SIMULATIONANDMODELINGOFCOMPRESSIBLEANDINCOMPRESSIBLE TURBULENTCHANNELFLOWSOVERROUGHWALLS By MostafaAghaeiJouybari ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2020 ABSTRACT SIMULATIONANDMODELINGOFCOMPRESSIBLEANDINCOMPRESSIBLE TURBULENTCHANNELFLOWSOVERROUGHWALLS By MostafaAghaeiJouybari Thee˙ectsofsurfaceroughnessonwall-boundedturbulent˛owsareimportantforfundamental turbulenceresearch,andturbulencemodelingandcontrol,inbothcompressibleandincompress- ibleregimes.Thisdissertationstudiesthesee˙ectsthroughstatisticalandstructuralanalysisof turbulence,andprovidespracticalinsightsformodelingofturbulenceinthepresenceofroughness forincompressible˛ows.Italsoproposesanimmersedboundarymethodtosimulatecompressible ˛owsoverroughwallswithcomplexgeometries,andstudiestheroughnesse˙ectsonsupersonic ˛owsoverwavywalls. Turbulencestatisticsinopenchannel˛owsoverasmoothwallandthreetypesofwallroughness: sand-grain,cuberoughnessandarealistic,multi-scaleturbine-bladeroughness,areexamined usingdirectnumericalsimulations.Transportofthemeanmomentum,normalcomponentsof theReynoldsstresstensor,andnormalcomponentsofthedispersivestresstensorareanalyzed. Theresultsshowhigherturbulenceisotropyfortheroughwallscomparedtothesmoothwall. Wakeproduction,themechanismthroughwhichenergyistransportedfromthewake˝eldtothe turbulence˝eld(andviceversa),isstronglyin˛uencedbythekindofroughwall.Forsynthetic roughwalls,thewakeproductionhasrelativelylargepositivevalues,whileitisnegativewitha smallermagnitude,fortheturbine-bladesurface.Theseresultsindicateastrongdependenceof turbulenceprocessesinthenearwallregionsontheroughnesstopography. Turbulentcoherentmotionsin˛owsoverroughwallsarealsoanalyzed.Two-pointvelocity correlations,lengthscales,inclinationangles,andvelocityspectraarestudied.Resultsfromlinear stochasticestimationsuggestthat,nearthewall,thequasi-streamwisevorticesobservedinsmooth- wall˛owarepresentinthelarge-scalerecessedregionsofmulti-scaleroughness,whereasthey arereplacedbyapairof`head-up,head-down'horseshoestructuresinthesandgrainandcube roughnesses,similartothoseobservedinthepreviousstudies.Thecon˝gurationofconditional eddiesnearthewallsuggeststhatthekinematicbehaviorofvorticesdi˙ersforeachkindofrough surfaces.Vorticesovermultiscaleroughnessareconjecturedtoobeyagrowthmechanismsimilar tothoseoversmoothwalls,whilearoundthecuberoughnessthehead-downhorse-shoevortices undergoasolid-bodyrotationontopoftheelementonaccountofthestrongshearlayer.This shortensthelongitudinalextentofthenear-wallstructuresandpromotesturbulenceproduction. DeepNeuralNetworks(DNN)andGaussianProcessRegression(GPR)areusedtopropose ahigh-˝delitypredictionoftheNikuradseequivalentsandgrainheight,( k s ),whichisfrequently usedinturbulencemodelingof˛owsoverroughwalls.Toprovideagooddatabase,45widely di˙erentsurfacegeometriesaregeneratedandsimulatedatfrictionalReynoldsnumberof1000, whicharealsoaccompaniedby15fullyroughexperimentaldata.ThedesignedDNNandGPR modelspredict k s with err rms < 10% and err max < 30% whichismuchmoreaccuratethanthe modelssuggestedinpreviousstudies. Finally,anewimmersedboundarymethodisproposedtosimulate˛owovercomplexgeometries insub-andsupersonicregimes.Themethodusesalevel-set˝eldtoimposeappropriateboundary conditionsattheinterfaceofthe˛uidandsolidcells.Di˙erentturbulencestatisticsarethenanalyzed andcomparedinsupersonic˛owsovertwo2-dimensionalandtwo3-dimensionalsurfaces,andthe resultsrevealastrongdependenceoftheturbulence˝eldontheroughnesstopographiesandthe associatedshockpatterns. Copyrightby MOSTAFAAGHAEIJOUYBARI 2020 Thisthesisisdedicatedtomyparents. v ACKNOWLEDGEMENTS Mysinceregratitudegoestomysupervisors,ProfessorGilesJ.BreretonandProfessorJunlinYuan, fortheirinvaluableguidanceandadvicethroughoutmyPhDprogram.Itrulyappreciatetheirhelp, kindnessandpatience. IthankthemembersofmyPhDcommittee,Prof.FarhadA.JaberiandProf.MichaelS.Murillo, fortheirthoughtfulandconstructivecommentsondi˙erentpartsofthethesis. Ialsowouldliketothankmyparentsandmysistersforthesacri˝cesanddedicationstheyhave madeforme;althoughIknowthereisnothankgreatenoughtocompensatefortheirendlesslove andsupports. vi TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ....................................... xi KEYTOSYMBOLSANDABBREVIATIONS ........................ xv CHAPTER1INTRODUCTION ............................... 1 1.1Literaturereview...................................2 1.2Researchoutline....................................4 CHAPTER2ROUGHNESSTOPOGRAPHICALEFFECTSONMEANMOMEN- TUMANDSTRESSBUDGETSINDEVELOPEDTURBULENTCHAN- NELFLOWS .................................. 6 2.1Introduction......................................6 2.1.1Literaturereview...............................7 2.1.2Objectives...................................8 2.2Methodology.....................................9 2.2.1Governingequations.............................9 2.2.2Surfaceroughness...............................10 2.2.3Simulationparameters............................12 2.2.4Statisticsofmean˛owandturbulence....................13 2.3Results.........................................16 2.3.1Meanmomentumbalance...........................16 2.3.2TransportofthenormalReynoldsstresses..................18 2.3.3Transportofthenormaldispersivestresses..................22 2.3.4Di˙erentparametersineddyviscositymodels................22 2.4Concludingremarks..................................25 CHAPTER3TURBULENCESTRUCTURESOVERREALISTICANDSYNTHETIC WALLROUGHNESSINOPENCHANNELFLOWATRE ˝ = 1000 ... 27 3.1Introduction......................................27 3.1.1Literaturereview...............................27 3.1.2Objectives...................................28 3.2Results.........................................29 3.2.1Two-pointvelocitycorrelations........................29 3.2.2Lengthscalesandinclinationangles.....................32 3.2.3Velocityspectra................................36 3.2.4Vorticityandhelicity.............................40 3.2.5Instantaneousvorticalmotionsandconditionaleddies............42 3.2.6Akinematicprocessofvorticesinlocalshearlayers.............47 3.3Concludingremarks..................................51 vii CHAPTER4DATA-DRIVENPREDICTIONOFTHEEQUIVALENTSAND-GRAIN HEIGHTINROUGH-WALLTURBULENTFLOWS ............ 54 4.1Introduction......................................54 4.2Problemformulation.................................56 4.2.1Surfaceroughness...............................57 4.2.2Simulationparameters............................61 4.3Results.........................................65 4.3.1Post-processedresults.............................65 4.3.2MLpredictionsoftheequivalentsand-grainheight.............69 4.3.3Uncertaintyestimation............................74 4.3.4Sensitivityanalysis..............................74 4.3.5ComparisonbetweenMLalgorithmsandpolynomialmodels........78 4.4Concludingremarks..................................80 4.5Supplementarymaterials...............................82 CHAPTER5COMPRESSIBLEFLOWSOVERROUGHWALLS ............. 83 5.1Introduction......................................83 5.1.1Literaturereviewonimmersedboundary(IB)methods...........84 5.1.2Literaturereview:physicsof˛owsoverroughness..............86 5.1.3Objectives...................................87 5.2Problemformulation.................................87 5.2.1Governingequations.............................87 5.2.2DetailsofthepresentIBmethod.......................88 5.2.3Surfaceroughnessesandsimulationparameters...............89 5.2.4ValidationofthenumericalmethodandtheproposedIBmethod......93 5.3Results.........................................94 5.3.1Meanandturbulencevariables........................96 5.3.2BudgetsoftheReynoldsstresses.......................100 5.4Concludingremarks..................................105 APPENDICES ......................................... 106 APPENDIXAIntrinsicAreaFiltering ........................ 107 APPENDIXBCorrectionofNeumannboundaryconditionusinglevel-setmethod . 115 BIBLIOGRAPHY ........................................ 117 viii LISTOFTABLES Table2.1:Geometricparametersofroughsurfaces.Ra = k k isthe˝rst-ordermo- mentofheight˛uctuations; k rms ,root-mean-squareofheight; s k ,skewness; k u ,kurtosis;ES x i ,e˙ectiveslopeinthe x i direction................11 Table2.2:Simulationparameters.Superscript + indicatesnormalizationinwallunits, = š u ˝ ,frictionvelocity u ˝ = p ˝ w š ˆ , L y = ,zero-planedisplacement heightis d ,dragcoe˚cient C d = ¹ u ˝ š U b º 2 ,Re b = U b š andthebulkvelocity is U b .Re ˝ = u ˝ š = 1000 inallsimulations....................12 Table4.1:Statisticalparametersofroughnesstopographyandtheequivalentsand-grain height k s foreachroughnessgeometry. R a , k a vg , k c , k t , k rms and k s values fromDNSarenormalizedbythechannelhalfheight ,whilecorresponding experimentalvaluesaregivenin mm . k s isnotlistedforcasesthoughttobe transitionallyrough.................................59 Table4.2:PartI:StreamwiseandspanwisevaluesofthesurfaceTaylormicro-scale T . PartII:Flow-relatedparametersobtainedfromDNS.The˛owisassumedfully roughif b k + s & 50 ,inwhichcase k s isequalto b k s ..................64 Table4.3:Errorsin k s predictionbyDNNandGPRcomparedtoerrorsoftheempirical correlations: err B 1 (equation4.15), err B 2 (equation4.17), err B 3 (equation 4.16)and err B 4 (equation4.18).Thefourlargesterrors(inmagnitude)for eachcolumnarecoloredinred.Theerrorsarepercentages............73 Table4.4:Errorsin k s predictionbyexcludingoneortwofeatures.Thebaseprediction includesallprimaryvariables.Thefourlargesterrors(inmagnitude)foreach columnarecoloredinred.Theerrorsarepercentages...............77 Table5.1:Statisticalparametersofroughnesstopography. k a vg = 1 A t x ; z kdA isthe averageheight, k rms = q 1 A t x ; z ¹ k k a vg º 2 dA istheroot-mean-square(RMS) ofroughnessheight˛uctuation, R a = 1 A t x ; z j k k a vg j dA isthe˝rst-order momentofheight˛uctuations, E x i = 1 A t x ; z @ k @ x i dA isthee˙ectiveslopein the x i direction, S k = 1 A t x ; z ¹ k k a vg º 3 dA . k 3 rms istheheightskewness,and K u = 1 A t x ; z ¹ k k a vg º 4 dA . k 4 rms istheheightkurtosis;where k ¹ x ; z º isthe roughnessheightdistributionand A f ¹ y º and A t arethe˛uidandtotalplanar areas.Valuesof k c , k a vg , k rms and R a arenormalizedby ............91 ix Table5.2:Postprocessingdata. u ˝; s = p ˝ w ; s š ˆ r and u ˝; r = p ˝ w ; r š ˆ r ,where ˝ w ; s = w d < u > d y y = 2 and ˝ w ; r = k c 0 F 1 ; ibm T d y .Here F i ; ibm = ˆ u i t isthe correspondingbodyforceduetoIBM( u i isthevelocitydi˙erenceof u i afterandbeforetheIBMcorrectionstep). T isasimpleplanaraveraging operatorthatincludesallthesolidand˛uidcells.Re ˝ = ˆ r u ˝; a vg š w , C f = 2 ¹ u ˝; a vg š U r º 2 and u 2 ˝; a vg = u 2 ˝; s + u 2 ˝; r š 2 ..................99 x LISTOFFIGURES Figure1.1:Theonsetofroughnessontheleadingedgeofawindturbinebladedueto surfaceerosion[1].................................1 Figure1.2:Moodydiagram[2].Copyright1944ASME....................2 Figure2.1:Simulatedroughsurfaces.The˝guresshowaportionof inthe x - z plane.10 Figure2.2:SurfaceheightpowerspectraforCB( left ),SGandTB( right )roughnesses....12 Figure2.3:Meanstreamwisevelocitypro˝lesinwallunits.Smooth-wallexperimental dataarefromSchultzandFlack[3].........................14 Figure2.4:Pro˝lesofnormal(a,b,d)andshear(d)Reynoldsstresstensorcomponentsin wallunits......................................15 Figure2.5:Meanmomentumequation.Thedashedverticallinesarethecrestlocations...17 Figure2.6:Meanmomentumintegralequation.Thedashedverticallinesarethecrest locations.......................................18 Figure2.7:Balanceofdi˙erenttermsinReynoldsstressestransportequation.Thedashed verticallinesarethecrestlocations.........................19 Figure2.8:Contoursof P + w ; uu at y = d forSG( a,b )andTB( c,d ).Regular( a,c )and superposedwithunderlyingroughnessshape( b,d ).................20 Figure2.9:Contoursof P + w ; tke at y = d forCB.........................21 Figure2.10:Balanceofbudgetsindispersivestressestransportequation.Thedashed verticallinesarethecrestlocations.........................21 Figure2.11:JointPDFsof ¹ e u ; e v º at y = d ............................23 Figure2.12:Comparisonofactual C versusthespeci˝ed C in k - and v 2 - f models....23 Figure2.13:Comparisonofactual t ,versusthemodeled t of k - and v 2 - f models.....24 Figure3.1:Pro˝lesof R uu ; n ¹ y ; r 1 º at y locationsfrom y ˇ 0 ( black )to y š = 0 : 3 ( blue )...30 xi Figure3.2:Pro˝lesof R uu ; n ¹ r 1 º ( black )and R ww ; n ¹ r 1 º ( blue ),at y š = 0 : 15 (a),at y š = 0 : 3 (b)....................................30 Figure3.3:Averagedstreamwise ¹ L 11 ; 1 º andspanwise ¹ L 11 ; 3 º energy-containinglength scales........................................32 Figure3.4:Turbulence ¹ a º andKolmogorov ¹ b º lengthscales.................32 Figure3.5:Contoursof R uu ; n ¹ r 1 ; r 2 º ( left )and R ww ; n ¹ r 1 ; r 2 º ( right )centeredat y = d for SG,TBandCBroughness,andat y + = 15 forthesmoothcase.Thecontour levelrangeis[0.41.0]withastepsizeof0.2....................34 Figure3.6:Contoursof R uu ; n ¹ r 1 ; r 2 º ( left )and R ww ; n ¹ r 1 ; r 2 º ( right )centeredat y š = 0 : 3 . Thecontourlevelrangeis[0.41.0]withastepsizeof0.2.............35 Figure3.7:One-dimensionallongitudinalvelocityspectra E 11 at y š = 0 : 3 .Linesin black areobtainedusing(3.3)andlinesin blue using(3.4).Experimental dataarefrom[4].Plotsincirclesanddiamondsareforboundarylayer˛ow atRe = 1500 andwake˛owatRe = 23 respectively.Thethindashedline describes E 11 = 0 : 49 5 š 3 1 2 š 3 ...........................37 Figure3.8:One-dimensionallongitudinalpremultipliedvelocityspectra, E ij ,andsurface- heightpowerspectra, E s . E 11 ; E 22 ; E 33 ; E 12 ; (thin) E s . E s isnormalizedtogiveamaximumvalueof1................39 Figure3.9:One-dimensionaltransversepremultipliedvelocityspectra. E 11 ; E 22 ; E 33 ; E 12 ; (thin) E s . E s isnormalizedtogiveamaximum valueof1......................................40 Figure3.10:Pro˝lesofRMSvorticityinwallunits.Thinverticallinesshowthecorre- spondingroughnesscrestelevations........................41 Figure3.11:JointPDFsofinstantaneousvaluesof u 0 and ! 0 ,normalizedbytheirre- spectiveRMSvalues;dataarefromthewall-parallelplaneat y + = 15 for smooth-wall˛owand y = d forSG,TBandCBroughness............43 Figure3.12:Instantaneousvorticalmotions,visualizedbyiso-surfacesof ci = 0 : 2 ci ; max , coloredaccordingtodistancefromthewall....................44 Figure3.13:Conditionaleddiesbasedonaneventof c > 0 ,atthree y locations;from bottomtotop, y + = 15 , y + = 40 , y š = 0 : 1 forSM;and y = d , y = k c , y = 0 : 2 forSG,TBandCBroughness.Plotsareiso-surfacesof ci = 0 : 4 obtainedfromtheconditionalvelocity˝eld....................45 xii Figure3.14:( a )Sketchofevolutionofaspanwisevortexapproachingacubeelement,and ( b )Meanspanwisevorticity, ! 3 = d v š dx d u š d y ,normalizedby u ˝ and . Thehorizontalplaneisat y = 0 : 75 k c ........................47 Figure3.15:Pro˝lesof V r ; z ( a ),and z ( b ),normalizedby u ˝ and .Thevertical linesarethecorrespondingcrestlocations.....................48 Figure3.16:Contoursof( a ) V r ; z ,( b ) z ,( c )time-averagedshearproductionofTKE, P s = u 0 i u 0 j @ u i š @ x j ,and( d )TKE.CaseCB.Normalizationisdoneusing u ˝ and .Thehorizontalplanesareat y = 0 : 75 k c ...................49 Figure4.1:Roughnessgeometrieseachplotisasectionofsize 0 : 5 inthe x - z plane.CasesC43toC45arefromsimulationswithregulardomainsizes[5,6].58 Figure4.2:Pro˝lesofstreamwisedouble-averagedvelocityplottedagainstazero-plane- displacementshiftedlogarithmic y abscissa.Thedashedlinesare u + = y + and u + = 2 : 5ln ¹ y d º + + 5 : 0 .Thereddot-dashlineinplotC46isthatofC21.66 Figure4.3:Pairplotsofgeometricalparametersand k s ,with k s plotsinthebottomrow andthe˝rstcolumn,DNSdata( blue ),experimentaldata( red )..........67 Figure4.4:(a,d)Scatterplotoftrue k s andpredicted k s ,(b,e)scatterplotoftrue k s andrelativeerror,(c,f)pdfsofrelativeerrorfor(a-c)DNNand(d-f)GPR predictions,withDNSdata( blue ),experimentaldata( red )............71 Figure4.5:Con˝denceinterval(CI)ofpredictionswiththeGPRmethod,withpredicted valuesof k s š R a in bluelines (called k sp )andtruevaluesof k s š R a in red dots .GPRpredictionsforbothtrainingandtestingdatasetsareshown k s and k sp areveryclosetoeachotherforthetrainingdatapoints,whilethey deviate(lessthan30%oferror)forsometestdatapoints.Linejaggedness isassociatedwithprojectionofahigh-dimensionalspacetoone-dimensional ones........................................75 Figure4.6:(a)Scatterplotoftrue k s andpredicted k s (denotedas k sp ),(b)scatterplotof true k s andrelativeerrorand(c)pdfofrelativeerrordistributionforprediction usingpolynomialfunctionde˝nedinequation(4.19),withDNSdata( blue ) andexperimentaldata( red ).............................79 Figure5.1:Surfaceroughnesses................................90 Figure5.2:Pro˝lesofmeanandturbulencevariablesforthesmooth-wall˛owatRe = 3000 and M = 1 : 5 .Presentsimulation( solidlines ),[7]( dashline ). ˝ w = 1 Re d u d y w .92 xiii Figure5.3:Contouroflevelset ˚ fortheIBmethod(a),meshgridfortheconformal setup(b).CaseC1.................................93 Figure5.4:PlotsofthemeanandturbulencevariablesforcaseC1,simulatedbyusingthe IBmethod(solidlines)andtheconformalmeshsetup(dashlines).Pro˝les ofdouble-averagedvelocity,temperatureanddensity(a),RMSofvelocity componentsinplusunits(roughnessside,b),timeandspanwiseaverageof velocityandtemperatureattheroughnesscrestandvalleylocations(c),and RMSoftemperature(d).Theverticaldot-dashlinesshow y = k c ........94 Figure5.5:Contoursofinstantaneous u ............................95 Figure5.6:Contoursofinstantaneous r u ..........................97 Figure5.7:Contoursofinstantaneous T ............................98 Figure5.8:Plotsofthemeanandturbulencevariablesforallcases.Pro˝lesofthe double-averagedstreamwisevelocity(a),componentsofReynoldsstressesin plusunits(roughnessside,b),double-averagedoftemperature(c),andRMS oftemperature(d).C1(solidlines),C2(dashlines),C3(dot-dashlines)and C4(dottedlines)..................................99 Figure5.9:BudgetsofTKE.Alltermsarenormalizedby U r and ,andaredouble- averagedintimeandthe x - z plane.........................102 Figure5.10:BudgetsofB11.Alltermsarenormalizedby U r and ,andaredouble- averagedintimeandthe x - z plane.........................103 Figure5.11:Contoursof P 11 .Itisnormalizedby U r and ...................104 FigureA.1:Schematicofthegeometry,auxiliaryCSandde˝nitions..............108 FigureA.2:Schematicoftwofundamentalviewpoints:(a)auxiliaryCSmoves,primary CS˝xed.(b)auxiliaryCS˝xed,primaryCSmoves.Inboth˝guressubscript m standsfor`moving', IC standsfor`inertialCS';and G servesas O a in auxiliaryCSand x inprimaryCS,inthemeantime................109 FigureA.3:Representationof x - dir velocity,goingfrom P 1 to P 2 ,in x - dir with x unit longinprimaryCS,willresultinchangeofabscissaofapointinauxiliary CS, J ,of x 0 = x ................................110 FigureB.1:2Dheatequationsolution,usingbody-conformalmesh (left) ,andCartesian mesh(IBmethodwithNeumannBCcorrection, right )..............115 xiv KEYTOSYMBOLSANDABBREVIATIONS DNS DirectNumericalSimulation VOF Volumeof˛uid NS Navier-Stokes RANS ReynoldsaveragedNSequations DANS DoubleaveragedNSequations FANS Space-˝lteredensemble-averagedNSequations IBM Immersedboundarymethod TKE Turbulencekineticenergy WKE Dispersive(wake)kineticenergy RMS Rootmeansquare ML MachineLearning DNN DeepNeuralNetwork GPR GaussianProcessRegression x , x 1 Streamwisedirection y , x 2 Wallnormaldirection z , x 3 Spanwisedirection t Time u , u 1 Velocitycomponentinthe x direction v , u 2 Velocitycomponentinthe y direction w , u 3 Velocitycomponentinthe z direction p Pressure ˆ Density P p š ˆ T (chapter5) Temperature E (chapter5) Totalenergy f 1 (chapter5) Bodyforceinthe x direction F i (chapter2) IBMbodyforce Dynamicviscosity Kinematicviscosity xv ˝ w Exerteddragforceonawall C f Dragcoe˚cient u ˝ Frictionalvelocity Channelhalfheight Viscouslengthscale Re Reynoldsnumber Re ˝ FrictionalReynoldsnumber Re b BulkReynoldsnumber M Machnumber Pr Prandtlnumber TKEdissipationrate k s Roughnessequivalentsandgrainheight k TKE k Roughnessheightdistribution d Displacementheight R u u Two-pointvelocitycorrelations E ij One-dimensionalpremultipliedvelocityspectra vonKarmanconstant (chapter5) Heatconductivitycoe˚cient ˚ (chapter5) VOF˝eld (chapter5) Level-set˝eld ! i Vorticitycomponents ci Theimaginarypartofthecomplexeigenvalueofthevelocitygradient tensor U + Roughnessfunction + Normalizationinwallunits Intrinsicareaaveraging(˝ltering)operator s Super˝cialareaaveraging(˝ltering)operator Timeaveragingoperator e (chapters2,3,4) Dispersive˛uctuation e (chapter5) Favreaveragingoperator 0 Reynolds(time)˛uctuation 00 (chapter5) Favre˛uctuation xvi CHAPTER1 INTRODUCTION Wallboundedturbulent˛owsareessentialtoturbulenceresearchduetotheirfrequentoccurrences innatural˛owsandindustrialapplications.Examplesincludeenvironmental˛ows,meteorological andcanopy˛ows,˛owsaroundships,oiltankersandpowergeneratingturbines,˛owsaround airfoilsandinsidejetengines,andinternal˛ows,suchasturbulent˛owswithinducts,channels andpipes. Wall`roughness'isaninevitablefeatureofmanyofthese˛ows.Canopyandurban˛ows areinherentlyassociatedwithroughnessduetopresenceofvegetationorbuildings.Thewetted areasofships,propellersandwind/hydroturbinebladescanbecomeroughduetosurfaceerosion, cavitation,growthofalgaeanddust/abrasiveairimpingement.Figure1.1showstheonsetof roughnessontheleadingedgeofawindturbinebladeduetosurfaceerosion.Forinternal˛ows suchaschannelandpipe˛ows,thesurfacematerial,ageandits˝nishingarethemaincausesof roughness[8].Inthepetroleumindustry,theinteractionofpipesmetallicsurfacewithcorrosive ˛uidssuchasoilincreasesthesurfaceroughness[9].River˛owsovergranular/sedimentbeds Figure1.1:Theonsetofroughnessontheleadingedgeofawindturbinebladeduetosurface erosion[1]. 1 Figure1.2:Moodydiagram[2].Copyright1944ASME. [10],andplaquebuildupwithinarteries[11]areexamplesofroughnessonnaturalandbiomedical ˛ows,respectively. 1.1Literaturereview Roughnesse˙ectsonwallboundedturbulent˛owscanbeexaminedfromdi˙erentperspectives, themostimportantofwhicharesummarizedbelow. Roughnessincreasesthehydrodynamicdrag[12].TheMoodydiagram,in˝gure1.2,charac- terizesthehydraulicfrictionfactorinturbulentpipe˛owsasafunctionof˛owReynoldsnumber (Re = Vd š ,where V isthebulkvelocity, d isthepipediameter, = š ˆ isthekinematicviscosity, isthe˛uiddynamicviscosityand ˆ isthe˛uiddensity),andtherelativeroughnesslengthscale š d (where istheroughnesslengthscale).Asthe˝gureshows,forsu˚cientlyhighRe,thefriction factordependsonlyontheroughnesslengthscale.Theextentofwhichroughnesscanin˛uence theboundarylayerischaracterizedbytheroughnessReynoldsnumber, k + s = k s u ˝ š (where k s istheroughnessequivalentsandgrainheightand u ˝ isthefrictionalvelocity).Accordingtothe reviewprovidedbyJiménez[13],for k + s . 4 thewallcanbeconsideredhydraulicallysmooth.For 2 thisrangeof k + s ,viscouse˙ectsaredominantanddissipateanyroughnesse˙ects,therefore,the dragcoe˚cientwilldependonlyonRe.Ontheotherhand,for k + s & 80 thesurfaceroughnesswill destroytheviscoussublayer,replacingitwitharoughnesssublayer.Asaresult,thewakebehind roughnesselementsandtheassociatedpressuredistributionaroundeachelementwillcontrolthe ˛owbehaviour,andthedragcoe˚cientwilldependonlyon k + s ( š d intheMoodydiagram).The ˛owiscalledfully-roughinthisregime.For 4 . k + s . 80 the˛owistransitionallyroughandthe dragcoe˚cientwilldependonbothReand k + s . Roughnessmodi˝esnearwallturbulencestructures[14].Thesestructuresareresponsiblefor di˙erentphysicalprocessessuchasmixingandtransportofmomentuminboundarylayers[15]. Therearemanystudiesofthee˙ectsofroughnessonnearwallcoherentmotions[16,17,18,19]. Roughnessenhancesscalarandmomentumtransportbymodifyingthesestructures[20,21],and makestheturbulencemoreisotropicbybreakingdownthelargescalestructurestothesmaller ones.Understandingthein˛uenceofroughnessoncoherentmotionsisofparticularinterestfor structure-basedturbulencemodelingapproaches[22,23],transportofairpollutioninurbanareas [24],andheattransferincanopyandroughwallboundarylayers[25,26,27]. Roughnessalterstheturbulenceresponseinnon-equilibrium˛ows[28].Spatiallyandtem- porallyaccelerating˛owsoccurinnatural˛owsandengineeringapplications.Examplesinclude ˛owoverairfoilsandintheatmosphericboundarylayer.Forsmoothwallboundarylayer˛ows,it iswell-knownthatthefavorablepressuregradient(asisthecaseinsink˛ows)dampsturbulence throughamechanismcalled`relaminarization'or`quasi-laminarization'[29,30].Roughness,on theotherhand,notonlydestroysthismechanism,butalsointensi˝estheturbulence˝eldthrough wakeproductionbehindtheroughnesselements,resultingtoa`rougher'surfaceinfavorable- pressure-gradient˛ows[31,28]. Inthesuper-andhypersonic˛ows,surfaceroughnesscanbeenhancedbyerosionandablation processes,duetothehighamountofheatloadintheseregimes.Theroughness(mostlyina distributedform)interactswiththeboundarylayer,causesacousticdisturbancesandearlytransition toturbulence,andenhancestheheattransferrate[32,33].Mostofthesee˙ectshavebeenanalyzed 3 experimentallytodate,duetodi˚cultiesassociatedwiththestabilityofnumericalsolversandwith thegenerationofbody-˝ttedgridsinthepresenceofcomplexsurfaceroughness. 1.2Researchoutline Understandingthee˙ectsofroughnessonwall-boundedturbulent˛owswillexpandourknowl- edgeofthephysicsofthese˛ows,andwillhelpustoimprovepredictionoftheirbehaviorand controltheme˚ciently.Thisdissertationanalyzesthesee˙ects,andthefollowingsummarizesthe mostimportantofthem. Chapter2isconcernedwithturbulencestatisticsinturbulent˛owsoverroughwalls.Itincludes acomprehensivestudyondi˙erentialandintegralformsofthemeanmomentumequation,and budgetsofReynoldsanddispersivestresses.Turbulencestatisticsarecomparedforopenchannel ˛owsoverthreeverydi˙erentroughwalls(asexplainedbrie˛ybelowandthoroughlyinsection 2.2.2)andasmoothwall.Thee˙ectsofroughnessondi˙erenttermsinthestandard k - model andtheellipticrelaxationmodelofDurbin( k - - v 2 - f ,[34])arealsoanalyzedtoprovideagood approachformodelingofturbulenceinthepresenceofroughness. Inchapter3,turbulencestructuresinfully-developedchannel˛owsoverroughwallsareana- lyzed.Themainobjectivesare˝rsttoexplorehowroughnessmodi˝esturbulentcoherentmotions in˛owsoverroughwalls,andsecond,tocomparethebehaviorofthesemotionsin˛owovera realisticroughsurfacewiththosein˛owsoversyntheticroughnesses.Thesesurfacesaresameas thoseusedinchapter2.Therealisticsurfaceischaracterizedbymultiplescales/wavelengthsof roughnesswhilethesyntheticroughsurfaces,madeofdistributedelementsofsimilarshapes,are describedbyasingleornarrowsetofdominantwavelengths.Flowsinboththeroughnesssublayer andtheouterlayerareexaminedandtwo-pointvelocitycorrelations,lengthscales,inclination angles,velocityspectra,vorticities,andhelicitiesareanalysed.Linearstochasticestimationisem- ployedtoexploretheaveragebehavioroftheinstantaneousvorticalmotionsandtheirdependence ontheroughnesstopography. Inchapter4,theroughnessequivalentsandgrainheightismodeledusingMachineLearning 4 techniques.Nikuradse[35]introduced k s ,toevaluatetheroughnessfunction U + ,whichrepresents thehydrodynamicdraginboundarylayer˛ows.Hisroughnessfunctionis U + = 1 ln k + s 3 : 5 (where = 0 : 41 isvonKarman'sconstantand + isnormalizationinwallunitsusing u ˝ and = š u ˝ ).Theapproachusesonlyalengthscaletodescribethehydraulicdragassociated withtheroughness.But,whatisthislengthscale?Inherpaper,`movingbeyondMoody?',Flack [36]makesitclearthatthislengthscaleisnotknownaprioriandnonofthegeometricallength scales,suchasaverageorroot-mean-square(RMS)ofroughnessheight,candescribeituniquely. Infact,itiswell-establishedthat k s dependsonmanygeometricalparameters,suchase˙ective slope[37,5]andskewnessofroughnessheightdistribution[38].Thischaptertriestoansweralong standingquestionaboutroughnesswhatistheequivalentsandgrainheight,givenaroughness topography?Toanswerthis,MachineLearningmethodsofDeepLearningandGaussianProcess Regressionwereemployedusing45datapoints.Thedatasetconsistsof30pointsfromdirect numericalsimulationsand15pointsfromexperiments,allofwhichareconsideredtobefully rough. Inchapter5,animmersedboundary(IB)methodisproposedtoenablesimulationofsupersonic ˛owsoverroughwallswithcomplexgeometries.Tothisend,asimulationoverasmoothwall (atMachnumberof1.5andbulkReynoldsnumberof3000)isperformed˝rst,anditsresultsare validatedwiththestudyofColemanetal.[7].Second,theIBmethodisde˝nedandisvalidatedby comparingdi˙erentmeanandturbulencestatisticsofasupersonic˛owoverawavywall,which, forvalidationpurposes,hasbeensimulatedtwice:withtheIBmethodandwithoutit(witha conventionalbody-˝ttedapproachsuitableforsimplegeometries).Inthethirdstep,˛owsover fourroughnesstopographies,two2-dimensionalandtwo3-dimensionalsurfaces,aresimulatedat aMachnumberof1.5andabulkReynoldsnumberof3000usingtheIBmethod,andthee˙ects ofwallroughnessonthemeanandturbulence˝elds,embeddedshockpatterns,andtheReynolds stressbudgetsarecomparedandanalyzed. 5 CHAPTER2 ROUGHNESSTOPOGRAPHICALEFFECTSONMEANMOMENTUMANDSTRESS BUDGETSINDEVELOPEDTURBULENTCHANNELFLOWS 2.1Introduction Turbulencestatisticsandtheirconnectiontotheroughnesstopographyareimportantforunder- standingphysicsofturbulenceoverroughwalls,andwillpavethewaytoextendexistingturbulence modelstocaptureroughnesse˙ects,ortointroducenewmodelscapableofpredictingturbulence behaviorin˛owsoverbothsmoothandroughwalls. Inthepresenceofroughness,thegoverningNavier-Stokes(NS)equationsaretemporallyand spatiallyaveraged(inaprocesscalleddouble-averaging)torepresenttheturbulent˛ow˝eld statistically.Theresultedequationsarecalleddouble-averagedNavier-Stokes(DANS)equations. TheprocessofobtainingDANSequationsarefullyexplainedinthestudiesof[39,40,41],andour newmethodofderivingtheseequationsisprovidedinappendixA.Therearetwomainchallenges associatedwiththeroughnessspatialheterogeneityinderivationofDANSequations.First,one needstousetripledecompositiontoseparatemeanand˛uctuation˝elds.Bydoingso,along withturbulence˝eld(˛uctuationsassociatedwiththeensemble-averageoperator),oneidenti˝es thewake˝eld(˛uctuationsassociatedwiththespatial-averageoperator)emerges,whichisrelated tothewakeregionbehindtheroughnesselements.Interactionsofthewake˝eldwiththeboth meanandturbulence˝eldsareimportantformomentumtransfer[14]into/outoftheroughness sublayer.Second,thespatial-averageoperatorandderivativeoperatordonot`commute'becauseof thespatialheterogeneityoftheroughness[39,42].TheuseofLeibniz'sintegralruleinaddressing thisissue,introducesnewformtermsinDANSequations,whicharethemainsourcesofdrag productionin˛owsoverroughness,anditisnecessarytoincludetheire˙ectsinthemodelingof turbulence.ThedetailsofderivationofDANSequations,tripledecompositionandformtermsare explainedinsection2.2andappendixA. 6 2.1.1Literaturereview Severalstudiesfocusedonthee˙ectsofroughnessondi˙erentturbulencestatistics.Raupach andShaw[39]analyzedbudgetsoftheReynoldsanddispersivestressestoexplainthephysicsof turbulent˛owsovervegetationcanopies.Usingempiricaldata,theyshowedthatwakeproduction (thetermthatconnectsturbulence˝eldtothewake˝eld)isoftheorderoftheshearproduction withinthecanopies.Mignotetal.[43]analyzedturbulencestatisticsinchannel˛owsovergravel- beds.Theyshowedthatmacro-scaleroughnesselementscontributesigni˝cantlytoturbulence kineticenergy(TKE)production(upto30%)despitetheirinfrequentoccurrence(about11%).Yuan andPiomelli[44]comparedturbulencebudgetsforsandgrain-typeroughnessesintransitionally andfullyroughconditions.Theyshowedthatthewake˝elddynamicallya˙ectsthewall-normal turbulence˛uctuation v 0 v 0 forfullyroughsurfaces,whileitse˙ectsaresmalleronthetransitionally roughsurfaces.Busseetal.[45]analyzedthee˙ectof`surfaceanisotropyratio'onthemean andtheReynoldsstresses.Theyhaveshownthatroughnessfunction, U + ,isstronglydependent onthisratio,andspanwise-anisotropicsurfaceshavemuchhigher U + (about200%increase) comparedtothestreamwise-anisotropicsurfaces.Thein˛uenceofthewakeproductioninthe roughnesssublayerforurban˛owwasanalyzedbyGiomettoetal.[46],whereitisshownthatthe wakeproductioncontributesupto50%totheTKE,andthee˙ectofpressuretransportissigni˝cant innearwallregions. Severalstudiestriedtoincorporatethee˙ectsofroughnessindi˙erentturbulencemodels.Suga etal.[47]introducedananalyticalwallfunctiontosimulateturbulent˛owsoverroughwallsand theassociatedheattransfer.Intheirwallfunction,thee˙ectsofroughnessareaccountedforusing theNikuradseequivalentsandgrainheight[ k s ,48].Theirnumericalresultsshowgoodagreements withexperimentaldataforbothequilibriumandnon-equilibrium˛ows.Lee[49]o˙eredaboundary treatmentfortheshear-stress-transport k - ! (SST- k ! )modeltoimparttheroughnesse˙ects.Inthis work,thee˙ectofroughnessisincorporatedusing k s andthetreatmentisappliedtotheboundary conditionsofTKEandvorticity. Thelocallyisotropicmodels,suchasthestandard k - model,failtopredict˛owbehavior 7 accuratelyevenforsmoothwalls,duetoturbulenceinhomogeneityinwallbounded˛ows.To remedythisissue,severalapproachesareemployedtobringthenon-localitye˙ectsintoturbulence modeling[22,50]ingeneral,andwall-boundedturbulencemodeling[34]inparticular.Methods basedonellipticrelaxation[34]haveshownpromisingcontributioninmodelingwallbounded ˛ows,wherethenon-localitye˙ectsaremodeledbysolvinganextraellipticequation.The equationsanddi˙erentlength/timescalesintheoriginalellipticrelaxationmethodofDurbin[34] solvesfourequationsfor k (TKE), (TKEdissipationrate), v 2 (wall-normalReynoldsstress)and f (theellipticrelaxationfunction).Thecoe˚cientsoftheoriginalmodelaretunedforthesmooth walls.Georgeetal.[51]usedexperimentalroughwalldataandfoundthat,forroughwall˛ows, themaximaof f occursneartheroughnesscrest,andthepeakvaluedecreasesasroughnessheight increases.Usingtheseresults,theymodi˝eddi˙erenttermsinthe f equationandthelengthand timescalestoimparttheroughnesse˙ectsintothemodeling. 2.1.2Objectives Thischapterincludesacomprehensivestudyondi˙erentialandintegralformofthemeanmo- mentumequations,transportoftheReynoldsanddispersivestressesandtransportofTKEand dispersive(wake)kineticenergy(WKE).Theresultscomparedi˙erentturbulencestatisticsfor openchannel˛owsoverthreewidelydi˙erentroughwalls(asexplainedinsection2.2.2)with thoseofasmoothwall.Roughnesssublayeristhecrucialregion,whereroughnesse˙ectsare moreintense,andmomentumistransportedto/fromtheinnerlayer.Thisregionisthefocusofthe presentstudy,althoughtheturbulencebehaviorintheouterlayerisalsoanalyzed. Thee˙ectsofroughnessondi˙erenttermsinthestandard k - modelandtheellipticrelaxation modelofDurbin( k - - v 2 - f )arealsoexaminedforabettermodelingofturbulenceinthepresence ofroughness. 8 2.2Methodology 2.2.1Governingequations TheNSequationsforaconstant-propertyNewtonian˛uid,weresolvedviadirectnumerical simulation(DNS).Theseequationsarewritteninindicialnotationas @ u i @ x i = 0 ; (2.1a) @ u i @ t + @ u i u j @ x j = @ P @ x i + @ 2 u i @ x j @ x j + F i ; (2.1b) where i ; j = 1 ; 2 ; 3 , x 1 ; x 2 and x 3 (or x ; y ; z )arethestreamwise,wall-normalandspanwisecoor- dinates,withcorrespondingvelocitycomponentsof u 1 ; u 2 and u 3 (or u ; v ; w )and P isde˝nedas p š ˆ ,where p isthepressureand ˆ isthe˛uiddensity; isthekinematicviscosity.Animmersed boundary(IB)method[52]wasusedtoenforcethe˝ne-grainedroughnessboundaryconditionson anon-conformalCartesiangrid.Thecorrespondingbodyforces F i areaddedtothetherighthand sideofthemomentumequationstoimposeano-slipboundaryconditionatthe˛uid-roughness interface.Tosolvetheequations,second-ordercentraldi˙erencingwasusedforspatialdiscretiza- tionsandsecond-orderAdams-Bashforthsemi-implicittimeadvancementwasemployed.The numericalsolverwasparallelizedusingthemessagepassinginterface(MPI)method[53]. Adouble-averagingdecomposition[39]wasusedtoresolveturbulenceanddispersivecompo- nentsof˛owvariablesinthepresenceofroughness.Inthisdecomposition,anyinstantaneous˛ow variable maybedecomposedintothreecomponents,as ¹ x ; t º = ¹ y º + 0 ¹ x ; t º + e ¹ x º (2.2) wherethetime-averagingoperatoris andtheintrinsicspatial-averagingoperatoris = 1 A f x ; z dA ( A f istheareaoccupiedbythe˛uid).TheReynoldsanddispersive˛uctuating componentsarethen 0 = and e = respectively. h i iscalledthedouble-averaged component.Wealsointroducethesuper˝cialspatial-averagingoperatoras s = 1 A t x ; z dA ( A t isthetotalplanararea),whichhasimplicationsinsomeoftheresultspresentedhere.Onesimply notices s = A f A t . 9 Figure2.1:Simulatedroughsurfaces.The˝guresshowaportionof inthe x - z plane. Thewallshearstress(includingbothviscousandpressuredragcontributionsonaroughwall) wasdeterminedbyintegratingthetime-averaged F 1 as ˝ w = ˆ L x L z V F 1 ¹ x ; y ; z º dxd y dz ; (2.3) where V representsthesimulationdomainvolumebelowtheroughnesscrestand L x i isthedomain lengthinthe x i direction.Readersarereferredto[52,44]fordetailsoftheimplementationand validationoftheIBmethodandthe ˝ w calculation. 2.2.2Surfaceroughness Thethreetypesofwallroughnessconsideredinthisstudyareshownin˝gure2.1.Forallrough cases,the y originissetatthelowesttroughofthesurface.Thesand-grainroughness(SG) [54]comprisesdenselypacked,randomlyorientedellipsoidalelementsofthesameshape,with semi-axislengthsof ¹ 1 ; 2 ; 3 º = ¹ 1 : 0 ; 0 : 7 ; 0 : 5 º k c ,where k c istheroughnesscrestheight.The turbinebladeroughness(TB)isthesurfaceS4ofYuanandPiomelli[5],mirroredoncewithrespect toboththe x and z directionstoaccommodateperiodicboundaryconditions.Itfeaturessurface 10 Surface k c š Ra š k rms š Ra s k k u ES x ES z SG 0 : 090 : 0141 : 050 : 482 : 970 : 430 : 44 TB 0 : 120 : 0141 : 170 : 203 : 490 : 100 : 08 CB 0 : 070 : 0140 : 362 : 457 : 01 -- Table2.1:Geometricparametersofroughsurfaces.Ra = k k isthe˝rst-ordermomentof height˛uctuations; k rms ,root-mean-squareofheight; s k ,skewness; k u ,kurtosis;ES x i ,e˙ective slopeinthe x i direction. lengthscalesin x and z whichexceedthechannelhalf-height .Thecuberoughness(CB)was generatedbyhomogeneousduplicationofacubeelementinthe x and z directionswithstepsof CB = 3 k c toyielda k -typeroughness[55]. Characteristicgeometricparametersofthesethreeroughsurfacesaredisplayedintable2.1. Eachsurfacehasthesame˝rst-ordermomentofheight˛uctuationRa = 0 : 014 ,acrestheightof roughly 0 : 1 ,andapositivelyskewedheightdistribution( s k > 0 ).Thee˙ectivesurfaceslopes ES x i [56]arede˝nedas ES x i = 1 L x L z L x ; L z @ k ¹ x ; z º @ x i dzdx ; (2.4) where k ¹ x ; z º isthelocalheightofthesurface,anddi˙ersappreciablyforthesethreesurfaces. ThevalueofES x i forTBroughnessishalfofthatforSGroughness.TheESvaluesfortheCB roughnessarenotcomparedduetotheirbeinglocallyin˝nite. Theprobabilitydensityfunctions(PDFs)ofthelocalsurfacegradients @ k š @ x i ,forSGandTB roughness,aregiveninYuanandAghaeiJouybari[57],whereitisshownthatgradientssteeper than 14 ininclinationtothehorizontalplaneoccurmorefrequentlyinSGroughnessthaninTB roughness.TheCBroughnesshasthesteepestlocalsurfacegradients.Streamwiseandspanwise powerspectraofthesesurfaceheightvariationsareshownin˝gure2.2.Theyfeaturespikesat r = 2 ˇ š CB forCBroughness,distinctpeaksat r = 2 ˇ š 1 forSGroughness,andanextended regionwithaslopeofroughly 2 forTBroughness.WethereforecharacterizetheCB,SGandTB topographiesasspiky,narrowbandwidthandfractal[58]surfacesrespectively,intermsoftheir spectralfeatures,otherthanphysicalones. 11 Figure2.2:SurfaceheightpowerspectraforCB( left ),SGandTB( right )roughnesses. Re b k + s y R š d š ¹ L x ; L z ºš ¹ n i ; n j ; n k º¹ x + ; y + min ; z + º C d 10 3 SM 20000 ¹ 6 ; 3 º¹ 512 ; 256 ; 512 º¹ 11 : 7 ; 0 : 3 ; 5 : 8 º 2 : 5 SG 11200780 : 120 : 044 ¹ 6 ; 3 º¹ 1024 ; 236 ; 512 º¹ 6 : 0 ; 0 : 7 ; 6 : 0 º 8 : 0 TB 14400240 : 140 : 058 ¹ 13 ; 13 º¹ 1024 ; 259 ; 1024 º¹ 13 : 0 ; 0 : 8 ; 13 : 0 º 4 : 8 CB 10900960 : 100 : 039 ¹ 6 ; 3 º¹ 1024 ; 373 ; 512 º¹ 6 : 0 ; 0 : 8 ; 6 : 0 º 8 : 4 Table2.2:Simulationparameters.Superscript + indicatesnormalizationinwallunits, = š u ˝ , frictionvelocity u ˝ = p ˝ w š ˆ , L y = ,zero-planedisplacementheightis d ,dragcoe˚cient C d = ¹ u ˝ š U b º 2 ,Re b = U b š andthebulkvelocityis U b .Re ˝ = u ˝ š = 1000 inallsimulations. 2.2.3Simulationparameters Theparametersusedinthesesimulationsareshownintable2.2.Allsimulationswerecarried outatfrictionalReynoldsnumber(Re ˝ = u ˝ š )ofRe ˝ = 1000 using n i ; n j and n k gridpoints instreamwise( x ),wall-normal( y )andspanwise( z )directions,withcorrespondingdomainsizes of L x , L y and L z .ThechoiceofthesamevalueofRe ˝ forallsimulationsprovidedagoodbasis forcomparisonbetweenrough-andsmooth-wall˛owsatamoderateReynoldsnumber.Allrough casesareinthefully-roughregimeaccordingtotheirrespectiveroughnessReynoldsnumber k + s (tobequanti˝edlater,+denotesnormalizationinwallunitsusing u ˝ and = š u ˝ ),andbased onthe˝ndingsofYuanandPiomelli[5]forSGandTBroughness,andthe k -typeroughnessstudy ofBandyopadhyay[59,fullyroughwhen k + s > 50 ]forCBroughness.Thisindicatesthatthe 12 near-wallmean-˛owpatternsarefullydeveloped,notmodi˝edbyafurtherincreaseof k + s .As such,thedi˙erenceinturbulentstructuresidenti˝edinthisstudyisconsideredpredominantlya resultofthedi˙erenceinroughnessgeometry,otherthanthedi˙erencein k + s .Thegridpointswere distributeduniformlyinthestreamwiseandspanwisedirections.Inthewall-normaldirection,they werespaceduniformlyfor y < k c butstretchedfor y > k c withlargergridsizesfartherfromthe wall.Forallsimulations,thegridsizes ¹ x ; y ; z ºš ,where istheKolmogorovlengthscale (see˝gure3.4b),werelessthan11inthe x and z directionsandmuchsmallerinthewall-normal direction.BasedontheobservationofMoserandMoin[60],scalessmallerthan 15 contributed predominantlytoturbulencedissipation.Thereforethegridresolutionsusedinthecurrentstudy aresmallenoughtoresolvethedissipativescales.Inthesehalf-channelsimulations,periodic boundaryconditionswereimposedinthe x and z directions,andsymmetryandno-slipboundary conditionswereappliedatthetopandbottomboundariesrespectively.Aftertheinitialtransientof thesimulationwascompleted,datawerecollectedforstatisticalanalysisoverasimulationtimeof 10-50large-eddyturnovertimes( T ˇ š u ˝ ). 2.2.4Statisticsofmean˛owandturbulence Theroughnesssublayerheight y R isde˝nedastheentirenear-walllayerwithnon-negligibleform- inducedvelocities. y R isquanti˝edasthelocationwhere e u 2 0 : 5 ˝rstmeets 0 : 06 u ,similarto themethodproposedbyPokrajacetal.[61]. y R 0 : 15 forallpresentsimulations. Meanstreamwisevelocitypro˝les,normalizedinwallunits,areshownin˝gure2.3.Each pro˝lehasadistinctlogarithmicregionovertherange 30 < ¹ y d º + < 200 ,indicatingconformity withthelawofthewall.Itfollowsthatanequivalentsand-grainheight k s canbeinferredfromthe dataof˝gure2.3andNikuradse'scorrelation(equation2.5)[48]inthelogarithmicregion: u + = 1 ln y d k s + 8 : 5 ; (2.5) where d isthedisplacementheightde˝nedasthelocationwherethemeandragappears[62], and = 0 : 41 isthevonKarmanconstant.Intable2.2,theinferredvaluesof k + s aregivenas 13 Figure2.3:Meanstreamwisevelocitypro˝lesinwallunits.Smooth-wallexperimentaldataare fromSchultzandFlack[3]. 24,78and96forTB,SGandCBroughnessrespectively,renderingallroughcasesfullyrough. Thecomparisonofthedragcoe˚cient C d givenintable2.2appearstofollowthatof k + s ,with C d ; TB < C d ; SG < C d ; CB .Thistrendmaybeexplainedasaresultoftheverydi˙erente˙ective slopevalues[56], ES TB < ES SG < ES CB .Itwillbeshowninthefollowingsectionsthatthe e˙ectsofroughnessonvariousturbulencestructuralparametersalsofollowsuchrelation,with˛ow characteristicsintheTBcasegenerallyintermediatebetweenthoseinthesmoothandSGcases, whilethee˙ectofCBroughnesstendtobemoresigni˝cantthanSGandTBroughnesses. Figure2.4showspro˝lesofthenormalandshearcomponentsoftheReynoldsstresstensorin wallunits.Overall,theouterlayersimilarityissatis˝ed.Intheroughnesssublayer,theroughness predictablychangestheReynoldsstresses,resultinginaloweranisotropyforroughwallscompared tothesmoothone.Consequently,thee˙ectofsurfacetopographyisgreateronthestreamwise normalstressthanontheotherthree. Theslightouter-layermismatchbetweensmooth-wallvaluesof h u 0 v 0 i + andthevalueson theroughwallsasshownin˝gure2.4(d)isduetonon-zerodispersiveshearstress, h e u e v i + in theouterlayer.Thesenon-zerovaluesof e u and e v resultfromvery-large-scalemotionswith loworhighmomentum,elongatedinthestreamwisedirections.Thesemotionsarethoughtto 14 Figure2.4:Pro˝lesofnormal(a,b,d)andshear(d)Reynoldsstresstensorcomponentsinwall units. originatefromthespanwiseheterogeneityofroughsurfaceswithspanwisewavelengths z whichare comparableto ,sameasthesurface-inducedmeansecondary˛ows˝rstobservedforturbine-blade roughness[63,64]andlatersystematicallystudiedbyVanderwelandGanapathisubramani[65]and YangandAnderson[66],amongothers,onorganizeddistributedroughness.Consistentwiththese studies,forthepresentSGandTBsurfacesitisfoundthatthespanwiselocationsofhighsurface drag(averagedin x )coincidewiththelocationsofthehigh-momentumpathways(notshown), evenfortheSGroughnesswhichisnotcharacterizedbyapredominantspanwisewavelengthof o ¹ º .However,thepresentsimulationsetupusingstreamwiseperiodicchannelsinevitablyleadsto `spanwiselocking'[67]ofcoherentmotionswithlengthslongerthanthestreamwisedomainsize ( 6 ,muchshorterthanthevery-large-scalemotions).Consequently,theseobservedpathwaysare 15 in˛uenced,ifnotsigni˝cantlymodi˝ed,bythesimulationsetupandthusnotdiscussedindepth herein. 2.3Results 2.3.1Meanmomentumbalance ThestreamwiseDANSequation(seeappendixA)reducesto d P s dx @ u 0 v 0 s @ y @ e u e v s @ y + @ 2 u s @ y 2 + f v + f p = 0 ; (2.6) where, f = A t B int @ u @ x j v j ^ n dl f p = 1 A t B int P v 1 ^ n dl : (2.7) f p and f aretheformviscousandpressuredragterms,fullyexplainedinappendixA.Figure 2.5showsthecontributionbetweendi˙erenttermsinequation2.6.Inthe˝gure, ˙ showsthe residualofthebalance,whichisapproximatelyzeroforallcases.Equation2.6and˝gure2.5 areimportantformodelingpurposes,especiallyasfarasthepeaklocationofeachtermandthe pro˝lessharpness/bluntnessareconcerned.OnenoticestheviscoussublayerinSMisreplacedby theroughnesssublayerinroughcasesandconsequentlytheviscoussinkterminSMissubstituted bytheformdragterms, f v + f p .TheReynoldsstresstermisthesourceterminallcases.Pro˝les oftheReynoldsstressaresmoothedoutinroughcasescomparedtotheSM,asaresultof˛ow isotropyinthepresenceofroughness.Thepeaklocationisaround y + = 15 forSMandliesinthe bu˙erlayer.However,thislocationscalesbetterwith k c intheroughcasesandisabout k c š 2 for SGandTB,andthevery k c forCB. Instationaryturbulentchannel˛ows,themeanintegraloftheu-momentumequationreduces to ˝ tot ˝ w = 1 y ; (2.8) 16 Figure2.5:Meanmomentumequation.Thedashedverticallinesarethecrestlocations. where ˝ tot @ u s @ y u 0 v 0 s e u e v s + y ¹ f v + f p º d y | {z } ˝ f : (2.9) Thisequationshowsthebalanceofdi˙erentshearstresses.ForSMtheviscousshearis dominantnearthewall( y + < 15 ),whilefarfromthewall,theonlycontributingfactoristhe Reynoldsstressterm.Forroughwallstheformdragtermisthedominanttermnearthewall.The surfaceporosityiszeroat ¹ y = 0 º forSGandTBcase,thereforetheformdragequalsto1at ¹ y = 0 º forthesecases.However,CBhasanon-zeroporosity(itispartiallyroughandpartiallysmooth)at ( y = 0 ).Therefore,theformdragislessthan1atthewallforCB.Thesmallviscousdrag,resulting fromtheshearoverthesmoothparts,compensatesthebalanceofthemeanmomentumintegral equationforCB.Thedispersivestressisnon-negligibleintheroughnesssublayer,althoughitis notthemostprominenttermatall. 17 Figure2.6:Meanmomentumintegralequation.Thedashedverticallinesarethecrestlocations. 2.3.2TransportofthenormalReynoldsstresses TransportequationsofnormalcomponentsoftheReynoldsstresstensorarederivedinthestudy of[57]as 0 = 2 u 0 v 0 s @ u @ y | {z } P s 2 ˝ š u 0 u 0 j @ e u @ x j ˛ s | {z } P w 2 u 0 u 0 j ˝ @ e u @ x j ˛ s | {z } P m ˝ @ @ x j ] u 0 u 0 e u j ˛ s | {z } T w ˝ @ @ x j u 0 u 0 u 0 j ˛ s | {z } T t + ˝ @ 2 u 0 u 0 @ x j @ x j ˛ s | {z } T v 2 ˝ u 0 @ P 0 @ x ˛ s | {z } 2 ˝ @ u 0 @ x j @ u 0 @ x j ˛ s | {z } ; (2.10) wheretheproductiontermsduetomeanshear,wake,andmeanwake-shearare P s , P w and P m , respectively.Thewake,turbulenceandviscoustransporttermsaredenotedas T w , T t and T ,the pressureworkas anddissipationas . Figure2.7depictsdi˙erenttermsinequation2.10normalizedinwallunits.TheReynoldsstress budgetsforsmoothwallhavebeenextensivelyanalyzedin[68].Forthiscase,inthestreamwise 18 Figure2.7:Balanceofdi˙erenttermsinReynoldsstressestransportequation.Thedashed verticallinesarethecrestlocations. direction,theshearproductionisthemainsourceofturbulenceproductionandtheenergyis redistributedtothewall-normalandspanwisedirectionsthroughthepressurework,whichisthe mainsourceofenergyinthosedirections.Thesametrendsapplyfortheroughwalls,exceptthatthe roughness-relatedterms,thewakeproduction P w andthemeanwake-shear P m ,haveasubstantial in˛uenceonthetransportbalance.Thevaluesof P w intheplotof B + < u 0 u 0 > s arepositiveforSG andCB,whilenegativeforTB.Tounderstandthephysicalreasonofthistrend,thecontoursof P w , beforeapplyingspatial-averagingoperator,areshownin˝gure2.8(a,c)forSGandTBand˝gure 2.9forCB.Inallthese˝gures,thepositivevaluesof P w occurinthewakebehindtheroughness elements,whilethenegativevaluesoccurintheregionsthat˛owispseudo-laminarized.Oneof theseregionsisimmediatelyupstreamofeachroughnesselementsforCB,SGandTB.Another region,noticeableinSGandTB,isinthelocationsfarfromtheupstreamelements(enclosedby 19 Figure2.8:Contoursof P + w ; uu at y = d forSG( a,b )andTB( c,d ).Regular( a,c )andsuperposed withunderlyingroughnessshape( b,d ). redcurvesinthe˝gures).Toestablishalinkbetweenroughnessandtheseregions,theunderlying roughnesstopographiesforSGandTBhavebeensuperposed(withatransparentcolor)tothe contoursof P w in˝gures2.8(b,d).Asthe˝guresshow,thosenegativeregionscorrespondtothe crestlocationoftheunderneathroughnesselements.Therefore,onecaninferthatpositivevalues of P w aremostlyduetothelarge-scaleroughnesselements,andnegativevaluesof P w aremostly duetotheunderlyingsmallscaleelements. 20 Figure2.9:Contoursof P + w ; tke at y = d forCB. Figure2.10:Balanceofbudgetsindispersivestressestransportequation.Thedashedvertical linesarethecrestlocations. 21 2.3.3Transportofthenormaldispersivestresses Following[39,44]thebudgetofdispersivestressesinnormaldirectionscanbewrittenas 0 = 2 e u e u j s @ u @ x j | {z } P s + 2 ˝ š u 0 u 0 j @ e u @ x j ˛ s | {z } P w 2 ˝ @ @ x j š u 0 u 0 j e u ˛ s | {z } T RS u j ˝ @ e u 2 @ x j ˛ s + ˝ @ @ x j e u e u e u j ˛ s | {z } T u + ˝ @ 2 e u e u @ x j @ x j ˛ s | {z } T v 2 ˝ e u @ e P @ x ˛ s | {z } 2 ˝ @ e u @ x j @ e u @ x j ˛ s | {z } ; (2.11) where P s istheshearproduction, T RS isthetransportduetotheReynoldsstress, T u isthesummation oftheconvectiveterm(almostzero)andthetriplewaketransport, T istheviscoustransport, is theworkdonebythepressure,and isthedissipationterm. Figure2.10showsthecontributionofdi˙erenttermsforroughwalls.Forallcases,themain sourcetermisthepressurework andthemainsinktermis T RS .Inthestreamwisedirection, themagnitudeofshearproductionislargeforSGandCB,whileitisnegligibleforTB.The contributingfactorin P s ,instreamwisedirection,isduetosheardispersivestress e u e v .Figure2.11 depictsjointPDFof e u and e v fortheroughcases.The e u - e v distributionismostlyconcentratednear theorigin,whileforSGandCBithappensmostlyinthesecondandforthquadrants( e u e v < 0 ).As aconsequenceitisexpectedtohaveaconsiderableshearproduction, P s ˇ 2 e u e v s @ u š @ y ,for SGandCBandnegligibleshearproductionforTB. 2.3.4Di˙erentparametersineddyviscositymodels Inthesimpli˝edReynolds-averaged-Navier-Stokes(RANS)equationsforstationaryturbulentchan- nel˛ows,theshearReynoldsstressismodeledasequation2.12ineddyviscositymodels, u 0 v 0 = t @ u @ y ; (2.12) 22 Figure2.11:JointPDFsof ¹ e u ; e v º at y = d . Figure2.12:Comparisonofactual C versusthespeci˝ed C in k - and v 2 - f models. 23 Figure2.13:Comparisonofactual t ,versusthemodeled t of k - and v 2 - f models. where t isthedynamiceddyviscosity.Therefore,theclosureproblemreducestothemodelingof t inanaccurateway.Twoeddyviscositymodelsareanalyzedtheclassical k - model,and theellipticrelaxation( k - - v 2 - f ,orsimply v 2 - f )modelofDurbin[34]. Thedynamiceddyviscosityismodeledasequations(2.13a)and(2.13b)for k - modeland k - - v 2 - f model,respectively. t = C ; k k 2 ; (2.13a) t = C ; v 2 f k v 2 ; (2.13b) where C ; k = 0 : 09 and C ; v 2 f = 0 : 20 . Figure2.12comparesthemodelcoe˚cient C obtainedfromDNSdatawiththesepre-speci˝ed values.Asthe˝gureindicates,thepre-speci˝edvaluesareaccurateformostpartsofthechannel, exceptforthenearthewallregions.Thisisanimportantissue,becauseregionsclosetothewallare themainsourceofshearthatcausessigni˝cantturbulenceinhomogeneityforwallbounded˛ows. Forsmoothwalls,itiswell-establishedintheliteraturethatisotropiceddyviscositymodels,such astheconventional k - ,areincapableofpredictingturbulencebehaviorforwall-bounded˛ows becausetheyignorethenon-locale˙ectsassociatedwithwallblocking.Manye˙ortshavebeen 24 conductedtoremedythisproblem,eitherbymodifyingthestandard k - model,orbydevisingnew modelscapableofcapturingthe˛owanisotropy. Figure2.13comparestheactualdynamiceddyviscosity,obtainedfromDNSdatausing t = u 0 v 0 š¹ @ u š @ y º ,withthatof k - and v 2 - f modelsasspeci˝edinequation2.13.Asthe˝gure shows,the k - modelover-predictsthevalueof t .Thiswasexpectedgiventhatitassumesan isotropicturbulence. Onecaneasilyrectifytheseissuesbytuningthemodelscoe˚cientsasfunctionsofroughness geometricalproperties.Thisisnotthefocusofthischapter,butitcanbeeasilydonebyusing regularcurve-˝ttingapproachesorbyusingMachineLearningtechniques,similartotheprediction of k s discussedinchapter4. 2.4Concludingremarks Inthischapter,turbulencestatisticsareanalyzedandcomparedbetweenasmoothwalland threeroughwallswithdi˙erenttopographies.Thesimulationsareperformedforfullydeveloped turbulentchannel˛owsat Re ˝ = 1000 .Theroughwallsaretwosyntheticsurfacesofsandgrain andcube-roughnessandarealisticturbinebladesurface.ItisshownthattheReynoldsstressesare moreisotropicinroughwallscomparedtothesmoothwall.AnalysisoftheReynoldsstressbudgets revealedthatthewakeandthemean-wakeshearproductions, P w and P m (manifestedbysurface roughness)playimportantrolesintransferringenergybetweenthewakeandthemean˛ow˝elds withtheturbulence˝eld. P w isstronglypositiveforsyntheticroughwalls,whileitisnegativefor therealisticsurface.Correlationbetweentemporal-averaged P w andsurfacegeometryindicated thatthemainsourceof P w isthewakebehindlarge-scaleprotuberances,whilethemainsinkof P w isassociatedwiththenear-walllaminarizationofturbulenceduetosmallscaleroughnesselements. Analysisofthedispersivestressesbudgetsrevealedthatthemainsourceofthedispersive stressesistheworkdonebythepressure,whichisalsoexpectedbecauseofthestrongassociation betweenwakebehindroughnesselementsandthepressurevariationaroundeachelement(whichis alsothemainsourceofhydraulicdrag).Itisshownthatthewakeshearproductionisalsopositive 25 forthesyntheticwalls,whileitisnegligiblefortherealisticsurface,duetodi˙erentdistributions of e u - e v .Thisdistributionisskewedtowards Q 2 and Q 4 quadrantsforthesyntheticwalls,whileitis centeredaroundzerofortherealisticsurface. Theresultsshowstrongdependenceofturbulencestatisticstotheroughnessgeometry,espe- ciallyintheroughnesssublayer.Theseareimportantforturbulencemodelingapplications,where theReynoldsstressesandtheirbudgetsneedtobemodeledforaccuratepredictionsofturbulence. 26 CHAPTER3 TURBULENCESTRUCTURESOVERREALISTICANDSYNTHETICWALL ROUGHNESSINOPENCHANNELFLOWATRe ˝ = 1000 3.1Introduction 3.1.1Literaturereview Theprocessesthatareessentialtosustainturbulenceinboundarylayerstakeplaceclosetothewall wherethemeanshearishigh.Theturbulentstructuresresponsibleforthesephysicalprocesses [15]havebeenexaminedextensively,whichhasledtoimprovedunderstandingofturbulence in˛owsoverbothsmoothandroughsurfaces.Forsmoothwalls,this˝eldofstudyiswell established,thoughitcontinuestodevelopinnewdirections.Someseminalstudiesinthe˝eld include,butarenotrestrictedto,analysisoftwopointcorrelations[69],linearstochasticestimation [70],instabilityanalysis[71],mechanismsforgenerationofcoherentpacketsofhairpins[72], evolutionarybehaviorofcoherentmotions[73],andturbulentspotidenti˝cation[74].Readersare referredtothemonogrambyTardu[75]foracomprehensivereviewofdi˙erentmethodsusedto analyzethesestructuresinturbulent˛ows. Turbulentstructuresoverroughsurfaceshavebeeninvestigatedextensivelyinpreviousstudies, foravarietyofdi˙erentroughnesstopographiessuchascubicalelements[16],turbineblade roughness[76],2-dimensional(2D)bars[17],pyramids[18]and2-dimensional(3D)sinusoidal surfaces[19].Foradistributedpyramidroughness,TalapatraandKatz[18]foundexperimental evidenceofinteractingU-shapedvorticesofthescaleofthepyramidintheroughnesssublayer. Chanetal.[19]showedthatfor3Dsinusoidallyvaryingsurfaces,theenergycontainedinlarger structureswasredistributedtothoseofthewavelengthofthesurface,inthevicinityofitscrests. KrogstadandAntonia[77]performedexperimentsonboundarylayer˛owsover k -typerough surfaces:thehighsensitivityofvaluesof v 0 rms and u 0 v 0 tothewalltextureledthemtopostulate thatactivemotionswerestronglydependentonthegeometryoftheroughwall. 27 Manyroughsurfacesfoundinnaturearemultiscaleorfractal-like[78].Severalrecentstudies onhowmultiscaleroughnesstexturesa˙ectturbulencehavefocusedonthefrictionaldragand turbulencestatisticswithintheroughnesssublayer.Ithasbeenfoundthatthedragscalespredomi- nantlyonwall-normalturbulence˛uctuationsattheelevationoftheroughnesscrest[79,80],and thatsurfaceroughnessatlongerwavelengthsdoesnotplayasigni˝cantroleindragproduction[81]. Forturbine-bladesurfaces[5],˝lteredgraphitesurfaces[82],andotherdissimilarrandomsurface roughnesses[80],ithasbeenshownthattheanisotropiesoftheReynoldsanddispersivestress tensorsaretexture-dependent.Ingeneral,Townsend'ssimilarityhypothesishasbeenobservedto applytothesestatistics,providedthattheratioofboundary-layerthicknesstoroughnessheightwas largeandthattheReynoldsnumberissu˚cientlyhigh. Formanyroughsurfaces,onlystatisticalinferencesofe˙ectsofroughnessonthe˛owhavebeen made,andthosestudiesonturbulencestructurefocusedmostlyonparticulartypesofroughness topography,withfewstudiesonmultiscaleroughness.Relativelylittleisunderstoodaboutturbulent structuresandcoherentmotionsoverdi˙erenttypesofroughsurfaces,andhowtheirmodi˝cation bysyntheticandrealisticwallroughnessesmightdi˙er. 3.1.2Objectives Thepresentstudyintendstoprovideamorecomprehensiveunderstandingofturbulencestructures infully-developedchannel˛owsoverroughwalls.Themainobjectivesare˝rsttoexplorehow roughnessmodi˝esturbulentcoherentmotionsin˛owsoverroughwalls,andsecondtocompare thebehaviorofthesemotionsin˛owoverarealisticroughsurfaceandin˛owovera`standard' roughness.Therealisticsurfaceischaracterizedbymultiplescales/wavelengthsofroughnesswhile astandardroughness,madeupofdistributedelementsofsimilarshapes,isdescribedbyasingle ornarrowsetofdominantwavelengths.Tothisend,turbulent˛owovera˛atsurfacewiththe roughnessofaproductionhydraulicturbinebladeissimulatedandcomparedwith˛owovera smoothwallandovertwootherquitedi˙erentroughsurfaces:sand-grainandcuberoughness.The surfacesaresameasthoseusedinchapter2.Flowsinboththeroughnesssublayerandtheouter 28 layerareexaminedandtwo-pointvelocitycorrelations,lengthscales,inclinationangles,velocity spectra,vorticities,andhelicitiesareanalysed.Linearstochasticestimationisemployedtoexplore theaveragebehaviorofinstantaneousvorticalmotionsinthoseregionsandtheirdependenceon roughnesstopography. Anunderstandingofhowroughnessmodi˝esturbulencestructurescanalsoprovideimportant insightsintomodelingandcontrolofturbulenceinbothequilibriumandnon-equilibrium(ac- celerating/decelerating)˛owsoverroughwalls,thoughitisrecognizedthat,innon-equilibrium turbulent˛ows,somee˙ectsofroughnesscanpropagatethroughouttheentireboundarylayer,in whichcaseequilibriumwall-similarityscalingsmaynotapply. Themethodologyofthischapterissimilartothatinchapter2.Pleaserefertosection2.2asfor de˝nitionofdi˙erentparametersandmathematicalsymbols. 3.2Results Inthefollowing,thecharacteristicsofturbulencestructurearediscussedusingtwo-point velocitycorrelation,variouslengthscales,energyspectra,vorticity,andbothinstantaneousand conditionally-averagedvorticalmotions. 3.2.1Two-pointvelocitycorrelations Thegeneralformofatwo-pointcorrelationofturbulencevelocitiesisde˝nedas R u u ¹ y ; r 1 ; r 2 ; r 3 º u 0 ¹ x ; y ; z ; t º u 0 ¹ x + r 1 ; y + r 2 ; z + r 3 ; t º : (3.1) Thenormalizedtwo-pointcorrelations(denotedbysubscript n )withaseparationin x is R u u ; n ¹ y ; r 1 º = R u u ¹ y ; r 1 ; 0 ; 0 ºš R u u ¹ y ; 0 ; 0 ; 0 º : (3.2) In˝gure3.1,thepro˝leof R uu ; n ¹ y ; r 1 º isshownatdi˙erentdistancesfromthewall.Itcanbe seenthat,closetothewall, R uu ; n ismosta˙ectedbylocalfeaturesofroughness,depictedbythe blacklines.Inanunperturbedturbulent˛ow˝eld,themagnitudeof R u u ; n decayssmoothlyfrom unityatzeroseparation,asshownin˝gure3.1for˛owoverasmoothwall.However,thee˙ect 29 Figure3.1:Pro˝lesof R uu ; n ¹ y ; r 1 º at y locationsfrom y ˇ 0 ( black )to y š = 0 : 3 ( blue ). Figure3.2:Pro˝lesof R uu ; n ¹ r 1 º ( black )and R ww ; n ¹ r 1 º ( blue ),at y š = 0 : 15 (a),at y š = 0 : 3 (b). 30 ofsurfaceroughnesscanbetoeitherenhanceorreducethevalueofthecorrelationatdi˙erent displacements,duetotheperiodicityandlengthscalesofroughness. Neararoughwall,thetwo-pointvelocitycorrelationatagivendisplacement r canbeattributed to: i )coherentmotionsofturbulenceofscale r ;and ii )periodicexcitationfromthesurface roughnessatwavelength r .Forexample,inthecaseofCBroughnessshownin˝gure3.1,nearthe wallthecorrelationatdisplacementsshorterthan 0 : 1 associatespredominantlytolocalstreaky motions,whilefor r 1 ofapproximately CB = 0 : 21 ,theperiodicspatialdistributionofthese streakymotions(smallerthan 0 : 21 )onaccountofperiodiccubearrangementscontributetothe correlation.Pronouncedlocalpeaksof R uu withaseparationofthewavelengthofdistributed roughnesssuchas2Dbarswasalsoobserved[83].Itwillbeshowninsection3.2.3thatthise˙ect resultsinpeaksandtroughsinvelocityspectrameasuredwithintheroughnesssublayer. Streamwiseandspanwisevelocitycorrelationsareplottedin˝gure3.2fortwoouter-layer locations, y š = 0 : 15 and y š = 0 : 3 .In˝gure3.2(a),itisshownthat R uu ; n isstilla˙ectedby surfaceroughnessat y š = 0 : 15 andin˝gure3.2(b),itisshownthat R uu ; n in˛owoverthesmooth wallandTBroughnesscollapseat y š = 0 : 3 ,whereasin˛owsoverwallswithSGandCBsurfaces theydoyexhibitalowerdegreeofcorrelationovertheentirechannel.Thespanwise velocitycorrelations R ww ; n yieldabetteroverallcollapseforthedi˙erentsurfacesthanstreamwise correlationsatbothlocations.AccordingtotheTownsend'souter-layersimilarityhypothesis[69], velocitycorrelationsatthislocationwouldbeexpectedtocollapsewhen š k (where k isthe roughnessheight)issu˚cientlylarge.Jiménez[13]hasproposedthatTownsend'ssimilarity hypothesisapplieswhen š k 50 .Inthepresentstudy,themaximumvalueof š k c is 14 (for CBroughness),whichmayexplaindeparturesfromthehypothesisforthetwo-pointcorrelation, althoughithasbeenshownthatwallsimilarityappliestosingle-pointstatisticssuchascomponents oftheReynoldsstresstensor.Itfollowsthatouter-layerstructuralcharacteristicsofturbulence (suchastwo-pointvelocitycorrelations)appearstobemoresensitivetoroughnessin˛uencethan single-pointstatistics. 31 Figure3.3:Averagedstreamwise ¹ L 11 ; 1 º andspanwise ¹ L 11 ; 3 º energy-containinglengthscales. Figure3.4:Turbulence ¹ a º andKolmogorov ¹ b º lengthscales. 3.2.2Lengthscalesandinclinationangles Thesizeoflarge-scaleenergy-containingcoherentmotionshavebeenpreviouslyquanti˝edin variousways:(1)alengthscalede˝nedbyacuto˙valueof R u i u j ; n [84,85,17],(2)integrallength scalesof R u i u j ; n [86,52],(3)thelengthscalesoflow-momentumregionsofalinear-stochastically- estimatedvelocity˝eld[87],and(4)thelengthscalesobtainedfromthespectralcoe˚cientsof thecorrelations[88,89].Inthepresentstudy,twotypesoflengthscalesareanalyzed:(1)the x - and z -extentoftheisocontourof R u i u j ; n = 0 : 5 [85,17], L ij ; k withseparationin x k ,asatwo-point 32 structuralproperty,and(2)theturbulencelengthscale, L trb k 3 š 2 š ,asasingle-pointstatistical property,where k istheturbulencekineticenergy(TKE)and istheTKEdissipationrate.The choiceof R u i u j ; n cuto˙valuein L ij ; k doesnotchangetheoverallcomparisonof L 11 ; 1 ,asshown bytheshapeof R u i u j ; n pro˝lesin˝gure3.2. Thesetwolengthscalesareplottedin˝gures3.3and3.4(a),respectively.Inboth˝gures,the lengthscalesshowlinearbehaviornearthewall( y + < 150 or y š < 0 : 15 ),whichimpliesthe self-similargrowthofthelarge-scalemotionsinanaveragesense.Thelinearincreaseofspanwise lengthscaleswerealsoobservedbyTomkinsandAdrian[87].Also,itisshownherethat L 11 is smalleraboveCBandSGsurfacesthanthoseabovethesmoothwallandTBsurface.Itshould benotedthat,although L trb (asadirection-freelengthscale)and L 11 ; 3 (asaspanwiselength scale)appeartoconformwiththeTownsend'souter-layersimilarityhypothesis, L 11 ; 1 and R uu ; n (˝gure3.2)donot.Thissuggeststhattheresponseoftheouterlayer˛owtowallroughnessis directionallysensitiveandthatthestreamwisestructurecanbein˛uencedstronglybytheroughness texture.TheexperimentalstudyofKrogstadandAntonia[84]onboundarylayer˛owsoversmooth androughwallswith š k ˇ 50 alsoshowedthat L 11 ; 1 (basedonacuto˙valueof R uu ; n = 0 : 3 )for roughwallswassmallerthan L 11 ; 1 forsmoothwallsintheouterlayer,and L 11 ; 3 wasalmostthe sameinbothcases. ThedeparturefromTownsend'ssimilarityinbothourresultsandthoseofKrogstadandAntonia [84]maybeduetolimitedReynoldsnumber,asarepeatedexperimentofturbulentboundary layer˛owoverasquarebarroughnessbyKrogstadandEfros[90]atahighReynoldsnumber ( Re = 32600 and š k = 131 )indicatedreducedroughnessin˛uenceontheouterlayer. Tocomparethesizeofenergy-containingmotionsmentionedabovewiththesizeofthe dissipative-scalemotions,theKolmogorovlengthscale = ¹ 3 š º 1 š 4 isplottedin˝gure3.4(b).It canbeseenthatpro˝lesof aresensitivetothesurfacetexturesonlybelowtheroughnesscrest. For y > k c ,all pro˝lescollapse,indicatingthatsmallscalestructuresbecomeindependentof thesurfacetextureeventhoughlarge-scalestructuresdonot.Thisobservationagreeswiththe ˝rsthypothesisofKolmogorov[91]that,atscalesoftheorderof ,turbulenceobeysauniversal 33 Figure3.5:Contoursof R uu ; n ¹ r 1 ; r 2 º ( left )and R ww ; n ¹ r 1 ; r 2 º ( right )centeredat y = d forSG,TB andCBroughness,andat y + = 15 forthesmoothcase.Thecontourlevelrangeis[0.41.0]witha stepsizeof0.2. equilibriumandislocallyisotropic. Toquantifytheinclinationanglesoflarge-scalecoherentmotions,contoursof R uu ; n ¹ y ; r 1 ; r 2 º and R ww ; n ¹ y ; r 1 ; r 2 º wereplottedinthe x - y plane,centeredattwotypesofelevations:(1)anear-wall locationintheroughnesssublayer( y = d )forroughcasesandinthebu˙erlayer( y + = 15 )forthe smoothcase,shownin˝gure3.5,and(2)anouter-layerlocation,at y š = 0 : 3 forallcases,shown in˝gure3.6.Thecharacteristicinclinationangleswereobtainedbyplottingthebest-˝ttedline (usinglinearleastsquaremethod),traversingthefarthestpointsfromtheoriginatcontourlevels of{0.4,0.6,0.8}.Theseanglesareexaminedinthecontextof: i )howtheangledependsonthe particularcorrelation,foragivensurface;and ii )howtheinclinationangledependsonthesurface texture. The˝guresshowthattheinclinationanglesof R uu ; n aremuchsmallerthanthoseof R ww ; n at boththenear-wallandouter-layerelevations.Aphysicalexplanationisthat w 0 ˛uctuationspartially originatefrom ! x motions,associatedwithindividualvorticalmotionssuchasquasi-streamwise vortices,whilethe u 0 motionsareassociatedwithstreakymotions.Asaresult,the x -extentof u 0 and w 0 motionsareverydi˙erentandcharacterizedbydi˙erentinclinationangles. 34 Figure3.6:Contoursof R uu ; n ¹ r 1 ; r 2 º ( left )and R ww ; n ¹ r 1 ; r 2 º ( right )centeredat y š = 0 : 3 .The contourlevelrangeis[0.41.0]withastepsizeof0.2. Forthenear-wallelevations,theanglesofinclinationdependsigni˝cantlyonthepresenceof roughnessanditstexture.Theseanglesareapproximately 3 for R uu ; n and 9 for R ww ; n nearthe smoothwallandmuchhigheronroughwalls,equaling 7 9 and 16 28 ,for R uu ; n and R ww ; n respectively(˝gure3.5).TheCBroughnessyieldsaninclinationangleof 28 for R ww ; n .This angleappearstocoincidewiththeangleformedbetweenthecubeheightandthecubespacing, tan 1 f k c š¹ CB k c ºg = 26 ;afullexplanationofthisassociationcannotyetbeprovidedandmay beatopicoffuturework.Nearthewall,YuanandPiomelli[52]alsoobservedhigherinclination anglesintheroughnesssublayerthanintheviscoussublayer,fortheSGroughnessinsink-˛ow boundarylayers.Intheouterlayer,allcasesshowsimilarinclinationsof 8 11 for R uu ; n and 26 - 31 for R ww ; n (˝gure3.6).Theouter-layervaluesareconsistentwiththosereportedbyVolinoet al.[17],whereanglesofturbulentstructuresforboundarylayer˛owsover3Dcubesand2Dbars werealsodeducedfrom R uu ; n contours.Theyfoundthatfor 0 : 2 < y š < 0 : 5 theinclinationangle of R uu ; n contourwassurface-textureindependentandrangedbetween 10 to 14 degrees.Similar 35 observationsweremadebyCocealetal.[16]forchannel˛owovercubes.Thesevaluesarelargely comparabletotheanglesfoundhereforverydi˙erentroughnessgeometries. 3.2.3Velocityspectra One-dimensionalvelocityspectraarede˝nedastwicetheFouriertransformof R u i u j ¹ y ; r 1 º ,as E ij ¹ y ; 1 º 1 ˇ 1 R u i u j ¹ y ; r 1 º e i 1 r 1 dr 1 : (3.3) Using(3.3)tocalculate E ij iscomputationallymoreexpensivethantheequivalentexpressionfrom cross-correlationtheory(theWiener-Khinchintheoremwhen i = j ): E ij = 2 F ¹ R u i u j º = 2 F ¹ u 0 i º F ¹ u 0 j º ; (3.4) where F istheFouriertransformoperatorand*indicatesthecomplexconjugate.Theright handsideof(3.4)ischeapertocomputeasitcanbeobtainedbyaFastFourierTransform(FFT) algorithm.Equation(3.4)wasalsousedtoobtain R ij bytakingtheinverseFouriertransformof itsrighthandside[92].Belowtheroughnesscrest,theinterpretationof E ij islessclearbecause ˛uiddomainsinthe x z planemaybemultiplyconnectedand,asyet,novelocityspectraappear tohavebeenreportedwhere y < k c .Theapplicabilityof(3.4)isalsoquestionablebecausethe spatialvelocitysignalsaresegmentedinthisregionandtheirFouriertransformisnotguaranteed toexist.Instead,(3.3)isusedforpower-spectralestimationbecause R u i u j iscontinuousinthis region,notwithstandingits`conditional'naturediscussedinsection3.2.1.Inthisstudy,thevelocity spectraatall y -locationsareobtainedusing(3.3),withaHannwindowfunctiontominimizethe Gibbsphenomenonatlargewavenumbers[93].Datawithinhalfofthe x and z domainsareusedto calculatethetwo-pointcorrelations;thus,noisesarepresentinthespectraifnowindowfunction isused. In˝gure3.7,theone-dimensionallongitudinalvelocityspectra E 11 areplottedat y š = 0 : 3 againstwavenumbermultipliedbytheKolmogorovscale.Itshowsauniversalbehaviorconsistent 36 Figure3.7:One-dimensionallongitudinalvelocityspectra E 11 at y š = 0 : 3 .Linesin black are obtainedusing(3.3)andlinesin blue using(3.4).Experimentaldataarefrom[4].Plotsincircles anddiamondsareforboundarylayer˛owatRe = 1500 andwake˛owatRe = 23 respectively. Thethindashedlinedescribes E 11 = 0 : 49 5 š 3 1 2 š 3 . withanenergycascadefromlargetosmalleddies[94].Theenergyspectracollapseonalineof slope 5 š 3 andobeyuniversalbehaviorinthepresenceofroughnessinhomogeneousdirections, evenoverthenarrowrangeofwavenumbersobtainedintheselow(comparedtoexperimental studies)Reynoldsnumbersimulations.ThisobservationisconsistentwiththoseofCocealetal. [16]for˛owovercubesofuniformsize,forwhichitwasnotedthatpre-multipliedspectraof 1 E 11 followeda 2 š 3 decaylaw.VelocityspectraobtainedbyFouriertransform(3.4),arealsoplotted in˝gure3.7forthepurposesofcomparison.Thetwomethodsforcalculatingenergyspectra, usingequations(3.3)and(3.4),yieldverysimilarspectrawhen y > k c andthevelocitysignalsare continuous. Thepremultipliedvelocityspectra E 11 , E 22 , E 33 ,and E 12 areplottedin˝gures3.8and3.9in the 1 and 3 directions,respectively.Theyarecomparedwiththepowerspectraofsurfaceheight ˛uctuationstoexploreapossiblerelationbetweenthelengthscalesofturbulenceandthoseofthe 37 roughsurface.Threetypesofelevationsareofparticularinterest.First,at y = d ,therough-case resultsarecomparedwiththoseinthesmoothcaseinsidethebu˙erlayerat y + = 15 .Secondly,the loweredgeofthelogarithmicregion(around y = k c ,see˝gure2.3)intheroughcasesiscompared with y + = 40 inthesmoothcase,whichisalsoattheloweredgeofthelogarithmicregion.And thirdly,theouterlayerat y š = 0 : 3 iscomparedamongallcases,wherethe˛owisexpectedto beindependentofsurfaceconditionsifTownsend'ssimilarityhypothesisapplies.Inboth˝gures 3.8and3.9,itcanbeseenthatat y = d and y = k c the E ij componentsaredependentonthekind ofsurfacechosenand,as y increases,theybecomeprogressivelymoreindependentofthesurface condition.Forexample,at y š = 0 : 3 ,thelocationoftheenergypeaksandthegeneralshapesof E ij aresimilarforeachkindofwallroughness,consistentwithTownsend'ssimilarityhypothesis. As y decreases,thepeakof E 11 movestowardhigherwavenumbersinboththe 1 and 3 directions,indicatingshrinkingofenergy-containingscalesasthewallisapproached.Theenergy carriedbythelow-wavenumber u 0 motionsis,inallroughcasesatboth y = d and y = k c , muchlessthaninthesmooth-wallcaseatthecorrespondingelevations,becausesurfacerough- nessbreaksdownthebu˙er-layerstreakymotionsbyintroducingvoritcalstructureswithhigh three-dimensionality.Inaddition,atthenear-walllocations,thepeakvaluesof E ij components, normalizedby u ˝ ,showhigheranisotropiesinSMandTBcases,consistentwiththeReynolds stresscomparisonin˝gure2.4. Atthenear-wallelevation,theconnectionbetweentheenergyspectraandthesurfacelength scalesisanalyzedforeachkindofroughness.ForSGroughness,itappearsthatthepeaksof E 22 ¹ 1 º and E s ¹ 1 º coincideatthesamewavenumber.Similarly,thepeakofthe E 11 ¹ 3 º coincides withboththatof E s ¹ 3 º ,thoughsuchwavelengthisalsowherethesmooth-wall E 11 ¹ 3 º peakis located.ForTBroughness,thefractalnatureofthesurface-heightdistributiondoesnotyielda dominantsurfacewavenumber;its E ij pro˝lesaresimilartothoseinthesmoothcase.ForCB roughness,theenergyspectraarestronglycorrelatedwiththesurfacepowerspectrum,whichhas spikesat = 2 ˇ š CB anditsmultiples.At y = d ,in˝gure3.8itcanbeseenthat E 11 and E 33 havetoughsandpeaks,respectively,atthesame 1 valuesasthesespikes.Theseextremaappear 38 Figure3.8:One-dimensionallongitudinalpremultipliedvelocityspectra, E ij ,andsurface-height powerspectra, E s . E 11 ; E 22 ; E 33 ; E 12 ; (thin) E s . E s isnormalizedto giveamaximumvalueof1. 39 Figure3.9:One-dimensionaltransversepremultipliedvelocityspectra. E 11 ; E 22 ; E 33 ; E 12 ; (thin) E s . E s isnormalizedtogiveamaximumvalueof1. tobeaconsequenceofthesurfaceperiodicitydescribedinsection3.2.1. 3.2.4Vorticityandhelicity Theroot-mean-square(RMS)valuesofvorticity˛uctuations,normalizedinwallunits,areplotted in˝gure3.10.Intheouterlayer,irrespectiveofthesurfacetype,allcasescollapseshowingthe isotropyofthevorticitytensorresultobservedpreviouslyforbothsmooth[95]androughwalls [96].Itisknownthat,nearasmoothwall ! 0 3 ; rms ismostlycausedbytheintensesheargenerated 40 Figure3.10:Pro˝lesofRMSvorticityinwallunits.Thinverticallinesshowthecorresponding roughnesscrestelevations. by u 0 betweenthewallandthestreaks, ! 0 2 ; rms representsmostly @ u 0 š @ z intheregionsbetween adjacenthighandlow-speedstreaks,and ! 0 1 ; rms displaysalocalminimumandalocalmaximum at y + = 5 and20respectively,attributedtothequasi-streamwisevortices.Roughnessa˙ectsthe anisotropyofthevorticitytensornearthewall.Thepeaksof ! 0 2 ; rms and ! 0 3 ; rms aresigni˝cantly lowerin˛owoverroughwallscomparedtosmoothones,implyingthatroughnessmodi˝esthe organizedmotionsbybreakingdownthenear-wallstreaksandpromotingthethree-dimensionality ofthevorticalmotions.SGandCBroughnesseshaveasimilare˙ectonproducingsigni˝cantly moreisotropicvorticity˛uctuations.Incontrast, ! 0 i neartheTBroughnessmaintainsahigh anisotropysimilartothesmoothwall.Thisisbecause,forTBroughness,low-speedstreakswith intense u 0 arestillpresentbelow y = d intherecessedroughnessregion(duetothelargest x and z wavelengthsofthesurface).ThebehaviorofTBroughnessisagainshowntobeintermediateto thatofsmoothwallandSGroughness. Thejointprobabilitydensityfunctions(PDFs)of u 0 and ! 0 ,whichcontributetothelocal helicity u 0 ! 0 ,arecomparedin˝gure3.11.Dataareplottedat y = d forSG,TBandCBroughness andat y + = 15 forthesmoothsurface.Threetypesofturbulenteddieshavebeende˝nedby Kassinos[97]as:jetal( u 0 , 0 , ! 0 ˇ 0 );vortical( u 0 ˇ 0 , ! 0 , 0 );andhelical( u 0 , 0 , ! 0 , 0 ). ThejointPDFforthe˛owoverthesmoothwallisskewedtowardsjetalmotionswith u 0 < 0 and ! 0 1 ˇ 0 ,whichisthecharacteristicofthelow-speedstreaks.Roughness,irrespectiveofitstexture, reducesthesigni˝canceofthesemotionsandmakesthejointPDFcontoursmoreconcentric.In 41 thecaseofCBroughness,aslighttendencyexiststowardspositivevaluesof u 0 ,whichmaysuggest slightlymorepronouncedsweepingeventsat y ˇ d .ThejointPDFsof v 0 and ! 0 2 showsthat, at y + = 15 onthesmoothwall,the v 0 motionsarepredominantlyassociatedwithjetalsweeps ( v 0 < 0 , ! 0 2 ˇ 0 ),inactivemotionsoriginatedfromabove,andhelicalejectionmotions( v 0 > 0 , ! 0 2 , 0 ),representingthecounter-rotatingpairsofquasi-streamwisevortices.Nearroughwalls, incontrast,alltypesofmotionsareroughlyequallypossible,indicatinghighlythree-dimensional shapeofcoherentmotions.ThejointPDFsof w 0 and ! 0 3 showasymmetricbehaviorforallcases, onaccountofthesymmetryduetotheboundaryconditionsinthisdirection.JointPDFswere alsoobtainedintheouterlayerat y š = 0 : 3 andwereallverysimilar(notshown),consistentwith Townsend'ssimilarityhypothesis. 3.2.5Instantaneousvorticalmotionsandconditionaleddies Instantaneouscoherentmotionsarevisualizedusingiso-surfacesofswirlstrength,quanti˝edby theimaginarypartofthecomplexeigenvalueofthevelocitygradienttensor ci [72].Theresults areshownin˝gure3.12for y š < 0 : 25 ,inwhichitcanbeseenthatinclinedquasi-streamwise vorticesexistinthevicinityofbothsmoothandroughwalls,withhigherinclinationangles(in the x - y plane)andtiltangles(inthe x - z plane),aswellasmoreirregularshapeforSGandCB roughnesses.Largenumberofspanwise-alignedvortexsegmentsarealsovisibleontherough surfaces.OnSGandCBsurfaces,twoarch-shapevorticalstructuresarehighlighted,whichmay beexamplesofthesolid-bodyrotationofanoriginallyhead-downhorse-shoevortexonaccountof stronglocalshearlayer,tobediscussedinSection3.2.6. Linearstochasticestimation(LSE)[70]wasusedtocomparethevortexshapeoverboththe smoothwall[72,98]androughones[18,99].TheLSEisanaveragevelocity˝eld(the conditional eddies ),conditionedongiveneventsatspeci˝edlocationsinthe˛owdomain;theprocedureis fullydescribedbyAdrian[100].FollowingTalapatraandKatz[18],weuseaneventbasedon thevortex-identi˝cationparameter,here ci > 0 .Theeventisascalar.Therearetwoadvantages forthisapproac˝rst,theconditionaleddyobtainedhereiniscomposedofallpossiblephysical 42 Figure3.11:JointPDFsofinstantaneousvaluesof u 0 and ! 0 ,normalizedbytheirrespective RMSvalues;dataarefromthewall-parallelplaneat y + = 15 forsmooth-wall˛owand y = d for SG,TBandCBroughness. eddies(since,byde˝nition,foraneddy ci isgreaterthanzero);anoverallpicturecanbeformed onhowalltypesofrelevanteddiesaremodi˝edbytheroughness,whichisespeciallyhelpfulfora qualitativecomparisonofvortexshapearoundrandomroughnessgeometries.Second,compared toavectorevent,whichisdependentonthedirectionofthespeci˝edevent,suchasthe Q 2 event usedbyZhouetal.[72],the ci > 0 eventwillbeindependentonthedirection.Itisalsonotedthat thelimitationofsuchapproachisthatitdoesnoto˙erlocation-speci˝cinformationontheshape ofrelevanteddies;however,thisisnotcriticalasherewefocusontheaverageeddyshape. 43 Figure3.12:Instantaneousvorticalmotions,visualizedbyiso-surfacesof ci = 0 : 2 ci ; max , coloredaccordingtodistancefromthewall. Theconditionalvelocity˝eldatdisplacement r fromthepoint x isdeterminedas D u 0 ¹ x + r º 0 ci ¹ x º > 0 E 0 ci = 0 ci ; s = u 0 ¹ x + r º 0 ci ¹ x º 0 ci ¹ x º 0 ci ¹ x º 0 ci ; s ; (3.5) with ci > 0 beingtheevent.Here, 0 ci ; s isequaltooneandactsasadummyvariable,since theresultsshowninthevisualizationsarenormalized.Weusethedataofall ¹ x ; z º locationsata speci˝edeventelevation,tocalculateequation(3.5).Thereforetheshapeofconditionaleddieswill befunctionof y only.Theconditionaleddiesarevisualizedastheiso-surfaceofswirlingstrength oftheobtainedconditionalvelocity˝eld.Theyareshownin˝gure3.13atvariouselevationsfor allcases.Asensitivityanalysiswascarriedouttoensurethattheoverallcomparisonofvortex shapeisindependentofthethresholdsused.Thenear-wallelevationscompare y + = 15 and40in thesmoothcasewith y = d and y = k c intheroughcases.Theouter-layerelevationislocated at y š = 0 : 2 .Notethat,in˝gure3.13,theroughsurfacesareplottedtoillustratethesizeand elevationoftheeddiesonlyandnottoshowtheexact ¹ x ; z º locationsofsuchcoherentmotions. 44 Figure3.13:Conditionaleddiesbasedonaneventof c > 0 ,atthree y locations;frombottomto top, y + = 15 , y + = 40 , y š = 0 : 1 forSM;and y = d , y = k c , y = 0 : 2 forSG,TBandCB roughness.Plotsareiso-surfacesof ci = 0 : 4 obtainedfromtheconditionalvelocity˝eld. Inthebu˙erlayer( y + = 15 )ofthesmoothcase,theconditionaleddycomprisespairsof inclinedstreamwise-alignedvortices,withabifurcation.Thisisbecausethequasi-streamwise vorticesarenotstrictlyalignedin x buttiltedtowardbothpositiveandnegative z -directionswith equalprobabilities[71].Theconditionaleddyat y = d forTBroughnessissimilartothesmooth casebecausethequasi-streamwisevorticesarealsopresentinthetroughsof -scaleundulationsof thisroughness.At y = d ,boththeSGandCBcasesshowapairofvorticeswiththree-dimensional shapes.ForSGroughness,itisevidentthatahead-upandahead-downU-shapevorticesare bothpresent.Thelowervortexappearstorepresentthevortexformedduetoshear-layerroll-up wrappingaroundtherecirculationregiondownstreamofaroughnesselement[101].Theupper vortexappearssimilartothehead-downhorse-shoevortexobservedbyTalapatraandKatz[18], whichisformedasbothendsofanincomingspanwise-alignedvortexundergostretchingin x (due tomean-˛owchannelingonbothsidesofthecube)andlifting(duetointeractionswiththeadjacent vorticesofthesamekind). 45 At y + = 40 ,thesmooth-wall˛owin˝gure3.13hasaninfanthairpinstructure,whichis producedbyquasi-streamwisestructuresatthislocation,andgrowsself-similarlyinthelogarithmic regiontoamaturehairpinstructureat y + = 100 .AsimilarprocessseemstotakeplaceforTB roughness,wherequasi-streamwisestructures,generatedintheundulatingpartsoftheroughness, mergetogetherandproduceastructurethatconformswiththeroughnessshapeandalsogrows self-similarlytoamaturestructureat y š = 0 : 2 .Thematurestructureinheritsfeaturesofboththe hairpinstructure(legsandthering,visibleinthematurehairpinofSMat y + = 100 )andthesurface topography.ThesimilarityofthenearwallconditionaleddiesofSMandTBimpliesthatcoherent structuresin˛owoverTBroughnessobeyasimilargrowthmechanismtothoseinsmooth-wall ˛ows,suchasthe streaktransientgrowth mechanismofSchoppaandHussain[102]. TheconditionaleddyforCBroughnessat y = k c consistsofwhatappearstobeapairofarches, thehorizontaloneofwhichmaybeexplainedbytheroll-overmechanisminSection3.2.6.This mechanismisthoughttocausesigni˝cantsweeping(Q4)eventsandtoshortenthevorticalmotions inthestreamwisedirection.Theirshortenedstreamwiselengthscale(con˝rmedby˝gure3.3)in ˛owoverCBroughnessisinheritedbythestructureobservedat y š = 0 : 2 ,whereahairpin-type structurewithsmalllegsisvisible.TheshapeoftheconditionaleddyonSGroughnessat y = k c seemstoevolvethroughacombinationofthemechanismsonbothTBandCBroughnesses. Intheoverlapregionsandabove( y 40 forthesmoothcaseand y y R fortherough cases),theconditionaleddiesofallcasesdisplayaconnectionbetweenthepairofparallelvortices, comprisingspanwise-alignedportionsthatresemblethehorse-shoeheads.Acomparisonofthe sizeoftheconditionaleddiesshowsthat,farfromthewall,spanwisespatialextentsoftheeddies areallverysimilar,whilethestreamwiseextentsareshorterforSGandCBroughnessescompared toTBroughnessandthesmoothwallduetotheshortenedlegs.Nearthewall,shorter x extents ofthevorticalmotionsarealsoobservedonSGandCBcomparedtoTBandthesmoothwall, possiblyresultingfromthedi˙erenceineddyshapeandorientation.The z extents,however,are largerinthecasesofTBandCBroughnesses,asthesetworoughnessgeometriesimpartlarger spanwiselengthscalestothe˛ow. 46 Figure3.14:( a )Sketchofevolutionofaspanwisevortexapproachingacubeelement,and( b ) Meanspanwisevorticity, ! 3 = d v š dx d u š d y ,normalizedby u ˝ and .Thehorizontalplaneis at y = 0 : 75 k c . 3.2.6Akinematicprocessofvorticesinlocalshearlayers Inthissection,wedescribeakinematicrotationalmechanism,referredhereasa rollover mecha- nism,whichmaybeundergonebyasigni˝cantnumberofvorticalstructuresinCBand,toalesser extent,inSGroughness.ThismechanismisthoughttocontributetolocalTKEproductionand toashorter x -extentofcoherenceinthesecases,comparedtothesmoothandTBsurfaces.A sketchisshownin˝gure3.14(a).Asthehead-downhorse-shoevortexwrappingaroundacube element(vortexB)developsfromanincomingspanwisevortex˝lament(vortexto theprocessdescribedbyTalapatraandKatzheadofthisvortexislifted(byconvection) bythemean-˛owejectionimmediatelyupstreamofthecube(where v > 0 ),whilethelegsare convectedbothdownstream(duetochannelingphenomenon, e u > 0 onbothsidesofthecube) andupward(duetomutualinductionbetweenadjacentlegs).Theresultantvortextakesashape similartothatofVortexC.AstheheadportionofvortexCmovesdownstreamnearthetopsurface ofthecube,itissubjecttointensetime-meanspanwisevorticity ! z ofanegativesign(shownin ˝gure3.14b).Consequently,theheadportionundergoesclockwisesolid-bodyrotationandrolls overcubeelements.Meanwhile,thelegportionsofvortexC,astheyareinclinedupward,undergo 47 Figure3.15:Pro˝lesof V r ; z ( a ),and z ( b ),normalizedby u ˝ and .Theverticallinesare thecorrespondingcrestlocations. stretchingbypositive @ h u iš @ y andquicklybreakdownduetonon-linearinteractionwithother vortices.Theresultisasigni˝cantlyshorterstreamwiseextentofthecoherentmotions(vortex D).Intheaveragededdyconditionedatelevationof k c shownin˝gure3.13,theaforementioned vorticalmotionmaymanifestitselfasanarch-shapedstructureroughlyorientedinthe ¹ x ; z º plane. Suchaprocessisaresultofarelatively˛attopsurfaceofahorizontallysizableroughness element;thelocalelevationofthissurfaceintothe˛ow(andthestronglocal ! z initsvicinityas aresult)leadstotheaforementionedsolid-bodyrotationofthevortex.FortheTBroughnesswith mostlysharpprotuberances,thisprocessisprobablylesspronounced.Itshouldbepointedout thatthesketchrepresentsidealprocessesleadingtotheroll-overmechanism;inrealitysomeof thestructuresmaypartiallyrollovertheroughnesselements,ormayonlyhaveone-sidedfeatures. Thosestructureshighlightedin˝gure3.12maybeexamplesofinstantaneousvorticesthatarein stageD. Inthefollowing,evidenceisprovidedtosupporttheexistenceofsuch z -alignedvorticeslocated inthevicinityofthetopsurfaceofacube.Suchamechanismrequirestheexistenceof z -aligned vortextubesinregionsofstrong ! z oncubesurfaces.Tothisend,theeddy'saxis-of-rotation aroundthecubeisquanti˝ed,inanaveragesense,bytwoidenti˝ers.First, V r ,whichisinferred 48 Figure3.16:Contoursof( a ) V r ; z ,( b ) z ,( c )time-averagedshearproductionofTKE, P s = u 0 i u 0 j @ u i š @ x j ,and( d )TKE.CaseCB.Normalizationisdoneusing u ˝ and .The horizontalplanesareat y = 0 : 75 k c . fromtheeddy-visualizationmethodofZhouetal.[72],andisintroducedhereas V r ¹ x ; t º = v r ¹ x ; t º ci ¹ x ; t º ; (3.6) where v r ,aunitvector,isthenormaleigenvectorofthevelocity-gradienttensorcorrespondingto therealeigenvalue(theothertwoeigenvaluesbeingcomplexconjugatesbyde˝nitionofswirling motion).And,second, ( = ci ! šj ! j ),whichisthethree-dimensionalextensionofthemethod usedinAndersonetal.[103,theyusedsignedswirlingstrengthin x i -direction, ci ! i šj ! i j to accountfortheeddy'saxisofrotationinthatdirection]. Both V r and weighthelocalaxis-of-rotationofaneddywithitslocalswirlingstrength ci toaccountforthee˙ectofeddystrengthinthestatistics.Therefore,ahighmagnitudeofthetime- averagedvalueoftheir i th component, j V r ; i j¹ x º or j r ; i j¹ x º ,indicateslocalstrongswirlingmotions 49 aroundthe x i axisinanaveragesense.Thewall-normalpro˝lesof h V r ; z i and h z i areshown in˝gure3.15(a)and(b),respectively,andarecomparedamongallcases.Wehaveintroduced thenewidenti˝er V r ,sincethe ,basedonthe ! 0 z ˝eld,canbequantitatively`contaminated'by non-vorticalstructures,suchasstreakymotions,duetothestrong @ u 0 š @ y theygenerate.Thisis evidentinthepro˝leofSMin˝gure3.15(b),wherethelargevaluesof jh z ij nearthewall(at y + ˇ 15 )isattributedtothedominanceofstreakymotionsinthisregion.Althoughitisexpected thatduetotheroughnesse˙ect,thespanwiserotationismoresigni˝cantforroughwallsthanthe smoothwall,comparisonbetweensmoothandroughwallspro˝lesof jh z ij donotre˛ectsuch atrend.The V r identi˝erisbasedonthevortexvisualizationmethod;therefore,non-vortical motionswouldhavesmallcontributiontoit.In˝gure3.15(a),onenoticesthemagnitudeof jh V r ; z ij forroughwallsmatchesthatofsmoothwall,consistentwiththeaforementionedtrend.Usingeither oftheseidenti˝ers,thefollowingdiscussionswouldremainunchanged. In˝gure3.15(a),thehigh-magnitudepeakof h V r ; z i (withanegativesign)locatedatthe crestlevelofCBroughness,whichisgreaterthanotherlocationsformorethan40%,indicates signi˝cantstrengthof z -aligned,clockwise-rotatingvorticesatthiselevation.Inaddition,the spatialdistributionof V r ; z aroundCBroughnessin˝gure3.16(a)showsthat z -alignededdieswith highswirlingstrengthsarefoundintheregionsimmediatelyupstreamofacubeelementandnearits topsurface;sucheddiesarealsoprevalentinthewakeregionofacube.Moreover,in˝gure3.14(b), itisshownthatregionsofstrong ! z ofthenegativesigns,coincidewiththeregionwithstrong j V r ; z j (˝gure3.16a)nearthetopsurfaceofacube.Theseevidences,alongwiththeresultsofconditional eddiesdiscussedinSection3.2.5,supporttheexistenceoftheaforementionedsolid-bodyrotation processofthehead-downhorse-shoevortexhead,whichleadstoashortened x -extentofcoherent motions. Anotherimportantaspectofthisprocessisthatitisexpectedtoyieldsigni˝cantTKEproduction atthetopsurfaceofacube.Atthislocationthespanwise-alignedheadofvortexCandvortexD in˝gure3.14(a)togetherwiththetwolegsinduceintense Q 4 motionsThisyieldssigni˝cantlocal shearproductionofTKEdrivenbythestrong @ u š @ y valueswithinthethinshearlayer.Thespatial 50 distributionoflocalshearproductionofTKE, P s = u 0 i u 0 j @ u i š @ x j (summedover i ; j = 1 ; 2 ; 3 )is plottedin˝gure3.16(c),showinghighvaluesinthevicinityoftheuppersurfaceofacubeelement. Thisalsoresultsinregionsofintensi˝edturbulencekineticenergyaboveeachroughnesselement, asdepictedby˝gure3.16(d). 3.3Concludingremarks Turbulencestructuresinfully-developedchannel˛owsat Re ˝ = 1000 havebeenanalyzedfor ˛owoverasmoothwall(SM),and˛owsoverarealisticturbine-bladesurface(TB),asandgrain surface(SG),anda k -typecube(CB)surface.Theresultspresentedhereappeartobeconsistent withthefollowingqualitativedescriptionofturbulentboundarylayersoverroughsurfaces.When comparedto˛owoverasmoothsurface,thebulkdragforceovereachroughsurfaceconsidered here(table2.1)issigni˝cantlylarger.Itisbecausetheincreaseinformdragcausedbytheparticular size,shape,anddistributionofroughnesselementsexceedsthecorrespondingreductioninviscous dragonaccountofroughness-induceddisruptionoftheviscoussublayer[13].Themagnitude oftheincreaseinformdragdependsupontheindividualsurfaceroughnesstopographyand,in thisstudy,isapproximatelytwiceaslargeforSGandCBroughnessasitisforTBroughness. Thesecondprincipale˙ectofsurfaceroughnessistomodifynearwallcoherentmotionsand, consequently,tomodifyturbulenceprocessesinthisregion.Thepenetrationextentofroughness e˙ectsismostlyintheroughnesssublayer.Howeveritcangobeyondthisregiondependingon thetypeoftheroughness,duetothelimited š k c ratio.Thesigni˝cantmodi˝cationstonearwall coherentmotions,causedbydi˙erentroughnessgeometriespresentedhere,are: i )thesmooth-wall quasi-streamwisevorticesareretainedoverTBroughnesswithin -scaleundulations,whereasthey arereplacedbyapairof`head-up,headdown'horse-shoevorticesoverSGandCBroughsurfaces; ii )ThelongitudinalextentofnearwallvorticalstructuresoversmoothwallisretainedintheTB roughness,butitisshortenedsigni˝cantlyforSGandCBroughsurfaces,whichisinpartdueto asolid-bodyrotationprocessofthehead-downhorse-shoevorticesonaccountofthestrongshear layerabovearoughnessprotuberance.Thise˙ectisinheritedinvorticalmotionsintheroughness 51 sublayerandbeyond.EvidencesupportingtheexistenceofsuchprocessincaseCBisprovided byanalysesoflinearstochasticestimationandeddy-axis-of-rotation.Theresultssuggestthatthe processconcernsaconsiderablenumberofeddiesattheroughnesscrest,andmayhaveasubstantial in˛uenceonturbulenceintensitiesandshearproductionofturbulencekineticenergy. iii )Thee˙ect ofshortenedstructuresin˛owsoverSGandCBistoreducetheenergylevelatlowwavenumbers (energycontainingeddies),whileincreasingenergylevelathigherwavenumbers,whicharecloser tothedissipativescales.Instationaryturbulent˛ows,theenhanceddissipationrequiresenhanced productionofturbulenceenergy(requiredbyequilibriumcondition),leadingtoincreaseindrag work,onaccountofsurfaceroughness. Thestrongdependenceofthenear-wall˛owonsurfacetextureisalsorevealedbypro˝lesof R u i u j ; n ,whichexhibitsurfaceperiodicity,andvelocityspectrawhichcontrastthedi˙erentcascades ofenergyfromlarge-scaletosmall-scalemotionsforroughandsmoothwall˛ows.JointPDF pro˝lesof ¹ u 0 ;! 0 º showthatroughnessweakenslow-speedstreaksandresultsinroughlyequal possibilitiesforalltypesofmotions,yieldingmoreisotropic˛owinthisregion.Roughnessalso increasestheinclinationanglesoflarge-scalestructuresnearthesurface.At y = d ,conditional eddiesobtainedbylinearstochasticestimationareverysimilarfor˛owoversmoothwallsand TBroughness:theyaretwobifurcatedstreamwisevortices.For˛owoverSGandCBroughness, theyareapairofhorse-shoestructures,oneontopoftheother.Theloweroneisconjecturedto beproducedbyshear-layerroll-upinthewakebehindroughnesselements,andtheupperoneis similartotheU-shapestructureobservedbyTalapatraandKatz[18],whichisproducedbyvortex stretchingdueto˛owchanneling. Intheouterboundarylayer,Townsend'ssimilarityappearstoapplytothesingle-pointstatistics, theaverageinclinationanglesofenergy-containingcoherentmotions,velocityspectra,helicity characteristics(jointPDFof u 0 , ! 0 )andtheaverageshapeofturbulenteddies.Howeverstreamwise two-pointvelocitycorrelationsandassociatedlengthscalesaresurface-texturedependentinthis region,probablyduetothelimited š k c usedherein.Streamwiselengthscales,obtainedbypro˝les of R u i u j ; n andconditionaleddies,forSGandCBroughnessareshorterthanthoseoversmooth 52 wallsandTBroughness. TheresultsdiscussedhereareforthevaluesofturbulenceReynoldsnumberand š k c permitted bytoday'scapabilitiesindirectnumericalsimulations.Amoredetailedunderstandingofthenear wallphysicsanditsconnectiontotheouter-layerregionfor˛owsoverroughwallsathighervalues ofRe ˝ and š k c ,remainsaninterestingchallengeforthefuture.Otherfutureworkmightinclude structuralcomparisonwithvariousroughnesstextureswithmatching k + s (orthedrag). 53 CHAPTER4 DATA-DRIVENPREDICTIONOFTHEEQUIVALENTSAND-GRAINHEIGHTIN ROUGH-WALLTURBULENTFLOWS 4.1Introduction Atsu˚cientlyhighReynoldsnumbersallsurfacesarehydrodynamicallyrough,asisalmost alwaysthecasefor˛owspastthesurfacesofnavalvehicles.Reviewsofroughnesse˙ectsonwall- boundedturbulent˛owsareprovidedbyRaupachetal.[14]andJiménez[13].Themostimportant e˙ectofsurfaceroughnessinengineeringapplicationsisanincreaseinthehydrodynamicdrag[36], whichisduepredominantlytothepressuredraggeneratedbythesmall-scalerecirculationregions associatedwithindividualroughnessprotuberances. Fortheforeseeablefuture,themostpracticalapproachtomakingpredictive˛owcalculations formanyrealisticapplicationsistouseengineeringone-pointclosuresofturbulence,suchas two-equationturbulenceeddy-viscositymodelstotheReynolds-averagedNavier-Stokes(RANS) equations.Existingrough-wallcorrectionstothistypeofclosuretypicallymodeltheincreasein hydrodynamicdragonasinglelengthequivalentsand-grainheight[48] k s physicallyresolvingthesurfaceorchangingthegoverningequations.Inthefullyrough˛ow regime,wherethewallfrictiondependsontheroughnessaloneandisindependentoftheReynolds number, k s wasobservedtoquantifytheincreaseinhydrodynamicdragthroughtheempirical relationwiththeroughnessfunction, U + (de˝nedastheo˙setofthelog-linearvelocitypro˝le ofarough-wall˛owrelativetothatofasmooth-wallone): U + = 1 ln k + s 3 : 5 ; (4.1) where = 0 : 41 isvonKármán'sconstantand + representsnormalizationinwallunits. Auniversallengthscale(e.g. k s inNikuradse'srelation,or intheMoodydiagram[2])that canpredictaccuratelythesurfacedragcoe˚cientisnotknownapriorianddoesnotappeartobe equivalenttoanysinglegeometricallengthscale,suchasanaverageoraroot-mean-square(RMS) 54 ofroughnessheight[36].Itisalsowell-establishedthat k s candependonmanygeometrical parameterssuchasthee˙ectiveslope[37,5]andtheskewnessoftheroughnessheightdistribution [38].ReadersarereferredtoFlack&Schultz[38]andBons[104]forextensivereviewsonthis topic.Empiricalexpressionsfor k s basedonasmallnumberofgeometricalroughnessparameters include,amongothers: k s = c 1 k a vg 2 rms + c 2 rms ; k s = c 1 k a vg c 2 s ; and k s = c 1 k rms 1 + S k c 2 ; (4.2) proposedbyBonsetal.[105],vanRijetal.[106]andFlack&Schultz[38]respectively.Here k a vg istheaverageheight, isthelocalstreamwiseslopeangleand s = S š S f S f š S s 1 : 6 (where S , S f , S s are,respectively,theplatformarea,thetotalfrontalarea,andthetotalwindwardwettedarea oftheroughness)while k rms and S k arethermsandskewnessoftheroughnessheight˛uctuations, and c 1 and c 2 areconstants. Thehydrodynamiclengthscale k s appearstobecorrelatedwithdi˙erentsetsofgeometrical parametersforeachtypeofroughsurfaceandnouniversalcorrelationcurrentlyexistsfor˛owover surfacesofarbitraryroughness.Forexample,forsyntheticroughnesscomprisingcloselypacked pyramids[107]andrandomsinusoidalwaves[37],ithasbeenshownthat k s scalesonthee˙ective slopewhenthesurfaceslopeisgentle(i.e.withinthe`waviness'regime),whereastheskewness andrmsheight,but not slopemagnitude,becomeimportantwhentheslopeissteeper(i.e.within the`roughness'regime).Theboundarybetweenthesetworegimeshasbeenshowntobesurface dependent[5]. Somemorerecentstudiesof k s correlationsaresummarizedbelow.Thakkaretal.[108]carried outDNSoftransitionally-roughturbulent˛owsfordi˙erentirregularroughnesstopographies.They foundthattheroughnessfunctionisin˛uencedbysolidity,skewness,thestreamwisecorrelation lengthscaleandthermsofroughnessheight.Flacketal.[109]performedseveralexperiments tosystematicallyinvestigatethee˙ectsoftheskewnessandamplitudeofroughnessheightonthe skinfriction.Theyfoundthatthermsandskewnessofroughnessheight˛uctuationsareimportant scalingparametersforpredictionofroughnessfunction;however,thesurfaceswithpositive, negativeandzeroskewnessvaluesneededdi˙erentcorrelations.Also,Chanetal.[110]simulated 55 turbulentpipe˛owsoversinusoidalroughnessgeometriesandcon˝rmedstrongdependenceof roughnessfunctionontheaverageheightandstreamwisee˙ectiveslope. Inpreviousstudies,thesmallnumberofroughnessparametersusedtodevise k s correlations tendedtolimittheirapplicationtoanarrowrangeofsurfaceroughnesses.Sinceitappearsthat manygeometricalparameters,suchasporosity,momentsofroughnessheight(e.g.rms,skewness andkurtusis),e˙ectiveslope,andsurfaceinclinationanglemighta˙ect k s ,itisusefultoemploya datascienceapproachsuitedtomodelinglargemulti-variate/multi-outputsystems. Speci˝cally,weuseMachineLearning(ML)toexplore k s -predictionapproachesthatdepend onalargesetofsurface-topographicalparameters,withtheexpectationthattheresultingmodels maybeappliedaccuratelytoawiderrangeofsurfaces.Sincethepredictionof k s fromsurface topographyisessentiallyalabeledregressionproblem,supervisedMLoperationswereperformed usingDeepNeuralNetworks(DNN)andGaussianProcessRegressions(GPR).Bothmethodsare explainedthoroughlyinsection4.3.ReadersarereferredtothemonographbyRasmussen& Williams[111]andthereviewprovidedbyLeCunetal.[112]fordetaileddescriptionsofthese methods. Aninitialensembleof60setsofdataon k s asafunctionoftopographicaldirect numericalsimulation(DNS)resultsand15experimentalasconsidered.Allexperimental datasetsarefullyrough,andoftheDNSdata,30areconsideredfully-rough˛ows;allfully-rough caseswereusedforMLtrainingandtesting.Tothebestofourknowledge,thisensembleof roughnessgeometriesisthemostextensiveusedfordevelopinga k s -predictionmethod. Inthischapter,we˝rstpresentsimulationparametersanddi˙erentroughnesstopographies andthendiscussthepost-processedDNSresultsusedtocalculate k s foreachsurface.Finally,we describetheMLmodels,theirpredictionsof k s andtheiruncertainty. 4.2Problemformulation Themethodologyofthischapterissimilartothatinchapter2.Pleaserefertosection2.2asfor de˝nitionofdi˙erentparametersandmathematicalsymbols. 56 4.2.1Surfaceroughness In˝gure4.1,surfaceplotsofthe45roughnessgeometriesusedinthesesimulationsaredisplayed; theirstatisticalpropertiesaregivenintable4.1.Eachcasenamein˝gure4.1andtable4.1begins withtheletterCorE,whichdenoteswhetherthedataiscomputationalorexperimental,followed byanidentifyingindexforthatparticularsurface.Forcomputationalcases,thisindexisfollowed by:acharacteristiclengthscale(asapercentageof )usedforroughnesssynthesis;anidenti˝er ofwhetherthesurfaceroughnessisregular(reg)orrandom(rnd);and˝nallyanidenti˝erforone additionalsurfacefeatureanditspositioninaseriesofsurfaceswithdi˙erentsizesofthatfeature. Thesefeatureswere:thestreamwiseinclinationangle I x insurfacesC01toC12;theporosity P o insurfacesC13toC24;andthestreamwisee˙ectiveslope E x insurfacesC25toC30.Forthe experimentaldatatwoindiceswereassignedtoeachsurface.The˝rstdenotestheyearinwhich thedatawerepublishedandthesecondisthesurfacedesignationinthatpublication.Thussurfaces withindex16arefromFlacketal.[113],thosewithindex18arefromBarrosetal.[114],and thosewithindex19arefromFlacketal.[109].Notethattheseexperimentaldatawereobtained fromfully-developedchannel˛ows,wherethedragwasmeasuredthroughthepressuredropalong thechannel.Thustheirresultsareexpectedtobemoreaccuratethanthoseofboundarylayer studieswherethedragisusuallyinferred. SurfacesC01throughC24werecreatedusingellipsoidalelements[54]ofdi˙erentsize,aspect ratioandinclination.For regular roughness,eachelementhadthesameorientationandsemi-axis lengths, ¹ 1 ; 2 ; 3 º = ¹ 1 : 0 ; 0 : 7 ; 0 : 5 º k c ,where k c isthepeak-to-troughheight(alsocalledthecrest height).For random roughness,theelementshadrandomorientationsandsemi-axislengths(with uniformdistributionsoftherandomvariables).Theaverageorientationandsemi-axislengthsfor random roughnesswerethesameasthecorresponding regular surface.SurfacesC25throughC30 comprisedsinusoidalwavesinthe x direction,ofthesamemagnitudebutdi˙erentwavelengths, togeneratedi˙erentvaluesofe˙ectiveslope E x .Thewavelengthswere 3 š 4 , 3 š 8 and š 6 . SurfacesC31andC37comprisedtherandomsand-grainroughnessofScotti,whichwereproduced byrandomlyorientedellipsoidalelementswith˝xedsemi-axesof ¹ 1 : 0 ; 0 : 7 ; 0 : 5 º k c .SurfacesC32 57 Figure4.1:Roughnessgeometrieseachplotisasectionofsize 0 : 5 inthe x - z plane.Cases C43toC45arefromsimulationswithregulardomainsizes[5,6]. 58 Casename k avg k c k t k rms R a I x I z P o E x E z S k K u k s C01,r4,reg,inc1 0.026 0.043 0.043 0.013 0.011 -0.801 -0.089 0.535 0.584 0.510 -0.544 2.177 C02,r4,reg,inc20.0300.0590.0590.0210.0190.0120.0320.6091.0290.562-0.2651.597 C03,r4,reg,inc3 0.025 0.043 0.043 0.013 0.011 0.821 -0.078 0.537 0.600 0.485 -0.459 2.052 C04,r6,reg,inc10.0320.0640.0640.0220.019-0.9780.0160.5970.5950.590-0.1671.6010.064 C05,r6,reg,inc2 0.038 0.088 0.088 0.033 0.030 0.025 0.064 0.654 0.916 0.643 0.109 1.436 0.124 C06,r6,reg,inc30.0310.0640.0640.0220.0190.9550.1210.5990.5880.558-0.0871.5900.059 C07,r4,rnd,inc1 0.025 0.086 0.084 0.022 0.019 -0.860 0.033 0.774 0.511 0.559 0.560 2.244 0.136 C08,r4,rnd,inc20.0270.1160.1150.0300.025-0.0070.0480.8190.8610.6040.8702.6270.322 C09,r4,rnd,inc3 0.025 0.083 0.081 0.021 0.018 0.829 0.002 0.753 0.517 0.482 0.514 2.292 0.131 C10,r6,rnd,inc10.0260.1250.1200.0300.025-0.957-0.0190.8350.4980.5780.9672.8740.269 C11,r6,rnd,inc2 0.033 0.172 0.169 0.044 0.037 0.076 0.138 0.842 0.758 0.543 1.150 3.176 0.536 C12,r6,rnd,inc30.0320.1270.1210.0320.0270.9230.0320.7840.5080.4710.7582.6420.272 C13,r4,reg,por1 0.038 0.059 0.059 0.018 0.015 0.024 0.067 0.498 1.043 0.523 -0.820 2.508 C14,r4,reg,por20.0180.0590.0590.0220.0200.0210.0380.7760.6130.4560.7081.8400.141 C15,r4,reg,por3 0.010 0.059 0.059 0.019 0.014 0.022 0.063 0.877 0.334 0.253 1.646 4.094 0.157 C16,r6,reg,por10.0510.0890.0890.0300.0260.0410.1490.5291.1370.534-0.5381.8730.077 C17,r6,reg,por2 0.022 0.089 0.089 0.031 0.027 0.041 0.080 0.801 0.537 0.403 0.982 2.308 0.260 C18,r6,reg,por30.0130.0890.0890.0260.0200.0570.1260.8860.3070.2301.8494.8390.247 C19,r4,rnd,por1 0.027 0.112 0.108 0.021 0.017 0.025 -0.107 0.806 0.487 0.486 0.732 3.422 0.158 C20,r4,rnd,por20.0130.0950.0870.0170.0140.032-0.6460.8960.3110.3231.3434.1260.106 C21,r4,rnd,por3 0.009 0.098 0.094 0.016 0.012 0.321 -0.741 0.929 0.219 0.233 2.168 7.728 0.103 C22,r6,rnd,por10.0350.1390.1390.0290.024-0.070-0.2450.7910.4560.4990.5912.8300.277 C23,r6,rnd,por2 0.017 0.123 0.111 0.025 0.020 -0.672 -0.841 0.885 0.305 0.325 1.467 4.347 0.175 C24,r6,rnd,por30.0140.1520.1450.0270.0190.189-0.0560.9260.2540.2572.3718.7400.260 C25,r4,reg,ES1 0.020 0.040 0.040 0.014 0.013 0.046 0.006 0.510 0.106 0.009 -0.032 1.503 C26,r4,reg,ES20.0210.0400.0400.0140.0130.039-0.0010.5100.2120.020-0.0711.5050.065 C27,r4,reg,ES3 0.023 0.040 0.040 0.014 0.012 0.006 -0.023 0.510 0.609 0.032 -0.214 1.544 C28,r6,reg,ES10.0300.0590.0590.0210.0190.0440.0180.5040.1580.015-0.0311.4990.071 C29,r6,reg,ES2 0.031 0.059 0.059 0.021 0.019 0.028 -0.069 0.504 0.316 0.022 -0.071 1.503 0.112 C30,r6,reg,ES30.0340.0590.0590.0200.0180.015-0.0690.5050.9170.048-0.2031.5430.064 C31,r4,rnd,SGR 0.025 0.059 0.059 0.011 0.009 0.104 -0.039 0.648 0.370 0.398 0.378 2.784 0.049 C32,r4,rnd,RND10.0400.0750.0720.0130.0100.1170.1080.4790.0680.169-0.0692.991 C33,r4,rnd,RND2 0.041 0.088 0.084 0.013 0.011 0.109 0.078 0.553 0.117 0.308 0.004 2.763 C34,r4,rnd,RND30.0420.0800.0710.0100.0080.0700.0510.5080.1750.458-0.0023.031 C35,r4,rnd,RND4 0.043 0.077 0.066 0.008 0.007 0.039 0.042 0.488 0.218 0.558 0.013 2.941 C36,r4,rnd,RND50.0450.0840.0670.0090.0070.0350.0370.5350.3780.8410.0753.018 C37,r6,rnd,SGR 0.037 0.088 0.088 0.018 0.015 0.312 0.180 0.640 0.428 0.463 0.323 2.686 0.109 C38,r6,rnd,RND10.0600.1060.0910.0160.0120.0450.0280.4440.0770.183-0.2203.258 C39,r6,rnd,RND2 0.061 0.098 0.095 0.012 0.009 0.111 0.057 0.400 0.108 0.285 -0.020 3.267 C40,r6,rnd,RND30.0640.1210.1120.0160.0130.0610.0220.5120.2800.7600.0372.9770.050 C41,r6,rnd,RND4 0.065 0.130 0.130 0.015 0.012 0.045 0.037 0.546 0.374 0.989 0.028 3.036 C42,r6,rnd,RND50.0680.1180.1160.0130.0100.0370.0250.5030.5471.2040.0522.933 C43,SG 0.036 0.089 0.087 0.017 0.014 0.288 0.156 0.649 0.425 0.441 0.476 2.970 0.093 C44,TB0.0550.1250.0880.0180.0140.007-0.0060.5690.0970.0810.2003.4930.024 C45,CB 0.010 0.070 0.070 0.023 0.016 0.420 0.508 0.878 0.249 0.247 2.101 5.569 0.150 C46,r4,rnd,por3,FS0.0090.0980.0940.0160.0120.321-0.7150.9290.2190.2342.1687.7280.104 E01,16,2 0.138 0.261 0.254 0.020 0.016 -0.005 0.011 0.472 0.720 0.835 -0.711 3.843 0.052 E02,16,30.1430.2520.2520.0210.016-0.0210.0100.4320.7400.868-0.3383.1590.050 E03,16,7 0.133 0.365 0.254 0.019 0.014 -0.038 0.000 0.638 0.618 0.705 -1.169 5.292 0.058 E04,16,80.1260.2980.2270.0170.013-0.0340.0090.5790.5870.682-1.4455.4210.056 E05,16,9 0.112 0.308 0.167 0.018 0.014 -0.031 0.015 0.637 0.636 0.753 -0.738 3.714 0.043 E06,16,150.0810.1910.1910.0130.010-0.0270.0030.5780.6210.713-0.6873.8540.035 E07,18,1 0.121 0.241 0.227 0.026 0.021 -0.013 -0.183 0.500 0.181 0.188 0.107 2.941 0.053 E08,18,20.1430.2760.2550.0320.025-0.0190.1940.4830.1620.1640.0932.9670.034 E09,19,1 0.204 0.398 0.344 0.046 0.036 0.042 -0.096 0.487 0.227 0.230 -0.080 2.989 0.065 E10,19,20.3890.7630.6890.0880.0700.0460.0020.4920.4470.452-0.0652.9250.200 E11,19,3 0.477 0.730 0.679 0.088 0.070 -0.029 -0.245 0.348 0.434 0.432 -0.660 3.274 0.160 E12,19,40.4590.7510.7100.0890.071-0.0520.0360.3910.4550.459-0.3513.0410.180 E13,19,5 0.292 0.732 0.650 0.090 0.072 -0.058 -0.004 0.602 0.445 0.452 0.346 3.051 0.245 E14,19,60.2020.7110.6040.0870.0690.004-0.0100.7160.3910.4000.8123.5590.435 E15,19,7 0.522 0.967 0.894 0.114 0.092 -0.050 -0.235 0.462 0.557 0.562 -0.066 2.794 0.230 Table4.1:Statisticalparametersofroughnesstopographyandtheequivalentsand-grainheight k s foreachroughnessgeometry. R a , k a vg , k c , k t , k rms and k s valuesfromDNSarenormalized bythechannelhalfheight ,whilecorrespondingexperimentalvaluesaregivenin mm . k s isnot listedforcasesthoughttobetransitionallyrough. 59 throughC36andC38throughC42weregeneratedasthelow-order(the˝rst5,10,20,30and 50)modesofFouriertransformsofwhitenoiseinthestreamwiseandspanwisedirections;they thereforedescriberandomsurfaceswithlarge-tosmall-wavelengthroughness.CasesC43,C44 andC45areDNSresultsfromfull-spanchannelcomputationsof˛owoversurfacesof:random sand-grainroughness;theroughnessfoundonaturbineblade[57];andarraysofcubes[from thestudyof6]respectively.CaseC46isafull-spanDNSofcaseC21,generatedtovalidatethe minimal-channelapproachoftheprecedingcases.Abaselinesmooth-wall˛owwasalsosimulated usingafull-spanchannel[57]. Thegeometricparametersreportedforeachsurfaceintable4.1are:roughnesspeak-to-trough height(alsotermedcrestheight) k c (i.e.distancebetweenthehighestandthelowestsurfacepoints); meanpeak-to-troughheight k t (i.e.theaverageofpeak-to-troughheightsobtainedfromsurface tilesofsize ,similartoForooghietal.[115]);meanroughnessheight k a vg ;˝rst-ordermoment ofheight˛uctuations R a ;root-mean-square k rms ,skewness S k andkurtosis K u oftheroughness height˛uctuations;surfaceporosity P o ;e˙ectiveslopeinthe x i direction E x i ;andinclinationangle (inradians)inthe x i direction I x i ,togetherwiththehydrodynamiclengthscale k s deducedfrom themeanvelocity˝eldusingequation(4.1). Thesegeometricalparametersarede˝nedas: k a vg = 1 A t x ; z kdA ; (4.3) R a = 1 A t x ; z j k k a vg j dA ; (4.4) k rms = s 1 A t x ; z ¹ k k a vg º 2 dA ; (4.5) S k = 1 A t x ; z ¹ k k a vg º 3 dA ˚ k 3 rms ; (4.6) K u = 1 A t x ; z ¹ k k a vg º 4 dA ˚ k 4 rms ; (4.7) 60 E x = 1 A t x ; z @ k @ x dA ; (4.8) E z = 1 A t x ; z @ k @ z dA ; (4.9) P o = 1 A t k c k c 0 A f d y ; (4.10) I x = tan 1 ˆ 1 2 S k @ k @ x ˙ ; (4.11) I z = tan 1 ˆ 1 2 S k @ k @ z ˙ ; (4.12) where k ¹ x ; z º istheroughnessheightdistributionand A f ¹ y º and A t arethe˛uidandtotalplanar areas. S k ¹ @ k š @ x i º istheskewnessof @ k š @ x i distribution.Intable4.1, k a vg , k c , k rms and k s arethennormalizedbythe˝rst-ordermomentofheight˛uctuations R a andwereincorporated intheMLalgorithmsinthisform.Allsurfacesconsideredwereintheranges k c š 0 : 17 and R a š 0 : 04 . 4.2.2Simulationparameters Directnumericalsimulationwasusedtocalculatethevelocityandpressure˝eldsinturbulent open-channel˛owsover45di˙erentroughsurfacesandonesmoothone,ataconstantfrictional ReynoldsnumberRe ˝ = u ˝ š = 1000 ,where u ˝ isthefrictionvelocityand isthechannel half-height.Inthesesimulations,thedomainsizeswere ¹ L x ; L y ; L z º = ¹ 3 ; 1 ; 1 º .Theoriginofthe y axiswastheelevationofthelowesttroughforeachroughsurface.Thenumberofgridpoints was ¹ n x ; n y ; n z º = ¹ 400 ; 300 ; 160 º .Auniformmeshwasusedinthe x and z directions,yielding gridsizesof x + = 7 : 5 and z + = 6 : 3 ,where + denotesnormalizationinwallunits.Forallcases, themeshwasstretchedinthe y directionwithahyperbolictangentfunction,withthethirdgrid pointfromtheoriginat y + < 1 .Fortherough-wallcases,attheroughnesscrest, y š k c 0 : 017 , 61 withthisratiotakingitshighestvalueforCaseC11.Themaximumgridsizewas y + max = 9 : 5 at thechannelcenterline,wheretheKolmogorovlengthscale + ˇ 6 .Moin&Mahesh[116]have proposedthatonerequirementforobtainingreliable˝rst-andsecond-order˛owstatisticsisthat thegridresolutionbe˝neenoughtocaptureaccuratelymostofthedissipation,whileMoser& Moin[60]notedthatmostofthedissipationincurvedchannel˛owoccursatscalesgreaterthan 15 (basedonaveragedissipation).ItfollowsthatforDNScomputationsofthesekindsof˛ow statisticsinchannelandboundary-layer˛ows, x š and z š aretypicallychosenbetween7to 15,and4to8respectively(see,forexample,[117],[118]and[44]).Thegridsizesinthisstudy werechosenaccordinglyandwere: x š < 7 : 5 , y š < 4 : 0 ,and z š < 6 : 5 . Periodicboundaryconditionswereimposedinthestreamwiseandspanwisedirections,with no-slipandsymmetryboundaryconditionsatthebottomandtopboundariesrespectively.After eachsimulationhadreachedstatisticalstationarity,datawerecollectedforensembleaveragingover 10large-eddyturn-overtimes( š u ˝ ).Inthesesimulations,thetimestep ˝ + 0 : 04 andsowas signi˝cantlysmallerthanthelargestacceptableoneof ˝ + ˇ 0 : 2 recommendedbyChoi&Moin [119]forDNS. ThesurfaceTaylormicro-scales T ; x and T ; z ,inthe x and z directions,wereusedtoevaluate theadequacyofthegridresolutionforresolvingdetailsof˛owintheroughnesssublayer,following Yuan&Piomelli[52].Thesegeometricmicro-scaleswereobtainedby˝ttingaparabolatothetwo- pointautocorrelationofthesurfaceheight˛uctuationintherespectivedirection.Theyrepresent thesizeofanequivalent`roughnesselement'inthecontextofrandommultiscaleroughness.The streamwiseandspanwisevaluesof T ,rescaledby u ˝ š as + T ,andtherespectivegridsizesare givenintable4.2(partI).Foreachcase, + T ; x i isoforder 10 to 10 2 ,indicatingthattheaverage sizeoftheroughnesselementislargeinviscousunits.Onaverage,roughnesselementswerewell resolvedbythegrid,withtypically4to12gridpointsper T ; x i microscaleineachdirection.For referencepurposes,Yuan&Piomelli[5]reportedaresolutionof T ; x š x ˇ 4 intheirlarge-eddy simulationsofchannel˛owoversurfaceswithsand-grainroughness.Thecasesintable4.2for which T wasnotwellresolvedinatleastonedirection( T ; x š x < 3 or T ; z š z < 3 )mayalso 62 nothavebeenfully-rough˛ows(asdiscussedinthefollowingsection),andsowerenotincluded intheensembleof˛owsforMLtrainingandtesting. Inrough-wall˛ows,thepressuredragiscausedprimarilybythelocal˛owstructuresand separationinthevicinityofindividualroughnessprotuberances,whicharepredominatelynear- wallphenomena.Tocarryoutthe46separateDNSsimulationsfordetermining k s e˚ciently, withsu˚cientnear-wallresolution,asmall-spanchannelsimulationapproachwasemployed.The conceptofminimal-spansimulationwasintroducedbyJiménezetal.[120].Chungetal.[121] andMacDonaldetal.[122]carriedoutanalysesoftheperformanceofDNSoversmallspanwise domainsforfullandopenchannel˛owsonroughandsmoothwallsandshowedthatminimal-span simulationscapturedtheessentialnear-walldynamicsandyieldedaccuratecomputationsofwall friction,andofmeanvelocitiesandReynoldsstressesasfarfromthewallas y ˇ 0 : 3 ,whenthe followingconstraintsweremet: L x max 1000 ; 3 L z ; r ; x ; (4.13a) L y k c š 0 : 15 ; (4.13b) L z max 100 ; k c š 0 : 4 ; r ; z ; (4.13c) where = š u ˝ and r ; x i isthecharacteristicroughnesswavelengthinthe x i direction.Alterna- tively,thesurfaceTaylormicroscalemaybeusedasthelengthscaleinthisconstraint.Conditions (4.13 a,c )weresatis˝edbychoosingdomainsizes L + x and L + z of3000and1000respectively,while condition(4.13 b )wasmetforallcasesexceptC11,whichfellbelowthe L y k c š 0 : 15 constraint byabout10%.C11isacasewithrandomgeometry;protuberancesbeyond 0 : 15 existbutarerare. Thecriteriaof(4.13)weredevelopedoriginallyforsimulationsof˛owoversurfaceswith uniformlydistributedroughnesselements.Inthisstudy,therandomroughnessgeometriesused requireanadditionalcriteriononthesu˚ciencyofthedomainsize:thearea L x L z shouldbelarge enoughtoachievestatisticalconvergenceofsurfaceparameters,suchas k rms and E x i ,andofthe ˛owparameter k s .Tochecktheadequacyofthechosendomainsize,anadditionalsimulation wascarriedoutofCaseC21,thesurfacecomprisingthelargestdominantspatialwavelength(and 63 PartIPartII Casename + T ; x T ; x š x + T ; z T ; z š zd š b k + s C01,r4,reg,inc119.72.621.13.40.03219.4 C02,r4,reg,inc220.42.733.15.30.04649.7 C03,r4,reg,inc319.82.622.93.70.03331.0 C04,r6,reg,inc127.73.728.44.50.03864.4 C05,r6,reg,inc231.64.239.16.20.057124.4 C06,r6,reg,inc329.94.030.04.80.04558.9 C07,r4,rnd,inc133.84.526.74.30.036136.2 C08,r4,rnd,inc226.13.532.75.20.052322.3 C09,r4,rnd,inc335.54.730.14.80.039131.1 C10,r6,rnd,inc138.25.129.74.80.042268.9 C11,r6,rnd,inc238.15.147.07.50.070536.4 C12,r6,rnd,inc347.96.440.26.40.053271.7 C13,r4,reg,por117.82.432.75.20.04741.4 C14,r4,reg,por227.53.734.25.50.032140.6 C15,r4,reg,por331.54.239.46.30.028157.1 C16,r6,reg,por125.63.446.17.40.06676.7 C17,r6,reg,por240.15.347.87.60.044259.8 C18,r6,reg,por344.45.954.88.80.039246.5 C19,r4,rnd,por132.74.431.15.00.042158.2 C20,r4,rnd,por235.64.731.35.00.026105.7 C21,r4,rnd,por337.45.034.25.50.027102.7 C22,r6,rnd,por144.65.935.35.60.053276.8 C23,r6,rnd,por247.16.339.76.40.038175.1 C24,r6,rnd,por347.16.344.47.10.045260.3 C25,r4,reg,ES189.011.90.02425.6 C26,r4,reg,ES266.58.90.02665.3 C27,r4,reg,ES327.13.60.03545.5 C28,r6,reg,ES190.612.10.03371.2 C29,r6,reg,ES266.88.90.040112.0 C30,r6,reg,ES327.23.60.05464.0 C31,r4,rnd,SGR27.83.725.04.00.03248.7 C32,r4,rnd,RND1131.217.554.18.70.0418.4 C33,r4,rnd,RND296.312.842.16.70.04317.6 C34,r4,rnd,RND356.47.522.43.60.04522.5 C35,r4,rnd,RND439.55.315.82.50.04618.3 C36,r4,rnd,RND525.13.311.41.80.05123.4 C37,r6,rnd,SGR36.54.931.95.10.046108.8 C38,r6,rnd,RND188.511.872.611.60.06012.0 C39,r6,rnd,RND293.812.535.75.70.06217.1 C40,r6,rnd,RND357.07.622.83.60.07050.4 C41,r6,rnd,RND440.55.415.62.50.07348.7 C42,r6,rnd,RND524.53.311.31.80.07643.8 C43,SG35.26.033.55.70.04493.0 C44,TB132.110.4168.513.20.05824.1 C45,CB25.74.525.54.40.039149.9 C46,r4,rnd,por3,FS37.65.034.65.50.027104.2 Table4.2:PartI:StreamwiseandspanwisevaluesofthesurfaceTaylormicro-scale T .PartII: Flow-relatedparametersobtainedfromDNS.The˛owisassumedfullyroughif b k + s & 50 ,in whichcase k s isequalto b k s . 64 consequentlythemostlimitedsamplingofrandomgeometricalcomponentswiththiswavelength) andalong-tailedheight-˛uctuationpdfwithakurtosisofaround8.Inthisvalidationsimulation, denotedCaseC46,thedomainsizesweredoubledin x and z ,byduplicatingC21inthesedirections. Thedouble-averagedvelocitypro˝les U + = h u i + ¹ y + º forCasesC21andC46areinaverygood agreementoverthelog-linearregion,asshownin˝gure4.2.Eachsurfacestatisticdi˙ersbyno morethan3%,withthegreatestdiscrepancyfoundin I z ,whiletheequivalentsandgrainroughness height k s isalmostequalinthetwocases.Thechosendomainsizewasthereforeconsidered su˚cientforaccuracyandconvergenceofstatisticsdescribing˛owovertherandomroughness geometriesofthisstudy. 4.3Results 4.3.1Post-processedresults In˝gure4.2,thestreamwisedouble-averagedvelocitypro˝lescomputedinthesesimulationsare shown.Thepro˝lesinthelogarithmicregionaredescribedforthesmooth-wallcaseandthe fully-roughrough-wallcasesas u + = 1 ln ¹ y + º + 5 : 0 ; and(4.14a) u + = 1 ln y d k s + 8 : 5 (4.14b) respectively,where d isthezero-planedisplacement,obtainedasthelocationofthecentroidofthe wall-normalpro˝leoftheaverageddragforce[62].Theshiftinthe y coordinateby d accountsfor the˛owblockagebysurfaceroughnesselements,andthevaluesof d aregivenintable4.2(part II). Todeterminewhetheraparticular˛owwaswithinthefullyroughregime,equation(4.14b) wasappliedtothecomputedlogarithmicvelocitypro˝letoyieldatestvalueof k s ,denotedas b k s intable4.2(partII).With b k s determinedforallcases,thosewith b k + s greaterthanathreshold valueof50weredeemedtobeinthefullyroughregime(30surfaces),inwhichcase k s wassetto equal b k s .Thosebelowthethresholdwerepossiblytransitionallyrough(15surfaces)andsowere 65 Figure4.2:Pro˝lesofstreamwisedouble-averagedvelocityplottedagainsta zero-plane-displacementshiftedlogarithmic y abscissa.Thedashedlinesare u + = y + and u + = 2 : 5ln ¹ y d º + + 5 : 0 .Thereddot-dashlineinplotC46isthatofC21. 66 Figure4.3:Pairplotsofgeometricalparametersand k s ,with k s plotsinthebottomrowandthe ˝rstcolumn,DNSdata( blue ),experimentaldata( red ). 67 notincludedinMLpredictionsinthisstudy.Thethresholdvalueof k + s lowerendofthe fullyroughbeenobservedtovarysigni˝cantlyfordi˙erenttypesofroughnessand istypicallybetween20and80.Forexample,thethresholdvaluesforsurfacesC43andC44are roughly80and20[5],and50forsurfaceC45[59]. Thethresholdvalueof k + s whichsigni˝esthebeginningofthefullyroughregimewasnot determinedmorepreciselybecauseofthecostofcarryingout,foreachsurface,simulationsat successivelyhighervaluesof k + s until k s š R a becameinvariantwiththeReynoldsnumber.Inthe GPRprediction,potentialuncertaintiesin k s whichmightarisethroughtreatingall˛owswith k + s > 50 asfullyrough,andothersourcesofpossibleerror,werecompensatedforbyincorporating anassumed10%noiselevelinthelearningstageofthepredictionof k s ,asdiscussedinsection 4.3.2.Thevaluesof k + s = 50 asthethresholdforfullyrough˛owsandtheassumednoiselevel werechosenaspartofatrade-o˙tomaximizethenumberofusabledata,toavoidover˝tting,while acknowledgingpossibleuncertaintiesinthemodelingdata. In˝gure4.3,pairplotsofthedi˙erenttopographicroughnessparametersareshownasscatter plots(lowerleft),jointpdfs(upperright),anddistributionpdfs(diagonal).Pairscatterplotsfor thetrue(DNSandexperimental)valueof k s andotherroughnessparametersarealongthebottom rowofthis˝gure.Itcanbeseenthat,fortheroughnesscaseschosen,thereissomecorrelation betweenkurtosisandrmsroughness(column1,row6),kurtosisandskewness(column5,row6), andskewnessandporosity(column2,row5).Therelationshipbetweenothersappearstobemore random.Fromthegraphsinthebottomrow,itcanbeseenthat k s š R a scalesonporositytosome power,albeitwithsomescatter(column2,row7).Italsoappearsthat k s š R a mightdecreasewith skewnessforsurfaceswith S k < 0 andincreasewithskewnessincaseswith S k > 0 (column5,row 7).Surfaceswithpositiveskewnessyieldedhighervaluesof k s comparedtothosewithnegative skewness,consistentwiththeobservationofFlacketal.[109].Beyondtheseobservations,there doesnotappeartobeaclearlinearcorrelationbetween k s andanyindividualroughnessparameter, whichmakesthesearchforafunctionaldependenceof k s ontheseparametersaproblemwellsuited toML.Themeasuresofinclination, I x and I z ,showednoclearcorrelationwithothervariablesor 68 with k s š Ra . 4.3.2MLpredictionsoftheequivalentsand-grainheight TheMLtechniquesofDNNandGPRwereemployedtopredict k s fromthedatasetsdescribed intheprevioussection.Theobjectivesofthisexerciseweretogenerateandcollectdata,and makequalitativecomparisonsbetweenMLpredictionsandthosefromconventionalcorrelations, ratherthanevaluatingandcomparingtheperformanceofvariousMLproceduresperse.DNNand GPRapproacheswereusedbecauseourexperiencewasthattheypredicted k s withhighaccuracy, notwithstandingtheirsimplicity.OtherapproachessuchastheSupportVectorMachine(SVM) techniquewereconsideredinitially,buttheirpreliminarypredictionswerenotasaccurateasthose foundusingDNNandGPRapproaches. ThemaincharacteristicsofDNNandGPRmethodsaredescribedbelow: ‹ Theinputsforbothtechniqueswere17roughnessgeometricalparameters,8ofwhichwere theprimaryvariables k rms š R a , I x , j I z j , P o , E x , E z , S k and K u (de˝nedinequations4.3to 4.12).Theother9wereproductsoftheprimaryvariables,whichwereaddedtoimprovethe e˚ciencyofeachlearningstage.Theywere p 1 = E x Ez , p 2 = E x S k , p 3 = E x K u , p 4 = E z S k , p 5 = E z K u , p 6 = S k K u , p 7 = E 2 x , p 8 = E 2 z and p 9 = S 2 k .Theseparticularproductswere chosenbecauseoftheirperceivedimportanceforcertaintypesofroughness. ‹ Thedatabaseconsistedof45di˙erentsets:30DNSofturbulentchannel˛owsoverdi˙erent surfacesatRe ˝ = 1000 ,and15experimentaldatasetsathigherReynoldsnumbers,withall datasetsinthefully-roughturbulent-˛owregime. ‹ TheDNNarchitecturewasa MultiLayerPerceptron ,withthreehiddenlayers(with18,7and 7neuronsrespectively).Theactivationfunctionsatallnodeswereofthe Recti˝edLinearUnit kind,andkernelregularizationwasusedtoavoidover˝tting.Thenetworkhad521trainable weightsintotal.Thepresetparameterstothealgorithmwereoptimizedbasedonavailable data,througha hyper-parametertuning process.Speci˝cally,270con˝gurationswere˝rst 69 generatedwithdi˙erentlengths(representingthenumberoflayers)andwidths(representing thenumberofneurons).Foreachcon˝guration,theDNNcompilerwasperformed1000 timeswithrandomselectionsoftraining(70%oftotal)andtesting(30%oftotal)datasetsto identifythebestperformanceofthecon˝guration.Thecon˝gurationthatyieldedthebest resultswasconsideredastheoptimalone,theresultsofwhicharepresentedhere.Thecost ofdata˝ttingforoneiteration(outof1000)foreachDNNcon˝gurationwasaboutone second.Intotal,ittookabout75hourstoobtaintheoptimalDNNnetwork.Thisarchitecture wasfoundtoprovidesuitableaccuracyinmodelingwithoutover˝tting,forthisparticular multivariatelabeledregressionproblem. ‹ TheGPRprocedureused RationalQuadratic kernelstorepresent k s asasuperpositionof scaledGaussianfunctionsoftheindependentvariablesofthemodelingproblem.Similarto theDNNmethod,thetrainingandtestingdatawerechosenrandomly,withrespectiveratios of70%and30%ofthetotaldatapoints.Thepresetparameters(e.g.kerneltype,number ofiterations,etc.)werealsotunedwiththeavailabledatabyrunningtheGPRcompilerfor about8000times.Ittookabout35hourstoobtaintheoptimal˝t.TheGPRmethodhasthe capabilityofincorporatinguncertaintyornoiseinthedeterminationofmodelparameters inthelearningstages.Suchnoisemightarisethrough:numericalanddiscretizationerrors; uncertaintyintheformandmodelcoe˚cientsofequation(4.1);theapplicabilityand˝tting rangeofequation(4.1)(whichwasdeducedfromhighReynoldsnumberexperiments)to simulationsatmuchlowerReynoldsnumbers;andthepossibilitythatsomeofthetraining datamayhavebeenfromsimulationsinwhichthe˛owwasnotquitefullyrough.Anoise levelof10%in k s š R a valueswaschosenasanupperestimateofthelikelyuncertaintyfrom thesesources.Noiselevelsof5%and15%werealsotested,butlittlesensitivityofthe k s predictionwasfoundtotheassumednoiselevelwithinthetestedrange. Thevaluesof k s predictedfromthesurfacetopographyparameters,henceforthcalled k sp ,are comparedtotheactual k s valuesin˝gure4.4,fortheDNNandGPRmethodsrespectively.Scatter 70 Figure4.4:(a,d)Scatterplotoftrue k s andpredicted k s ,(b,e)scatterplotoftrue k s andrelative error,(c,f)pdfsofrelativeerrorfor(a-c)DNNand(d-f)GPRpredictions,withDNSdata( blue ), experimentaldata( red ). plotsof k sp andthetruevalueof k s in˝gures4.4(a)and(d)revealatightclusteringofdataalong the y = x diagonal,withonlyafewoutlyingpoints.Thisveryhighdegreeofcorrelationbetween k sp and k s impliesthatbothtechniqueshavebeenappliedwithequalsuccesstothisprediction problem.Theerrorrange,˝gures4.4(b)and(e),islessthan 30 %( L 1 norm)andtheaverage error( L 1 norm)islessthan8%,forbothtechniques. Theconsistencybetweenboththe k s predictionsanderrorbandsfortwoquitedi˙erentML techniquessuggeststhattheyarebothwell-suitedtothiskindofproblem,andpossiblyclosetoan optimumforthisclassofMLapproach. Theerrorvaluesaspercentages,fortheDNNandGPRmethods,aregivenintable4.3,together withtheerrorintheempiricalrelation k s = 2 : 91 k rms ¹ 2 + S k º 0 : 284 ; (4.15) proposedbyFlacketal.[113],and k s = 1 : 07 k t ¹ 1 e 3 : 5 E x º¹ 0 : 67 S 2 k + 0 : 93 S k + 1 : 3 º ; (4.16) 71 givenbyForooghietal.[115],aswellastheirrespectiverecalibratedcorrelations: k s = 1 : 11 k rms ¹ 2 + S k º 0 : 74 ; (4.17) k s = 0 : 04 k t ¹ 1 e 5 : 50 E x º¹ S 2 k + 2 : 57 S k + 9 : 82 º : (4.18) whenextendedtoallcasesinthecurrentdatabase.Itisinterestingtonotethattheformof equation(4.15)waschosenforsurfacesgeneratedbygritblasy-packed,random,three- dimensionalroughnesseswithawiderangeofscales(E01-E06),whilemanyofthesimulated surfacesaretwo-dimensional,somearecharacterizedbydiscreteelementsofsimilarsizes,while othersaresparseorwavy(characterizedbylowslopes).Equation(4.16),ontheotherhand,includes aslopeparameterandwascalibratedfornumericallygeneratedsurfacesconsistingelementsof randomsizesandaprescribedshape. Formostcases,theerrorsfromtheDNNandGPRmethodswereofthesameorderofmagnitude andmuchsmallerthantheerrorinusingequation(4.15)or(4.16).IntheDNNandGPRpredictions ofsimulationcases,thegreatesterrors(about25%-30%)aroseincasesE07andE08.Thesurfaces associatedwiththesecasesarecharacterizedbyfractalfeatures(withspectralslopesof-0.5and -1.0,respectively[114]).Thesizeoftheerrorsforthesecasesmightbeattributedtothesmall numberofsurfaceswiththisfeatureusedinthetrainingset(asopposedtothemanysurfacesthat aremostlycharacterizedbysingle-scaleelements).Acloseexaminationofthepredictionerrorsfor theDNScasesshowedasubtletrendbetweenrelativelyhigherrorsandlowroughnesssolidity(or low E s andinsigni˝cantwakesheltering),in,forexample,casesC28andC44.Boththesecases arecharacterizedbylarge-wavelength,wavyfeatures,suggestinganunder-representationofsparse roughnessinthedataset.Beyondthisobservation,noclearcorrelationwasfoundbetweenthe errorandotherprimaryroughnessparametersincludedhereinorsurfacecategorizations(2D/3D, random/regular). Theerrorsassociatedwithusingequation(4.15)aresmallforsurfacesE01throughE06,which wereusedtocalibratethisrelation.Theerrorsinusingequations(4.15)and(4.16)overallsurfaces 72 Casename err DNN err GPR err B 1 err B 2 err B 3 err B 4 C04,r6,reg,inc14.04.1-16.7-40.98.6-63.9 C05,r6,reg,inc20.710.3-38.3-49.52.0-71.7 C06,r6,reg,inc34.27.5-10.4-33.624.5-59.8 C07,r4,rnd,inc110.5-4.7-63.5-63.610.0-73.5 C08,r4,rnd,inc2-0.6-4.8 -80.1 -77.6 -4.1-81.7 C09,r4,rnd,inc36.0-1.5-63.4-64.28.3-73.4 C10,r6,rnd,inc10.2-2.7-76.3-72.511.8-77.8 C11,r6,rnd,inc22.6-6.1 -82.9 -78.9 4.1 -82.2 C12,r6,rnd,inc3-1.0-18.7-74.7-72.7-2.3-78.7 C14,r4,reg,por25.30.0-66.2-64.2-8.7-80.3 C15,r4,reg,por34.2-3.1-76.7-66.529.0-78.8 C16,r6,reg,por11.85.43.5-41.721.5-59.4 C17,r6,reg,por2-0.6-1.3-74.5-70.3-10.9 -82.7 C18,r6,reg,por3-5.1-5.1-79.1-68.435.5-78.8 C19,r4,rnd,por11.8-2.3-71.4-69.444.9-67.8 C20,r4,rnd,por21.117.2-67.0-56.682.1-66.3 C21,r4,rnd,por30.0-1.8-69.6-50.0254.1-46.2 C22,r6,rnd,por1-7.1-7.9-77.0-76.7-10.6-78.4 C23,r6,rnd,por20.23.4-70.9-60.480.8-67.9 C24,r6,rnd,por3-0.1-6.7 -80.5 -66.3136.7-66.5 C26,r4,reg,ES2-5.4-12.7-48.6-61.6-57.6 -83.8 C28,r6,reg,ES19.69.8-29.2-45.9-51.9-81.2 C29,r6,reg,ES2-2.6-9.8-54.7-66.2-53.2 -83.2 C30,r6,reg,ES3-1.53.4-21.8-45.78.1-65.7 C31,r4,rnd,SGR-0.63.3-46.7-50.765.1-53.8 C37,r6,rnd,SGR-1.5-7.9-61.3-65.011.9-68.6 C40,r6,rnd,RND3-3.19.1-23.6-39.698.3-30.8 C43,SG5.52.1-58.6-60.146.3-62.0 C44,TB-3.3 22.7 77.651.931.5-51.6 C45,CB1.8-16.5-70.4-52.079.3-72.8 E01,16,2-2.13.56.2-47.5370.263.0 E02,16,32.35.23.3-33.7 429.4 79.5 E03,16,7-2.31.2-2.2-69.1368.138.6 E04,16,8-3.9-5.71.3 -78.8 412.4 27.6 E05,16,9-3.312.410.9-46.3262.127.3 E06,16,15 -16.0 -2.5-3.0-51.1 405.4 79.9 E07,18,1 -29.8 -25.8 17.3-4.0208.311.2 E08,18,2 28.1 26.1 120.7 79.4 388.8 80.0 E09,19,16.29.469.225.9312.556.9 E10,19,2-8.90.65.8-20.7258.920.6 E11,19,38.97.447.4-24.1247.432.2 E12,19,4-6.62.124.1-21.0258.432.2 E13,19,56.7 19.4 -16.6-23.8287.26.6 E14,19,65.38.9-56.8-52.5177.2-38.2 E15,19,7 22.3 9.419.8-10.2342.643.0 L 1 5.47.847.652.8133.860.6 L 1 29.826.1120.779.4429.483.8 Table4.3:Errorsin k s predictionbyDNNandGPRcomparedtoerrorsoftheempirical correlations: err B 1 (equation4.15), err B 2 (equation4.17), err B 3 (equation4.16)and err B 4 (equation4.18).Thefourlargesterrors(inmagnitude)foreachcolumnarecoloredinred.The errorsarepercentages. 73 inthedatabaseare120%and430%respectively.However,whenrecalibratedagainstthefull database,equations(4.17)and(4.18)haveasigni˝cantlysmallererrorbandwithmaximumvalues of79%and84%.Thehigherrorvaluesoftheempiricalcorrelations,comparedtoDNNorGPR prediction,areattributedtothesmallnumberofgeometricalvariablesusedintheircalibrations andtherestrictedrangeofthemodels'parameters. 4.3.3Uncertaintyestimation Inadditiontopredictionsofequivalentsand-grainheight,theGPRmethodprovidescon˝dence marginsasfunctionsofeachinputparameter.Thesemarginscanbeusefulforindicatingthekinds ofsurfacesforwhichadditionaltrainingdatacouldimprovecon˝denceinpredictions.Thisfeature oftheGPRapproachmakesitveryattractiveforstudiesofthiskind,sinceDNSandexperimental generationofdatacanbeexpensive. Thecon˝denceintervalsdeterminedbytheGPRtechniqueareshownasfunctionsofthe normalizedsurfacermsroughnessheight,e˙ectiveslope,porosityandskewnessin˝gure4.5. Widerintervalsindicatehigherestimatedvaluesofpredictiveerror,suchasatroughnessporosity of0.68,andskewnessesof-1.5and2.0.Surfacesofroughnesswithsimilarvaluesofporosityand skewnesswouldthenbeprioritiesforadditionalsimulationsorexperiments. 4.3.4Sensitivityanalysis ThedependenceofDNNpredictionsof k s onindividualroughnessparametersisexploredby determiningthechangeintheerrornormswheneachoftheprimarysurfaceparametersisremoved fromthedatafromwhichtheDNNpredictionwasmade.Intable4.4,theactualerrorforeach surface,andthevaluesofthe L 1 and L 1 normsoferrorsinthepredictionof k s overthe45surfaces, arereportedwhentheparameter(s)inthe˝rstrowis(are)theexcludedone(s).Theerrorsofthe baseprediction(whichincludesall8primaryparameters)arelistedinthesecondcolumn.Inthe followingdiscussion,wefocusonthe L 1 normforeaseofcomparisonoverall45cases. 74 Figure4.5:Con˝denceinterval(CI)ofpredictionswiththeGPRmethod,withpredictedvaluesof k s š R a in bluelines (called k sp )andtruevaluesof k s š R a in reddots .GPRpredictionsforboth trainingandtestingdatasetsareshown k s and k sp areveryclosetoeachotherforthetraining datapoints,whiletheydeviate(lessthan30%oferror)forsometestdatapoints.Linejaggedness isassociatedwithprojectionofahigh-dimensionalspacetoone-dimensionalones. 75 Whenthevaluesof L 1 areconsidered,therelativeimportanceofthesesurfaceparametersfor predicting k s is: E x , I x , j I z j , E z , P o , k rms š R a , S k ,andofleastimportance, K u .The L 1 -normerror issmallwhenallparametersareincluded(7.4%).Excludinganysingleoneoftheseparameters increasesthe L 1 -normerroruptoaround9%.Ontheotherhand,theexclusionof K u fromtheinput parametersdoesnotworsenpredictionsof k s signi˝cantly.Instead,thisobservationappearstobe aconsequenceofcorrelationbetween K u andothersurfaceparameterslike k rms š R a (see˝gure 4.3).Whensuchcorrelationsexistandonecorrelatingparameterisexcluded,theDNNprocess redistributestheweightingsgiventoothercorrelatedparameters,withlittlelossinpredictive accuracy. Toreducethecorrelationbetweentheexcludedparametersandtheremainingones,onemay excludegroupsofparametersthatarethoughttocharacterizethesametypeofsurfacefeature.For thisreason,asensitivityanalysiswascarriedoutonthee˙ectofgroupsofvariablesonprediction of k s .Thecharacteristicsofsurfaceslope,elementinclinationangle,porosity,andintensityof height˛uctuations,arecontainedinpairsof( E x , E z ),( I x , I z ),( P o , S k )and( k rms , K u ),respectively. Parameterswithineachpairhavebeenshowntobecorrelatedtosomedegreein˝gure4.3.Table 4.4showshowtheaccuracyof k s predictionisa˙ected,ifanyoneofthesepairsisexcluded. Accordingtothetable,thepredictionof k s issensitivetoallfourpairs,butwithgreatersensitivities tothesurfaceporosity(describedby P o , S k )andthesurfaceslope(describedby E x and E z ).As expected,theeliminationofbothparametersofapairworsensthepredictionmorethanremoving eithersingleparameter(fromaround7-9%errorstoupto14%). Accordingtothesensitivityanalysis,allparametersconsideredareofsomeimportanceinthe predictionof k s .Thee˙ective x -slope E x androughnessheightskewness S k havebeensuggestedas especiallysigni˝cantinearlierstudies[37,38,5].Theinclinationangleinthestreamwisedirection I x makesasigni˝cantcontributiontothe k s predictionbecause,physically, I x characterizesthe averageaerodynamicshapeoftheroughnesselements.Surfaceswith I x > 0 areaerodynamically blu˙bodieswhencomparedwithsurfacesofthesamesizebutwith I x = 0 ,andsurfaceswith I x < 0 tendtobemorestreamlinedandhenceproducelessdrag. 76 None E x E z E x ; E z k rms K u k rms ; K u S k P o S k ; P o I x I z I x ; I z C042-23-1-1-215-1313-120-3 C055-81163 -22 -408-4-6-2-11 C06010-1101025151868 C0713-1210 -23 01-6-11319 C08-15-14-1-4 -19 -24 -19 -2 -23 -36 -4-7-9 C09184630361-265118 C1001-161-140-1-122-1315011 C11-12-3-3-23-2-2-5-12-1-291-2-2 C120-4-40-18-3-4-1-7-2-30-2 C14045515 26 38-6360 C15165002900-11-24-54 C161-2-1 24 -2-2-33-266-114 C17-4817171481513-4353 C18-1-6-10-11-2-3-11-3 -21 -17-10 -25 -16 C19-10-15-11-12-3456-4-11-1-2-11 C201334332430 23 25 13 C219213112-1000814 C22-3-3-8-9-2-6-8-3-8-9-9-20-12 C230-2-100-5-17-10-12-32 C240 -21 -1-1-1100040-4-7 C26-6-17-12-9-8-5 -19 -15-13-5-13-14-10 C28181921 26 1718-3161632 21 14 20 C29-9-19-8-22-6-5-13 -25 -11-22-18-17-19 C30-4611 25 -1006 24 0-8265 C31 22 20 819 24 0-218-1-149-19 C37-2-8-7-310-4-5-1-5-1-9-8-12 C40-3-6 -27 -21-6-5-70-12-10-8-18 C433-4-4616120723-15-1-12 C44-615117131420-6-12-2-16 -21 C45121-4-65-1-1111529 E011244-92-3-11511-101-3-3 E02-136-6-7-2121-210-9137-2 E0315-60-54-6-437-32212 E040-15-9-9-2-6-6-3-52-240 E0551751749975288513 E06-5-3-6-3-10-9-10-6-7-9-10-10-5 E07 -21 -21 -24 -18-16-21-18-17 -23 -41 -25 -25 -24 E08 22 22 25 22 19 18 25 24 724 21 22 24 E095-315 27 -1 22 26 21 -2-21-322 E10-18-19-5-8 -25 -4-51-14 38 -148-2 E11-1-15 -23 -19-71612-29290-50 E12-9-360-102-2-15-1028-15-22-4 E1311817617287 21 -1514 25 15 E14 22 6106421 25 33 95-5 E1501818-411915111932192316 L 1 7.48.98.29.77.67.17.97.38.014.28.88.69.1 L 1 22222727252426252541252524 Table4.4:Errorsin k s predictionbyexcludingoneortwofeatures.Thebaseprediction includesallprimaryvariables.Thefourlargesterrors(inmagnitude)foreachcolumnarecolored inred.Theerrorsarepercentages. 77 Animportant˝ndingfromthisstudyisthatthee˙ective z -slope E z isofsimilarimportance toaccurate k s predictionas S k or E x .Theexclusionof E z adverselya˙ectsthepredictionfora largenumberofroughsurfaces.Physically, E z describeswhetherthesurfaceisclosetoatwo- dimensional(2D)roughnesswith E z = 0 (suchasatransversebarroughness)orathree-dimensional (3D)roughnesswith˝nite E z .Itisknownthatak-type2Droughnessproducesahigherdragthan a3Droughnesswiththesameheightduetothelargerspanwiselengthscalethatthe2Droughness impartstothe˛ow[123]. 4.3.5ComparisonbetweenMLalgorithmsandpolynomialmodels Explicitalgebraicdatarepresentations,suchaspolynomialfunctions,canalsobedeterminedfor thedatasetsofthisstudy,using˝ttingorminimizationprocedures.Insuchmethods,asetofbasis functionsisproposedforamodel,theunknowncoe˚cientsofwhicharethenoptimizedaccording tospeci˝edconstraints.Theyareageneralizationofthemodelsofequation(4.2),whichwerebased onexperimentalobservationsofthedependenceof k s onasmallnumberofsurfaceparameters.A 30-degree-freedompolynomialbasiswasproposedasa`white-box'modelfor k s ,analogoustoa low-orderTaylorseriesexpansionfor k s : k s š R a = 0 + 1 ¹ k rms š R a º 2 + 3 I x + 4 j I x j 5 + 6 j I z j + 7 j I z j 8 + 9 P 10 o + 11 E 12 x + 13 E 14 z + 15 S k + 16 j S k j 17 + 18 ¹ K u 3 º + 19 j K u 3 j 20 + 21 ¹ k rms š R a º 22 P 23 o + 24 ¹ k rms š R a º 25 E 26 z + 27 P 28 o E 29 z ; (4.19) where a i ( i = 0 ; 1 ; ; 29 )arethemodelcoe˚cients.Tokeepthismodelassimpleaspossible andtobringthee˙ectsofallcontributingfactorsintoaccount,weusedtermsas i j foratest variable thattakeonlypositivevalues(e.g. k rms ),andtermsas i + j j j k forthosevariables thattakebothpositiveandnegativevalues(e.g. S k ).Forthelatter,thepowerof inthe˝rsttermis ˝xed(atone)insteadof˝tted,toeliminatethepossibilityofanimaginarynumber.Combinations ofsixparameters( E x , E z , P o , S k , k rms š Ra and K u ),takeninpairs,werealsoincluded.Since,for 78 Figure4.6:(a)Scatterplotoftrue k s andpredicted k s (denotedas k sp ),(b)scatterplotoftrue k s andrelativeerrorand(c)pdfofrelativeerrordistributionforpredictionusingpolynomialfunction de˝nedinequation(4.19),withDNSdata( blue )andexperimentaldata( red ). thepresentcollectionofsurfaces,strongcorrelationswereobservedbetweenindividualvariables withinthethreepairsof ¹ E x ; E z º , ¹ P o ; S k º and ¹ k rms š Ra ; K u º ,shownin˝gure4.3,onlyone variablefromeachpairwasusedforthecombinationtermsinequation(4.19).Usingtheother variablefromanyofthesepairsinsteadwouldnotleadtoasigni˝cantchangeintheprediction usingequation(4.19). Thehigh-dimensionalspaceof a i ispoorlysuitedtocurve-˝ttingandminimizationprocedures whichusestochasticgradientdescentalgorithms.However,itiswellsuitedtorobustminimization methodslikethedi˙erentialevolutionalgorithm[124],withwhichglobalminimacanoftenbe founde˚cientlyinspacesofhighdimension.Inthiscase,itisusedtodeterminethevaluesofthe coe˚cients a i whichminimizethe L 1 norm. In˝gure4.6,thepredictionqualityofthiswhite-boxmodelwithoptimizedcoe˚cientvalues isshown.Thismethodyieldsanaveragepredictionerrorof12%andamaximumoneof51% whenusingall45fully-roughdatasets(togivethebestpossiblepredictionaccuracy)forthemodel training. Theoptimizedvaluesof a i 'sare 79 0 =5.312, 1 =-1.172, 2 =4.264, 3 =0.050, 4 =-1.283, 5 =8.393, 6 =-0.347, 7 =-5.771, 8 =1.785, 9 =7.919, 10 =4.058, 11 =-0.979, 12 =3.414, 13 =6.380, 14 =1.354, 15 =1.023, 16 =2.969, 17 =1.273, 18 =-0.946, 19 =-0.762, 20 =0.056, 21 =1.647, 22 =-8.176, 23 =3.523, 24 =-9.472, 25 =-5.656, 26 =0.580, 27 =-5.425, 28 =0.283, 29 =7.177. Thepredictiveaccuracyofthisoptimizedexplicitmodelequationisconsiderablylowerthan thatoftheDNNandGPRmethods.Onereasonforthisreducedaccuracyisthatlow-orderfunctions ofgeometricalparametersdonotfaithfullyrepresentthedependenceof k s onsurfaceparameters becauseeachcoe˚cientinthemodelisrequiredtotakethesamevalueovertheentiresurface- parameterspace.InMLapproaches,suchrestrictionsneednotapplyastheyarenotconstrainedto low-orderpolynomialfunctionsbutinsteadadoptamethodicalsearchforthebestrepresentationof k s asafunctionofthesurfaceparameters.Thissearchiscarriedoutthrough`featureselection'in the˝rstlayersofDNNandthepropertiesofthebasisfunctionsadoptedinGPR,eachofwhichare designedtoyieldthesamemeanandstandarddeviationof k s š R a asintheoriginaldataset[111]. 4.4Concludingremarks Theconstructionofapredictivemodelfromalargeensembleofdatasetfortheequivalent sandgrainheight k s ofasurfaceofarbitraryroughness,asafunctionofmanydi˙erentmeasures ofsurfacetopography,isalabeledregressionproblemthatiswell-suitedtomachinelearning techniques.Inthischapter,datafrom45di˙erentroughsurfaces(infullyrough˛ows)wereusedto deviseDNNandGPRpredictionsfor k s asfunctionsof8di˙erentsurface-roughnessparameters. Bothmodelswereabletopredict k s forthe45surfaceswithanaverageerrorbelow10%,with thelargesterrorforanyonesurfacelessthan30%.Thesepredictionsweresigni˝cantlybetterthan thoseofexistingformulas,andofa30degree-of-freedompolynomialmodel˝ttedtothesamedata, wherethegreatesterrorforanysurfacewasabout50%. Sensitivityanalysesrevealedthatinclusionofnearlyallthesurfaceroughnessdescriptive parameterswasnecessarytominimizetheaveragepredictionerror,butthatexclusionofeither 80 measuresofporosityormeasuresofthesurfaceslopeincreasedthemaximumpredictionerror moresigni˝cantlythanomittingotherparameters. Machinelearningtechniquesarewellsuitedtothismodelingproblembecause: i )itiscomplex insofarasdi˙erentkindsofsurfaceroughnessyielddi˙erent˛owphenomenawhicharemodeled mostaccuratelyindi˙erentways,makingtheprospectofageneralphysicalmodelveryremote; and ii )thedependentsurface-roughnessvariablesuponwhich k s ismodeledarealargenon- orthogonalsetforwhichrobustmultivariableregressiontechniquesarerequired.Asmachine learningmethods,theytakenoaccountofphysicalmodelingconceptsorobservedphenomena withinroughnesssublayers,suchasrecirculationregions,enhancedturbulenceproductioninthe wakeofroughnesselements,assumedscalingsfordrag etc. ,eachofwhichisapplicableto˛ows oversomeroughsurfacesbutnotothers.Noraretheyhinderedbythelackoforthogonalityof thesurfaceroughnessparametersasthedependentvariablesof k s .Thetechniquesusedcanbe con˝guredreadilytomimicmodelswithverymanydegreesoffreedomand,whencomparedto polynomialmodels,theirfeatureselectionpropertiesprovidetheequivalentofdi˙erentvaluesfor polynomialcoe˚cientsindi˙erentregionsofthesurface-parameterspace.Inthisapplication, bothapproachesofDNNandGPRyieldedmodelswithverysimilarpredictiveaccuracy,even thoughthetechniquesthemselveswereverydi˙erent.Wethereforeconcludethattheyyieldhigh- ˝delitypredictionsoftheequivalentsand-grainroughnessheightforturbulent˛owsoverawide rangeofroughsurfaces,asasigni˝cantimprovementoverothermethods.Improvedprediction mightbeachievedbyenlargingthedatabasetoincluderough-wall˛owswithsurfaceparameters whichcorrespondtotherelativelylowpredictioncon˝denceintheGPRmethod,andbyincluding additionalroughnessparametersasinputswhichmightdescribesparsenessandtwo-dimensionality, suchasthesolidity,correlationlengthscalesandothertwo-pointsurfacestatistics. Inadditiontothe k s predictiondescribedhere,theDNSdatabaseandtheMLtechniquesin generalcanalsobeusedtouncoverrelationsbetweenroughnessgeometryandphysics-related quantities,suchasthe˛owpatternaroundroughnessprotuberances,˛owseparationlocations, characteristicsoftheshearlayersassociatedwiththeseparationbubbles,thewakesheltering 81 volume, etc. Speci˝cally,aMLnetworktrainedtocorrelatethese˛owcharacteristics(asoutputs) totheroughnessgeometry(asinputs)maybeane˚cienttoolfordeterminingthesetsofroughness geometricalfeatureswhichareimportantforcharacterizingthesee˙ects.Knowledgeofsuchaset ofsigni˝cantroughnessparametersmayalsoguidetheconstructionofrough-surfacedatabases thatyieldmoree˚cientandmorewidelyapplicablepredictionsof k s orotherquantities. 4.5Supplementarymaterials Therough-wall˛owdatabase(including k s ,surfaceheightmapandsurfaceparameters)and thetrainedDNNandGPRnetworks,calledPredictionoftheRoughnessEquivalentSandgrain Height(PRESH),canbeaccessedonlineinmyGitHubrepositoryat https://github.com/MostafaAghaei/Prediction-of-the-roughness-equivalent-sandgrain-height. Withthispackageofdataandprograms,interestedresearcherscan:i)usetheMLnetworks describedinthischaptertomakepredictionsof k s forsurfacesoftheirownroughnesstopography; ii)downloadthecodeandtrainnewDNNandGPRnetworkstopredict k s foradi˙erentset ofsurfacesofarbitrarytopography;andiii)usethedatabaseof45rough-wall˛owsforother applications.ItisrecommendedtousetheMLcon˝gurationsdescribedinthischapterforsurfaces withparametersinsidetherangesspeci˝edin˝gure4.3.Extrapolations(usinginputswhichare beyondthespeci˝edrange)willleadtoadditionaluncertainty. 82 CHAPTER5 COMPRESSIBLEFLOWSOVERROUGHWALLS 5.1Introduction Thee˙ectsofwallroughnessonphysics,control,andmodelingofcompressible˛ows(subsonic, sonic,super-andhypersonic)arenotwellunderstoodtoday.Inhighspeed˛owstudies,roughness istypicallyan isolated (e.g.steps,joints,gaps,etc.),ora distributed (e.g.screwthreads,surface ˝nishing,andablation)e˙ect.Themaine˙ectsofroughnessonsupersonic˛ightvehiclesare primarilytoincreasethedragcoe˚cientandsecondarilytoadvanceboundary-layertransitionto turbulence( earl y transition ),whichincreasestheheattransfercoe˚cient.Anunderstandingof thesee˙ectsisimportantfor˛ightcontrolandthermalmanagement(throughthermalcoatings), especiallyforreentryapplicationsandreusablelaunchvehicles.Reda[125]andSchneider[32] havereviewedthee˙ectsofroughnessonboundarylayertransition,basedonexperimentalwind- tunnelandin-˛ighttestdataof˛owsinsupersonicandhypersonicconditions.Radeztskyet al.[126]analyzedthee˙ectsofroughnessofacharacteristicsizeof1- m(atypicalsurface ˝nish)ontransitionsinswept-wing˛ows,andLatinetal.[127]investigatede˙ectsofroughness onsupersonicboundarylayersusingroughsurfaceswith k s = O ¹ 1 mm º ( 100 < k + s < 600 ) and Re ˝ ˇ 40000 .Experimentalstudiesofdistributedroughnesse˙ectsoncompressible˛ows, boundarylayertransition,andheattransferincludethoseof[128,129,130]and[131]. Mostnumericalstudieshavefocusedonisolatedroughness[seee.g.132]oridealdistributed roughnesssuchaswavywalls[seee.g.133],duetothesimplicityinmeshgenerationandnumerical procedures.However,complexdistributedroughnessisofprimaryimportanceandmorerelevantto ˛ightvehicles,sinceinhigh-speed˛owsventhemostwell-controlledsurfacewillappearrough astheviscousscalebecomessu˚cientlysmall"[134].Also,accordingtoSchneider vehiclesoftendevelopsurfaceroughnessin˛ightwhichisnotpresentbeforelaunch.This ˛ight-inducedroughnessmaybediscretestepsandgapsonsurfacesfromthermalexpansion,or 83 distributedroughnessinducedbyablationortheimpactofdust,water,oricedroplets.Studiesof thiskinddemonstratetheneedforacompressiblesolverthatcanhandlecomplexdistributedrough surfaces. 5.1.1Literaturereviewonimmersedboundary(IB)methods Athoroughreviewondi˙erentIBmethodsisgivenin[135].Irrespectiveofthecompressibility e˙ects,IBmethodscanbedividedindi˙erentcategories,themostimportantofwhichare: 1. Continuousforcingapproach[penaltyIBmethodofreference136,amongmanyothers], wheree˙ectsofsolidboundariesareaccountedforinsertinganadditionalforcetermin theNSequations.Thismethodiswellsuitedforelasticboundaries,andrigidbodies[ifa feedbackforcingisprovided,seee.g.137].Solidboundariesarecaptureddi˙usivelyinthis approach,andthemethodrequiresaLagrangiangridfordeformablesolidparts.Theforcing termisa simpli˝edmodel representingtheboundaries[135].Inordertopreventsti˙nessof numericalsolverforrigidboundaries,evenbymeansoffeedbackforcing,averylowCFL numberof O ¹ 10 3 10 2 º isrequired,whichfor3-dimensional(3D)problemsmightbe cumbersome[138]. 2. Discretizedforcingapproach[138],whereDirichlet'sBCisimposedbyexertingadiscretized forceinthefollowingformat u l + 1 u l t = RHS l + 1 š 2 + f l + 1 š 2 f l + 1 š 2 = RHS l + 1 š 2 + V l + 1 B u l t ; wheretheRHScontainstheconvective,pressureandviscousterms.Thismethoddoesnot su˙erfromnumericalsti˙ness,andcanalsobeusedformoving3Dsolidsurfaces. 3. GhostcellIBmethod[seee.g.139,amongmanyothers],whereboundaryconditionsare imposedbymeansofghostpoints(thosepointsinthesolidcells,whichareinthevicinity 84 ofthe˛uiddomain).Thismethodiswellsuitedforstationaryrigidbodies,andcancapture solidboundariessharply.TheDirichletandNeumannBCscanbeimposedexactly(with- outincorporationofanymodels);however,extendingthismethodtomovingboundaries needsspecialconsiderations.Treating3Dcomplexobjectssuchasroughness,wherecusp points,concave/convexcurvaturesandothertypesofsingularitiesexistinthedomain,isnot straightforwardinthismethod.Issuesarisewhentherearemultipleimagepointsforaghost cell,ortherearenone.Luoetal.[140]addressedsomeoftheseissuesin2-dimensional (2D)domains.Alsointerpolationschemesaredependentontheghostpointlocationinthe soliddomain,whichdemandsatleast3di˙erenttypesofinterpolationfor2Ddomains.The situationismorecomplicatedfor3Dones. Thefollowingcomprisesabriefdescriptionofdi˙erentstudiesonIBmethods,withemphasis onthecompressible˛owregimes. Ghiasetal.[141]usedghostcellmethodtosimulate2Dviscoussubsoniccompressible˛ows. TheyimposedDirichlet'sBCfor u and T ,theequationofstatefor P andextrapolationfor ˆ (see section5.2asforde˝nitionofdi˙erentparameters).Theirmethodaccuracywassecondorder, locallyandglobally.Vitturietal.[142]usedadiscretizedforcingapproach,basedona˝nite volumesolver,tosimulate2D/3Dviscoussubsonicmultiphasecompressible˛ows.Theyimposed Dirichlet'sBCfor u and T ,theequationofstatefor P and˛uxcorrectionfor ˆ and E .Chaudhuri etal.[143]usedcombinedghostcellanddirectforcingmethodstosimulate2Dinviscid,sub- supersoniccompressible˛ows.Theyusedtheequationofstatefor P ,anddirectforcingfor ˆ , u and E equations.Theykeptthe˝fthorderaccuracyofWENOshock-capturingschemebyusingtwo layersofghostcells.Wangetal.[144]usedcontinuousforcing(penaltyIBmethod)tosimulate ˛uid-structureinteractionwith2Dcompressible(sub,super,andhypersonic)multiphase˛ow. Yuan&Zhong[145]usedghostcellmethodtosimulate2D(sub-supersonic)compressible˛ows aroundmovingbodies. 85 5.1.2Literaturereview:physicsof˛owsoverroughness Tyson&Sandham[133]analyzedsupersonicchannel˛owsover2DsinusoidalroughnessatMach number( M )of M = 0 : 3 , 1 : 5 and 3 tounderstandcompressibilitye˙ectsonmeanandturbulence propertiesacrossthechannel.Theyusedbody-˝ttedgridstoperformthesimulationsandfound thatthevaluesofvelocityde˝citdecreasewithincreasingtheMachnumber.Theirresultssuggest strongalternationofmeanandturbulencestatisticsbytheshockpatternsassociatedwiththewall roughness. Ekotoetal.[146]experimentallyinvestigatedthee˙ectsofsquareanddiamondroughness elementsonthesupersonicturbulentboundarylayers.Theobjectiveoftheirstudywastounderstand howroughnesstopographyaltersthe localstrain-ratedistortion , d max ,whichhasadirecte˙ecton turbulenceproduction.Theirresultsindicatethatthesurfacewith d-type squareroughnessgenerated weakbowshocksupstreamofthecubeelements,causingasmallvalueof d max ( ˇ 0 : 01 ),and thesurfacewithdiamondelementsgeneratedstrongobliqueshocksandexpansionwavesnearthe elements,causingalargevariationin d max (rangingfrom 0 : 3 to 0 : 4 acrosstheelements).These valuesof d max ledtoacanonicalrough-wallboundarylayertrendforthesquareroughnessand regionswithlocalizedextraturbulenceproductionforthediamondsurface. StudiesofLatinetal.([127],[147]and[148])includeacomprehensiveinvestigationon supersonicturbulentboundarylayersoverroughwalls.Fiveroughsurfaces(including2Dbar,3D cube,andthreedi˙erentsandgrainroughness)havebeenanalyzedat M = 2 : 9 .E˙ectsofwall roughnessonmean˛ow,turbulence,energyspectraand˛owstructuresarestudiedtounderstand thephysicsof˛ow,toexpandexperimentaldatabase,andtoevaluatealgebraicnumericalmodels for˛owsinthisregime.Theirresultsshowstronglineardependenceofturbulencestatisticsonthe surfaceroughness,andalso,strongdependenceofturbulentstructureslengthscalesandinclination anglesontheroughnesstopographies. Muppidi&Mahesh[149]analyzedtheroleofidealdistributedroughnessontransitionto turbulenceinsupersonicboundarylayers.Theyhavefoundthatcounter-rotatingvortices,generated bytheroughnesselements,breaktheoverheadshearlayerup,whichleadstotransitiontoturbulence 86 morequickly.AsimilarstudywasmadebyBernardinietal.[132],whoinvestigatedtheroleof isolatedcubicalroughnessonboundarylayertransition.Theirresultssuggesttheinteractionof hairpinstructures,shedbytheroughnesselement,withtheshearlayerexpeditestransitionto turbulence,regardlessoftheMachnumber. 5.1.3Objectives Inthisstudywe˝rstintroducealevel-setbasedimmersedboundarymethodthatcanaccurately capturethemeanandturbulence˝elds.Thenweanalyzethe˛owphysicsinsupersonicchannel ˛owsat M = 1 : 5 andbulkReynoldsnumberof3000overtwo2-dimensional(2D)andtwo 3-dimensional(3D)sinusoidalsurfaces.Di˙erentmean,turbulenceandenergyquantitiesare analyzed,and˝nally,thetransportequationsofturbulencekineticenergy(TKE)andnormal Reynoldsstressinthestreamwisedirectionareexamined. 5.2Problemformulation 5.2.1Governingequations Thenon-dimensionalformofcompressibleNavier-Stokesequationare @ˆ @ t + @ @ x i ¹ ˆ u i º = 0 ; (5.1a) @ˆ u i @ t + @ @ x j ˆ u i u j + p ij 1 Re ˝ ij = f 1 i 1 ; (5.1b) @ E @ t + @ @ x i u i ¹ E + p º 1 Re u j ˝ ij + 1 ¹ 1 º PrRe M 2 q i = f 1 u 1 ; (5.1c) where x 1 , x 2 , x 3 (or x , y , z )arecoordinatesinthestreamwise,wall-normalandspanwisedirections, withcorrespondingvelocitiesof u 1 , u 2 and u 3 (or u , v and w ).Density,pressure,temperatureand dynamicviscosityaredenotedby ˆ , p , T and ,respectively. E = p š¹ 1 º + ˆ u i u i š 2 isthetotal energy, ˝ ij = @ u i @ x j + @ u j @ x i 2 3 @ u k @ x k ij istheviscousstresstensor,and q i = @ T @ x i isthethermalheat ˛ux. f 1 isabodyforcethatdrivesthe˛owinthestreamwisedirection,analogoustothepressure 87 gradient.ThereferenceReynolds,MachandPrandtlnumbersare,respectively,Re ˆ r U r L r š r , M U r š p RT r ,andPr r C p š ,wheresubscript r standsforreferencevalues.Thegasconstant R andthespeci˝cheats C p and C v areassumedtobeconstantthroughoutthedomain(calorically perfectgas).Theyarerelatedby R = C p C v ,andtheratioofspeci˝cheats C p š C v isassumed tobe1.4.Theheatconductivitycoe˚cientisdenotedby . Thesetofequationsin(5.1)isclosedthroughtheequationofstate,whichforaperfectgasis p = ˆ T M 2 : (5.2) Equations(5.1)and(5.2)aresolvedusinga˝nite-di˙erencemethodinaconservativeformat andageneralizedcoordinatesystem.A˝fth-ordermonotonicity-preserving(MP)shock-capturing schemeandasixthordercompactschemeareutilizedforcalculatingtheinviscidandviscous˛uxes respectively.ThesolveruseslocalLax-Friedriches(LLF)˛ux-splittingmethodandemploysan explicitthird-orderRunge-Kuttaschemefortimeadvancement.ReadersarereferredtoLi&Jaberi [150]forextensivedetailsandexplanationsaboutthecompressiblesolver. 5.2.2DetailsofthepresentIBmethod Weusedacombinationoflevel-setandvolume-of-˛uid(VOF)methods.Thelevel-set˝eldis obtainedbysolving ˝ = sign ¹ º¹ 1 j r jº ; (5.3) where ˝ isa˝ctitioustimecontrollingthewidthoftheinterface.Thisapproachissuitablefor stationaryinterfacesandassuresthat issign-distancedinthevicinityof = 0 level[151,152]. Theinitiallevel-set˝eldisde˝nedas 0 = 8 > > > > > > > >< > > > > > > > > : 1 for˛uidcells, 0 forinterfacecells, 1 forsolidcells. (5.4) 88 Itissu˚cienttomarchin(˝ctitious)timeuntil ˝ = O ¹ º , beingthe(relative)widthofthe interface,tohaveasmoothandconvergedsolutionfor .ThentheVOF˝eld, ˚ ,isde˝nedas ˚ ¹ 1 + ºš 2 ; (5.5) and ˚ = 0 , 0 <˚< 1 ,and ˚ = 1 ,respectively,correspondtosolid,interfaceand˛uidcells.The normaldirectionattheinterfaceiscalculatedusing b n = r = r ˚ šj r ˚ j ; (5.6) positiveintothe˛uid. Toimposethedesiredboundaryconditionforatestvariable ,wesimplycorrectitsvalueat theboundariesbeforeeachtimestep.Itreadsas ! ˚ + ¹ 1 ˚ º b ; (5.7) forDirichlet'sBC,and @ @ n = r b n = @ @ n b (5.8) forNeumann'sBC.ImplementationofDirichlet'sBCisstraightforward.Readersarereferredto appendixBfordetailsofimplementingNeumann'sBC.Thesubscript b inequations(5.7)and(5.8) denotestheboundaryvalues. Bothcorrectionsinequations(5.7)and(5.8)use˝rst-orderinterpolationstoimposethebound- aryconditions.Asaconsequence,thesolveraccuracywouldlocallyreduceto˝rst-orderinthe vicinityofroughness.Itisworth-notingthatthisisafamiliarproblemforIBmethods,even forthosebasedontheghost-cellandfeed-backforcingapproaches,astheymostlyuse˝rst-order interpolationstocorrectfortheboundaryconditions. 5.2.3Surfaceroughnessesandsimulationparameters Fullydeveloped,periodiccompressiblechannel˛owsaresimulatedusingfourroughnesstopogra- phies.Thechannelsareroughenedonlyatonesurface(bottomwall)andtheothersurface 89 Figure5.1:Surfaceroughnesses. issmooth.Areferencesmooth-wallchannelisalsosimulatedforvalidationandcomparison purposes.Thechannelsdimensionsinstreamwise,wall-normalandspanwisedirectionsare,re- spectively, L x = 12 , L y = 2 and L z = 6 ,where isthechannelhalf-heightandequalsto 1. Figure5.1showsfourroughnesstopographiesusedforthepresentsimulations.Allcasesshare thesamecrestheight, k c = 0 : 1 .Thetroughlocationissetat y = 0 .CaseC1andC2are 2-dimensional(2D)sinusoidalsurfaceswithstreamwisewave-lengthsof x = 2 and x = , respectively.Theroughnessheights, k ¹ x ; z º ,forthesesurfacesareobtainedby k ¹ x ; z º = 0 : 05 1 + cos ¹ 2 ˇ x š x º : (5.9) CaseC3andC4are3-dimensional(3D)sinusoidalsurfaceswithstreamwiseandspanwise wave-lengthsof ¹ x ; z º = ¹ 2 ; 2 º forC3,and ¹ x ; z º = ¹ ; º forC4.Theroughnessheightsfor themareobtainedby k ¹ x ; z º = 0 : 05 1 + cos ¹ 2 ˇ x š x º cos ¹ 2 ˇ z š z º : (5.10) 90 Case k c k a vg k rms R a E x E z S k K u C10.10.050.0350.0320.1000.0000.01.50 C20.10.050.0350.0320.2000.0000.01.50 C30.10.050.0250.0200.0640.0640.02.25 C40.10.050.0250.0200.1270.1270.02.25 Table5.1:Statisticalparametersofroughnesstopography. k a vg = 1 A t x ; z kdA istheaverage height, k rms = q 1 A t x ; z ¹ k k a vg º 2 dA istheroot-mean-square(RMS)ofroughnessheight ˛uctuation, R a = 1 A t x ; z j k k a vg j dA isthe˝rst-ordermomentofheight˛uctuations, E x i = 1 A t x ; z @ k @ x i dA isthee˙ectiveslopeinthe x i direction, S k = 1 A t x ; z ¹ k k a vg º 3 dA . k 3 rms is theheightskewness,and K u = 1 A t x ; z ¹ k k a vg º 4 dA . k 4 rms istheheightkurtosis;where k ¹ x ; z º is theroughnessheightdistributionand A f ¹ y º and A t arethe˛uidandtotalplanarareas.Valuesof k c , k a vg , k rms and R a arenormalizedby . Table5.1summarizessomestatisticalpropertiesofthesurfacesandtheirde˝nitions.These statisticsaredi˙erentmomentsofsurfaceheight,surfacee˙ectiveslopesandporosity. Foratestvariable ,thetime,Favreandspatialaveragingoperatorsareshownrespectivelyby , e = ˆ š ˆ and (sumsoverhomogeneousdirections x and z ),withcorresponding˛uctuation componentof 0 , 00 and 000 .Therefore = + 0 = e + 00 = + 000 : (5.11) PeriodicBCsareusedinthestreamwiseandspanwisedirections.Ano-slipiso-thermalwall BCisimposedatbothtopandbottomwalls,assuming u w = 0 and T w = 1 : 0 ( w denotesthewall values).ThereisnoneedtoimposeaBCfordensity,andequation(5.1a)canbesolvedusing one-sideddi˙erentiationtoupdatethedensityvaluesattheboundaries.Thisapproachissimilar tootherwall-boundedcompressible˛owstudies[seee.g.7,133].Thepressureattheboundaries iscalculatedthroughtheequationofstate. Thereferencedensityandvelocityarethoseofbulkvalues,de˝nedas ˆ r = 1 2 2 0 ˆ d y and U r = 1 2 2 0 u d y .Bothoftheseparametersareconstrainedtobeequalto1,andthetime- 91 Figure5.2:Pro˝lesofmeanandturbulencevariablesforthesmooth-wall˛owatRe = 3000 and M = 1 : 5 .Presentsimulation( solidlines ),[7]( dashline ). ˝ w = 1 Re d u d y w . dependentbodyforce f 1 inNSequation(5.1)isdeterminedsoastoimposetheseconstraintsat eachtimestep.Thereferencelengthandtemperaturescalesare and T r = T w ,respectively.The simulationsareconductedatRe = 3000 and M = 1 : 5 ,assumingPr = 0 : 7 and = T 0 : 7 (noteboth and T arealreadynon-dimensionalized). Therespectivegridsizesinthe x , y and z directionsare n x = 800 , n y = 200 and n z = 400 . ForthepresentchannelsizeandReynoldsnumberofthesimulations,thegridcorrespondsto x + , y + max and z + lessthan3.0,whichis˝neenoughforDNS.Alsothe˝rst3gridpointsinthe wall-normaldirectionareinthe y + < 1 : 0 region.Thesimulationsaresu˚cientlyrunintimeto reachthestationaryturbulenceandthereafterthestatisticsareaveragedoverapproximately20large eddyturnovertime. 92 Figure5.3:Contouroflevelset ˚ fortheIBmethod(a),meshgridfortheconformalsetup(b). CaseC1. 5.2.4ValidationofthenumericalmethodandtheproposedIBmethod Thenumericalmethodisvalidatedbysimulatingsupersonicturbulentchannel˛owoverasmooth wallat M = 1 : 5 andRe = 3000 .ThesamesetupwasalsoemployedbyColemanetal.[7],which isusedhereasthebenchmarkstudy. Figure5.2comparesmeanandturbulencepropertiesofthepresentsimulationwiththoseof Colemanetal.[7].Asthe˝gureshows,thetwosimulationsareinagoodagreementforall depictedvariables.Thisveri˝esthatournumericalsolverisimplementedcorrectly. TovalidatetheproposedIBmethod,wehavesimulatedcaseC1intwoways:oneusestheIB methodandtheothersolvestheconventionalNSequationsonaconformalbody-˝ttedmeshsetup. ThecontouroflevelsetfortheIBmethodandthemeshoftheconformalsetupareshownin˝gure 5.3. Figure5.4showspro˝lesofdi˙erentmeanandturbulencevariables,includingmeanvelocity, temperatureanddensityaswellasReynoldsstressesandvarianceoftemperature.Alltheseplots indicateaverygoodmatchbetweenthesimulationwithIBmethodandthesimulationwiththe 93 Figure5.4:PlotsofthemeanandturbulencevariablesforcaseC1,simulatedbyusingtheIB method(solidlines)andtheconformalmeshsetup(dashlines).Pro˝lesofdouble-averaged velocity,temperatureanddensity(a),RMSofvelocitycomponentsinplusunits(roughnessside, b),timeandspanwiseaverageofvelocityandtemperatureattheroughnesscrestandvalley locations(c),andRMSoftemperature(d).Theverticaldot-dashlinesshow y = k c . body-˝ttedmesh.Therefore,weconcludetheproposedIBmethodhastheaccuracyneededfora validDNSanalysis. 5.3Results Contoursoftheinstantaneousstreamwisevelocity˝eldareplottedin˝gure5.5forallcases. Strongmodi˝cationsofthenearwallturbulence,especiallyintheroughnessside,arenoticeable fromthe˝gures.Themaindi˙erencebetweenthepresentgeometries,however,istheshock patternsgeneratedby2Dand3Droughnesses,whicharevisibleincontoursof r u ,shownin ˝gure5.6.Asthis˝gureshows,2Dsurfaces(casesC1andC2)impartstrongshockpatternsthat 94 Figure5.5:Contoursofinstantaneous u . 95 goallthewayuptotheuppersurfaceandre˛ectfromthiswalltothedomain.Theseshockpatterns exhibitthesamewavelengthoftheroughnessgeometries,andcanin˛uencethe˛owpropertiesin thewholechannel.Thisisobviousinthecontoursofinstantaneoustemperature˝eldsin˝gure5.7, wheretemperatureperiodicallychangesinthecompressionandexpansionregionsassociatedwith roughnessgeometriesinC1andC2.For3Dcasesalltheembeddedshockpatternsarebrokenand, consequently,replacedbythesmall-scaleshocklets,whicharedistributedoverthewholedomain withaveryslightdependenceontheroughnesswavelengths. 5.3.1Meanandturbulencevariables Figure5.8comparespro˝lesofthemeanandturbulencequantitiesbetweendi˙erenttestcases.The meanstreamwisevelocity(˝gure5.8a)anddensity(notshown)showasimilartrendforallcases, which,otherthanthenearwallregions,areweaklydependentontheroughnessgeometryacrossthe channelheight.Thetemperaturepro˝les(˝gure5.8c),ontheotherhand,dependontheroughness topographies,andarehigherfor2Dcases(C1andC2)comparedtothe3Dones(C3andC4).As explainedinsection(5.2.1),theconstraintsof U r = 1 and ˆ r = 1 wereimposedonthevelocityand density˝elds,respectively.Theseconstraintssubsequentlypreventmajordi˙erencesinvelocity anddensitypro˝lesbetweendi˙erentroughnesscasesandthatiswhytheyarealmostindi˙erent totheroughnessgeometries.However,suchaconstraintdoesnotexistforthetemperature˝elds, andsincestrongshockpatternsin2Dcasesinvolvemoreentropyinthedomainthanthe3Dcases, theirreversibleheatgenerationishigherforthesecasesandthereforethetemperatureishigherfor themthanforthe3Dsurfaces. Thevaluesoffrictionalvelocitiesinthesmoothandroughsidesaswellasthefrictional ReynoldsnumberRe ˝ anddragcoe˚cient C f aretabulatedintable5.2.BothofRe ˝ and C f showsimilartrendstothetemperaturepro˝les: i) theyarehigherfor2Dcasesthan3Dones, ii) theydecreasewithdecreasingroughnesswavelengthin2Dsurfaces,and iii) theyincreasewith decreasingroughnesswavelengthsin3Dsurfaces.Theassociatedshockpatternsarebelievedto alsoberesponsibleforthesetrendsasexplainedabove.Weaddthatthe˛owblockagedueto 96 Figure5.6:Contoursofinstantaneous r u . 97 Figure5.7:Contoursofinstantaneous T . 98 Figure5.8:Plotsofthemeanandturbulencevariablesforallcases.Pro˝lesofthe double-averagedstreamwisevelocity(a),componentsofReynoldsstressesinplusunits (roughnessside,b),double-averagedoftemperature(c),andRMSoftemperature(d).C1(solid lines),C2(dashlines),C3(dot-dashlines)andC4(dottedlines). Case u ˝; s š U r u ˝; r š U r u ˝; a vg š U r Re ˝ C f 10 3 C10.06520.07210.06872069.4 C20.06600.06750.06682008.9 C30.06500.05770.06151847.6 C40.06570.06200.06391918.2 Table5.2:Postprocessingdata. u ˝; s = p ˝ w ; s š ˆ r and u ˝; r = p ˝ w ; r š ˆ r ,where ˝ w ; s = w d < u > d y y = 2 and ˝ w ; r = k c 0 F 1 ; ibm T d y .Here F i ; ibm = ˆ u i t isthecorresponding bodyforceduetoIBM( u i isthevelocitydi˙erenceof u i afterandbeforetheIBMcorrection step). T isasimpleplanaraveragingoperatorthatincludesallthesolidand˛uidcells. Re ˝ = ˆ r u ˝; a vg š w , C f = 2 ¹ u ˝; a vg š U r º 2 and u 2 ˝; a vg = u 2 ˝; s + u 2 ˝; r š 2 . 99 roughnessdoesnotchangesigni˝cantlybydecreasingroughnesswavelengthfor2Dsurfaces(as theroughnesselementsarealreadyspannedintheentirewidthofthechannel),whileitincreases forthepresent3Dsurfaces.Thiscanfurtherexplainthetrendsof (ii) and (iii) ,consideringthatthe hydraulicdragisproportionaltothe˛owblockageduetosurfaceroughness. TheRMSofvelocitycomponentsareplottedin˝gure(5.8b)inviscousunits,wherethey arenormalizedby u ˝; r (seetable5.2forde˝nition)and ; r = w š¹ ˆ r u ˝; r º .Theplotsshowthat roughnesse˙ectsarecon˝nedtotheroughnesssublayerregionandoutsidethisregionthepro˝les almostcollapseforallvelocitycomponents.The v and w componentsalsoexhibitalmostthe sametrendforregionsinsidetheroughnesssublayers.However,the u componentsfor3Dcases showhigherpro˝lesthantheir2Dcounterpartsfor y k c .Thisisbecausetheycanimpartlarger coherentmotions(oftheordersgreaterthan x )inthestreamwisedirectionthanthe2Dsurfaces in y k c ,duetotheirbeingthreedimensional. TheRMSoftemperature,˝gure(5.8d),dependsstronglyontheroughnessgeometryinthe outerlayer.For2Dcases,theirregularlyshapedregionsinthe˝gureareassociatedwiththeshock patternsinthedomain,andoccurinthelocationswheretheshockwavescoincideandformthe nodesoftheshockdiamonds(thenodesthatareawayfromwalls).Theseshockdiamondsare alsovisiblein˝gure5.7(C1andC2).For3Dcasestheshockdiamondsaredeterioratedbythe roughnesses,thereforenoobviousbumpyregionoccursintheirpro˝lesoftheRMSoftemperature. 5.3.2BudgetsoftheReynoldsstresses Thetransportequationfordi˙erentcomponentsoftheReynoldsstresstensorreadsas[153] @ @ t ¹ ˆ u 00 i u 00 j º = C ij + P ij + D M ij + D T ij + D P ij + ij + ij + M ij ; (5.12) where i , j ={1,2,3}and C , P , D M , D T , D P , , and M ,are,respectively,meanconvection, production,moleculardi˙usion,turbulentdi˙usion,pressuredi˙usion,pressure-strain,dissipation, 100 andturbulentmass˛uxterms,andarede˝nedas C ij = @ @ x k ¹ ˆ u 00 i u 00 j e u k º P ij = ˆ u 00 i u 00 k @ e u j @ x k ˆ u 00 j u 00 k @ e u i @ x k D M ij = @ @ x k ¹ u 00 i ˝ kj + u 00 j ˝ ki º D T ij = @ @ x k ¹ ˆ u 00 i u 00 j u 00 k º D P ij = @ @ x k ¹ p 0 u 00 i jk + p 0 u 00 j ik º ij = p 0 @ u 00 i @ x j + @ u 00 j @ x i ij = ˝ ki @ u 00 j @ x k ˝ kj @ u 00 i @ x k M ij = u 00 i @ ˝ kj @ x k @ p @ x j + u 00 j @ ˝ ki @ x k @ p @ x i : (5.13) Thebudgestarecalculatedforallnon-zerocomponentsoftheReynoldsstresstensor, B 11 , B 22 , B 33 and B 12 aswellasforTKE.However,onlytheresultsofTKEand B 11 areshownherefor brevity(˝gures5.9and5.10).The˝guresarenormalizedbythereferenceunitsforcomparison purposes.Inthe˝gures, ˙ istheresidualofthebudgets,andislessthan1%inallcases.Thisnot onlyveri˝esthatbudgetsarecalculatedcorrectly,butalso,con˝rmsthatthenumericaldissipation (asaresultofthesolver's˛ux-splittingprocedureandtheIBmethod),issmall,whichisessential foranaccurateDNSanalysis. Thebudgettermsarestronglymodi˝edbytheroughnesselementsinall˝gures.Theproduction termsinTKEand B 11 budgets,arehigherfor2Dcasesthanthe3Dones.Beyondthestrongshear valuesinthewakesofthe2Delementswhichenhancestheshearproduction[57],therearetwo otherreasonsforthis.First,for2Dsurfaces,allstreamwiseturbulencestructuresarebrokenby theroughnesselementswhichspanthewidthofthechannel,thee˙ectsofwhichcangrowtothe boundarylayerthicknessevenintheincompressible˛ows[17].Thisresultsinsmallerlengthscales ofturbulencestructuresandenhancedpressuredrag.In3Dsurfaces,thestreamwisestructures 101 Figure5.9:BudgetsofTKE.Alltermsarenormalizedby U r and ,andaredouble-averagedin timeandthe x - z plane. wraparoundtheroughnesselements[99]andthereforereducedvortexbreakdownand,asaresult, turbulenceproductionoccursforthesecases.Thesecondreasonofenhancedturbulenceproduction for2Dsurfacesisthemutualinteractionofshockwaves.Thecontoursof P 11 aredepictedin˝gure 5.11toexplainthise˙ect.Asthe˝gureshows,for2Dsurfaces,theregionswhere2obliqueshock wavesimpingetogetherhaveenhancedturbulenceproduction,whetheritisintheroughorsmooth wallside. Thisisanimportantphenomenonandrepresentsafundamentaldi˙erencebetweensupersonic andsubsonicturbulent˛owsoverroughwallsforsubsonic˛owsmostoftheroughnesse˙ects arecon˝nedtonearwallregionsandtheouterlayerisexpectedtobeindependentofthewall condition,alsoknownasTownsend[69]outerlayersimilarityhypothesis.Thishasbeenveri˝edin numerousstudiesinthe˝eld[77,13,6].Butforthesupersoniccaseshere,asisobviousin˝gure 102 Figure5.10:BudgetsofB11.Alltermsarenormalizedby U r and ,andaredouble-averagedin timeandthe x - z plane. 5.11,thee˙ectsofwallroughness,viathegeneratedobliqueshocks,propagateacrossthechannel andmodifyturbulenceproductionintheupperwallregion.Thesameprocessoccursontherough wallside,wherethere˛ectedshocks,fromthesmoothside,impingetogetherneartheroughness crestandenhancetheturbulenceproductionintheseregions.Evenforthe3Droughnesses,itis conjecturedthatturbulenceprocesses,onbothsides,dependontheshockletsgeneratedbybreaking downtheshockpatterns,which,bythemselvesdependontheroughnesstopographies.Thisclearly showsthatTownsend'souterlayersimilarityhypothesisdoesnotapplytosuchsupersonicchannel ˛ows.Suchprocessesmaybeofpotentialusein˛owandturbulencecontrol,whereonecancontrol theturbulenceprocessesononesideofthechannelbyalteringthewallroughnessontheotherside. Similarto P ,otherturbulenceprocessesarealsoa˙ectedbywallroughnessandtheassociated shockpatterns,whicharealsoevidentin˝gures5.9and5.10.Inparticular,thepressure-strain 103 Figure5.11:Contoursof P 11 .Itisnormalizedby U r and . 104 term in B 11 ,reachesitsmaximummagnitudeneartheroughnesscrestlocation,whichindicates anexcessturbulenceproductioninthisregionfor vv and ww budgets,asthenegativeofthisterm actsasaprominentsourceterminthe B 22 and B 33 budgets. 5.4Concludingremarks Inthisstudyweproposedanimmersedboundarymethodtosimulatesupersonicturbulent ˛owsoverroughwalls.Tothisend,weusedalevel-setmethod,andthevelocityandtemperature ˝eldswerecorrectedtoimposeDirichletandNeumannboundaryconditionsattheinterface cells.Thedensityandpressurewerecalculatedusingthecontinuityandstateequations.The methodwasvalidatedbycomparingtheDNSresultsofasinusoidalwavywall,whichhasbeen simulatedusingboththeIBmethodandabody-˝ttedmesh.Thesimulationsresultsshowexcellent agreementbetweenmeanandturbulencequantitiescomputedwithbothmethods,whichcon˝rms thesuitabilityoftheIBmethodforaccurateDNSsimulations. Inthenextstep,wehavesimulatedsupersonic˛owsat M = 1 : 5 ,overfourroughnesstopogra- phies,twoofwhichwere2Dsinusoidalsurfacesandtwowere3Dsinusoidalsurfaces.Thesurfaces sharedthesameroughnessheight,buttheydi˙eredinthesurfacewavelengths.Ourresultsindicate strongmodi˝cationsofturbulence˝eldaswellasthemeanandRMSofthetemperature˝elds bytheroughnessgeometries.Speci˝cally,2Dsurfacesgeneratestrongobliqueshockpatterns throughoutthechannel,whichpredominantlyacttomodifytheturbulenceproductionterm, P . Contourplotsof P showthatroughnessenhances P notonlyontheroughnessside,butalsoon theinner-layeroftheotherwallofthechannel.Thischannel-˛owe˙ectisnotconsistentwiththe well-knownTownsend'souter-boundary-layersimilarityhypothesisforincompressibleturbulent ˛owsoversmoothandroughwalls. Ourresultsshowthatthe3Droughwallsbreakdowntheembeddedshockpatternsandgenerate randomlyorientedweakshocklets.Theseshockletsimpartlessentropyintothe˛ow˝eldthanthe strongshockwavesinthe2Dcases.Thiscauseslessirreversibleheatgenerationfor3Dsurfaces andcoolertemperature˝eldsinthesecases. 105 APPENDICES 106 APPENDIXA INTRINSICAREAFILTERING A.1Introduction Di˙erent˝eldsof˛uidmechanicssuchasmultiphase˛ow,˛owinporousmedia,meteoro- logical˛ows,˛owovercanopiesandroughness,whichhavesomesortofspatialandgeometrical anomalies,involveareaaveraging.Tostudyeachofthese˛owtypes,oneneedstoimplement areaaveragingtothegoverningequationsandwritetheaverageofderivativesbasedonderiva- tivesoftheaverage.Tostudyhomogeneousmultiphase˛owand˛owoverroughness,Howes& Whitaker[40]andRaupach&Shaw[39],respectively,o˙eredasolutiontocommuteaveraging andderivationoperators,andtheire˙ortswereextendedbyCushman[154],Giménez-Curtoetal. [155]andNikoraetal.[41]fornon-homogeneous˝elds.Thesourceofdi˚cultiesintheprevious studiesisthecommutingproblemwhichhasageneralandconcisesolution,whentheappropriate mathematicaltoolsareusedtodescribespatial-dependent˛owphysics. Becausearea-averagingisinherentlyarea-˝ltering,space-˝lteredensemble-averagedNavier Stokes(FANS)equationsarereintroduced.Thedouble-averagedNaveir-Stokes(DANS)equations canbeeasilyobtainedfromtheFANSequationsassumingtheaveragingareaislargeenoughthat theroughnesscouldbeassumedhomogeneous @ A r @ x = @ A r @ z = 0 ,where A r istheplanararea occupiedbytheroughness . A.2Problemformulation A.2.1Intrinsicarea˝ltering Intrinsicarea˝lteringofatestfunction ˚ isde˝nedby ˚ ¹ x º 1 A D ¹ O a ; B º ˚ ¹ x 0 ; z 0 º dA 0 : (A.1) 107 FigureA.1:Schematicofthegeometry,auxiliaryCSandde˝nitions. FigureA.1showstheschematicoftheproblem, D isthe˛uidoccupieddomainand A isits area; x = ¹ x ; y ; z ; t º determinesaparticlepositioninspaceandtime(inaright-handedcoordinate system)and˝lteringisperformedinthe x - z plane.Thegoalisthecalculationof @˚ @ x i asa functionof @<˚> @ x i (tocommuteaveragingandderivationoperators);todoso,oneneedstode˝ne anauxiliarycoordinatesystemtocalculatethevalueofintegral.Therearetwofundamentally di˙erentapproachestode˝ningthiscoordinatesystem(CS).The˝rstapproach,usedinprevious studies,isto˝xtheprimarydomainanddisplacethe˝lteringarea(see˝gureA.2a);inthiscase, wehave O a B ¹ x ; z º ; B = B x 0 ¹ y ; t º ; z 0 ¹ y ; t º ; ¹ A : 2 a ; b º where O a istheauxiliaryCSoriginand B determinestheboundariesofthe˝ltereddomain, D ,in thissystem.Inroughnessandporousmediastudies B isthesolid-˛uidboundaryandinmultiphase ˛ow B canbetheboundarybetweendi˙erentphases;therefore,asfaras B isconcerned, second phase and roughness willbeusedinterchangeablyhenceforth. Although @˚ @ x i x iscalculableusingthe˝rstviewpoint,itrequiresdeliberatediscussionsand advancedmathematicalmethods.The time coordinatealsoneedsspecialtreatment. Aswillbeshown,thereisageneralandstraightforwardmethodtocalculate @˚ @ x i .Wemust changeourviewoftheproblemandusethesecondapproach i : e : to˝xthe˝lteringareaanddisplace 108 FigureA.2:Schematicoftwofundamentalviewpoints:(a)auxiliaryCSmoves,primaryCS˝xed. (b)auxiliaryCS˝xed,primaryCSmoves.Inboth˝guressubscript m standsfor`moving', IC standsfor`inertialCS';and G servesas O a inauxiliaryCSand x inprimaryCS,inthemeantime. theprimarydomain(˝gureA.2b).Inthiscase,wehave O a B ¹ 0 ; 0 º ; B = B x 0 ¹ x ; y ; z ; t º ; z 0 ¹ x ; y ; z ; t º : ¹ A : 3 a ; b º Bydoingso,andassumingthat ˚ hascompactsupportand B issmoothenough(asisthe caseforanyphysicalprocess),allconditionsofLeibniz'sintegralrulearesatis˝edand @˚ @ x i x is calculableusingthisruleas D @˚ @ x i E = 1 A D @˚ @ x i dA 0 = 1 A @ D ˚ dA 0 @ x i B ˚ v i ^ n dl ! = 1 A @ A ˚ @ x i 1 A B ˚ v i ^ n dl ; (A.4) where v i israteofdilatationof˛uidboundaryin i th direction(including t ).Wecanseparate theboundaryintotwoparts:theinterior, B int ,andtheexterior, B ext .Tostudyregulargeometries, onecan˝xtheexternalboundary ¹ v i B ext = 0 º andtheequationwillbe D @˚ @ x i E = 1 A @ A ˚ @ x i 1 A B int ˚ v i ^ n dl : (A.5) 109 FigureA.3:Representationof x - dir velocity,goingfrom P 1 to P 2 ,in x - dir with x unitlongin primaryCS,willresultinchangeofabscissaofapointinauxiliaryCS, J ,of x 0 = x . ^ n isunitnormalvector,positiveoutofthe˛uid;andtheintegralshouldbetraversedcounter clockwiseon B int accordingtothetheorem. A.2.2Calculationof v i time : v t isthevelocityoftheboundaryat x - z plane v t = @ x 0 @ t ; @ z 0 @ t = v b x zplane : (A.6) x - dir :considering˝gureA.3,andassumingweareatpoint P 1 andwanttocalculate @˚ @ x at point P 2 ,weshoulddisplacetheprimaryplanein i º directionwithlengthof x = x P 2 x P 1 . Inthemeantime,thevalueof x 0 atanarbitrarypointontheboundary, J ,wouldchangeby x 0 = x P 2 x P 1 º = x intheauxiliarycoordinatesystem.Onecaneasilyseethat v x = @ x 0 @ x ; @ z 0 @ x = i : (A.7) z - dir :wecansimilarlydeducethat v z = k . y - dir : v y canberepresentedinvariousformsaccordingtothetypesofcoordinatesystem.Here, wecalculatedboth v y andthe y - dir integraltermof(A.5)inCartesianandcylindricalcoordinate systems. CartesianCS:inthissystem v y = @ x 0 @ y ; @ z 0 @ y ,andaccordingtotheconventionfor ^ n andthedirection 110 oftraversingthecurvature,inaright-handedcoordinatesystem,wehave ^ n dl = ¹ dz 0 ; dx 0 º . Therefore B int ˚ v y ^ n dl = B int ˚ @ x 0 @ y dz 0 B int ˚ @ z 0 @ y dx 0 : (A.8) CylindricalCS:inthissystem,apointontheboundaryofmanifoldcanberepresentedby x j b = r = r ^ r ;and ^ n dl = ^ r rd + ^ dr ,thus v y = @ ¹ r ^ r º @ y = @ r @ y ^ r + r @ ^ r @ y = @ r @ y ^ r + r @ @ y ^ ; (A.9) B int ˚ v y ^ n dl = B int ˚ r @ @ y dr B int ˚ r @ r @ y d : (A.10) A.2.3GeneralformofcontinuityandFANSequations ImplementingArea-FilteringtotheReynolds-Averagedcontinuityequation,oneobtainstheconti- nuityequationinthefollowingform @ U j @ x j = U j A @ A @ x j + 1 A B int U j v j ^ n dl : (A.11) 111 StartingwithRANSequations,onecanobtainthegeneralformofFANSequationsasfollows @ U i @ t + @ U i U j @ x j = 1 ˆ @ P @ x i + @ 2 U i @ x j @ x j + g i @ u 0 i u 0 j @ x j @ R ij @ x j U i A @ A @ t P A ˆ @ A @ x i + U i U j u 0 i u 0 j R ij + @ U i @ x j ! 1 A @ A @ x j + @ U i @ x j 1 A @ A @ x j + U i 1 A @ 2 A @ x j @ x j 9 > > > > > > > >= > > > > > > > > ; i A B int @ U i @ x j v j ^ n dl + 1 A ˆ B int P v i ^ n dl + 1 A B int U i v t ^ n dl + 1 A B int U i U j v j ^ n dl A @ ¹ B int U i v j ^ n dl º @ x j + 1 A B int u 0 i u 0 j v j ^ n dl 9 > > > > > > >= > > > > > > > ; = i ; (A.12) where R ij U i U j U i U j isthedispersivestressterm. Equation(A.12)isthemostgeneralformoftheEnsemble-Averaged,Area-FilteredIncompress- ibleNavierStokesequationswhichcanbeimplementedtostudyawiderangeof˛uidsphysics including,butnotrestrictedto:immisciblemulti-phasemulti-components˛uids(bywritingthe equationforeachphaseandregardingintegraltermsascouplingagentsbetweenphases),mete- orological,atmosphericandspecies˛ows(inwhich˛ow˝eldisa˙ectedbyspecies˝eld),˛ow inporousmedia,and˛owovercanopiesorroughsurfaces.Theequationscanbeusedtostudy mixturesofthese˛owsaswell( e : g : multiphase˛owoverroughsurfaces).Inthissense,theproce- durepresentedhereuni˝esallaforementionedstudies.Wewillelaborateon˛owoverroughness insubsequentsections. Alltermsinthe˝rstlineof(A.12)areclosed,butthesecondlinecontainswellknownReynolds anddispersivestresses,whicharetobemodeled. Theimportant,butunstudiedtermsin(A.12)are i and = i .Similartermsexistinmass conservationequation(seeA.11)aswell. i representsthee˙ectofareadistribution,inall 112 directions,onthe˛ow˝eld.Thistermisaconservationofmomentum(ormassinA.11)due toareachange.Theterm`conservation'waschosenbecause isduetoareachangeandnotdue toanexplicitexternalforce.Iftheareadistributionisknown,whichisthecaseinstudyof˛ow overmostroughsurfaces, i isclosedanditdoesnotneedanymoremodi˝cations.Ifthearea distributionisunknown(suchasinmultiphase˛ows),thenthedeterminationof i willdepend uponthemodelingofareagradients.Inthestudyof˛uidphysicsthathavesomesortofinteraction withsolidmaterials,like˛owoverroughnessandcanopiesor˛owinporousmedia, i isanexplicit e˙ectofthegeometryofthesolidon˛owdynamics,anditintroduces˛uidareadistribution,or interchangeably,solidareadistribution,asaprimarymeansbywhichthesolida˙ectsthe˛uid dynamics. = i in(A.12)isthebalanceofmomentum(ormassinA.11)duetointeractionwithotherphases, orsolidparts.Thistermrepresentsdi˙erentprocessesinwhichonephaseinteractswithothers, amongwhichthepressureandviscousdragterms(˝rstlinein = i )arethebestknownones.The term @ ¹ B int U i v j ^ n dl º @ x j isaviscouse˙ectduetothegradientofthebulkexpansion-contraction oftheboundaries. A.2.4FANSequationsforfully-developedchannel˛ows Forfully-developedchannel˛owsoverroughwalls,onenotices v s = w s = 0 and @ u s @ x = @ u s @ z = 0 .Assuminghomogeneousroughness @ A r @ x = @ A r @ z = 0 ,thenthestreamwiseFANS equationreducesto d P s dx @ u 0 v 0 s @ y @ e u e v s @ y + @ 2 u s @ y 2 + f v + f p = 0 ; (A.13) where, f = A t B int @ U 1 @ x j v j ^ n dl f p = 1 A t B int P v 1 ^ n dl ; (A.14) 113 aretheformviscousandpressuredragterms,respectively.EquationA.13isthestreamwiseDANS equationforfully-developedchannel˛owsoverhomogeneousroughwalls. 114 APPENDIXB CORRECTIONOFNEUMANNBOUNDARYCONDITIONUSINGLEVEL-SET METHOD DirectforcingapproachesaredevisedforDirichlet'sboundarycondition,andtheyposedsome issueswithimplementationofNeumann'sboundarycondition,whicharemostlybecausethe sourceterminright-hand-sideofNSequationscannotbede˝nedsimplytosatisfyNeumann's BCinthismethod.Thisproblemcanbetreatedbyusingalevel-setfunctionandintroducing anothercorrectionstepinthesolver,whichcorrectstheboundaryvaluestoaccountfor˛uxesat theboundary,suchthat @ P @ n = r P r ˚ r ˚ = q where P isageneral˝eldparameter, n isthenormaldirection(positiveinto˛uid), ˚ isthelevel-set function(0inthesolid,1inthe˛uid),and q isthe˛uxof P attheboundary. Numerically,oneneedstouseone-sideddiscretization(usedatafrom˛uidcellsonly)for r P , inordertocuttheconnectionofsolidand˛uidzones,thiscanbeachievedsimplybyusingthe directionof r ˚ .Forexample,ina2D˛ow,inthelocationswhere ˚ x < 0 ;˚ y > 0 ,thecorrection FigureB.1:2Dheatequationsolution,usingbody-conformalmesh (left) ,andCartesianmesh(IB methodwithNeumannBCcorrection, right ). 115 stepwillreduceto P ij = ˚ i + 1 ; j ˚ i 1 ; j º P i 1 ; j + 2 ¹ ˚ i ; j + 1 ˚ i ; j 1 º P i ; j + 1 ˚ i + 1 ; j ˚ i 1 ; j º + 2 ¹ ˚ i ; j + 1 ˚ i ; j 1 º (B.1) where = x y .Pleasenoteuseofbackwarddiscretizationfor P x ,andforwarddiscretizationfor P y . FigureB.1comparesthesolutionofaheatequation T = 0 (withDritchet'sBCfortheexternal boundariesandNeumann'sBCfortheinternalboundary)obtainedbythismethod,withthe solutionofthesameproblemobtainedbyabody-conformalmeshinFluentsoftware.Asone notices,Neumann'sBCissatis˝edproperlyusingtheIBmethod. ThediscretizationusedinequationB.1is˝rst-orderaccurate.Extendingthisapproachto high-order(one-sided)di˙erentiationschemesisstraightforward. 116 BIBLIOGRAPHY 117 BIBLIOGRAPHY [1] L.Rempel.Rotorbladeleadingedgeerosion-reallifeexperiences. WindSystemsMagazine , pages2012. [2] L.F.Moody.Frictionfactorsforpipe˛ow. ASMETrans. ,66:671684,1944. [3] M.P.SchultzandK.A.Flack.Reynolds-numberscalingofturbulentchannel˛ow. Phys. Fluids ,2013. [4] S.G.SaddoughiandS.V.Veeravalli.Localisotropyinturbulentboundarylayersathigh Reynoldsnumbers. J.FluidMech. ,1994. [5] J.YuanandU.Piomelli.Estimationandpredictionoftheroughnessfunctiononrealistic surfaces. J.Turbul. ,2014. [6] M.AghaeiJouybari,G.J.Brereton,andJ.Yuan.Turbulencestructuresoverrealisticand syntheticwallroughnessinopenchannel˛owatRe ˝ = 1000 . J.Turbul. , December2019. [7] G.N.Coleman,J.Kim,andR.D.Moser.Anumericalstudyofturbulentsupersonic isothermal-wallchannel˛ow. J.FluidMech. ,1995. [8] G.S.WilliamsandA.Hazen. Hydraulictables;theelementsofgagingsandthefriction ofwater˛owinginpipes,aqueducts,sewers,etc.asdeterminedbytheHazenandWilliams formulaandthe˛owofwateroversharp-edgedandirregularweirs,andthequantity discharged,asdeterminedbyBazin'sformulaandexperimentalinvestigationsuponlarge models .J.Wiley&sons,NewYork,1909. [9] F.Farshad,H.Rieke,andJ.Garber.Newdevelopmentsinsurfaceroughnessmeasurements, characterization,andmodeling˛uid˛owinpipe. J.Pet.Sci.Eng. ,29(2):139150,2001. [10] F.Canovaro,E.Paris,andL.Solari.E˙ectsofmacro-scalebedroughnessgeometryon˛ow resistance. WaterResour.Res. ,43(10),2007. [11] K.S.CunninghamandA.I.Gotlieb.Theroleofshearstressinthepathogenesisof atherosclerosis. Lab.Investig. ,2005. [12] F.M.White. FluidMechanics .McGraw-Hill,NewYork,7thededition,2010. [13] J.Jiménez.Turbulent˛owsoverroughwalls. Annu.Rev.FluidMech. ,2004. [14] M.R.Raupach,R.A.Antonia,andS.Rajagopalan.Rough-wallboundarylayers. Appl. Mech.Rev. ,1991. [15] S.K.Robinson.Coherentmotionsintheturbulentboundarylayer. Annu.Rev.FluidMech. , 1991. 118 [16] O.Coceal,A.Dobre,T.G.Thomas,andS.E.Belcher.Structureofturbulent˛owover regulararraysofcubicalroughness. J.FluidMech. ,2007. [17] R.J.Volino,M.P.Schultz,andK.A.Flack.Turbulencestructureinboundarylayersover periodictwo-andthree-dimensionalroughness. J.FluidMech. ,2011. [18] S.TalapatraandJ.Katz.Coherentstructuresintheinnerpartofarough-wallchannel˛ow resolvedusingholographicpiv. J.FluidMech. ,2012. [19] L.Chan,M.MacDonald,D.Chung,N.Hutchins,andA.Ooi.Secondarymotioninturbulent pipe˛owwiththree-dimensionalroughness. J.FluidMech. ,2018. [20] M.R.RaupachandA.S.Thom.Turbulenceinandaboveplantcanopies. Annu.Rev.Fluid Mech. ,1981. [21] A.Christen,E.vanGorsel,andR.Vogt.Coherentstructuresinurbanroughnesssublayer turbulence. Int.J.Climatol. ,2007. [22] S.C.Kassinos,W.C.Reynolds,andM.M.Rogers.One-pointturbulencestructuretensors. J.FluidMech. ,2001. [23] J.Yuan,G.J.Brereton,G.Iaccarino,A.A.Mishra,andM.Vartdal.Single-pointstructure tensorsinrough-wallturbulentchannel˛ow.In Proceedingsofthe2018SummerProgram , 2018. [24] C.Klipp.Winddirectiondependenceofatmosphericboundarylayerturbulenceparameters intheurbanroughnesssublayer. J.Appl.Meteorol.Climatol. ,2007. [25] SchumannU.andSchmidtH.Heattransferbythermalsintheconvectiveboundarylayer. AdvancesinTurbulence2 ,1989. [26] A.Sera˝movich,C.Thomas,andT.Foken.Verticalandhorizontaltransportofenergyand matterbycoherentmotionsinatallsprucecanopy. Bound.-Lay.Meteorol. , 2011. [27] Y.Z.Zhang,C.Sun,Y.Bao,andQ.Zhou.Howsurfaceroughnessreducesheattransportfor smallroughnessheightsinturbulentRayconvection. J.FluidMech. ,836:R2, 2018. [28] J.YuanandU.Piomelli.Numericalsimulationofaspatiallydevelopingacceleratingbound- arylayeroverroughness. J.FluidMech. ,2015. [29] P.R.Spalart.Numericalstudyofsink-˛owboundarylayers. J.FluidMech. , 1986. [30] R.NarasimhaandK.R.Sreenivasan.Relaminarizationinhighlyacceleratedturbulent boundarylayers. J.FluidMech. ,1973. [31] H.W.Coleman,R.J.Mo˙at,andW.M.Kays.Theacceleratedfullyroughturbulent boundarylayer. J.FluidMech. ,1977. 119 [32] S.P.Schneider.E˙ectsofroughnessonhypersonicboundary-layertransition. J.Spacecr. Rockets ,2008. [33] B.D.Kocher,C.S.Combs,P.A.Kreth,J.D.Schmisseur,andS.J.Peltier. Investigationofthe E˙ectsofDistributedSurfaceRoughnessonSupersonicFlows .2017. [34] P.A.Durbin.Near-wallturbulenceclosuremodelingwithoutdampingfunctions. Theor. Comput.FluidDyn. ,1991. [35] J.Nikuradse.Strömungsgesetzeinrauhenrohren. VDI-Forsch. ,1933. [36] K.A.Flack.MovingbeyondMoody. J.FluidMech. ,2018. [37] E.Napoli,V.Armenio,andM.DeMarchis.Thee˙ectoftheslopeofirregularlydistributed roughnesselementsonturbulentwall-bounded˛ows. J.FluidMech. ,2008. [38] K.A.FlackandM.P.Schultz.Reviewofhydraulicroughnessscalesinthefullyrough regime. J.FluidsEng. ,2010. [39] M.R.RaupachandR.H.Shaw.Averagingproceduresfor˛owwithinvegetationcanopies. Bound.-Lay.Meteorol. ,1982. [40] F.A.HowesandS.Whitaker.Thespatialaveragingtheoremrevisited. Chem.Eng.Sci. , 1985. [41] V.Nikora,I.McEwan,S.McLean,S.Coleman,D.Pokrajac,andR.Walters.Double- averagingconceptforrough-bedopen-channelandoverland˛ows:theoreticalbackground. J.Hydr.Eng. ,2007. [42] J.Finnigan.Turbulenceinplantcanopies. Annu.Rev.FluidMech. ,2000. [43] E.Mignot,E.Barthelemy,andD.Hurther.Double-averaginganalysisandlocal˛owcharac- terizationofnear-bedturbulenceingravel-bedchannel˛ows. J.FluidMech. , 2009. [44] J.YuanandU.Piomelli.Roughnesse˙ectsontheReynoldsstressbudgetsinnear-wall turbulence. J.FluidMech. ,760:R1,2014. [45] A.BusseandT.O.Jelly.In˛uenceofsurfaceanisotropyonturbulent˛owoverirregular roughness. FlowTurbul.Combust. ,pages2020. [46] M.G.Giometto,A.Christen,C.Meneveau,J.Fang,M.Krafczyk,andM.B.Parlange. Spatialcharacteristicsofroughnesssublayermean˛owandturbulenceoverarealisticurban surface. Bound.-Lay.Meteorol. ,pages2016. [47] K.Suga,T.J.Craft,andH.Iacovides.Ananalyticalwall-functionforturbulent˛owsand heattransferoverroughwalls. Int.J.HeatFluidFl. ,2006. [48] J.Nikuradse.Lawsof˛owinroughpipes. NACATechnicalMemorandum1292 ,1933. 120 [49] C.H.Lee.Roughboundarytreatmentmethodfortheshear-stresstransportk- ! model. Eng. Appl.Comput.FluidMech. ,2018. [50] M.Samiee,A.Akhavan-Safaei,andM.Zayernouri.Afractionalsubgrid-scalemodelfor turbulent˛ows:Theoreticalformulationandaprioristudy. Phys.Fluids ,32(5):055102, 2020. [51] J.George,A.DeSimone,G.Iaccarino,andJ.Jimenez.Modelingroughnesse˙ectsin turbulentboundarylayersbyellipticrelaxation.In Proc.SummerProgram,Centerfor TurbulenceResearch ,pages2010. [52] J.YuanandU.Piomelli.Numericalsimulationsofsink-˛owboundarylayersoverrough surfaces. Phys.Fluids ,2014. [53] A.Keating. Large-eddysimulationofheattransferinturbulentchannel˛owandinthe turbulent˛owdownstreamofabackward-facingstep .PhDthesis,UniversityofQueensland, 2004. [54] A.Scotti.Directnumericalsimulationofturbulentchannel˛owswithboundaryroughened withvirtualsandpaper. Phys.Fluids ,2006. [55] S.Leonardi,P.Orlandi,andR.A.Antonia.Propertiesofd-andk-typeroughnessina turbulentchannel˛ow. Phys.Fluids ,19(12):125101,2007. [56] E.Napoli,V.Armenio,andV.DeMarchis.Thee˙ectoftheslopeofirregularlydistributed roughnesselementsonturbulentwall-bounded˛ows. J.FluidMech. ,2008. [57] J.YuanandM.AghaeiJouybari.Topographicale˙ectsofroughnessonturbulencestatistics inroughnesssublayer. Phys.Rev.Fluids ,2018. [58] A.MajumdarandB.Bhushan.Roleoffractalgeometryinroughnesscharacterizationand contactmechanicsofsurfaces. J.Tribol. ,1990. [59] P.R.Bandyopadhyay.Rough-wallturbulentboundarylayersinthetransitionregime. J. FluidMech. ,1987. [60] R.D.MoserandP.Moin.Thee˙ectsofcurvatureinwall-boundedturbulent˛ows. J.Fluid Mech. ,1987. [61] D.Pokrajac,L.J.Campbell,V.Nikora,C.Manes,andI.McEwan.Quadrantanalysisof persistentspatialvelocityperturbationsoversquare-barroughness. Exp.Fluids , 2007. [62] P.S.Jackson.Onthedisplacementheightinthelogarithmicvelocitypro˝le. J.FluidMech. , 1981. [63] R.Mejia-AlvarezandK.T.Christensen.Low-orderrepresentationsofirregularsurface roughnessandtheirimpactonaturbulentboundarylayer. Phys.Fluids , 2010. 121 [64] J.M.BarrosandK.T.Christensen.Observationsofturbulentsecondary˛owsinarough-wall boundarylayer. J.FluidMech. ,748:R1,2014. [65] C.VanderwelandB.Ganapathisubramani.E˙ectsofspanwisespacingonlarge-scale secondary˛owsinrough-wallturbulentboundarylayers. J.FluidMech. ,774:R2,2015. [66] J.YangandW.Anderson.Numericalstudyofturbulentchannel˛owoversurfaceswith variablespanwiseheterogeneities:Topographically-drivensecondary˛owsa˙ectouter-layer similarityofturbulentlengthscales. FlowTurb.Combust. ,2018. [67] W.Munters,C.Meneveau,andJ.Meyers.Shiftedperiodicboundaryconditionsforsimula- tionsofwall-boundedturbulent˛ows. Phys.Fluids ,28:025112,2016. [68] N.N.Mansour,J.Kim,andP.Moin.Reynolds-stressanddissipation-ratebudgetsina turbulentchannel˛ow. J.FluidMech. ,1988. [69] A.A.Townsend. Thestructureofturbulentshear˛ow .CambridgeUniversityPress,1976. [70] R.J.Adrian.Ontheroleofconditionalaveragesinturbulencetheory. Turbul.Liquids ,pages 1975. [71] J.Jeong,F.Hussain,W.Schoppa,andJ.Kim.Coherentstructuresnearthewallinaturbulent channel˛ow. J.FluidMech. ,1997. [72] J.Zhou,R.J.Adrian,S.Balachandar,andT.M.Kendall.Mechanismsforgenerating coherentpacketsofhairpinvorticesinchannel˛ow. J.FluidMech. ,1999. [73] A.Lozano-DuránandJ.Jiménez.Time-resolvedevolutionofcoherentstructuresinturbulent channels:characterizationofeddiesandcascades. J.FluidMech. ,2014. [74] X.Wu,P.Moin,J.M.Wallace,J.Skarda,A.Lozano-Duran,andJ.-P.Hickey.T turbulentspotsandspotsinboundarylayers. Proc.Natl.Acad.Sci.USA , pages2017. [75] S.Tardu. TransportandCoherentStructuresinWallTurbulence .Wiley&Sons,Inc.,2014. [76] Y.WuandK.T.Christensen.Spatialstructureofaturbulentboundarylayerwithirregular surfaceroughness. J.FluidMech. ,2010. [77] P.-Å.KrogstadandR.A.Antonia.Surfaceroughnesse˙ectsinturbulentboundarylayers. Exp.Fluids ,1999. [78] P.Passalacqua,F.Porté-Agel,E.Foufoula-Georgiou,andC.Paola.Applicationofdynamic subgrid-scaleconceptsfromlarge-eddysimulationtomodelinglandscapeevolution. Water Resour.Res. ,2006. [79] P.Orlandi,S.Leonardi,andR.A.Antonia.Turbulentchannel˛owwitheithertransverseor longitudinalroughnesselementsononewall. J.FluidMech. ,2006. 122 [80] P.Forooghi,A.Stroh,P.Schlatter,andB.Frohnapfel.Directnumericalsimulationof˛ow overdissimilar,randomlydistributedroughnesselements:Asystematicstudyonthee˙ect ofsurfacemorphologyonturbulence. Phys.Rev.Fluids ,3:044605,2018. [81] J.M.Barros,M.P.Schultz,andK.A.Flack.Measurementsofskin-frictionofsystematically generatedsurfaceroughness. Int.J.HeatFluidFlow ,72:17,2018. [82] A.Busse,M.Lützner,andN.D.Sandham.Directnumericalsimulationofturbulent˛ow overaroughsurfacebasedonasurfacescan. Comput.Fluids ,2015. [83] S.Leonardi,P.Orlandi,L.Djenidi,andR.A.Antonia.Structureofturbulentchannel˛ow withsquarebarsononewall. Int.J.HeatFluidFlow ,2004. [84] P.-Å.KrogstadandR.A.Antonia.Structureofturbulentboundarylayersonsmoothand roughwalls. J.FluidMech. ,1994. [85] K.T.ChristensenandY.Wu.Characteristicsofvortexorganizationintheouterlayerof wallturbulence.In Proc.FourthInt.Symp.onTurbulenceandShearFlowPhenomena , volume3,pagesWilliamsburg,Virginia,2005. [86] J.Sabot,I.Saleh,andG.Comte-Bellot.E˙ectsofroughnessontheintermittentmaintenance ofReynoldsshearstressinpipe˛ow. Phys.Fluids ,1977. [87] C.D.TomkinsandR.J.Adrian.Spanwisestructureandscalegrowthinturbulentboundary layers. J.FluidMech. ,2003. [88] J.C.delÁlamo,J.Jiménez,P.Zandonade,andR.D.Moser.Scalingoftheenergyspectra ofturbulentchannels. J.FluidMech. ,500:135144,2004. [89] S.HoyasandJ.Jiménez.Reynoldsnumbere˙ectsontheReynolds-stressbudgetsinturbulent channels. Phys.Fluids ,2008. [90] P.-ÅKrogstadandV.Efros.Aboutturbulencestatisticsintheouterpartofaboundary layerdevelopingovertwo-dimensionalsurfaceroughness. Phys.Fluids , 2012. [91] A.N.Kolmogorov.Thelocalstructureofturbulenceinincompressibleviscous˛uidforvery largeReynoldsnumber. Dokl.Akad.Nauk.SSSR ,1941. [92] B.Ganapathisubramani,N.Hutchins,W.T.Hambleton,E.K.Longmire,andI.Maru- sic.Investigationoflarge-scalecoherenceinaturbulentboundarylayerusingtwo-point correlations. J.FluidMech. ,2005. [93] J.Jiménez,S.Hoyas,M.P.Simens,andY.Mizuno.Turbulentboundarylayersandchannels atmoderateReynoldsnumbers. J.FluidMech. ,2010. [94] S.B.Pope. TurbulentFlows .CambridgeUniv.Press,Cambridge,U.K.,2000. [95] R.D.Moser,J.Kim,andN.N.Mansour.Directnumericalsimulationofturbulentchannel ˛owupto Re ˝ =590. Phys.Fluids ,1999. 123 [96] K.Bhaganagar,J.Kim,andG.Coleman.E˙ectofroughnessonwall-boundedturbulence. FlowTurbul.Combust. ,2004. [97] S.C.Kassinos. Astructure-basedmodelfortherapiddistortionofhomogeneousturbulence . PhDthesis,StanfordUniversity,1995. [98] R.J.Adrian.Hairpinvortexorganizationinwallturbulence. Phys.Fluids ,19:041301,2007. [99] J.Hong,J.Katz,andM.P.Schultz.Near-wallturbulencestatisticsand˛owstructuresover three-dimensionalroughnessinaturbulentchannel˛ow. J.FluidMech. ,2011. [100] R.J.Adrian. StochasticEstimationoftheStructureofTurbulentFields ,pages SpringerVienna,Vienna,1996. [101] M.Manhart.Vortexsheddingfromahemisphereinaturbulentboundarylayer. Theor. Comput.FluidDyn. ,1998. [102] W.SchoppaandF.Hussain.Coherentstructuregenerationinnear-wallturbulence. J.Fluid Mech. ,2002. [103] W.Anderson,J.M.Barros,K.T.Christensen,andA.Awasthi.Numericalandexperimental studyofmechanismsresponsibleforturbulentsecondary˛owsinboundarylayer˛owsover spanwiseheterogeneousroughness. J.FluidMech. ,2015. [104] J.P.Bons.Stand c f augmentationforrealturbineroughnesswithelevatedfreestream turbulence. J.Turbomach. ,2002. [105] J.P.Bons,R.P.Taylor,S.T.McClain,andR.B.Rivir.Themanyfacesofturbinesurface roughness. J.Turbomach. ,2001. [106] J.A.vanRij,B.J.Belnap,andP.M.Ligrani.Analysisandexperimentsonthree-dimensional, irregularsurfaceroughness. J.FluidsEng. ,2002. [107] M.P.SchultzandK.A.Flack.Turbulentboundarylayersonasystematicallyvariedrough wall. Phys.Fluids ,2009. [108] M.Thakkar,A.Busse,andN.D.Sandham.Surfacecorrelationsofhydrodynamicdragfor transitionallyroughengineeringsurfaces. J.Turbul. ,2017. [109] K.A.Flack,M.P.Schultz,andJ.M.Barros.Skinfrictionmeasurementsofsystematically- variedroughness:Probingtheroleofroughnessamplitudeandskewness. FlowTurbul. Combust. ,pages2019. [110] L.Chan,M.MacDonald,D.Chung,N.Hutchins,andA.Ooi.Asystematicinvestigationof roughnessheightandwavelengthinturbulentpipe˛owinthetransitionallyroughregime. J.FluidMech. ,2015. [111] C.E.RasmussenandC.K.I.Williams. GaussianprocessesforMachineLearning .MIT Press,Cambridge,MA,2006. 124 [112] Y.LeCun,Y.Bengio,andG.Hinton.Deeplearning. Nature ,521(7553):436,2015. [113] K.A.Flack,M.P.Schultz,J.M.Barros,andY.C.Kim.Skin-frictionbehaviorinthe transitionally-roughregime. Int.J.HeatFluidFlow ,2016. [114] J.M.Barros,M.P.Schultz,andK.A.Flack.Measurementsofskin-frictionofsystematically generatedsurfaceroughness. Int.J.HeatFluidFlow ,2018. [115] P.Forooghi,A.Stroh,F.Magagnato,S.Jakirlic,andB.Frohnapfel.Towardauniversal roughnesscorrelation. J.FluidsEng. ,2017. [116] P.MoinandK.Mahesh.Directnumericalsimulation:atoolinturbulenceresearch. Annu. Rev.FluidMech. ,1998. [117] J.Kim,P.Moin,andR.D.Moser.Turbulencestatisticsinfullydevelopedchannel˛owat lowReynoldsnumber. J.FluidMech. ,1987. [118] P.R.Spalart.Directsimulationofaturbulentboundarylayerupto R = 1410 . J.Fluid Mech. ,1988. [119] H.ChoiandP.Moin.E˙ectsofthecomputationaltimesteponnumericalsolutionsof turbulent˛ow. J.Comput.Phys. ,1994. [120] J.JimenezandP.Moin.Theminimal˛owunitinnear-wallturbulence. J.FluidMech. , 1991. [121] D.Chung,L.Chan,M.MacDonald,N.Hutchins,andA.Ooi.Afastdirectnumerical simulationmethodforcharacterisinghydraulicroughness. J.FluidMech. , 2015. [122] M.MacDonald,D.Chung,N.Hutchins,L.Chan,A.Ooi,andR.García-Mayoral.The minimal-spanchannelforrough-wallturbulent˛ows. J.FluidMech. ,2017. [123] R.J.Volino,M.P.Schultz,andK.A.Flack.Turbulencestructureinaboundarylayerwith two-dimensionalroughness. J.FluidMech. ,2009. [124] R.StornandK.Price.Di˙erentialevolutionasimpleande˚cientheuristicforglobal optimizationovercontinuousspaces. J.Glob.Optim. ,1997. [125] D.C.Reda.Reviewandsynthesisofroughness-dominatedtransitioncorrelationsforreentry applications. J.Spacecr.Rockets ,2002. [126] R.H.Radeztsky,M.S.Reibert,andW.S.Saric.E˙ectofisolatedmicron-sizedroughness ontransitioninswept-wing˛ows. AIAAJournal ,37:13701377,1999. [127] R.M.Latin. TheIn˛uenceofSurfaceRoughnessonSupersonicHighReynoldsNumber TurbulentBoundaryLayerFlow .PhDthesis,SchoolofEngineeringoftheAirForceInstitute ofTechnologyAirUniversity,1998. 125 [128] A.L.BraslowandE.C.Knox.Simpli˝edmethodfordeterminationofcriticalheightof distributedroughnessparticlesforboundary-layertransitionatMachnumbersfrom0to5. Technicalreport,NationalAdvisoryCommitteeforAeronautics,1958. [129] E.ReshotkoandA.Tumin.Roleoftransientgrowthinroughness-inducedtransition. AIAA Journal ,42:766770,2004. [130] Y.Ji,K.Yuan,andJ.N.Chung.Numericalsimulationofwallroughnessongaseous˛ow andheattransferinamicrochannel. Int.J.HeatMassTransf. ,49(7):13291339,2006. [131] D.C.Reda,M.C.Wilder,D.W.Bogdano˙,andD.K.Prabhu.Transitionexperimentson bluntbodieswithdistributedroughnessinhypersonicfree˛ight. J.Spacecr.Rockets ,45:210 215,2008. [132] M.Bernardini,S.Pirozzoli,andOrlandi.P.Compressibilitye˙ectsonroughness-induced boundarylayertransition. Int.J.HeatFluidFlow ,35:4551,2012. [133] C.J.TysonandN.D.Sandham.Numericalsimulationoffully-developedcompressible˛ows overwavysurfaces. Int.J.HeatFluidFlow ,41:215,2013. [134] I.Marusic,B.J.McKeon,P.A.Monkewitz,H.M.Nagib,A.J.Smits,andK.R.Sreenivasan. Wall-boundedturbulent˛owsathighReynoldsnumbers:Recentadvancesandkeyissues. Phys.Fluids ,22(6):065103,2010. [135] R.MittalandG.Iaccarino.Immersedboundarymethods. Annu.Rev.FluidMech. , 261,2005. [136] Y.KimandC.S.Peskin.Penaltyimmersedboundarymethodforanelasticboundarywith mass. Phys.Fluids ,19:053103,2007. [137] D.Goldstein,R.Handler,andL.Sirovich.Modelingano-slip˛owboundarywithanexternal force˝eld. J.Comput.Phys. ,1993. [138] E.A.Fadlun,R.Verzicco,P.Orlandi,andJ.Mohd-Yusof.Combinedimmersed-boundary ˝nite-di˙erencemethodsforthree-dimensionalcomplex˛owsimulations. J.Comput.Phys. , 2000. [139] Y.H.TsengandJ.H.Ferziger.Aghost-cellimmersedboundarymethodfor˛owincomplex geometry. J.Comput.Phys. ,192:593623,2003. [140] K.Luo,C.Mao,Z.Zhuang,J.Fan,andN.E.LHaugen.Aghost-cellimmersedboundary methodforthesimulationsofheattransferincompressible˛owsunderdi˙erentboundary conditionspart-ii:Complexgeometries. Int.J.HeatMassTran. ,104:98111,2017. [141] R.Ghias,R.Mittal,andH.Dong.Asharpinterfaceimmersedboundarymethodfor compressibleviscous˛ows. J.Comput.Phys. ,225:528553,2007. [142] M.de'MichieliVitturi,T.EspostiOngaro,A.Neri,M.V.Salvetti,andF.Beux.An immersedboundarymethodforcompressiblemultiphase˛ows:applicationtothedynamics ofpyroclasticdensitycurrents. Comput.Geosci. ,2007. 126 [143] A.Chaudhuri,A.Hadjadj,andA.Chinnayya.Ontheuseofimmersedboundarymethods forshock/obstacleinteractions. J.Comput.Phys. ,230(5):17311748,2011. [144] L.Wang,G.M.D.Currao,F.Han,Neely.A.J.,J.Young,andF.B.Tian.Animmersed boundarymethodfor˛uid-structureinteractionwithcompressiblemultiphase˛ows. J. Comput.Phys. ,346:131151,2017. [145] R.YuanandC.Zhong.Animmersed-boundarymethodforcompressibleviscous˛owsand itsapplicationinthegas-kineticBGKscheme. Appl.Math.Model. ,55:417446,2018. [146] I.W.Ekoto,R.D.W.Bowersox,T.Beutner,andL.P.Goss.Supersonicboundarylayers withperiodicsurfaceroughness. AIAAJournal ,46:486497,2008. [147] R.M.LatinandR.D.W.Bowersox.Flowpropertiesofasupersonicturbulentboundary layerwithwallroughness. AIAAJournal ,38:18041821,2000. [148] R.M.LatinandR.D.W.Bowersox.Temporalturbulent˛owstructureforsupersonic rough-wallboundarylayers. AIAAJournal ,40:832841,2002. [149] S.MuppidiandK.Mahesh.Directnumericalsimulationsofroughness-inducedtransition insupersonicboundarylayers. J.FluidMech. ,2012. [150] Z.LiandF.A.Jaberi.Ahigh-order˝nitedi˙erencemethodfornumericalsimulationsof supersonicturbulent˛ows. Int.J.Numer.Meth.Fl. ,2012. [151] M.Sussman,P.Smereka,andS.Osher.Alevelsetapproachforcomputingsolutionsto incompressibletwo-phase˛ow. J.Comput.Phys. ,114(1):146159,1994. [152] F.Gibou,R.Fedkiw,andS.Osher.Areviewoflevel-setmethodsandsomerecentapplica- tions. J.Comput.Phys. ,353:82109,2018. [153] M.A.Vyas,D.A.Yoder,andD.V.Gaitonde.Reynolds-stressbudgetsinanimpinging shock-wave/boundary-layerinteraction. AIAAJournal ,2019. [154] J.H.Cushman.Multiphasetransportequations:I-generalequationformacroscopicstatis- tical,local,space-timehomogeneity. Transp.TheoryStat.Phys. ,1983. [155] L.Giménez-CurtoandA.CornieroLera.Oscillatingturbulent˛owoververyroughsurfaces. J.Geophys.Res. ,1996. 127