velocity profiles o ver the permeable w all, along with fitted velocity profiles. ϕ c , y i = − 3 H/ 20 ), U i is the double-a veraged v elocity at y i and U D is the velocity inside the homogeneous porous medium. Both fitted correlations are presented in Fig. 8 along with v olume-a veraged v elocity profiles. The re- sultant v alues of the parameters are gi v en in T ab . 1 . Although, the law of the w all gi ven by eq. ( 4 ) de- scribes the time-a veraged v elocity , since the filter size is small in comparison to δ p , the filtering operation does not change the velocity profile to a great e xtent and ⟨ u ⟩ p + can be represented well by eq. ( 4 ). The obtained v alues of κ , d 0 and h r correspond relati v ely well to the trends observed in the literature (see for e x- ample, Brugem et al. , 2006; Suga et al. , 2010; K uwata and Suga, 2016) and sho w weak dependence on the permeability Reynolds number . The exponential fit for the v elocity at the PFI gi ven by eq. ( 6 ) also fits the velocity profile with remark- able accuracy , howe ver , we hav e observ ed that better match is present for the higher Re cases. The v alues of α increase as the permeability Reynolds number is de- creased, in line width observations by Breugem et al. (2006) that the v alue of α =1 should be the limit in creeping flo w . The same correlation for ⟨ u ⟩ was also tested by Kuw ata and Suga (2016), who ha v e reported that it was not suitable to properly represent their data, which was a v eraged in spanwise and streamwise di- rections, i.e., the authors ha v e not performed spatial filtering in the wall-normal direction. W e would like to remark, that the filtering procedure was necessary for eq. ( 6 ) to fit gathered data well. 5 Conclusions and outlook The results of highly resolved Lar ge Eddy Simu- lations of flo ws o ver porous w alls provide an insight into the Reynolds number dependence of the turb u- lent flo w at the porous-fluid interface. Due to the presence of the permeable wall, the flow field is ob- served to sustain turb ulence ev en at a relati v ely lo w Reynolds number , which is possibly a result of large scale K elvin–Helmhotz instability originating from the porous-fluid interface and magnifying w all-normal fluctuations near the top wall of the channel. The prop- erties of the boundary layer ov er permeable w all hav e also been examined, including the space-time av er- aged velocity fluctuations and the shape of the mean velocity profile near the interf ace. In the future, the log-law coef ficients gathered in current study , along with the data already present in the literature, and aug- mented with results from geometries with dif ferent v alues of ϕ c and K c , will be used to improv e the cur - rent characterization of the turb ulent flo ws ov er porous surfaces. Acknowledgments This work w as funded by the Deutsche Forschungs- gemeinschaft (DFG, German Research Foundation) – Project-ID 422037413 - TRR 287. Refer ences Breugem, W .P . and Boersma, B.J. (2005), Direct numerical simulations of turb ulent flow o v er a permeable wall using a direct and a continuum approach. Phys. Fluids Breugem, W . P ., Boersma, B. J. and Uittenbogaard, R. E. (2006), The influence of wall permeability on turb ulent channel flo w . J. Fluid Mec h. Celik, I. B., Cehreli, Z. N., and Y avuz, I. (2005). Index of Resolution Quality for Large Eddy Simulations. J. Fluids Eng. , 127(5), 949–958. Celik, I., Klein, M., and Janicka, J. (2009). Assessment Measures for Engineering LES Applications. ASME. J . Flu- ids Eng. March 2009; 131(3): 031102. Chapman, D. R. (1979). Computational Aerodynamics De- velopment and Outlook. AIAA Journal, 17(12), 1293–1313. Kuw ata, Y . and Suga, K. (2016), Lattice Boltzmann direct numerical simulation of interface turb ulence o ver porous and rough walls. Int. J . Heat Fluid Flow . Monke witz, P . A. (2024). On the dif ficulty of determin- ing K ´ arm ´ an “constants” from direct numerical simulations. Phys. Fluids , 36(4), 045162. Pope, S. B. (2000). T urb ulent flo ws. Cambridge Univ ersity Press. Sado wski, W ., Sayyari, M., di Mare, F . and Marschall, H. (2023a), Large eddy simulation of flo w in porous media: Analysis of the commutation error of the double-a veraged equations. Phys. Fluids Sado wski, W ., Sayyari, M. and di Mare, F . (2023b), Large- eddy simulation of a channel flo w ov er an irre gular porous matrix. Pr oc. Appl. Math. and Mech. Sagaut, P . (2006). Large eddy simulation for incompressible flo ws: An introduction (3rd ed). Springer . Suga, K., Matsumura, Y ., Ashitaka, Y ., T ominaga, S., and Kaneda, M. (2010). Ef fects of wall permeability on turb u- lence. Int. J. Heat Fluid Flow , 31(6), 974–984. Tsukahara, T ., Seki, Y ., Kawamura, H. and T ochio, D. (2014), DNS of turbulent channel flo w at very lo w Reynolds numbers. arXiv:1406.0248 . Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 38 38 10 0 10 1 10 2 10 3 ( y + d ) p + 0 2 4 6 8 10  u  p + Re K = u p τ √ K c /ν 3 . 7 5 . 4 6 . 9 10 . 0 12 . 4 Figure 8: The v elocity profiles o v er the permeable w all, along with fitted v elocity profiles. ϕ c , y i = − 3 H/ 20 ), U i is the double-a v eraged v elocity at y i and U D is the v elocity ins ide the homogeneous porous medium. Both fitted correlations are presented in Fig. 8 along with v olume-a v eraged v elocity profiles. The re- sultant v alues of the parameters are gi v en in T ab . 1 . Although, the la w of the w all gi v en by eq. ( 4 ) de- scribes the time-a v eraged v elocity , since the filter size is small in comparison to δ p , the filtering operation does not change the v elocity profile to a great e xtent and ⟨ u ⟩ p + can be represented well by eq. ( 4 ). The obtained v alues of κ , d 0 and h r correspond relati v ely well to the trends observ ed in the literature (see for e x- ample, Brugem et al. , 2006; Sug a et al. , 2010; K uw ata and Sug a, 2016) and sho w weak dependence on the permeability Re ynolds number . The e xponential fit for the v elocity at the PFI gi v en by eq. ( 6 ) also fits the v elocity profile with remark- able accurac y , ho we v er , we ha v e observ ed that better match is present for the higher Re cases. The v alues of α increase as the permeability Re ynolds number is de- creased, in line width observ ations by Breugem et al. (2006) that the v alue of α =1 should be the limit in creeping flo w . The sam e correlation for ⟨ u ⟩ w as also tested by K uw ata and Sug a (2016), who ha v e reported that it w as not suitable to properly represent their data, which w as a v eraged in spanwise and streamwise di- rections, i.e., the authors ha v e not performed spatial filtering in the w all-normal direction. W e w ould lik e to remark, that the filtering procedure w as necessary for eq. ( 6 ) to fit g athered data well. 5 Conclusions and outlook The results of highly resolv ed Lar ge Eddy Simu- lations of flo ws o v er porous w alls pro vide an insight into the Re ynolds n um ber dependence of the turb u- lent flo w at the porous-fluid interf ace. Due to the presence of t he permeable w all, the flo w field is ob- serv ed to sustain turb ulence e v en at a relati v ely lo w Re ynolds number , which is possibly a result of lar ge scale K elvin–Helmhotz instability originating from the porous-fluid interf ace and magnifying w all-normal fluctuations near the top w all of the channel. The prop- erties of the boundary layer o v er permeable w all ha v e also been e xamined, including the space-time a v er - aged v elocity fluctuations and the shape of the mean v elocity profile near the interf ace. In the future, the log-la w coef ficients g athered in current study , along with the data already present in the literature, and aug- mented with results from geometries with dif ferent v alues of ϕ c and K c , will be used to impro v e the cur - rent characterization of the turb ulent flo ws o v er porous surf aces. Ackno wledgments This w ork w as funded by the Deutsche F orschungs- gemeinschaft (DFG, German Researc h F oundation) – Project-ID 422037413 - TRR 287. Refer ences Breugem, W .P . and Boersma, B.J. (2005), Direct numerical simulations of turb ulent flo w o v er a permeable w all using a direct and a continuum approach. Phys. Fluids Breugem, W . P ., Boersma, B. J. and Uittenbog aard, R. E. (2006), The influence of w all permeability on turb ulent channel flo w . J . Fluid Mec h. Celik, I. B., Cehreli, Z. N., and Y a vuz, I. (2005). Inde x of Resolution Quality for Lar ge Eddy Simulations . J . Fluids Eng . , 127(5), 949–958. Celik, I., Klein, M., and Janicka, J. (2009). Assessment Measures for Engineering LES Appli cations. A SME. J . Flu- ids Eng . March 2009; 131(3): 031102. Chapman, D. R. (1979). Computational Aerodynamics De- v elopment and Outlook. AIAA Journal, 17(12), 1293–1313. K uw ata, Y . and Sug a, K. (2016), Lattice Boltzmann direct numerical simulation of interf ace turb ulence o v er porous and rough w alls. Int. J . Heat Fluid Flow . Monk e witz, P . A. (2024). On the dif ficulty of determin- ing K ´ arm ´ an “constants” from direct numerical simulations. Phys. Fluids , 36(4), 045162. Pope, S. B. (2000). T urb ulent flo ws. Cambridge Uni v ersity Press. Sado wski, W ., Sayyari, M., di Mare, F . and Marschall, H. (2023a), Lar ge eddy simulation of flo w in porous media: Analysis of the commutation error of the double-a v eraged equations. Phys. Fluids Sado wski, W ., Sayyari, M. and di Mare, F . (2023b), Lar ge- eddy simulation of a channel flo w o v er an irre gular porous matrix. Pr oc. Appl. Math. and Mec h. Sag aut, P . (2006). Lar ge eddy simulation for incompressible flo ws: An introduction (3rd ed). Springer . Sug a, K., Matsumura, Y ., Ashitaka, Y ., T ominag a, S., and Kaneda, M. (2010). Ef fects of w all permeability on turb u- lence. Int. J . Heat Fluid Flow , 31(6), 974–984. Tsukahara, T ., Seki, Y ., Ka w amura, H. and T ochio, D. (2014), DNS of turb ulent channel flo w at v ery lo w Re ynolds numbers. arXiv:1406.0248 . P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 38 RANS MUL TI - SCALE TURBULENCE MODEL LING AND ITS APPLICA TION TO HIGH FREESTREAM TURB ULENCE BOUND AR Y LA YERS S. Coroama 1 , J. V aquero 1 , N. Renard 1 and F . Chedev ergne 2 1 D AAA, ONERA, Institut P olytec hnique de P aris, 92190, Meudon, F rance 2 DMPE, ONERA, Univ ersit ´ e de T oulouse, 31000, T oulouse, F rance stefan.cor [email protected] Abstract The present paper aims at describing the de- velopment rationale behind a tw o-scale k − ω RANS (Reynolds-A veraged Na vier -Stokes) turb u- lence model. This model is designed to cope with the dif ficulties of classical single-scale models on the prediction of turb ulent boundary layers under the in- fluence of high free-stream turb ulence (FST). From a detailed literature re vie w , two physical phenomena are spotted and careful modelling is proposed by using the possibilities of fered by the multi-scale RANS frame- work. The numerical results obtained with the new model on a flat plate under strong FST with a wide range of turb ulence le vels demonstrate the impro v ed skin friction predictions attained when compared to a k − ω single-scale model. 1 Intr oduction T urb ulent boundary layers (TBLs) e volving under the influence of a highly turb ulent free-stream are commonly encountered in industrial flo w config- urations such as turbomachinary flo ws. Indeed, high pressure turbine blades usually operate with a free-stream turb ulence (FST) intensity up to 30% (Macieje wski and Mof fat (1992)). From no w on, the FST intensity is quantified by the parameter T u ∞ =  u ′ 2 ∞ /U ∞ where u ′ 2 ∞ and U ∞ denote the free-stream streamwise Reynolds stress and streamwise velocity respecti vely . The effects of the interaction between a TBL and a lar ge-scale, high-intensity FST ha ve been e xtensi v ely studied in the literature and it has been reported that the skin friction and heat transfer are greatly increased when compared to a canonical TBL (Thole and Bogard (1996), Esteban et al (2017)). Furthermore, it has been observed that the inner -scaled mean streamwise velocity profile U + , where U is the mean streamwise velocity and the superscript + refers to the classical inner -scaling using the friction velocity u τ ( =  τ w /ρ with τ w the wall-shear stress and ρ the density), is unchanged in the viscous sublayer , b uf fer layer and logarithmic re gions of the TBL no matter the strength of the FST whereas the wak e region is strongly af fected (Sharp et al (2009), Dogan et al (2016)). The penetration of high intensity FST in the TBL has also a considerable impact on the diagonal Re ynolds stresses which tend to greatly increase in the wak e region (Stefes and Fernholz (2004)). Moreov er , there is a strong interaction between the inner and outer region for the streamwise stress u ′ 2 as this component is seen to increase e v en in the inner layer (Dogan et al (2016)) similarly to what is observ ed for canonical TBLs at high Reynolds numbers (Smits et al (2021)). Further insights about the beha viour of the dif ferent turb ulent scales in volv ed in the interaction between the TBL and the FST is gained by performing a spectral analysis of the streamwise Reynolds stress. Such studies highlight the distinct beha viour of the small and lar ge turbulent scales of w all turb ulence in volv ed in TBL ’ s dev eloping under strong FST . First, it has been pointed out that an outer peak with high intensity emer ges in the u ′ 2 spectra when a high le vel of FST is present and that this peak is associated with high wa v elength turb ulent structures (Hearst et al (2018), Jooss et al (2021)). Second, the inner peak, associated with small wa velengths, w as found to remain unchanged by the FST intensity while a subsequent increase in the contrib ution of large scales in the inner region e xists. Finally , the scale separation computed by Dogan et al (2016), by applying a sharp spectral cut-of f filter , further demonstrates the distinct beha viour of small and large scales. Indeed, their study indicates that the large-sc ale turbulent structures coming from the intense lar ge-scale FST seem to penetrate into the TBL all the way to the w all while no influence of the FST is found on the small-scale velocity v ariance. Accurate numerical predictions of the skin friction and wall heat transfer coef ficient enhancement for TBLs under a high FST intensity are highly v alu- able for design purposes of high pressure turbine blades. The multi-scale RANS (Reynolds-A veraged Na vier -Stokes) approach to turb ulence modelling appears to of fer a sound frame work to cope with the dif ficulties of classical one-scale turb ulence models Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 39 39 Figure 1: Left - profiles of the inner -scaled streamwise Reynolds stress u ′ 2 + in the TBL for dif ferent v alues of T u ∞ . Middle - small-scale ( λ + x < 4000 ) contrib ution to u ′ 2 + . Right - large-scale ( λ + x > 4000 ) contrib ution to this Reynolds stress. Data taken from Dogan et al (2016). to predict such flo ws (Iyer and Y avuzkurt (1999), Bush et al (2019)). These difficulties arise from the stringent approximations single-scale turb ulence models are based on such as K olmogorov’ s hypothesis which ultimately implies that the turb ulence spectrum can be determined by unique length and velocity scales. T wo-scale models a void this crude hypothesis by enabling modelling of the scale segre gation in small and lar ge-scale contributions. The choice of a two-scale turb ulence model to tackle the interaction between TBLs and strong FST is then moti v ated by the experimental observ ations concerning the dif ferent beha viour of the small and lar ge scales in the TBL which cannot be reproduced by a single-scale turb ulence model. 2 Scale separation and inactiv e charac- ter of the large scales Numerous experimental studies ha ve shed light on the strong scale separation observed in TBLs under the influence of a lar ge-scale and intense FST (Sharp et al (2009), Dogan et al (2016), Hearst et al (2018)). A better glimpse at the respecti ve contrib utions of small and large scales to u ′ 2 + is gi ven by Dogan et al (2016) where partial inte gration of the streamwise Reynolds stress spectrum w as carried out. The spectral cut-of f was fixed at a w a velength λ + x = 4000 ( λ x /δ ≈ 1 − 2 with δ the boundary layer thickness). Figure 1 illustrates the results obtained in their study and it is clearly sho wn that the small-scale contrib utions to u ′ 2 + are relati vely uni versal with respect to the FST le vel whereas the lar ge-scale com- ponent is greatly increased throughout the whole TBL. Although the inner -scaled streamwise Reynolds stress substantially increases in the entire TBL when it is subjected to a high FST , the inner -scaled mean velocity and Re ynolds shear stress u ′ v ′ + (where v ′ is the wall-normal fluctuating v elocity component) in the b uf fer and logarithmic layers are not influenced by the le v el of FST . These e xperimental observ ations may be linked with the concept of active and inactive motions introduced by T ownsend in 1961 (T o wnend (1961)) which recei ved attention in recent research concern- ing high-Reynolds-number TBLs (Deshpande et al (2021)). Originally , this hypothesis was formulated to conciliate the experimental observ ations about the uni- versality of u ′ v ′ + while u ′ 2 + sho ws Re τ -dependence in canonical TBLs (with Re τ = ρ w δu τ /µ w the fric- tion Reynolds number where µ is the dynamic vis- cosity and the subscript w refers to wall quantities). The distinct Reynolds stresses Re τ -dependence can be better grasped with the Attached Eddy Hypothe- sis (T ownsend (1976), Perry et al (1986)) which pre- dicts a logarithmic la w for the wall-parallel Re ynolds stresses while u ′ v ′ + and v ′ 2 + are constant in the log- arithmic layer . This is attributed to the wall-blockage kinematic ef fect of the solid boundary on the wall nor - mal fluctuating velocity v ′ . The pressure-strain corre- lation then redistrib utes energy from the w all-normal Reynolds stress to the w all parallel ones (Bradsha w (1994)). The analysis from Deshpande et al (2025) suggests that because of the wall-blockage ef fect ex- erted on the wall-normal v elocity component, e very attached structure only contrib utes to v ′ in the vicinity of its height while the same structure enhances u ′ and w ′ , w ′ being the transverse v elocity fluctuation, all the way to the w all (in viscid theory). As the TBL dev elops and its friction Reynolds number increases, additional taller structures appear and add contrib utions to the wall-parallel Re ynolds stresses all the way to the w all while only mar ginally contributing to u ′ v ′ (see figure 3 of Deshpande et al (2025)). As such, turbulent motion in the TBL can be decomposed in an active and inac- tive component, the first being Re ynolds-number in- dependent as opposed to inactive motion (Deshpande et al (2021)). Panton (2007) proposes the follo wing Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 40 40 Figure 1: Left - profiles of the inner -scaled streamwise Re ynolds stress u ′ 2 + in the TBL for dif ferent v alues of T u ∞ . Middle - small-scale ( λ + x < 4000 ) contrib ution to u ′ 2 + . Right - lar ge-scale ( λ + x > 4000 ) contrib ution to this Re ynolds stress. Data tak en from Dog an et al (2016). to predict such flo ws (Iyer and Y a vuzkurt (1999), Bush et al (2019)). These dif ficulties arise from the stringent approximations single-scale turb ulence models are based on such as K olmogoro v’ s h ypothesis which ultimately implies that the turb ulence spectrum can be determined by unique length and v elocity scales. T w o-scale models a v oid this crude h ypothesis by enabling modelling of the scale se gre g ation in small and lar ge-scale contrib utions. The choice of a tw o-scale turb ulence model to tackle the interaction between TBLs and strong FST is then moti v ated by the e xperim ental observ ations concerning the dif ferent beha viour of the small and lar ge scales in the TBL which cannot be reproduced by a single-scale turb ulence model. 2 Scale separation and inacti v e charac- ter of the lar ge scales Numerous e xperimental studies ha v e shed light on the s trong scale separation observ ed in TBLs under the influence of a lar ge-scale and intense FST (Sharp et al (2009), Dog an et al (2016), Hearst et al (2018)). A better glimpse at the respecti v e contrib utions of small and lar ge scales to u ′ 2 + is gi v en by Dog an et al (2016) where partial inte gration of the streamwise Re ynolds stress spectrum w as carried out. The spectral cut-of f w a s fix ed at a w a v elength λ + x = 4000 ( λ x /δ ≈ 1 − 2 with δ the boundary layer thickness). Figure 1 illustrates the results obtained in their study and it is clearly s ho wn that the small-scale contrib utions to u ′ 2 + are relati v ely uni v ersal with respect to the FST le v el whereas the lar ge-scale com- ponent is greatly increased throughout the whole TBL. Although the inner -scaled streamwise Re ynolds stress substantially increases in the entire TBL when it is subjected to a high FST , the inner -scaled mean v elocity and Re ynolds shear stress u ′ v ′ + (where v ′ is the w all-normal fluctuating v elocity component) in the b uf fer and log arithmic layers are not influenced by the le v el of FST . These e xperimental observ ations may be link ed with the concept of active and inactive motions introduced by T o wnsend in 1961 (T o wnend (1961)) which recei v ed attention in recent research concern- ing high-Re ynolds-number TBLs (Deshpande et al (2021)). Originally , this h ypothesis w as formulated to conciliate the e xperimental observ ations about the uni- v ersality of u ′ v ′ + while u ′ 2 + sho ws Re τ -dependence in canonical TBLs (with Re τ = ρ w δu τ /µ w the fric- tion Re ynolds number where µ is the dynamic vis- cosity and the subscript w refers to w all quanti ties). The distinct Re ynolds stresses Re τ -dependence can be better grasped with the Attached Eddy Hypothe- sis (T o wnsend (1976), Perry et al (1986)) which pre- dicts a log arithmic la w for the w all-parallel Re ynolds stresses while u ′ v ′ + and v ′ 2 + are constant in the log- arithmic layer . This is att rib uted to the w all-blockage kinematic ef fect of the solid boundary on the w all nor - mal fluctuating v elocity v ′ . The pressure-strain corre- lation then redistrib utes ener gy from the w all-normal Re ynolds stress to the w all parallel ones (Bradsha w (1994)). The analysis from Deshpande et al (2025) suggests that because of the w all-blockage ef fect e x- erted on the w all-normal v elocity component, e v ery attached st ructure only contrib ute s to v ′ in the vicinity of its height while the same structure enhances u ′ and w ′ , w ′ being the transv erse v elocity fluctuation, all the w ay to the w all (in viscid theory). As the TBL de v elops and its friction Re ynolds number increases, additional taller structures appear and add contrib utions to the w all-parallel Re ynolds stresses all the w ay to the w all while only mar ginally contrib uting to u ′ v ′ (see figure 3 of Deshpande et al (2025)). As such, turb ulent m otion in the TBL can be decomposed in an active and inac- tive component, the first being Re ynolds-number in- dependent as opposed to inactive motion (Deshpande et al (2021)). P anton (2007) proposes the follo wing P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 40 decomposition :      u ′ = u ′ a + u ′ ia , w ′ = w ′ a + w ′ ia , v ′ = v ′ a (1) such that u ′ v ′ = u ′ a v ′ a . In (1), the subscript “ia” and “a” respecti v ely refer to inactive and active motions. By inserting this decomposition in the correlation fac- tor R u ′ v ′ , one obtains: R u ′ v ′ = − u ′ v ′  u ′ 2  v ′ 2 = − u ′ a v ′ a +  u ′ 2 a + + u ′ 2 ia +  v ′ 2 a + (2) by neglecting non-linear interactions such as ampli- tude modulation (Mathis et al (2009)). Equation (2) along with the pre vious analysis suggests that if the correlation factor decreases with the Re ynolds number or the FST le v el, it is attrib uted to an increase in the magnitude of inactive motions. Plots of the correlation factor R u ′ v ′ (eq (2)) are sho wn in figure 2 for dif fer - ent FST intensities and it is observed that this factor decreases as T u ∞ rises. Figure 2: Profiles of the correlation factor R u ′ v ′ for dif fer - ent FST intensities. Data at T u ∞ = 0% , 2.6% and 5.8% is taken from Hancock and Bradsha w (1989). Profiles at T u ∞ =8 . 3% and 12.7% are reproduced from Dogan et al (2019). The decrease in correlation factor with the FST in- tensity observed in figure 2 along with the increase in the lar ge-scale contribution with T u ∞ while the small scales are unmodified by the FST (figure 1) suggest that the lar ge-scale turbulent structures pen- etrating into the TBL contrib ute only partially to the Reynolds shear stress (Dog an et al 2019). The major flaw of one-scale turb ulence models for predicting the ef fect of strong FST on TBLs could then be associ- ated with their inability to capture the inactive compo- nent of the turb ulent kinetic energy carried by the large scales of the flo w . Thus, the small/large scale separa- tion operated by the two-scale turb ulence model de vel- oped in this work is used to capture this phenomenon. 3 A two-scale turb ulence model Multiple-scale turb ulence models are b uilt by partitioning the turb ulent kinetic energy spectrum in multiple parts and then integrating the spectrum on each wa v enumber interv al (Hanjalic et al (1980)). Each of this wa v enumber interv al is then characterised by its partial kinetic ener gy k ( m ) , the spectral fluxes F ( m ) from lar ger to smaller turbulent scales (figure 3) as well as a partial eddy viscosity µ ( m ) t . The analytical work from Schiestel et al (1987) sho ws that a model transport equation can be deri v ed for k ( m ) and dif ferent model v ariants are a v ailable according to the nature of the second transported partial v ariable. For instance, Gleize et al (1996) deriv ed a two-scale model (named MKFLC2) by using the spectral fluxes as the second turb ulent v ariable and obtained very satisfactory results for nonequilibrium boundary layers. Nonetheless, this model was not designed for TBLs interacting with strong FST so a ne w two-scale model has been de v eloped. Figure 3: T urbulent kinetic ener gy spectrum partitioning. The proposed two-scale model is based on a k − ω formulation where the characteristic turb ulent fre- quencies ω ( m ) are defined as  F (1) = β (1) k (1) ω (1) , F (2) = β (2) kω (2) (3) where β ( m ) are model constants and k is the total tur - b ulent kinetic energy k = k (1) + k (2) . This model relies on the Boussinesq hypothesis which assumes a linear relation between the Reynolds stresses and the strain-rate tensor . For multi-scale models, this assumption is extended to yield, using Fa vre mass- a veraged quantities noted  · and Einstein’ s con v ention : − ρ  u ′′ i u ′′ j ( m ) + 2 3 ρk ( m ) δ ij =2 µ ( m ) t  S ij − 1 3 ∂  U l ∂x l δ ij  (4) where δ ij is the Kronecker symbol, µ ( m ) t is the par - tial eddy viscosity and S ij is the strain-rate tensor for the Fa vre-av eraged velocity . The total eddy viscosity Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 41 41 (a) Po wer spectral density at x/L =0 . 5 and y /D =0 . V ertical dashed lines: Rossiter modes according to Heller et al (1971). (b) Evolution at y/D =0 . Figure 4: Sound pressure le v el at the bottom wall of the ca vity . Symbols: experimental measurements from ONERA S8Ch wind tunnel, : ZDES mode 2 (2020), : ZDES mode 3. (a) ZDES mode 2 (2020) (b) ZDES mode 3 Figure 5: SPL at the bottom wall of the cavity ( z /D = − 1 ). state vector q considered, ˆ Ψ contains the eigen vec- tors which correspond to the SPOD modes and ˆ Λ the corresponding eigen v alues. The matrix W is a weight matrix that defines the inner product such that for two dif ferent states vectors, it is verified ⟨ q 1 , q 2 ⟩ = q 1 H Wq 2 , where H denotes the Hermitian transpose. The reader may find more details in Schmidt and Colo- nius (2020). W elch’ s method has been employed for the PSD estimations and the number of blocks for W elch’ s av eraged periodogram method has been set to 8. It is a good compromise between the number of blocks and still capturing the lo west frequencies in the flo w . In this study , the SPOD has been ap- plied to the streamwise and spanwise velocity fields at the horizontal surface z /D =0 bounded by all the edges of the ca vity , the bottom wall being located at z /D = − 1 . A global e xploration of the SPOD ener gy at dif ferent Rossiter modes has e videnced the great dominance of the leading SPOD mode for the second and third Rossiter modes of the ca vity , con- taining about 95% of the ener gy and therefore leaving very little ener gy for the remaining SPOD modes (not sho wn). Some combinations of SPOD and Rossiter modes carefully chosen are illustrated in figures 6 and 7 for the streamwise and spanwise velocity components re- specti v ely . For the streamwise v elocity component, first and second SPOD modes are plotted at the fre- quency corresponding to the second Rossiter mode (the most dominant one, figure 4a). For the spanwise velocity componenet, only the first SPOD mode is il- lustrated, but this is done for the second and the third Rossiter modes. Figures 6a and 6b depict the leading SPOD mode of the second Rossiter mode for the streamwise veloc- ity fluctuation. The SPOD mode is v ery close for both ZDES mode 2 (2020) and ZDES mode 3, which in- dicates that this mode does not seem to be very sen- siti v e to the turbulent incoming field. In addition, there is a clear and interesting completely asymmet- rical shape composed of some what oblique structures. When looking at the second SPOD mode in figures 6c and 6d, which contains much less ener gy as already mentioned, a quite dif ferent mode shape is obtained between both simulations. In particular , ZDES mode 3 sho ws a more important presence of this mode near the side edges of the ca vity at around x/L =0 . 4 and an al- ternating mode shape do wnstream. In contrast, ZDES mode 2 (2020) presents a mode which also sho ws an asymmetry of field, but seems essentially composed of two elongated oblique structures. Reg arding the SPOD analysis for the spanwise ve- locity fluctuation, the leading SPOD mode for the sec- ond Rossiter mode, figures 7a and 7b, appears to be very close between both simulations. Similarly to the observ ations of the SPOD of the streamwise v eloc- ity , this means that for this Rossiter mode, the leading SPOD mode seems quite insensiti v e to incoming tur- b ulence and rather dominated by the intrinsic cavity dynamics. Ne vertheless, there is a slight dif ference in the mode distrib ution, again near the side cavity edges. It is also in this case interesting to notice the highly Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 48 48 (a) Po wer spectral density at x/L =0 . 5 and y /D =0 . V ertical dashed lines: Rossiter modes according to Heller et al (1971). (b) Ev olution at y /D =0 . Figure 4: Sound pressure le v el at the bottom w all of the ca vity . Symbols: e xperi mental measurements from ONERA S8Ch wind tunnel, : ZDES mode 2 (2020), : ZDES mode 3. (a) ZDES mode 2 (2020) (b) ZDES mode 3 Figure 5: SPL at the bottom w all of the ca vity ( z /D = − 1 ). state v ector q considered, ˆ Ψ contains the eigen v ec- tors which correspond to the SPOD modes and ˆ Λ the corresponding eigen v alues. The matrix W is a weight matrix that defines the inner product such that for tw o dif ferent states v ectors, it is v erified ⟨ q 1 , q 2 ⟩ = q 1 H Wq 2 , where H denotes the Hermitian transpose. The reader m ay find more details in Schmidt and Colo- nius (2020). W elch’ s method has been emplo yed for the PSD estimations and the number of blocks for W elch’ s a v eraged periodogram method has been set to 8. It is a good compromise between the number of blocks and still capturing the lo west frequencies in the fl o w . In this study , the SPOD has been ap- plied to the streamwise and spanwise v elocity fields at the horizontal surf ace z /D =0 bounded by all the edges of the ca vity , the bottom w all being located at z /D = − 1 . A global e xpl o r ation of the SPOD ener gy at dif f erent Rossiter modes has e videnced the great dominance of the leading SPOD mode for the second and third Rossiter modes of the ca vity , con- taining about 95% of the ener gy and therefore lea ving v ery little ener gy for the remaining SPOD modes (not sho wn). Some combinations of SPOD and Ross iter modes carefully chosen are illustrated in figures 6 and 7 for the streamwise and spanwise v elocity components re- specti v ely . F or the streamwise v elocity component, first and second SPOD modes are plotted at the fre- quenc y corresponding to the se cond Rossiter mode (the most dominant one, figure 4a). F or the spanwise v elocity componenet, only the first SPOD mode is il- lustrated, b ut this is done for the second and the third Rossiter modes. Figures 6a and 6b depict the leading SPOD mode of the second R ossiter mode for the streamwise v eloc- ity fluctuation. The SPOD mode is v ery close for both ZDES mode 2 (2020) and ZDES mode 3, which in- dicates that this mode does not see m to be v ery sen- siti v e to the turb ulent incoming field. In addition, there is a clear and interesting completely asymmet- rical shape composed of some wh a t oblique structures. When looking at the second SPOD mode in figures 6c and 6d, which contains much less ener gy as already mentioned, a quite dif ferent mode shape is obtained between both simulations. In particular , ZDES mode 3 sho ws a more important presence of this mode near the side edges of the ca vity at around x/L =0 . 4 and an al- ternating mode shape do wnstream. In contrast, ZDES mode 2 (2020) presents a mode which also sho ws an asymmetry of field, b ut seems essentially composed of tw o elong ated oblique structures. Re g arding the SPOD analysis for the spanwise v e- locity fluctuation, the leading SPOD mode for the sec- ond Rossiter mode, figures 7a and 7b, appears to be v ery close between both simulations. Similarly to the observ ations of the SPOD of the streamwise v eloc- ity , this means that for this Rossiter mode, the leading SPOD mode seems quite insensiti v e to incoming tur - b ulence and rather dominated by the intrinsic ca vity dynamics. Ne v ertheless, there is a slight dif ference in the mode distrib ution, ag ain near the side ca vity edges. It is also in this case interesting to notice the highly P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 48 (a) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 2 (2020) (b) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 3 (WMLES) (c) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 2 (2020) (d) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 3 (WMLES) Figure 6: SPOD of the streamwise velocity component. (a), (b): SPOD leading mode; (c), (d): SPOD second mode. (a) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 2 (2020) (b) 2 nd Rossiter mode, St D =0 . 115 . ZDES mode 3 (WMLES) (c) 3 rd Rossiter mode, St D =0 . 181 . ZDES mode 2 (2020) (d) 3 rd Rossiter mode, St D =0 . 181 . ZDES mode 3 (WMLES) Figure 7: SPOD of the spanwise velocity component. SPOD leading mode for (a), (b) 2 nd and (c), (d) 3 rd Rossiter modes. non symmetric shape of the mode, featured again by lar ge oblique structures. These observations may also be applied to the leading SPOD mode of the third Rossiter mode illustrated in figures 7c and 7d. Indeed, the modes distrib utions are qualitati vely the same and the highly asymmetric shape is also observed. It is nonetheless worth mentioning some minor discrepan- cies locally located around x/L =0 . 8 . 5 Conclusions The ef fect of realistic turb ulent boundary layer fluctuations on a supersonic high-Reynolds number ca vity flo w is rarely adressed in the literature due to its complex implementation in an industrial solv er . T ur- b ulent inflo w conditions ha ve been properly generated in the present work thanks to the CDF for the WM- LES (ZDES mode 3) simulation and their ef fect on the ca vity do wnstream has been studied. Compar - isons to hybrid RANS/LES (ZDES mode 2 (2020)) ha ve been made, where inflo w fluctations are absent since the incoming flo w upstream of the ca vity is com- pletely treated in RANS approach. The av ailable e x- perimental data ha ve pro v en that both simulations re- produce accurately the dynamics of the ca vity flo w and in particular the SPL spectral distrib ution and its e v o- lution ov er the ca vity bottom wall. SPL at the bottom wall seems nonetheless some what sensitiv e to the in- flo w conditions. The SPOD analysis of the two domi- nant Rossiter modes has also e videnced a globally fair agreement between both simulations. Discrepancies appear to be confined to the second SPOD mode which carries much less ener gy than the leading mode, thus confirming the great dominance of the intrinsic ca vity dynamics. This study sho ws that e v en when generating proper turb ulent fluctuations their ef fect on this ca vity dynam- ics seem to be secondary . This is howe ver a promising result for hybrid RANS/LES approaches that enable a rapid de v elopment of instabilities such as ZDES mode 2 (2020). Besides, the consequent database generated Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 49 49 with both simulations and the dif ferences observed are worth further studying. These, together with the asym- metric feature of the mean flo w , compose an interest- ing aspect to keep e xploring for supersonic ca vity dy- namics. Acknowledgments The authors would lik e to thank all the people in volv ed in the de v elopment of the FLU3M solver . The colleagues from the AMES, MAPE, MASH and MSA T units of the D AAA department of ONERA are also ackno wledged, in particular Colin Leclercq and Lucas Boussard, for fruitful discussions, as well as the ONERA research project SUPERFLAP . Refer ences Baugher , S. K., Outten, B., Gaitonde, D. V ., and K umar , R. (2024). 3D ef fects of quasi-static door opening on super - sonic ca vity flo ws. In AIAA SCITECH 2024 F orum. Ameri- can Institute of Aeronautics and Astronautics. Beresh, S. J., W agner , J. L., Pruett, B. O. M., Henfling, J. F ., and Spillers, R. W . (2015). Supersonic flo w ov er a finite- width rectangular ca vity . AIAA J ournal , 53(2):296–310. Burg, J. P . (1978). Maximum Entropy Spectral Analysis. Modern Spectrum Analysis, edited by D. G. Childers, IEEE Pr ess, New Y ork , p. 34–41. Chang, K., Constantinescu, G., and P ark, S.-O. (2007). As- sessment of predicti ve capabilities of detached eddy simula- tion to simulate flo w and mass transport past open cavities. J ournal of Fluids Engineering , 129(11):1372–1383. Crook, S. D., Lau, T . C. W ., and K elso, R. M. (2013). Three- dimensional flo w within shallo w , narrow ca vities. J ournal of Fluid Mechanics , 735:587–612. Deck, S. (2012). Recent impro vements in the Zonal De- tached Eddy Simulation (ZDES) formulation. Theor etical and Computational Fluid Dynamics , 26:523–550. Deck, S. (2005). Numerical Simulation of T ransonic Buf fet ov er a Supercritical Airfoil. AIAA J ournal , 43:1556. Deck, S. and Renard, N. (2020). T o wards an enhanced pro- tection of attached boundary layers in hybrid RANS/LES methods. J ournal of Computational Physics , 400:108970. Deck, S., Renard, N., Laraufie R. and Sagaut, P . (2014). Zonal detached eddy simulation (ZDES) of a spatially de vel- oping flat plate turb ulent boundary layer ov er the Reynolds number range 3150 ≤ Re θ ≤ 14000 . Physics of Fluids , 26:025116. Deck, S., W eiss, P .-E., and Renard, N. (2018). A rapid and lo w noise switch from RANS to WMLES on curvilinear grids with compressible flo w solvers. J ournal of Compu- tational Physics , 363:231–255. DeMauro, E. P ., Casper , K. M., Beresh, S. J., W agner , J. L., Henfling, J., and Spillers, R. (2017). Study of the flo w within finite-span complex ca vities using particle image velocime- try and pressure sensiti ve paint. In 55th AIAA Aer ospace Sciences Meeting . American Institute of Aeronautics and Astronautics. Gloerfelt, X. and Berland, J. (2013). T urbulent boundary- layer noise: direct radiation at Mach number 0.5. J ournal of Fluid Mechanics , 723:318–351. Hamilton Smith, C. O., Lawson, N., and V io, G. A. (2024). History , revie w and summary of the cavity flo w phenomena. Eur opean Journal of Mec hanics - B/Fluids , 108:32–72. Heller , H. H., Holmes, D. G. and Cov ert, E. E. (1971). Flo w- induced pressure oscillations in shallo w ca vities. Journal of Sound and V ibration , 18:545-553. Laraufie, R., Deck, S., and Sagaut, P . (2011). A dynamic forcing method for unsteady turb ulent inflo w conditions. J ournal of Computational Physics , 230:8647–8663. Larche v ˆ eque, L., Sagaut, P ., L ˆ e, T .-H. and Comte, P . (2004). Large-eddy simulation of a compressible flo w in a three- dimensional open ca vity at high Reynolds number . J ournal of Fluid Mechanics , 516:265–301. Larche v ˆ eque, L., Sagaut, P ., and Labb ´ e, O. (2007). Lar ge- eddy simulation of a subsonic ca vity flo w including asym- metric three-dimensional ef fects. J ournal of Fluid Mechan- ics , 577:105–126. Liu, Q., Sun, Y ., Y eh, C.-A., Ukeiley , L. S., Cattafesta, L. N., and T aira, K. (2021). Unsteady control of supersonic turb ulent cavity flo w based on resolvent analysis. J ournal of Fluid Mechanics , 925:A5. Maia, I., Gojon, R., Bauerheim, M., Fiore, M., and Node- Langlois, T . (2023). W all-modelled LES of a high subsonic ca vity flo w at large Re ynolds number . In AIAA A VIA TION 2023 F orum . American Institute of Aeronautics and Astro- nautics. Maull, D. J. and East, L. F . (1963). Three-dimensional flo w in ca vities. Journal of Fluid Mec hanics , 16(4):620–632. Robison, Z. D. and Garmann, D. J. (2024). Comparativ e study of finite-width and spanwise-periodic ca vity flo ws. In AIAA SCITECH 2024 F orum . American Institute of Aero- nautics and Astronautics. Schmidt, O. T . and Colonius, T . (2020). Guide to Spectral Proper Orthogonal Decomposition. AIAA Journal , 58:1023- 1033. Spalart, P . R., Deck, S., Shur , M. L., Squires, K. D., Strelets, M. K., and Tra vin, A. (2006). A ne w version of detached- eddy simulation, resistant to ambiguous grid densities. The- or etical and Computational Fluid Dynamics , 20:181– 195. Sun, Y ., Liu, Q., Cattafesta, L. N., Ukeiley , L. S., and T aira, K. (2019). Effects of side-w alls and leading-edge blo w- ing on flo ws ov er long rectangular ca vities. AIAA J ournal , 57(1):106–119. T urpin, A. M., Speth, R. L., Sherer , S. E., and Granlund, K. O. (2021). Lo w- frequency , spanwise oscillation in a finite- width ca vity at Mach 1.5. Physics of Fluids , 33(7):076102. V aquero, J., Renard, N., and Deck, S. (2022). Outer layer turb ulence dynamics in a high-Reynolds-number boundary layer up to Re θ ≈ 24 , 000 reco vering from mild separation. J ournal of Fluid Mechanics , 942:A42. Zhuang, N., Alvi, F . S., Alkislar , M. B. and Shih, C. (2006). Supersonic Ca vity Flo w and Their Control. AIAA Journal , 44:2118-2128. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 50 50 with both simulations and the dif ferences observ ed are w orth further studying. These, together with the asym- metric feature of the mean flo w , compose an interest- ing aspect to k eep e xploring for supersonic ca vity dy- namics. Ackno wledgments The authors w ould lik e to thank all the people in v olv ed in the de v elopment of the FLU3M solv er . The colleagues from the AMES, MAPE, MASH and MSA T units of the D AAA department of ONERA are also ackno wledged, in particular Colin Leclercq and Lucas Boussard, for fruitful discussions, as well as the ONERA research project SUPERFLAP . Refer ences Baugher , S. K., Outten, B., Gaitonde, D. V ., and K umar , R. (2024). 3D ef fects of quasi-static door opening on super - sonic ca vity flo ws. In AIAA SCITECH 2024 F orum. Ameri- can Institute of Aeronautics and Astronautics. Beresh, S. J., W agner , J. L., Pruett, B. O. M., Henfling, J. F ., and Spillers, R. W . (2015). Supersonic flo w o v er a finite- width rectangular ca vity . AIAA J ournal , 53(2):296–310. Bur g, J. P . (1978). Maximum Entrop y Spectral Analysis. Modern Spectrum Analysis, edited by D. G. Childer s, IEEE Pr ess, Ne w Y ork , p. 34–41. Chang, K., Constantinescu, G., and P ark, S.-O. (2007). As- sessment of predic ti v e capabilities of detached eddy simula- tion to simulate flo w and mass transport past open ca vities. J ournal of Fluids Engineering , 129(11):1372–1383. Crook, S. D., Lau, T . C . W ., and K elso, R. M. (2013). Three- dimensional flo w within shallo w , narro w ca vities. J ournal of Fluid Mec hanics , 735:587–612. Deck, S. (2012). Recent impro v ements in the Zonal De- tached Eddy Simulation (ZDES) formulation. Theor etical and Computational Fluid Dynamics , 26:523–550. Deck, S. (2005). Numerical Simulation of T ransonic Buf fet o v er a Supercritical Airfoil. AIAA J ournal , 43:1556. Deck, S. and Renard, N. (2020). T o w ards an enhanced pro- tection of attached boundary layers in h ybrid RANS/LES methods. J ournal of Computational Physics , 400:108970. Deck, S., Renard, N., Laraufie R. and Sag aut, P . (2014). Zonal detached eddy simulation (ZDES) of a spatially de v el- oping flat plate turb ulent boundary layer o v er the Re ynolds number range 3150 ≤ Re θ ≤ 14000 . Physics of Fluids , 26:025116. Deck, S., W eiss, P .-E., and Renard, N. (2018). A rapid and lo w noise switch from RANS to WMLES on curvilinear grids with compressible flo w solv ers. J ournal of Compu- tational Physics , 363:231–255. DeMauro, E. P ., Casper , K. M., Beresh, S. J., W agner , J. L., Henfling, J., and Spillers, R. (2017). Study of the flo w within finite-span comple x ca vities using particl e image v elocime- try and pressure sensiti v e paint. In 55th AIAA Aer ospace Sciences Meeting . American Institute of Aeronautics and Astronautics. Gloerfelt, X. and Be rland, J. (2013). T urb ulent boundary- layer noise: direct radiation at Mach number 0.5. J ournal of Fluid Mec hanics , 723:318–351. Hamilton Smith, C. O., La wson, N., and V io, G. A. (2024). History , re vie w and summary of the ca vity flo w phenomena. Eur opean J ournal of Mec hanics - B/Fluids , 108:32–72. Heller , H. H., Holmes, D. G. and Co v ert, E. E. (1971). Flo w- induced pressure oscillations in shallo w ca vities. J ournal of Sound and V ibr ation , 18:545-553. Laraufie, R., Deck, S., and Sag aut, P . (2011). A dynamic forcing method for unsteady turb ulent inflo w conditions. J ournal of Computational Physics , 230:8647–8663. Larche v ˆ eque, L., Sag aut, P ., L ˆ e, T .-H. and Comte, P . (2004). Lar ge-eddy simulation of a compressible flo w in a three- dimensional open ca vity at high Re ynolds num ber . J ournal of Fluid Mec hanics , 516:265–301. Larche v ˆ eque, L., Sag aut, P ., and Labb ´ e, O. (2007). Lar ge- eddy si mulation of a subsonic ca vity flo w including asym- metric three-dimensional ef fects. J ournal of Fluid Mec han- ics , 577:105–126. Liu, Q., Sun, Y ., Y eh, C.-A., Uk eile y , L. S., Cattafesta, L. N., and T aira, K. (2021). Unsteady control of supersonic turb ulent ca vity flo w based on resolv ent analysis. J ournal of Fluid Mec hanics , 925:A5. Maia, I., Gojon, R., Bauerheim, M., Fiore, M., and Node- Langlois, T . (2023). W all-modelled LES of a high subsonic ca vity flo w at lar ge Re ynolds number . In AIAA A VIA TION 2023 F orum . American Institute of Aeronautics and Astro- nautics. Maull, D. J. and East, L. F . (1963). Three-dimensional flo w in ca vities. J ournal of Fluid Mec hanics , 16(4):620–632. Robison, Z. D. and Garmann, D. J. (2024). Comparati v e study of finite-width and spanwise-periodic ca vity flo ws. In AIAA SCITECH 2024 F orum . American Institute of Aero- nautics and Astronautics. Schmidt, O. T . and Colonius, T . (2020). Guide to Spectral Proper Orthogonal Decomposition. AIAA J ournal , 58:1023- 1033. Spalart, P . R., Deck, S., Shur , M. L., Squires, K. D., Strelets, M. K., and T ra vin, A. (2006). A ne w v ersion of detached- eddy simulation, resistant to ambiguous grid densities. The- or etical and Computational Fluid Dynamics , 20:181– 195. Sun, Y ., Liu, Q., Cattafesta, L. N., Uk eile y , L. S., and T aira, K. (2019). Ef fects of side-w alls and leading-edge blo w- ing on flo ws o v er long rectangular ca vities. AIAA J ournal , 57(1):106–119. T urpin, A. M., Speth, R. L., Sherer , S. E., and Granlund, K. O. (2021). Lo w- frequenc y , spanwise os cillation in a finite- width ca vity at Mach 1.5. Physics of Fluids , 33(7):076102. V aquero, J., Renard, N., and Deck, S. (2022). Outer layer turb ulence dynamics in a high-Re ynolds-number boundary layer up to Re θ ≈ 24 , 000 reco v ering from mild separation. J ournal of Fluid Mec hanics , 942:A42. Zhuang, N., Alvi, F . S., Alkislar , M. B. and Shih, C. (2006). Supersonic Ca vity Flo w and Their Control. AIAA J ournal , 44:2118-2128. P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 50 I MP A CT OF W ALL T EMPERA TURE ON T RANSONIC G AS T URBINES V ANE F L OW D YN AMICS : AW ALL -M ODELED LES A PPR O A CH F . De V anna 1 , ∗ , E. Benini 1 1 Dipartimento di Ingegneria Industriale, Univ ersit ` a degli Studi di P ado va, V ia V enezia, 1, 35121 ∗ fr [email protected] Abstract This study in vestigates ho w wall temperature af- fects the flo w dynamics of transonic gas turbine stators using W all-Modeled Large Eddy Simulations (WM- LES) and the Immersed Boundary Method. Sim- ulations cov er a range of w all-to-recov ery tempera- ture ratios from adiabatic to highly cooled conditions. Results re v eal that lo wer wall temperatures suppress near -wall turb ulence, alter Re ynolds stress distrib u- tions, and broaden the wake, leading to increased aero- dynamic losses. Spectral analysis sho ws ener gy shifts to ward lo wer frequencies as cooling intensifies, con- firming a disruption in the turb ulent energy cascade. The methodology provides high-fidelity insights at re- duced computational cost compared to WRLES. Com- parison with experiments v alidates the accuracy of the approach. Thus, the frame work is suitable for rapid design optimization in turbomachinery compo- nents and aeroengines. 1 Intr oduction Increasing comb ustion temperatures in gas tur - bines improv e ef ficiency b ut impose se vere thermal stress, accelerating material degradation and neces- sitating adv anced cooling strate gies. In transonic regimes, cooling profoundly af fects flo w dynamics through shock-wa v e/boundary-layer interactions (SB- LIs) and steep thermal gradients, complicating heat transfer predictions. Accurately modeling these ef- fects is crucial for optimizing performance and ensur - ing component durability . Scale-resolving approaches like DNS and LES capture unsteady flo w physics and transitional phe- nomena more ef fecti v ely than RANS. Ho we ver , the prohibiti v e cost of DNS and W all-Resolv ed LES (WR- LES) limits their practical use. W all-Modeled LES (WMLES) of fers a computationally viable alternati v e, resolving lar ge-scale structures while modeling near- wall turb ulence. This study examines the impact of w all tempera- ture on transonic gas turbine stators using WMLES and the Immersed Boundary Method, accelerated on multi-GPU architectures. Six wall-to-reco very tem- perature ratios, from adiabatic to highly cooled condi- tions, are analyzed. Results indicate that cooler w alls suppress turb ulence, modify w ake beha vior , and in- crease aerodynamic losses by reducing energy transfer to the boundary layer . This leads to wake e xpansion and reduced flo w stability . By lev eraging high-fidelity simulations, this research provides crucial insights into optimizing cooling strategies for enhanced gas turbine ef ficienc y and durability . 2 Gov er ning equations The present study utilizes URANOS (De V anna, F . et al (2023). De V anna, F . & Baldan G. (2024)), a DNS/LES solver de veloped at the Depart- ment of Industrial Engineering of the Uni v ersity of Pado v a. URANOS is tailored to solve the compress- ible Na vier -Stokes equations in a conserv ati ve formu- lation. Specifically , incorporating the F a vre filtering ( ˜ ϕ = ρϕ/ ¯ ρ ) for any flo w v ariable ϕ , where ¯ ϕ denotes the Reynolds-a veraged v alue, the model reads as fol- lo ws: ∂ ¯ ρ ∂t + ∂ ¯ ρ ˜ u j ∂x j =0 , (1a) ∂ ¯ ρ ˜ u i ∂t + ∂ (¯ ρ ˜ u i ˜ u j +¯ pδ ij ) ∂x j = ∂ ¯ τ ij ∂x j − ∂T S GS ij ∂x j + F i , (1b) ∂ ¯ ρ ˜ E ∂t + ∂  ¯ ρ ˜ H ˜ u j  ∂x j = ∂  ¯ τ ij − T S GS ij  ˜ u j  ∂x j − ∂ ¯ J j ∂x j − ∂E S GS j ∂x j + E . (1c) Here, ¯ ρ is the filtered density , ˜ u i the filtered veloc- ity in the i -th direction, and ¯ p the filtered pressure. The total enthalpy is gi ven by ˜ H = ˜ E +¯ p/ ¯ ρ , where ˜ E =˜ e +  u i u i / 2 represents the filtered total ener gy per unit mass, with ˜ e as the internal energy per unit mass. The filtered molecular heat flux in the j -th di- rection is denoted as ¯ J j . The system in Eqs. (1) is closed using the ideal gas la w , ¯ p =¯ ρR ˜ T , and the thermodynamic relation ˜ e = c v ˜ T , where ˜ T is the filtered temperature, R the specific gas constant, and Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 51 51 c v = R/ ( γ − 1) , c p = γ R/ ( γ − 1) the specific heats at constant v olume and pressure, respecti vely , with γ = c p /c v as the specific heat ratio. The viscous stress tensor ¯ τ ij and heat flux ¯ J j follo w Ne wtonian flo w as- sumptions and Fourier’ s la w , respecti vely . The molec- ular viscosity µ ( ˜ T ) and thermal dif fusi vity λ ( ˜ T )= c p µ ( ˜ T ) /P r are computed using Sutherland’ s law . The terms F i and E in Eqs. (1b) and (1c) account for spe- cific boundary conditions. The sub-grid scale (SGS) stress tensor , T S GS ij = ρu i u j − ¯ ρ ˜ u i ˜ u j , is modeled us- ing Boussinesq’ s hypothesis. This study employs the W all-Adaptiv e Lar ge-Eddy (W ALE) model (Nicoud, F ., and Ducros, F . (1999)) to compute the SGS viscos- ity , ensuring improv ed near-w all turb ulence represen- tation. W all turb ulence treatment T o address the challenge of insufficient resolution near solid boundaries, URANOS adopts a WMLES strategy . The solver implements an equilibrium-based wall model that assumes the instantaneous balance be- tween con vecti v e and pressure forces in the near-w all region. This model introduces two v ariables: U wm and T wm , which denote the wall-parallel v elocity and temperature a veraged along with the w all normal di- rection. The subscript wm distinguishes quantities de- ri v ed from the wall model as opposed to those resolved by LES. Under the equilibrium assumption, the simpli- fied momentum and ener gy equations take the form: d dy  µ tot dU wm dy  =0 (2a) d dy  λ tot dT wm dy  = − d dy  µ tot U wm dU wm dy  (2b) Here, the total viscosity and thermal dif fusi vity are defined as µ tot = µ wm + µ t,w m and λ tot = c p ( µ wm / Pr + µ t,w m / Pr t,wm ) , respecti vely . The spe- cific heat c p is assumed constant, and the turbulent Prandtl number is set to Pr t,w m =0 . 9 . Due to the thermal sensiti vity of the problem, the laminar vis- cosity µ wm is computed as a function of tempera- ture using Sutherland’ s law , i.e., µ wm = µ wm ( T wm ) . This allo ws for accurate representation of thermally dri v en viscosity changes, enhancing w all shear and heat flux predictions. The eddy viscosity in the model is calculated using µ t,w m = κρ wm u τ yD , with D denoting the V an Driest damping function. The von K ´ arm ´ an constant is fixed at κ =0 . 41 , and wall model density is obtained from the ideal gas law as ρ wm = p LES / RT wm , where p LES is the LES pressure at the interface. The damping function is gi ven by D = [1 − exp ( − y ∗ / A + )] 2 , with A + = 17 . T o bet- ter capture near -wall v ariations in thermal and mo- mentum transport, a semi-local scaling for turbulent viscosity is employed. The wall-normal coordinate y ∗ is defined as y ∗ = y  ρ wm τ w,wm / µ wm , where y is the wall-normal distance. Equations (2) are integrated ov er a dedicated 1D mesh embedded in the main LES grid. This auxiliary grid starts at the wall with pre- scribed no-slip and isothermal conditions and e xtends to y = h wall , where the wall model couples with the outer LES solution. At this interface, velocity , temper - ature, and pressure are matched via U wm = u ∥ , LES , T wm = T LES , and P wm = p LES . Additional informa- tion on the wall modeling frame work is av ailable in De V anna, F ., (2021) and De V anna, F ., et al. (2023). 3 Numerical methods and setup Numerical discr etization methods The gov erning equations are discretized using a sixth-order , fully split energy-preserving scheme for con vecti ve fluxes blended with a se venth-order WENO scheme for shock capturing. V iscous fluxes are com- puted using a semi-conserv ati v e sixth-order finite dif- ference method, follo wing the approach in De V anna, F ., (2021). T ime integration is performed using a lo w-storage total v ariation diminishing Runge–K utta method. The time step is dynamically adjusted at each iteration to maintain stability , based on the Courant–Friedrichs–Le wy (CFL) condition and the Fourier stability criterion. The CFL and Fourier num- bers are set to 0.5 and 0.1, respecti vely , with the smaller resulting time step selected to adv ance the so- lution. Computational domain and r esolution The computational domain is a three-dimensional box of size L x / c × L y / c × L z / c =3 × 0 . 85 × 0 . 1 , where c denotes the blade chord length. A structured grid with 768 × 384 × 80 points is employed. The grid is uniform in the spanwise and cross-sectional directions b ut refined along the streamwise axis near the blade to improv e resolution. The simulation is ex ecuted on four NVIDIA A100 GPUs on the Leonardo supercomputer , achie ving 0.11 seconds per Na vier -Stokes iteration. A total of 8.2 million iterations are performed, with 200,000 used for initial transients and 8 million for sta- tistical a veraging, requiring approximately 6,000 GPU hours. Boundary condition The boundary conditions are defined as follo ws: a subsonic stagnation state is imposed at the inlet, while a back-pressure condition is applied at the outlet. Specifically , the non-dimensional total pressure and temperature at the inlet are set to p 0 in /p ref =1 . 01584 and T 0 in /T ref =1 . 00450 , respecti vely . At the outlet, static pressure and density are fix ed at p out /p ref = 0 . 524191 and ρ out /ρ ref =0 . 630428 . This config- uration ensures unitary static v alues at the inlet, in agreement with the MUR47 operating point described in Arts, T ., (1990). Additionally , velocity compo- nents at the outlet are loosely constrained using results from a preliminary RANS simulation, with u out /u in = 0 . 284771 and v out /u in = − 1 . 03684 . A sponge layer near the outflo w gradually adapts the solution to the boundary condition ov er time, ef fecti vely damping tur - Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 52 52 c v = R/ ( γ − 1) , c p = γ R/ ( γ − 1) the specific heats at constant v olume and pressure, respecti v ely , with γ = c p /c v as the specific heat ratio. The viscous stress tensor ¯ τ ij and heat flux ¯ J j follo w Ne wtonian flo w as- sumptions and F ourier’ s la w , respecti v ely . The molec- ular viscosity µ ( ˜ T ) and thermal dif fusi vity λ ( ˜ T )= c p µ ( ˜ T ) /P r are computed using Sutherland’ s la w . The terms F i and E in Eqs. (1b) and (1c) account for spe- cific boundary conditions. The sub-grid scale (SGS) stress tensor , T S GS ij = ρu i u j − ¯ ρ ˜ u i ˜ u j , is modeled us- ing Boussinesq’ s h ypothesis. This study emplo ys the W all-Adapti v e Lar ge-Eddy (W ALE) model (Nicoud, F ., and Ducros, F . (1999)) to compute the SGS viscos- ity , ensuring impro v ed near -w all turb ulence represen- tation. W all turb ulence tr eatment T o address the challenge of insuf ficient resolution near solid boundaries, URANOS adopts a WMLES strate gy . The solv er implements an equilibrium-based w all model that assumes the instantaneous balance be- tween con v ecti v e and pressure forces in the near -w all re gion. This model introduces tw o v ariables: U wm and T wm , which denote the w all-parallel v elocity and temperature a v eraged along with the w all normal di- rection. The subscript wm distinguishes quantities de- ri v ed from the w all model as opposed to those resolv ed by LES. Under the equilibri um assumption, the simpli- fied momentum and ener gy equations tak e the form: d d y  µ tot d U wm d y  =0 (2a) d d y  λ tot d T wm d y  = − d d y  µ tot U wm d U wm d y  (2b) Here, the total viscosity and thermal dif fusi vi ty are defined as µ tot = µ wm + µ t ,w m and λ tot = c p ( µ wm / Pr + µ t ,w m / Pr t ,w m ) , respecti v ely . The spe- cific heat c p is assumed constant, and the turb ulent Prandtl number is set to Pr t ,w m =0 . 9 . Due to the thermal sensiti vity of the problem, the laminar vis- cosity µ wm is c o m puted as a function of tempera- ture using Sutherland’ s la w , i.e., µ wm = µ wm ( T wm ) . This allo ws for accurate representation of thermally dri v en viscosity changes, enhancing w all shear and heat flux predictions. The eddy viscosity in the model is calculated using µ t ,w m = κρ wm u τ yD , with D denoting the V an Driest damping function. The v on K ´ arm ´ an constant is fix ed at κ =0 . 41 , and w all model density is obtained from the ideal g as la w as ρ wm = p LES / RT wm , where p LES is the LES pressure at the interf ace. The damping function is gi v en by D = [1 − ex p ( − y ∗ / A + )] 2 , with A + = 17 . T o bet- ter capture near -w all v ariations in thermal and mo- mentum transport, a semi-local scaling for turb ulent viscosity is emplo yed. The w all-normal coordinate y ∗ is defined as y ∗ = y  ρ wm τ w, wm / µ wm , where y is the w all-normal distance . Equations (2) are inte grated o v er a dedicated 1D mesh embedded in the main LES grid. This auxiliary grid starts at the w all with pre- scribed no-slip and isothermal conditions and e xtends to y = h w all , where the w all model couples with the outer LES solution. At this int erf ace, v elocity , temper - ature, and pressure are matched via U wm = u ∥ , LES , T wm = T LES , and P wm = p LES . Additional informa- tion on the w all modeling frame w ork is a v ailable in De V anna, F ., (2021) and De V anna, F ., et al. (2023). 3 Numerical methods and setup Numerical discr etization methods The go v e rning equations are discretized using a sixth-order , fully split ener gy-preserving scheme for con v ecti v e flux es blended with a se v enth-order WENO scheme for shock capturing. V iscous flux es are com- puted using a semi-conserv ati v e sixth-order finite dif- ference method, follo wing the approach in De V anna, F ., (2021). T ime inte gration is performed using a lo w-storage total v ariation diminishing Runge–K utta method. The time step is dynamically adjusted at each iteration to mai ntain stability , based on the Courant–Friedrichs–Le wy (CFL) condition and the F ourier stability criterion. The CFL and F ourier num- bers are set to 0.5 and 0.1, respecti v ely , with the smaller resulting time step selected to adv ance the so- lution. Computational domain and r esolution The computational domain is a three-dimensional box of size L x / c × L y / c × L z / c =3 × 0 . 85 × 0 . 1 , where c denotes the blade chord length. A struc tured grid with 768 × 384 × 80 points is emplo yed. The grid is uniform in the spanwise and cross-sectional directions b ut refined along the streamwise axis near the blade to impro v e resolution. The simulation is e x ecuted on four NVIDIA A100 GPUs on the Le o na rdo supercomputer , achie ving 0.11 seconds per Na vier -Stok es iteration. A total of 8.2 million iterations are performed, with 200,000 used for initial transients and 8 million for sta- tistical a v eraging, requiring approximately 6,000 GPU hours. Boundary condition The boundary conditions are defined as follo ws: a subsonic stagnation state is imposed at the inlet, while a back-pressure condition is applied at the outlet. Specifically , the non-dimensional total pressure and temperature at the inlet are set to p 0 in /p ref =1 . 01584 and T 0 in /T ref =1 . 00450 , respecti v ely . At the outlet, static pressure and density are fix ed at p out /p ref = 0 . 524191 and ρ out /ρ ref =0 . 630428 . This config- uration ensures unitary static v alues at the inlet, in agreement with the MUR47 operating poi n t described in Arts, T ., (1990). Additionally , v elocity compo- nents at the outlet are loosely constrained using results from a preliminary RANS simulation, with u out /u in = 0 . 284771 and v out /u in = − 1 . 03684 . A sponge layer near the outflo w gradually adapts the solution to the boundary condition o v er time, ef fecti v ely damping tur - P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 52 Pressure side ←− −→ Suction side − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 s/c 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 p w /p in T rat RANS ( T rat ≃ 0 . 72) Exp. ( T rat ≃ 0 . 72) ((a)) W all pressure Pressure side ←− −→ Suction side − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 s/c 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 M is T rat RANS ( T rat ≃ 0 . 72) Exp. ( T rat ≃ 0 . 72) ((b)) Isentropic Mach number Pressure side ←− −→ Suction side − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 s/c 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 C f = | τ w | /q in T rat RANS ( T rat ≃ 0 . 72) ((c)) W all friction Figure 1: Comparison of wall pressure (a), isentropic Mach number (b), and wall friction (c) distrib utions with experimental data and steady-RANS calculations (symbols). Experimental data sourced from Ref. Arts, T ., (1990). b ulent structures and pre venting reflections. Periodic boundary conditions are imposed in both the y - and z -directions. Under this setup, the nominal Mach and Reynolds numbers at the inlet are Ma in =0 . 15 and Re in =2 . 3 × 10 5 , respecti vely . On the blade surfaces, no-slip, impermeable, and isothermal wall boundary conditions are enforced via two immersed boundary method blocks. Follo wing the IBM+WMLES frame work, velocity at the wall is set to zero, wall temperature is prescribed, and turb ulence quantities are locally adapted to ensure accurate pre- dictions of friction and heat flux. The numerical con- figuration is designed to assess the ef fects of wall cool- ing by systematically modifying the wall-to-reco very temperature ratio: T rat = T w T r (3) In the present study , T rat is v aried between 1, corre- sponding to adiabatic conditions, and 0.5, representing a strongly cooled wall. The recov ery temperature T r is defined as T r = T in  1+ γ − 1 2 Pr 1 / 3 in Ma 2 in  , (4) where T in is the total inlet temperature, Pr in is the Prandtl number at the inlet, and Ma in is the inlet Mach number . Reducing T rat to 0.5 implies that the wall tem- perature is approximately half of the stagnation tem- perature, creating significant thermal gradients. T o introduce realistic turbulence at the inlet, ve- locity fluctuations are generated using the digital filter approach. The imposed turb ulence intensity is Tu =  1 / 3  u ′′ i u ′′ i u in , (5) and is set to 3%. Consistent fluctuations in tempera- ture and density are also added at the inlet by applying the Reynolds analogy . 4 Comparison with experimental data The numerical frame work is compared to the e x- perimental setup in Arts, T ., (1990), which explored the aerothermal beha vior of a high-pressure turbine nozzle guide v ane arranged in a linear cascade config- uration. The experiments were carried out at the Isen- tropic Light Piston Compression T ube facility of the vo n K ´ arm ´ an Institute. Among the dif ferent test con- figurations presented in Arts, T ., (1990), the present study tar gets the MUR47 case for numerical simu- lation. This of f-design condition is characterized by an isentropic outlet Mach number of 1.020 and fea- tures complex phenomena such as shock-induced sep- aration, shock–boundary-layer interaction, and strong wak e de velopment. Figures 1(a), 1(b), and 1(c) show the distrib u- tions of normalized wall pressure p w /p in , isentropic Mach number , and friction coefficient C f = | τ w | /q in along both pressure and suction sides of the blade. Here, the reference dynamic pressure is defined as q in = 1 / 2 ρ in u 2 in . Results from WMLES (gray lines with markers) are benchmark ed against e xperimental measurements and steady-RANS outcomes (symbols). The analysis re v eals that wall cooling has a limited influence on wall pressure and isentropic Mach num- ber profiles, while it leads to a modest reduction in wall shear stress, especially on the suction side. The drop in C f is most e vident near its maximum, a region where the boundary layer remains laminar . In con- trast to WMLES, which predicts a gradual decrease in C f beyond s/c ≈ 0 . 5 , RANS maintains an al- most flat profile up to the trailing edge, reflecting its assumption of a fully turb ulent boundary layer . This assumption ov erlooks transitional ef fects and the local turb ulent patches captured by WMLES. On the pres- sure side, v ariations in cooling intensity produce neg- ligible changes in the friction coef ficient, suggesting that the boundary layer in this region is less af fected by thermal conditions. 5 Results Figure 2 presents the temperature distrib ution for Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 53 53 ((a)) T emperature ( T rat =1 . 0 ) ((b)) T emperature ( T rat =0 . 5 ) Figure 2: Instantaneous freestream-total-scaled temperature as a function of the w all-to-rec o v ery t emperature ratio. 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 75 0 . 80 0 . 85 0 . 90 0 . 95 1 . 00 p 0 /p 0 in x / c =0 . 6 T rat ((a)) W ak e total pressure ( x / c =0 . 6 ) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 75 0 . 80 0 . 85 0 . 90 0 . 95 1 . 00 p 0 /p 0 in x / c =0 . 8 T rat ((b)) W ak e total pressure ( x / c =0 . 8 ) Figure 3: Ensemble-a v eraged total pressure distrib ution at tw o dischar ge stations as a function of the w all-to-reco v ery temper - ature ratio. both the adiabatic case ( T rat =1 ) and the most cooled configuration ( T rat =0 . 5 ). As observ ed, the flo w is di- vided into three distinct re gions: an upstream area with nearly uniform temperature, a midsection where accel- eration lo wers temperat ure, and a do wnstream w ak e re gion. W all cooling has a significant impact on w ak e temperature. Specifically , at T rat =0 . 5 , the w ak e be- comes cooler than the surrounding irrotational flo w , in contrast to the adiabatic case, where it remains hotter . T o quantitati v ely asses the role of cooling, at first we w ant to highlight ho w w all temperature influences momentum losses. Figure 3 presents the total pressure distrib ution, normalized by freestream conditions, at tw o do wnst ream locations: x / c =0 . 6 and 0 . 8 . The total pressure is computed as ¯ p 0 =¯ p  1+ γ − 1 2 Ma 2  γ γ − 1 , (6) where ¯ p and Ma denote the static pressure and the ensemble-a v eraged Mach number , respecti v ely . At x / c =0 . 6 the total pressure e xhi bits a charac- teristic dip within the w ak e core, surrounded by nearly in viscid re gions where losses are minimal. As w all temperature decreases due to more aggressi v e cooling, the total pressure deficit in the w ak e deepe n s , and the e xtent of the loss-af fected re gion gro ws. Specifically , cooler w all conditions dampen turb ulence near the sur - f ace, reducing the momentum e xchange with the outer flo w , thereby enhancing losses and b r oadening the w ak e. Con v ersely , the non-turb ulent freestream zones remain relati v ely stable across all cases, retaining pres- sure le v els close to the undisturbed flo w . A similar yet more pronounced pattern emer ges further do wnstream at x / c =0 . 8 . W e conclude that lo wer w all tempera- tures intensify the pressure drop, widen the w ak e, and ultimately lead to higher aerodynamic penalties, un- derscoring the adv erse impact of cooling on turbine performance. Based on this finding, we look for a ph ysical in- terpretation of the phenomenon. Figure 4 reports the ensemble-a v eraged Re ynolds stress components nor - malized by the freestream dynamic pressure: ¯ τ ij = ρ  u ′′ i u ′′ j q in (7) As in the total pressure analysis, results are sho wn at x / c =0 . 6 . The focus is on the diagonal compo- nents ¯ τ 1 1 , ¯ τ 2 2 , and ¯ τ 3 3 , which are presented separately . The data re v eal significant turb ulence anisotrop y . The streamwise component ¯ τ 1 1 (Figure 4(a)) e xhibits a symmetric peak centered near y / t ≈ 0 . 4 , indicati v e of strong turb ulence production. The w all-normal com- ponent ¯ τ 2 2 (Figure 4(b)) also sho ws a b ump-lik e distri- b ution b ut with a leftw ard inflection (point P1) that be- P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 54 54 ((a)) T emperature ( T rat =1 . 0 ) ((b)) T emperature ( T rat =0 . 5 ) Figure 2: Instantaneous freestream-total-scaled temperature as a function of the w all-to-rec o v ery t emperature ratio. 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 75 0 . 80 0 . 85 0 . 90 0 . 95 1 . 00 p 0 /p 0 in x / c =0 . 6 T rat ((a)) W ak e total pressure ( x / c =0 . 6 ) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 75 0 . 80 0 . 85 0 . 90 0 . 95 1 . 00 p 0 /p 0 in x / c =0 . 8 T rat ((b)) W ak e total pressure ( x / c =0 . 8 ) Figure 3: Ensemble-a v eraged total pressure distrib ution at tw o dischar ge stations as a function of the w all-to-reco v ery temper - ature ratio. both the adiabatic case ( T rat =1 ) and the most cooled configuration ( T rat =0 . 5 ). As observ ed, the flo w is di- vided into three distinct re gions: an upstream area with nearly uniform temperature, a midsection where accel- eration lo wers temperat ure, and a do wnstream w ak e re gion. W all cooling has a significant impact on w ak e temperature. Specifically , at T rat =0 . 5 , the w ak e be- comes cooler than the surrounding irrotational flo w , in contrast to the adiabatic case, where it remains hotter . T o quantitati v ely asses the role of cooling, at first we w ant to highlight ho w w all temperature influences momentum losses. Figure 3 presents the total pressure distrib ution, normalized by freestream conditions, at tw o do wnst ream locations: x / c =0 . 6 and 0 . 8 . The total pressure is computed as ¯ p 0 =¯ p  1+ γ − 1 2 Ma 2  γ γ − 1 , (6) where ¯ p and Ma denote the static pressure and the ensemble-a v eraged Mach number , respecti v ely . At x / c =0 . 6 the total pressure e xhi bits a charac- teristic dip within the w ak e core, surrounded by nearly in viscid re gions where losses are minimal. As w all temperature decreases due to more aggressi v e cooling, the total pressure deficit in the w ak e deepe n s , and the e xtent of the loss-af fected re gion gro ws. Specifically , cooler w all conditions dampen turb ulence near the sur - f ace, reducing the momentum e xchange with the outer flo w , thereby enhancing losses and b r oadening the w ak e. Con v ersely , the non-turb ulent freestream zones remain relati v ely stable across all cases, retaining pres- sure le v els close to the undisturbed flo w . A similar yet more pronounced pattern emer ges further do wnstream at x / c =0 . 8 . W e conclude that lo wer w all tempera- tures intensify the pressure drop, widen the w ak e, and ultimately lead to higher aerodynamic penalties, un- derscoring the adv erse impact of cooling on turbine performance. Based on this finding, we look for a ph ysical in- terpretation of the phenomenon. Figure 4 reports the ensemble-a v eraged Re ynolds stress components nor - malized by the freestream dynamic pressure: ¯ τ ij = ρ  u ′′ i u ′′ j q in (7) As in the total pressure analysis, results are sho wn at x / c =0 . 6 . The focus is on the diagonal compo- nents ¯ τ 1 1 , ¯ τ 2 2 , and ¯ τ 3 3 , which are presented separately . The data re v eal significant turb ulence anisotrop y . The streamwise component ¯ τ 1 1 (Figure 4(a)) e xhibits a symmetric peak centered near y / t ≈ 0 . 4 , indicati v e of strong turb ulence production. The w all-normal com- ponent ¯ τ 2 2 (Figure 4(b)) also sho ws a b ump-lik e distri- b ution b ut with a leftw ard inflection (point P1) that be- P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 54 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 τ 11 /q in T rat T rat T rat T rat T rat T rat x / c =0 . 6 ((a)) τ 11 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 τ 22 /q in T rat T rat T rat T rat T rat T rat x / c =0 . 6 P1 ((b)) τ 22 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 y /t 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 τ 33 /q in T rat T rat T rat T rat T rat T rat x / c =0 . 6 P2 ((c)) τ 33 Figure 4: Ensemble-av eraged Re ynolds stress components and turbulent kinetic ener gy as functions of wall-to-reco v ery tem- perature ratio. 10 − 2 10 − 1 10 0 10 1 10 2 St = f c/u in f · S pp /σ 2 T rat T rat T rat T rat T rat T rat 0.2 1.3 45 120 ((a)) Probe 1 10 − 2 10 − 1 10 0 10 1 10 2 St = f c/u in f · S pp /σ 2 T rat T rat T rat T rat T rat T rat 1.5 85 ((b)) Probe 2 10 − 2 10 − 1 10 0 10 1 10 2 St = f c/u in f · S pp /σ 2 T rat T rat T rat T rat T rat T rat 6 70 ((c)) Probe 3 10 − 2 10 − 1 10 0 10 1 10 2 St = f c/u in f · S pp /σ 2 T rat T rat T rat T rat T rat T rat 10 85 ((d)) Probe 4 Figure 5: Power Spectral Density (PSD) analysis at dif ferent probe locations along the suction side. comes more prominent under increased cooling. The spanwise term ¯ τ 33 (Figure 4(c)) maintains lo wer ov er - all v alues, roughly half those of the other components, yet follo ws a similar inflection trend (point P2). Overall results demonstrate that cooling the w all suppresses turb ulence intensity . As the w all tempera- ture decreases, Reynolds stress le v els are reduced, in- dicating weaker ener gy transfer from the freestream to ward the near -wall zone. Con versely , higher wall temperatures enhance the peak v alues, suggesting in- tensified turb ulence acti vity and stronger coupling be- tween the outer flo w and boundary layer . In ¯ τ 22 , the influence of wall cooling is particularly pronounced, reinforcing the notion that thermal conditions signif- icantly shape turb ulence anisotropy and ener gy parti- tioning across velocity components. Thus, to finally substantiate the findings and ex- plaining the rationale behing the incresed lossed in cooled scenarios, Figure 5 displays the Power Spec- tral Density (PSD) of pressure signals acquired from selected wall-mounted probes. Pressure fluctuations are analyzed using Lagrangian statistics at four probes Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 55 55 along the suction side, as reported in Fig. 2. The PSD of a time signal x T ( t ) , ov er a duration T , is formally defined as: S xx = lim T →∞  1 T | ˆ x T ( f ) | 2  , (8) where | ˆ x T ( f ) | 2 denotes the squared magnitude of the Fourier transform of the signal, constructed via the con volution of x T ( t ) and its comple x conjugate x ∗ T ( − t ) , with f representing the frequency . In Fig- ure 5, individual gray curv es represent ra w PSD estimates computed using W elch’ s method, while smoothed en velopes (gray-shaded) are generated using the K onno–Ohmachi filter , which maintains a constant bandwidth in logarithmic space. Pressure signals were di vided into ten ov erlapping se gments (50% ov erlap), each tapered with a Hann windo w . The resulting spec- tra are obtained by a veraging the periodograms of each segment. PSDs are shown in a pre-multiplied form, f · S pp , so that the integral under the curv e represents the total ener gy at frequency f . T o enable direct com- parison, each spectrum is normalized by the square of the root-mean-square pressure fluctuation, σ 2 , and plotted against the Strouhal number St = f c/u in . For visualization purposes, spectra are vertically of fset by one order of magnitude for each decreasing wall-to- recov ery temperature ratio; consequently , the vertical axis v alues are omitted as they no longer carry abso- lute meaning. Across all four probes, pressure spectra re veal contrib utions distrib uted across lo w , mid, and high- frequency bands. At Probe 1, a lo w-frequenc y peak appears near St =0 . 2 , follo wed by a medium- frequency maximum at St =1 . 3 , and two distinct high-frequency peaks at St = 45 and St = 120 . Moving to Probe 2, the low-frequenc y feature van- ishes, the medium-frequenc y peak shifts slightly to St =1 . 5 , and the high-frequency content consolidates into a broader single peak around St = 85 . Further do wnstream on the suction side, Probes 3 and 4 sho w more prominent mid-frequency peaks at St =6 and St = 10 , respecti v ely . Howe ver , the high-frequency components, representati ve of finer -scale turbulence, progressi v ely weaken with distance from the leading edge. The ke y outcome of this spectral analysis is the at- tenuation of high-frequency pressure fluctuations with increased wall cooling. As w all temperature de- creases, small-scale peaks consistently lose intensity across all probes, while ener gy accumulates at lo wer frequencies. Notably , at T rat =0 . 5 , the high-frequency features at Probes 3 and 4 v anish entirely , suggesting a strong inhibition of ener gy transfer from large-scale pressure and momentum fluctuations to finer turb ulent motions. These observations corroborate trends found total pressure analysis and Reynolds stresses, reinforc- ing the idea that colder walls intensify thermal ef fects, which in turn disrupt the turb ulent energy cascade. 6 Conclusions This study highlights the ef fecti v eness of integrat- ing WMLES with IBM for analyzing transonic gas tur - bine stator aerothermodynamics. The approach pro- vides detailed insights into the impact of wall cool- ing on flo w beha vior , demonstrating that lo wer wall temperatures suppress near -wall turb ulence, alter the ener gy cascade, and modify w ake structures, ulti- mately increasing pressure losses. These findings of- fer v aluable guidance for optimizing cooling strategies to improv e turbine performance and ef ficienc y . The methodology is also computationally ef ficient, partic- ularly for mean flo w analysis. F or design applica- tions, it deliv ers results comparable to steady RANS at a fraction of the cost of WRLES, making it well- suited for rapid geometric optimization while main- taining high accuracy . Further details will be presented in the final paper and discussed during the conference. Refer ences Arts, T ., Lambert de Rouvroit, M., and Rutherford, A. W . (1990), Aero-thermal in vestigation of a highly loaded tran- sonic linear turbine guide v ane cascade: A test case for in vis- cid and viscous flo w computations, NASA STI/Recon T ech- nical Report N , V ol. 91, 23437. De V anna, F ., A v anzi, F ., Cogo, M., Sandrin, S., Bettencourt, M., Picano, F ., and Benini, E. (2023), URANOS: A GPU-accelerated Na vier-Stok es solver for compressible wall-bounded flo ws, Computer Physics Communications , V ol. 287, 108717, doi: https://doi.org/10.1016/j.cpc.2023.108717 De V anna, F ., & Baldan, G. (2024). URANOS-2.0: Im- prov ed performance, enhanced portability , and model ex- tension to wards e xascale computing of high-speed engineer - ing flo ws. Computer Physics Communications, V ol. 303, 109285, doi: https://doi.org/10.1016/j.cpc.2024.109285 De V anna, F ., Cogo, M., Bernardini, M., Picano, F ., and Benini, E. (2021), Unified wall-resolved and w all-modeled method for lar ge-eddy simulations of compressible wall- bounded flo ws, Phys. Rev . Fluids , V ol. 6, No. 3, 034614, doi: https://doi.org/10.1103/Ph ysRe vFluids.6.034614 De V anna, F ., Baldan, G., Picano, F ., & Benini, E. (2023, October). High-Reynolds compressible flo ws simulation with wall-modeled LES and immersed boundary method. In ERCOFT A C W orkshop Direct and Lar ge Eddy Simula- tion (pp. 203-208), doi: https://doi.org/10.1007/978-3-031- 47028-8 31 De V anna, F ., Benato, A., Picano, F ., and Benini, E. (2021), High-order conserv ati ve formulation of viscous terms for v ariable viscosity flo ws, Acta Mechanica , V ol. 232, pp. 2115–2133, doi: https://doi.org/10.1007/s00707- 021-02937-2 Nicoud, F ., and Ducros, F . (1999), Subgrid-scale stress mod- elling based on the square of the velocity gradient tensor , Flow , T urbulence and Comb ustion , V ol. 62, No. 3, pp. 183–200, doi: https://doi.org/10.1023/A:1009995426001 Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 56 56 along the suction side, as reported in Fig. 2. The PSD of a time signal x T ( t ) , o v er a duration T , is formally defined as: S xx = lim T →∞  1 T | ˆ x T ( f ) | 2  , (8) where | ˆ x T ( f ) | 2 denotes the squared magnitude of the F ourier transform of the signal, constructed via the con v olution of x T ( t ) and its comple x conjug ate x ∗ T ( − t ) , with f representing the frequenc y . In Fig- ure 5, indi vidual gray curv es represent ra w PSD estimates computed using W elch’ s method, while smoothed en v elopes (gray-shaded) are generated using the K onno–Ohmachi filter , which maintains a constant bandwidth in log arithmic s pace. Pressure signals were di vided into ten o v erlapping se gments (50% o v erlap), each tapered with a Hann windo w . The resulting spec- tra are obtained by a v eraging the periodograms of each se gment. PSDs are sho wn in a pre-multiplied form, f · S pp , so that the inte gral under the curv e represents the total ener gy at frequenc y f . T o enable direct com- parison, each spect rum is normalized by the square of the root-mean-square pres sure fluctuation, σ 2 , and plotted ag ainst the Strouhal number St = f c /u in . Fo r visualization purposes, spectra are v ertically of fset by one order of magnitude for each decreasing w all-to- reco v ery temperature ratio; consequently , the v ertical axis v alues are omitted as the y no longer carry abso- lute meaning. Across all four probes, pressure spectra re v eal contrib utions distrib uted across lo w , mid, and high- frequenc y bands. At Probe 1, a lo w-frequenc y peak appears near St =0 . 2 , follo wed by a medium- frequenc y maximum at St =1 . 3 , and tw o dist inct high-frequenc y peaks at St = 45 and St = 120 . Mo ving to Probe 2, the lo w-frequenc y feature v an- ishes, the medium-frequenc y peak shifts slightly to St =1 . 5 , and the high-frequenc y content consolidates into a broader single peak around St = 85 . Further do wnstream on the suction side, Probes 3 and 4 sho w more prominent mid-frequenc y peaks at St =6 and St = 10 , respecti v ely . Ho we v er , the high-frequenc y components, representati v e of finer -scale turb ulence, progressi v ely weak en with distance from the leading edge. The k e y outcome of this spectral analysis is the at- tenuation of high-frequenc y pressure fluctuations with increased w all cooling. As w all temperature de- creases, small-scale peaks consistently lose intensity across all probes, while ener gy accumulates at lo wer frequencies. Notably , at T rat =0 . 5 , the high-frequenc y features at Probes 3 and 4 v anish entirely , suggesting a strong inhibition of ener gy transfer from lar ge-scale pressure and momentum fluctuations to finer turb ulent motions. These observ ations corroborate trends found total pressure analysis and Re ynolds stresses, reinforc- ing the idea that colder w alls intensify thermal ef fects, which in turn disrupt the turb ulent ener gy cascade. 6 Conclusions This study highlights the ef fecti v eness of inte grat- ing WMLES with IBM for analyzing transonic g as tur - bine stator aerothermodynamics. The approach pro- vides detailed insights into the im pact of w all cool- ing on flo w beha vior , demonstrating that lo wer w all temperatures suppress near -w all turb ulence, alter the ener gy cascade, and modify w ak e struct u r es, ulti- mately increasi n g pressure losses. These findings of- fer v aluable guidance for optimizing cooling strate gies to impro v e turbine performance and ef ficienc y . The methodology is also computationally ef ficient, partic- ularly for mean flo w analysis. F or design applica- tions, it deli v ers results comparable to steady RANS at a fraction of the cost of WRLES, making it well- suited for rapid geometric optimization while main- taining high accurac y . Further details will be presented in the final paper and discussed during the conference. Refer ences Arts, T ., Lambert de Rouvroit, M., and Rutherford, A. W . (1990), Aero-thermal in v estig ation of a highly loaded tran- sonic linear turbine guide v ane cascade: A test case for in vis- cid and viscous flo w computations, N ASA STI/Recon T ec h- nical Report N , V ol. 91, 23437. De V anna, F ., A v anzi, F ., Cogo, M., Sandrin, S., Bettencourt, M., P icano, F ., and Benini, E. (2023), URANOS: A GPU-accelerated Na vier -Stok es solv er for compressible w all-bounded flo ws, Computer Physics Communications , V ol. 287, 108717, doi: https://doi.or g/10.1016/j.cpc.2023.108717 De V anna, F ., & Baldan, G. (2024). URANOS-2.0: Im- pro v ed performance, enhanced portability , and model e x- tension to w ards e xascale computing of high-speed engineer - ing flo ws. Computer Ph ysics Communications, V ol. 303, 109285, doi: https://doi.or g/10.1016/j.cpc.2024.109285 De V anna, F ., Cogo, M., Bernardini, M., Picano, F ., and Benini, E. (2021), Uni fied w all-resolv ed and w all-modeled method for lar ge-eddy simulations of compressible w all- bounded flo ws, Phys. Re v . Fluids , V ol. 6, No. 3, 034614, doi: https://doi.or g/10.1103/Ph ysRe vFluids.6.034614 De V anna, F ., Bal dan, G., Picano, F ., & Benini, E. (2023, October). High-Re ynolds compressible flo ws simulation with w all-modeled LES and immersed boundary method. In ERCOFT A C W orkshop Direct and Lar ge Eddy Simula- tion (pp. 203-208), doi: https://doi.or g/10.1007/978-3-031- 47028-8 31 De V anna, F ., Bena to, A., Picano, F ., and Benini, E. (2021), High-order conserv ati v e formulation of viscous terms for v ariable viscosity flo ws, Acta Mec hanica , V ol. 232, pp. 2115–2133, doi: https://doi.or g/10.1007/s00707- 021-02937-2 Nicoud, F ., and Ducros, F . (1999), Subgrid-scale s tress mod- elling based on the square of the v elocity gradient tensor , Flow , T urb ulence and Comb ustion , V ol. 62, No. 3, pp. 183–200, doi: https://doi.or g/10.1023/A:1009995426001 P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 56 W ALL C OOLING E F FECTS ON E NTR OPY P R ODUCTION IN S HOCK -B OUND AR Y L AY E R I NTERA CTIONS F . De V anna 1 , ∗ 1 Dipartimento di Ingegneria Industriale, Univ ersit ` a degli Studi di P ado va, V ia V enezia, 1, 35121 ∗ fr [email protected] Abstract This study in vestigates entrop y generation in shock–boundary layer interactions (SBLI) at Mach 1.1 under systematically v aried wall cooling. W all- resolved lar ge-eddy simulations (WRLES) and a second-law thermodynamic frame work are employed to quantify irre versible losses. Results re veal that w all cooling strongly modifies the entropy balance between viscous and thermal mechanisms. Three regimes are identified: cooling-dominated, friction-dominated, and entropy-balanced. Self-similar , non-dimensional coef ficients are introduced to isolate viscous ( C ∗ v ) and conducti v e ( C ∗ h ) contrib utions. These follo w robust po wer -law trends with momentum-thickness Re ynolds number . The frame work enables a predicti ve charac- terization of SBLI entropy beha vior . Implications ex- tend to aerospace and turbomachinery systems. 1 Intr oduction Entropy generation is a k ey f actor in engineer - ing inef ficiencies, directly linked to irre v ersibilities in ener gy-con verting and transport systems. It arises pri- marily from viscous dissipation due to shear stresses and thermal transport across temperature gradients, contrib uting to aerodynamic losses, pressure drops, and thermal inef ficiencies in ener gy de vices. Quan- tifying and minimizing entropy generation is essential for improving system performance, particularly in ap- plications where reducing losses can lead to significant gains in ef ficienc y and durability . In high-speed aerodynamics, shock wa v es and tur - b ulent boundary layers interact, leading to substantial irre v ersibilities. These ef fects are particularly critical in aerospace applications such as high-speed aircraft, rocket nozzles, and gas turbines. In such systems, wall cooling is a common strate gy to regulate mate- rial temperatures, especially in components exposed to extreme thermal loads, such as turbine blades and supersonic vehicle surf aces. Ho we v er , while cooling mitigates thermal stress, it also alters the structure of the boundary layer and entropy generation. The com- petition between these ef fects makes it essential to un- derstand ho w wall cooling influences entrop y produc- tion to optimize performance in v arious thermal-fluid applications. Pre vious studies ha ve e xtensi v ely analyzed entropy production in turb ulent boundary layers and shock- dominated flo ws, primarily under adiabatic condi- tions. Howe ver , the impact of wall cooling remains an open question, particularly in transonic and supersonic regimes where cooling modifies both shear -dri v en and heat conduction-related entropy sources. A cooled boundary layer experiences increased near -wall den- sity , which alters velocity gradients and turb ulence dy- namics. Simultaneously , cooling enhances thermal en- tropy generation, which may of fset reductions in vis- cous dissipation. The balance between these compet- ing mechanisms is complex and requires systematic in vestigation to characterize the ef fect of cooling in- tensity on entropy production. This study examines entrop y beha vior in a Mach 1.1 shock-boundary layer interaction using W all- Resolved Lar ge-Eddy Simulations (WRLES), provid- ing a frame work for e valuating entrop y generation in cooled compressible wall flo w configurations. In particular , this study adopts a second-la w thermody- namic perspecti v e to e v aluate shock–boundary-layer interaction. The results highlight how w all cooling reshapes the distrib ution of entropy generation be- tween the shock and the boundary layer , leading to three distinct regimes: a cooling-dominated re gime, where reduced wall temperature lo wers total entropy belo w in viscid shock predictions; a friction-dominated regime, where viscous dissipation in the boundary layer becomes the dominant source; and a transi- tional, entropy-balanced regime, where both mecha- nisms contrib ute comparably . Furthermore, the study introduces non-dimensional, self-similar entropy dis- sipation coef ficients to quantify the contrib utions from shear and thermal transport. The paper is structured as follo ws: Section 2 de- tails the model; Section 3 outlines the simulation setup and the numerical methods; Section 4 presents the main findings; and Section 5 summarizes key results and future outlook. Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 57 57                                                                                                                                                                                                                                                                                     ı                                                                                                                                                                                                                                                                                                                                                                                                                                                                      roplet motion is governed by Newton’s second law                𝐶𝐶 �                       𝐶𝐶 � �� 1 + � 2 �� + � 3 �� 2  (1)          � 1  � 2  � 3                         ��                                � � �� � (𝐶𝐶 ��� −𝐶𝐶 � �� )      � �                 � �                    𝐶𝐶 � ��    𝐶𝐶 � ��                       Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 64 64                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ı                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    r o p let m o tio n is g o v er n ed b y New to n ’ s s ec o n d law                                                      𝐶𝐶 �                                                             𝐶𝐶 � �� 1 + � 2 �� + � 3 �� 2  (1 )                     � 1  � 2  � 3                                                 ��                                                                                                                                            � � �� � (𝐶𝐶 � �� −𝐶𝐶 � �� )         � �                                        � �                                                  𝐶𝐶 � ��     𝐶𝐶 � ��                                                                    P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 64             � �      �ℎ ��  �ℎ �� � � � � � � � �� � 2 + 0.6�� � 12 ⁄ �� 13 ⁄  (3)   � �          � � ��                      ��           �� � � �� ���    � �                    �� �   �� � �� � ( � + �� ) −� � ( � ) � −� � � 2 � � �� � � ��     � � ��             ��                   � � � � � �� � �� � ℎ� � ( � � −� � ) − �� � �� ℎ ��     � � �         ℎ                 � �         ℎ ��             Proper ties  Water liquid  Mixture (Air + water vap or)  Molecular weight [ g/mol]  18  28.96 6 (air)  Density  [kg/m 3 ]  999  Ideal gas     0.00091- 0.001002  1.7894e- 05     0.072- 0.07275  -  A 2  5.11564  -  B 2  1687.537  -  C 2  230.17  -  T min – T max 3  [° C]  0.05 - 200.05  -  Thermal cond uctivity  [W/(m.K)]  0.61  0.024 2  Thermal capacity  [J/kg.K]  4180.01  1006.43  Latent h eat  [kJ/kg]  2260  -  D AB (T ref ) 4  [m 2 /s]  2.16 x 10 -5  -                     ı                                          � ���      � �� ( � � ) �� �� (� � �� )( � � � � �� ) 3 2 ( 101325 � � . ( 1+� � � )  (6)   � �� (� � �� )              � �               �            � �           � � � �� . � ��� � ���            ��� 10 � ��� � �− � 𝐶𝐶 +� �  (7)                     � �                                                                                                                            ı                                                                     DPM is suited for low particle loading (≤   –                                       Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 65 65                                      “S  2,” the bottom was a 0.203 m/s velocity inlet and      ı                                                                                                                                             –                                                                    2, w hich incl udes a 0. 203 m/s upward air-       ing dro plet dia mete r by ~16 .8% at 450 s . Despit e                                                                    Exper imental results of dataset “S  1 ”     ( red) and dataset “S  2”  ı                                                                                                                                                                   Exper imental results of dataset “A  1 ”     ( cyan), dataset “A  2 ”     “A  3” (gree n)                                                          Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 66 66                                                                                                                                        “S  2 , ” th e b o tto m was a 0 . 2 0 3   m /s v elo city in let an d                                 ı                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   –                                                                                                                                                                       2, w h i c h i n c l u de s a 0. 20 3  m / s upw a rd a i r -                                          i ng d r o pl e t di a m e t e r b y ~ 16 . 8% a t 45 0  s . D e s pi t e                                                                                                                                                                                                                                             E x p er im en tal r esu lts o f d ataset “S  1 ”                   ( r e d ) an d d ata s et “S  2 ”     ı                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  E x p er im en tal r esu lts o f d ataset “A  1 ”                  ( cy a n ) , d ata s et “A  2 ”                                      “A  3” ( gr e e n)                                                                                                                                                                               P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 66   size. For droplets und er 3 mm, CFD p redicts term inal                       abov e 5 mm, as seen in                                                                         0% at 1.17  mm to 7.4% at 3 mm and 23.3% at 5 mm.           velocity results of (a) dataset “F  2”     and (b) dataset “F  4”               v elocity results of dataset “F  1 ”                                                                                    ∼                                              Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 67 67             evapo ration results of ( a) dataset “F  6”    and (b) dataset “F  7”                                                                                                                                                 falli ng drople ts under 3 mm, support ing      For droplets over 3 mm, a non                                                                                                                                                                                                                                                                                                                                                                                                                                                                          –                                                 Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 68 68                                           ev ap o r atio n r esu lts o f ( a ) d ata s et “F  6 ”               an d ( b ) d ataset “F  7 ”                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     f a l l i n g dr o pl e t s unde r 3  m m , s upp or t i n g                           F o r dr opl e t s ove r 3  m m , a non                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     –                                                                                                                                                                                           P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 68          –                                                                           –                                               –                                                                                                  –                                                                 –                                    –                                                                                                                   –                                                –                                          –                                                                 Perry’s chemical engin eers’ han dbook. 7th ed .        Poling, B. E., Prausnitz, J. M., & O’Connell, J. P.                                                         –                                         –          ı  k, J., Džumb ová, L., Schwarz, J., & Ku lmala, M.                            –           ı  k, J., Džumb ová, L., Schwarz, J., & Ku lmala, M.                            –                        –                                    –                                                                                                     Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 69 69                                  –                                        –                                    –                                                                                                   –           Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 70 70                                                                                                           –                                                                                                                                                                             –                                                                                                                     –                                                                                                                                                                                                                                                                                                                                               –                                 P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 70                                                                                                                                                                                                                                                                                                                                                                                                                                      –                                                                                                      .        ω                                                                                                                                                                          ‘           ’                                                                                                                                                                                                                                                                                                                                                                               ‘ v o r t e x r eact o r ’                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         𝑢𝑢 �                                                                                                                                                                                                                 � �� � ) ∶ 𝑢𝑢 � = 2 �� �             � �� � ) ∶ 𝑢𝑢 � =2 �� � � 2 �                                                                                                                                                                                                                                                 –                                                                                                                                                                           –                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 71 71                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            0. 2 25  m                                                                                                                                                                                                                                                                                                                                                                                                                              –                                                                                                                                                                                                                                                                                                     θ                                                             �� � �� + � �� � ( � � � � ) =0     � �� ( � � � � ) + � �� � ( � � � � � � ) =− �� �� � + � �� � � � � [ �� � �� � + �� � � � � ] ) + � �� � (− � � � � ′    � � ′    )+ � � � � +� �      � �� ( � � � � )+ � �� � ( � � � � � � ) = 0     � � +� � =1                                                                 𝑢𝑢 � ′                                                                          ρ                                                                                                                                                                                                                                                                α                                                               α                                       α                                                                                                                  α                              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                                                                                                                                                                                                                                                                                                                                                           0. 2 25  m                                                                                                                                                                                                                                                                                                                                                                                                                              –                                                                                                                                                                                                                                                                                                     θ                                                             �� � �� + � �� � ( � � � � ) =0     � �� ( � � � � ) + � �� � ( � � � � � � ) =− �� �� � + � �� � � � � [ �� � �� � + �� � � � � ] ) + � �� � (− � � � � ′    � � ′    )+ � � � � +� �      � �� ( � � � � )+ � �� � ( � � � � � � ) = 0     � � +� � =1                                                                 𝑢𝑢 � ′                                                                          ρ                                                                                                                                                                                                                                                                α                                                               α                                       α                                                                                                                  α                                               −� � � � ′    � � ′                                                                                                                                       –ε                                                          –ω                                                                                                                                                                                                      z 1  z 2  z 3  z 4  z 5  z 6        P ro ceed in gs of th e 15th E R COFT A C S ymp osiu m on E n gin eerin g T u rb u len ce Mo d ellin g an d Measu remen ts Du b ro vn ic, Croatia, 22-24 S ep tem b e r 2025 d oi: 10.5281/zen o d o.17280519 72                                                                                                                         –ε                                                                                      –ω                                                                                                    –ε     – ω                                         –ω         –ω                                                                                              –ε    –ω                                                                                       –                                       –                                 −                                                       ε      ω                       ω                                                                                               –ω                                                                             ‘for ce d’               ‘free’ vortex                                                                  . I n the ce ntral ‘forc ed’ vo rtex re gio n, most                               –     –       ε      ω                     ω CC mo del                                 wall ‘f ree’ vortex region , the  ε                                     ω                                        ε                                                                    ε  turb u len c e mod e l f ail s to cap tu re th e ‘co m- bine d vorte x ’ mot io n observe d in                                                                                                            ω                              the ‘com bin ed vort ex’ shape.          ω                                            ‘      ’                                     ω        slightly better pred ictions in th e ‘for ced’ vor tex re-                                                                                                                          Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 73 73 large eddies filtered spectrum scale separation grid cutoff ∥ ϕ ∥ 2 ∥ ϕ ⋆ ∥ 2 ∥ ϕ ′ ∥ 2 sub grid Figure 3: Sketch of the ener gy spectra of the full solution and that of the lar ge eddies. The entire spectrum di vides into three parts. The subgrid part ∥ ϕ ′ ∥ 2 is cuttof f by the grid. The model aims to separate the supergrid ener gy ∥ ϕ ⋆ ∥ 2 from the energy ∥ ϕ ∥ 2 of the large eddies. as ϕ ⋆ = S  ϕ . The time rate of change of ϕ ⋆ is obtained by applying the dif ference operator S to Eq. (11): h∂ t ϕ ⋆ = ∇ S ( Φ + τ ) , (24) where the symmetric N × N matrix ∇ S is defined by ∇ S = 1 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ − 21 1 1 − 21  1 − 21 11 − 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ This matrix is neg ati ve-semidefinite (lik e a second deri v ati v e). Ho we ver , it takes on the nature of a gradi- ent if we do not consider it as a linear map of  ϕ b ut as one of ϕ ⋆ : ( ∇ S  ϕ ) i − 1 / 2 = − ϕ ⋆ i + ϕ ⋆ i − 1 . The flux Φ in Eq. (24) can be di vided into a con- vecti ve and a dif fusiv e part. So, we get h∂ t ϕ ⋆ = ∇ S ( c ( u , ϕ ) − 2 ν h ϕ ⋆ + τ ) (25) where the con vecti ve contrib ution is described by means of the vector c with components c i = u i ϕ i with i = 1 , ⋅⋅⋅ ,N . The con vecti ve term in Eq. (25) depends only on the density ϕ and the velocity u of the lar ge eddies. T o see what happens if no model is used, we take τ = 0 in Eq. (25). The residual density ϕ ⋆ is then dri v en by the large eddies, that is the lar ge eddies produce ϕ ⋆ and that’ s the only way the y can be produced. Thus ϕ ⋆ is subor dinate to ϕ . But if the lar ge eddies increase the amount of residual density , then the dynamics are not closed; in a closed system lar ge eddies should only transport and dissipate themselves. Ef fecti v ely the dissipation of the lar ge eddies falls short if the y produce ϕ ⋆ . In conclusion, if ϕ ⋆ is produced by the large eddies then τ should not be zero. Modeling the separation of supergrid scales Eq. (23) implies that the dissipation of energy breaks do wn into three parts: d t ∥ ϕ ∥ 2 2 = h d t ∥ ϕ ∥ 2 + h d t ∥ ϕ ⋆ ∥ 2 + d t ∥ ϕ ′ h ∥ 2 2 . (26) The last part, the subgrid dissipation, v anishes accord- ing to Eq. (19); hence the abov e right-hand side con- sists exclusi vely of super grid contrib utions. Now sup- pose the system gov erning the lar ge eddies is not well closed. Then the large eddies could produce smaller scales of motion. So it could be that d t ∥ ϕ ⋆ ∥ 2 > 0 . Eq. (26) then implies d t ∥ ϕ ∥ 2 2 > h d t ∥ ϕ ∥ 2 , i.e., the dissipation of the ener gy of the large eddies is then less than the dissipation of all scales of motion. In other words, the dissipation model τ is wrong: it provides too fe w dissipation. Hence, in this case τ is to be taken such that d t ∥ ϕ ⋆ ∥ 2 = 0 . (27) Eq. (26) then implies d t ∥ ϕ ∥ 2 2 = h d t ∥ ϕ ∥ 2 . No w the lar ge eddies dissipate the right amount of en- er gy , with the cav eat that this applies globally - since only the total ener gy is considered, no statement can be made about the local dissipation of ener gy . Of course it could also be that d t ∥ ϕ ⋆ ∥ 2 < 0 . (28) This can occur if (1) Eq. (25) is dominated by viscous dissipation or (2) ener gy is transferred from the resid- ual scales to the lar ge eddies. Case (1) hints that the residual scales are well-resolved; case (2) is kno wn as backscatter . As ϕ ⋆ is subordinate to ϕ backscatter can only be generated by τ . Backscatter is hard to model because the production of ϕ ⋆ is suppressed and there- fore there is no accurate description of ϕ ⋆ . Hence, feeding ϕ ⋆ back into ϕ introduces errors in the dy- namics of the lar ge eddies. So, in case inequality (28) holds, we take τ = 0 . The combination of inequality (28) and equality (27) guarantees that d t ∣∣ ϕ ⋆ ∣∣ 2 ≤ 0 , (29) for any time t , i.e., no r esidual density is pr oduced . Note that our reasoning implicitly relies on the 2- norm, though alternati ve norms could be considered. Interestingly , taking the 1-norm in Eq. (29) results into total v ariation diminishing (TVD) schemes. Inequality (29) can be re written using Eq. (24). Thus we get ∇ S ϕ ⋆ ⋅ ( Φ + τ ) ≤ 0 , (30) Note that we are considering periodic boundary con- ditions. As only a requirement is imposed on the com- ponent of τ in the direction of the vector ∇ S ϕ ⋆ , we take τ = − α ∇ S ϕ ⋆ , (31) where the coef ficient α ≥ 0 is taken as minimally as possible . This leads to α = ( ∇ S ϕ ⋆ ⋅ Φ ) + ∥ ∇ S ϕ ⋆ ∥ 2 , (32) Pro ceedings of the 15th ER COFT AC Symposium on Engineering T urbulence Mo delling and Measuremen ts Dubro vnic, Croatia, 22-24 Septem b er 2025 doi: 10.5281/zeno do.17280519 80 80 [Document text truncated for crawler view.]