Large eddy simulation of turbomachinery flows using a high-order implicit residual smoothing scheme

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Jean-Christophe HOARAU, Paola CINNELLA, Xavier GLOERFELT - Large eddy simulation of

turbomachinery flows using a high-order implicit residual smoothing scheme - Computers & Fluids

- Vol. 198, p.104395 - 2020

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Computers

and

Fluids

Large

eddy

simulation

of

turbomachinery

flows

using

a

high-order

implicit

residual

smoothing

scheme

J.-Ch.

Hoarau,

P.

Cinnella

∗

,

X.

Gloerfelt

DynFluid Laboratory - Arts et Métiers ParisTech - 151 boulevard de l’Hôpital, Paris 75013, France

Keywords:

LES

Implicit residual smoothing Turbomachinery

a

b

s

t

r

a

c

t

Arecentlydevelopedfourth-orderaccurateimplicitresidualsmoothingscheme(IRS4)isinvestigatedfor thelargeeddysimulationofturbomachineryflows,characterizedbymoderatetohighReynoldsnumbers andsubjecttosevereconstraintsonthemaximumallowabletimestepifanexplicitschemeisused.For structuredmulti-blockmeshes,theproposedapproachleadstotheinversionofascalarpentadiagonal systembymeshdirection,whichcanbedoneveryefficiently.Ontheotherhand,applyingIRS4ateach stageofanexplicitRunge–Kuttatimeschemeallowstoincreasethetimestepbyafactor5to10,leading tosubstantialsavingsintermsofoverallcomputationaltime.Withrespecttostandardsecond-orderfully implicitapproaches,theIRS4doesnotrequireapproximatelinearizationandfactorizationproceduresnor innerNewton-Raphsonsubiterations.Asaconsequence,itrepresentsabettercost-accuracycompromise forthe numerical simulationsof turbulentflows where the maximum timestepis controlledby the lifetimeofthesmallestresolvedturbulentstructures.Numerical resultsforthewell-documented high-pressureVKILS-89 planarturbinecascadeillustratethepotentialofIRS4forsignificantlyreducingthe overallcostofturbomachinerylargeeddysimulations,whilepreservinganaccuracysimilartotheexplicit solverevenforsensitivequantitiesliketheheattransfercoefficientandtheturbulentkineticenergyfield.

1. Introduction

Thedesignofmodernturbomachinerycomponentsrequires ad-vancedcomputationalfluiddynamics(CFD)tools.Turbomachinery flows aregenerallythree-dimensional,unsteady, andoften transi-tional.Additionally,they maybe characterizedby highMach and Reynolds numbers, especially for compressor and high-pressure turbine configurations. Finally, they involve complex geometries and moving elements, with interactions between the fixed and movingwheels.Despitetheintrinsiccomplexityofturbomachinery flows,CFDsolversarelargely employedinthedesignprocess and arepredominantlybasedonsteadyorunsteadyReynolds-averaged Navier–Stokes (RANS) models, due to their robustness and their moderatecomputational cost.However,RANS haveknown limita-tions in the prediction ofnon-equilibrium flows, and specifically flows withunsteadiness,separation,rotationaleffects, strong gra-dients and transition (see, e.g., [1]). High-fidelity, scale-resolving simulationapproacheslikedirectnumericalsimulation(DNS)and large-eddysimulation(LES)haverecentlyshowntheirpotentialfor drasticallyimprovingthepredictionofsomecrucial

turbomachin-∗ _{Corresponding author.}

Hoarau),

eryflowfeatures,likeheattransferandseparation[2–10]. Unfortu-nately,DNSstill remains prohibitively expensiveorjust unafford-ableatmoderatetohighReynoldsnumbers,likethoseoccurringin high-pressureturbinesand incompressor cascades.Onthe other hand,LES is still computationally too expensiveto be feasible in the regular design cycle. Part of the problem is, once again, the relativelyhighReynoldsnumberoftheflowover turbomachinery blades and the requirements for wall resolved LES of extremely fine meshes nearthe wall for capturing the energetic structures intheboundarylayer.ThisisespeciallycriticalforLESsolvers us-ing explicit time integration schemes [3], since fine wall resolu-tionleads tosevere restrictions on themaximum allowabletime step. The latter is much smaller than the time-step required to achieveasatisfactoryaccuracylevelofthesolution.Awayof relax-ingstabilityconstraintsconsistsinadoptingan implicittime inte-grationmethod.Unfortunately,thisgenerallyinvolvesmuchlarger computationalandmemorycosts.Asaconsequence,fullyimplicit schemesareprohibitivelyexpensivetouseandsomeformof par-tialimplicitationorapproximatecalculationoftheJacobianshasto beusedtoreducecomputationalcost toanamenablelevel. Addi-tionally,theaccuracyofsuchimplicitschemesisgenerallylimited tosecond order(the mostused approachbeingthesecond-order backwardtimediscretization),whichmayintroducesignificant dis-sipationanddispersionerrorsiflargetimestepsareused.Onthe

E-mail addresses: [email protected] (J.-Ch.

[email protected] (P. Cinnella), [email protected] (X. Gloerfelt).

otherhand,suchlargetimestepsarepreferredtocompensatethe overcostassociatedwiththeresolutionofanonlinearimplicit sys-tem of equations, including inner subiterations used to rule-out approximatelinearizationandfactorizationerrors.Nonetheless,for wall-resolvedLESandforDNS, themaximumallowabletimestep hasinanycasetobe sufficientlysmalltoresolve thelifetime of thetinieststructuresinthesimulations.

Inspection of the existing literature on LES and DNS applied to turbomachinery flows shows that the main time integrations strategiesinuseareexplicitRunge–Kuttaschemes(e.g.[6,7]) and second-orderimplicitschemeswithdualtimesteppingor Newton-Raphson subiterations [6,11]. Sometimes hybrid approaches are adopted[3],wherebyimplicitschemesareusedto relaxthetime stepconstraintsingridblocksclosetothesolidwalls,whereasless costlyexplicitschemesareusedtoadvancethecalculationingrid blocksdiscretizingthebladepassage.

In a recent work, a high-order accurate and efficient implicit scheme was proposed and applied to the LES and DNS of se-lectedgeometricallysimpleflowconfigurations,likehomogeneous isotropic turbulence and turbulent channel flows [12]. The ap-proach uses a high-order extension of the well-known Implicit Residual Smoothing (IRS) approach, initially proposed by Lerat etal. [13]and widely used inthe past to speed-up convergence of steady Euler and Navier-Stokes calculations based on Runge– Kuttatimestepping[14,15].Thehigh-orderIRSusesanimplicit bi-Laplaciansmoothing operator (of fourth-order accuracy) to filter-outhigh-frequencymodesoftheresidual,whichleadstothe solu-tionofpentadiagonalsystemsforeachspacedirectionandRunge– Kuttastage.Thanks to theefficient inversionofscalar pentadiag-onal matrices, the extra computational cost associated with the implicitoperatorwasshowntoremainmuchlowerthanstandard implicitschemesatleastfortheconsideredconfigurations.

The present paperaims atassessing the benefits ofthe high-order IRS scheme for the LES of turbomachinery flows, both in termsof accuracy and computational cost. In order to deal with turbomachinery geometries, the scheme of Cinnella and Content

[12] is generalized to multiblock curvilinear structured grids by using a finite-volume formulation. The proposed methodology is firstvalidatedon asimpleflow case(vortex advection)anda 2D turbinerotorgeometry, andsubsequentlyapplied tothe LESofa high-pressure turbine cascade configuration, specifically, the VKI LS89 cascade. The latter has been extensively investigated both experimentally and numerically [3,4,6–9,16,17]. The sensitivity of thenumericalresultstothe tuningparametersoftheIRS scheme andthe benefits compared to standard explicit Runge–Kutta and second-orderimplicitschemesareshownup.

The paperisorganizedasfollows: Section2presentsthe gov-erningequationsandthetimeandspaceintegrationschemesused inthisstudy.Section3 reportspreparatoryvalidations for under-lyingflowconfigurations.LESresultsfortheLS89cascadeare pre-sentedinSection4.Finally,Section5containsconcludingremarks andperspectivesforfuturework.

2. Numericalmethods

WeconsiderthecompressibleNavier–Stokesequationsintheir instantaneous, Reynolds-averaged orfiltered formulation. Afinite volume(FV)approachisusedtodealwithnon-Cartesianmesh ge-ometries,sothatthecorrespondingsystemofconservationlawsis writtenintheintegralform:

d dt  w d



+  ∂

φ

· n d



=0 (1)

withinitialconditions

w

(

x, y, z, 0

)

= w 0

(

x, y, z

)

where t is the time, x, y and z are Cartesian space coordinates, w=[

ρ

,

ρ

U,

ρ

V,

ρ

W,

ρ

E]T is the vector of conservative variables (with

ρ

thefluiddensity,U,V,WtheCartesiancomponentsofthe velocityvectorandEthespecifictotalenergy),



isaclosed con-trolvolume withboundary

∂



,

φ

isthephysicalfluxdensityand

n isthe unit outward normal.Theflux densitycontainsthe con-tributions ofboththeconvectiveandviscous fluxes:

φ

₌

φ

_c−

φ

_v, where

φ

cand

φ

varesmoothfunctionsofthevariables(w),andof

thevariablesandtheirspatialgradient(w,

∇

w),respectively. The viscous fluxesmay be set to zero (for inviscid flows),or maycontainthe contributionsofthe Reynoldsstresses(for RANS calculations) orthe contributionsof thesubgrid stresses(for LES calculations). In practice, the LES presented in the following are conducted without the introduction of an explicit model for the subgrid terms, i.e. we carry out an Implicit LES (ILES, [18,19]), wherebythedampingofunresolvedsubgridscalesisensured im-plicitly by the regularizing numerical dissipation term associated withthespatialscheme,asdiscussedlater.

2.1. Spatialdiscretization

Defineastructuredmeshcomposedofhexaedralcells



j,k,land

denotethecellfacesby



_j+1

2,k,l,



j,k+12,l or



j,k,l+12,suchthat:

∂



j,k,l =



j+1

2,k,l∪



j,k+12,l∪



j,k,l+12

∪



j−1

2,k,l∪



j,k−12,l∪



j,k,l−12

The cell volume is denoted by |



j,k,l| and an edge surface by

|



j+1

2,k,l

|

. For each cell face



_j+1 2,k,l,

we denote



_j+1 2,k,l

the oriented surface directed in the sense of increasing mesh in-dices. For each cell



j,k,l we identify the cell center, noted Cj,k,l,

by its coordinates (xj,k,l, yj,k,l, zj,k,l), and we denote its

maxi-mum dimension in each direction as

δ

x_j,k,l,

δ

y_j,k,l, and

δ

z_j,k,l, re-spectively. In this work, we consider cell-centered finite volume schemes, i.e. we choose to locate the unknown vector w at cell centers. Finally, we define a characteristic mesh size by h₌ max

(

max

j,k,l

δ

xj,k,l,maxj,k,l

δ

yj,k,l,maxj,k,l

δ

zj,k,l

)

.

Appliedtothecell



j,k,l,theconservationlaw(1)reads: d dt  j,k,l w d



+  ∈∂j,k,l  

φ

· n d



=0 (2)

Byintroducingsuitableapproximationsofthevolumeandsurface integrals,Eq.(2)canbewritten:

|



j,k,l

|

_dt d



V ˜w



j,k,l+S ˜

(

w j,k,l

)

=0 (3)

where_V˜ isa (linear)operatorapproximating thevolumeintegral



_˜ V w



_j ,k,l= 1

|



j,k,l

|

 j,k,l w d



+O

(

h p

)

(4)

and_S˜approximatesthesurfaceintegrals

˜ S

(

w _j,k,l

)

=  ∈∂j,k,l



 

φ

· n d



+

|



|

O

(

h p

₎



₍₅₎

foranyface



of



_j,k,l.IfEqs.(4)and(5)aresatisfied simultane-ously,theFVapproximation(3)issaidtobeaccurateatorderpin theFVsense[20].

In the present work, we use a five-point per direction spa-tial scheme [21] supplemented with nonlinear artificial dissipa-tion based on a blending of second and fourthorder derivatives

[22]. Such a scheme is third-order accurate in smooth flow re-gions onsufficientlyregular grids.Numericaldissipation through-out thedomain ismostlyensuredby the fourthderivatives,with

adissipation coefficient(denotedk4)that iskeptaslow as possi-ble whileensuring robustnessofthecalculation. The lower-order nonlinear dissipation is used to damp unphysical oscillations in flow regions characterized by discontinuities, so that the corre-sponding coefficient (denoted k2) may be set equal to zero for fullysubsonic flows. An analysisofthe spectral propertiesofthe spatial discretizationscheme for a linearscalar problem (not re-portedforbrevity, see[23]),showsthat, usingk2=0,k4=0.032 the scheme requires a resolution of 10 and 16 pointsper wave-length to achieve, respectively, a dispersion and a dissipation er-ror of 0.1% on a Fouriermode ofthe numerical solutionat each timestep.Sincethenumericalschemeintroducesalreadysufficient dampingof thesolutionatsmaller scales,andbased onour pre-vious studies [24,25],we choose an Implicit LES (ILES)modeling approach,whichreliesonthenumericalviscosityforsolution reg-ularization.

2.2. Timeintegrationschemes

Inthissectionwefirstrecallthebaselineexplicittime integra-tion scheme, andthen introduce the high-order implicitresidual smoothing procedure. The formulation ofthe second-order back-ward implicittime integrationscheme,usedforcomparison with the IRS in thefollowing sections,is also describedfor complete-ness.

Afterapproximatingthespaceintegralsbyasuitable discretiza-tionscheme,Eq.(1)maybeformallyrewritten:

dw dt

j,k,l + 1

|



j,k,l

|

R

(

w j,k,l

)

=0 (6)

where _R is the space approximation operator. The semi-discrete

Eq.(6)representsasetofordinarydifferentialequations, depend-ingonthenumberofdegreesoffreedom,controlvolumesorgrid pointscontainedineachgrid.

2.2.1. ExplicitRunge–Kuttascheme

Thebaselineexplicittimeintegrationschemeisthelow-storage six-stepoptimizedRunge–Kutta(RK6)methodofBogeyandBailly

[26],widelyusedintheliteratureforLESandDNScalculation.This maybewrittenincompactformas

⎧

⎪

⎪

⎨

⎪

⎪

⎩

w (_j,k,l0) = w n j,k,l



w (_jk_,k,l)= −ak



t

|



j,k,l

|

R

(

w (k−1) j,k,l

)

, k =1, . . . s w n_j,k,l+1 = w (_j,k,ls) (7)

where wn _{is the numericalsolution attime n}

_

_t,

_

_w(k)_=w(k)₋

w(0)isthesolutionincrementatthekthRunge–Kuttastage,s=6 isthe numberofstages,andak arethe optimizedscheme

coeffi-cients. The lattercanbe found in[26].The precedingRK6is for-mallyonlysecond-orderaccuratebutexhibitsverylowdispersion and dissipation errors up to the lowest resolved frequencyfor a giventime-step



t.

2.2.2. Implicitresidualsmoothingscheme

The stability domain of the explicit RK schemes can be en-largedbyusinganimplicitresidualsmoothing(IRS)technique.The main idea of IRS isto run the explicitscheme with a time step greaterthanitsstabilitylimit.Thecalculationisthenstabilizedby smoothingthe residualbymeans ofadissipativespatialoperator added to the left hand side of Eq.(7). MostIRS operators intro-duced in the past were only first or second order accurate (e.g.

[14,15]),andintroducedlargeadditionaldissipationanddispersion errorswithrespect tothebaseline timescheme.In thiswork we extendthestabilitydomainofthebaselineRK6bymeansofa re-centlyproposedhigh-orderIRSscheme[12],presentedhereafterin thefinite-volumeframework.

The IRS smoothsthe residuals by means of a bi-Laplacian in-crementateach Runge–Kuttastage. Incompact formthiswrites:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

w (0)_j,k,l = w n j,k,l



 d=j,k,l J d





w (_j,k,lk)= −ak



t

|



j,k,l

|

R

(

w (k−1) j,k,l

)

, k =1, . . . s w n+1 j,k,l = w ( s) j,k,l (8)

In the preceding equation, Jd denotes the implicit residual

smoothing operator in the dth direction. This is a conservative finite-volumeextensionofthefinite-differenceIRSoperatorofRef.

[12].Specifically,ford= j,Jjisdefinedas:



J j



w



j,k,l=



w j,k,l+

θ



t

_

j,k,l

×



σ

3 j+1 2,k,l

λ

4 j+1 2,k,l



δ

3 j



w



j+1 2,k,l −

σ

3 j−1 2,k,l

λ

4 j−1 2,k,l



δ

3 j



w



j−1 2,k,l



(9)

wherewe introducedthedifferenceoperatorinthegriddirection j

δ

j

(

•

)

j+1

2,k,l=

(

•

)

j+1,k,l−

(

•

)

j,k,l

suchthat,e.g.,

(

δ

3

j



w

)

j+1

2,k,l=



w j+2,k,l− 3



w j+1,k,l+3



w j,k,l−



w j−1,k,l

Thecoefficient

σ

,definedatcellface



_j+1

2,k,l isgivenby

σ

j+1 2,k,l=



t

|



j+1 2,k,l

|

with

|



_j+1 2,k,l

|

=



|



j+1,k,l

|

+

|



j,k,l

|



/2.Finally,

θ

isa tuning

pa-rameterand

λ

isthespectralradius(denoted

ρ

(•))oftheinviscid fluxJacobianalongthejthdirection:

λ

j+1 2,k,l=

ρ



d

φ

_c dw

j+1 2,k,l ·



j+1 2,k,l



wheretheinterfacevalue dφc dw

j+1 2,k,l iscomputedasanarithmetic averageoftheJacobiansatcellcentersC_j,k,landC_j+1,k,l.Similar for-mulasanddefinitionsholdformeshdirectionskandl.

Ananalyticalstudyoftheoptimumvalueof

θ

forunconditional stabilityisdifficult,butanumericalsearchfora1Dscalarproblem

[12]showsthatunconditionalstabilityisobtainedfor:

θ

0. 005

Fora 1D system ofconservation lawsanda regular Cartesian gridthe additionalerror introduced bythe IRS operator with re-specttotheexplicitschemeisoftheform:

−1 12

θ



t 4

_ρ

₍

_A

₎

4

∂

5

φ

c

∂

x 5 +O





t 4



_{, A =} d

φ

c dw (10)

ie.,the proposed IRS treatment introduces an additional error of O(



t4_{), with respect to the baseline RK6. For this reason, this} schemeisreferredtoasIRS4inthe following.Beingproportional toafifthderivativeofthefluxfunction

φ

c,thiserrorisrecognized

tohaveadispersivenature.Theerrorincreasesforlargervaluesof theCourant–Friedrichs–Lewy (CFL) numberandofthe smoothing coefficient

θ

.Asaconsequence,thelatterhastobetakenassmall aspossibletoensure stabilitywhilenot deterioratingaccuracy. A detailedstudy ofthe dissipation and dispersion errorintroduced by theIRS4 schemein theFourierspace hasbeen carriedout in

[12],showingthat the smoothingoperator doesnot alterthe ac-curacy of the baseline scheme in use significantly,provided that theCFLnumberisnottoohigh(typically,below10).Suchfindings were verified for a linearadvection problemand forthe DNS of

geometricallysimpleflowconfigurations,namelythedecayof ho-mogeneousisotropicturbulenceandaturbulentchannelflow.The aimofthiswork istoassessthevalidity oftheabove-mentioned theoreticalresultsformorerealisticandchallengingconfigurations. The factorized IRS4 operator Jd in Eqs. (8)-(9) leads to the

inversion of a pentadiagonal matrix per mesh direction at each Runge–Kuttastep.Such a matrix requiresspecial treatment close tothebordersof ameshblock. Differenttreatments are usedfor solid wallsandforpermeable boundaries(inlet, outlet, and inte-riorboundariesbetweenadjacentblocksformulti-block computa-tions).WhenDirichletboundary conditionsare applied,the solu-tionincrementattheboundary issetequalto 0.Forother kinds ofboundary conditions (ie. Neumann, mixed orperiodic) and at interiorboundaries, layers of ghost cells are used to make mesh blocksindependentfromeachotherandreducetherequired num-berofparallelcommunications.Aminimumoftwolayersofghost cellsisrequiredbythepresentfive-pointIRSoperator.Withinthe ghostcells, informationforcomputing Jd islacking, andthe IRS

operatorissimplysetequaltotheidentity,whichmeansthat the ghost cellsare advanced explicitly in time, while the right-hand sidesarecommunicatedfromtheneighboringblock.Sucha proce-dureleadstotheinversionofan

(

n+2

)

×

(

n+2

)

matrixineach meshdirection, nbeingthenumberofinnercells.Thissimplified treatmentmayhoweverleadtostabilityproblemsforhighvalues oftheCFL(largerthan10).Besides,itintroducesanadditional er-rorwithrespecttothesingle-blockIRSoperator,whichmaybe re-ducedbyusingaredundantnumberofghost cells.Increasingthe numberof ghostcells impliesa computational overcost in terms ofparallel communicationsamongmesh blocks,which varies ac-cordingtothespatialoperatorinuse.Thesimulationpresentedin thefollowingare basedon a 5-pointspatialdiscretization opera-tor,whichalsorequirestwolayersofghostcellsformultiple-block computations.Thus,ifonly twolayers ofghost cellsareused for theIRStreatment,theovercostassociatedwithparallel communi-cations to theghost cells israther small, andgood parallel scal-abilityis observed by using meshblocks with at least503 _cells. However,anumberofghostcellslargerthantwomayberequired forpreservingaccuracy,andthefinalchoicederivesfroma trade-off betweencomputationalcostandaccuracy.Theinfluenceofthe numberofghostcellsinvolvedintheIRStreatmentonboth accu-racyandcomputational efficiencyisinvestigated inthefollowing Section.

2.2.3. Backwarddifferenceschemes

InordertoassesstheperformanceoftheIRS4scheme,awidely usedsecond-order implicitscheme is alsoconsidered in some of the following simulations. Specifically, we consider the second-orderbackwardmultistepscheme(Gearscheme),whichwrites:

Fn+1 j,k,l = Dw n+1 j,k,l



t + 1

|



j,k,l

|

R

(

wn+1 j,k,l

)

=0, (11) with Dw n+1 j,k,l



t = 3



w n j,k,l−



wnj,k,l−1 2



t ,

and



wn_=wn+1_{− w}n_{.Likeall second-orderschemes,theleading}

errortermofGearschemeisofdispersivenature.

Eq. (11) represents a system of non linear equations that is solvedateachphysicaltimestepbymeansoftheNewton-Raphson method.Forthispurpose,theresidualislinearized,leadingto:

dF dw

n j,k,l



w n j,k,l=−Fj,k,ln , (12)

withtheJacobian:

dF dw

n j,k,l = 3 2



t I + 1

|



j,k,l

|

dR dw

n j,k,l

Inpractice,theJacobianofthespatialoperatorisnotcomputed exactly. As we use high-order schemes forthe evaluation of the explicit part,the full implicitation could not be performed with-outaconsiderablecomputationalcost periteration.Thisdifficulty iscircumventedbyapplyingadefectcorrectionapproach,inwhich afirst-orderRoe–Hartenoperatorisusedtoapproximatethe Jaco-bianmatrix.Finally,aKrylovsubspacemethod,namelyGMRES,is used tosolve Eq.(12) iteratively using theapproximate Jacobian. Duetothe useofaniterativetechniqueinsteadofadirectlinear solver,theresultingalgorithmiscategorizedasan”inexact” New-tonmethod.The readermayreferto[27] formoredetails onthe formulationoftheimplicitscheme.

Thenumberofsub-iterationsrequiredtoconvergeEq.(12) de-pends on the problem and on the properties of the spatial dis-cretizationscheme.Inthefollowingcalculations,thenumberof it-erationsoftheinnerloopissetequalto4andtheinexactNewton iteration is convergeduntil the residual isreduced by two order of magnitude with respect to the initial value or when a maxi-mum numberof inner iterationsisreached. Forinstance, forthe expensiveLESofSection4themaximumnumberofinexact New-tonsubiterationswaslimitedto10.

3. Preliminaryvalidations

InthissectiontheaccuracyandefficiencyoftheIRS4 are pre-liminarily assessed for two test cases preparatory to the turbine LESpresentedinSection4.Thefirsttestcaseallowsan investiga-tionoftheschemeimplementationformulti-blockgrids.The sec-ondone isapreliminaryapplicationtotheunsteadyflow around aturbinecascade,computedatthisstage bysolvingtheunsteady RANSequations.

All the numerical methods described in Section 2 are imple-mentedwithinthein-housestructuredfinite-volumecode DynHo-Lab[28].

3.1. Vortexadvection

The IRS4 is used tosimulate the advectionof a Taylorvortex byaninviscid uniformfield.Thistestcasehasbeenoftenusedto investigatethe accuracy of numericalmethods for computational aeroacousticsandisverysensitivetotheboundarytreatment[29]. TheTaylorvortexisdefinedbythefollowingvelocityandpressure fields:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

u=A y



y exp

(

α

R 2

₎

v

=

v

∞− A

_

x_x exp

(

α

R 2

)

p = p _∞−

ρ

∞ A 2 4

α



x



y exp

(

2

α

R 2

₎

(13)

whereR=



(

x− x0

)

2+

(

y− y0

)

2with

(

x0,y0

)

=

(

−12.5,12.5

)

the initialvortexposition,

α

=− ln2/b2_{andwherewechosethe} half-widthb₌10_×



x₌2_.5mandthevortexstrengthA₌10m/s.The vortex is embedded in a constant mean flow with Mach num-ber M=

v

∞/c∞=−0.5, i.e.,it isadvected downwards inthe ver-tical direction. The vortex is set in a rectangular computational domain of sides Lx=Ly=50m. The domain is discretized by a

set offour uniform Cartesian grids with a numberof cells vary-ing between 100 × 100 and1600 × 1600, subdivided into four equally sized blocks. The time-step used forthe computations is reducedaccordingtospatialmeshrefinement,andvariesbetween

Fig. 1. Vortex advection problem: L ∞ norm of the error E with respect to the exact

solution at final time, CFL = 2.5, and various numbers of ghost cells, as a function of the mesh size x .



t=2.2× 10−3_and

_

_{t=0.137× 10}−3_{s,respectively,} correspond-ingtoamaximumCFLnumberofabout2.5.Duringtheadvection, thevortexcrossesthesubdomain border,locatedaty₌0.Several computationsarecarriedoutbyvaryingthenumberofghostcells (gh),andtheresultsare comparedtothose obtainedusinga sin-glemeshblock.Forallthecomputations,thesmoothingcoefficient is

θ

=0.01andthe artificialdissipationcoefficientsofthe spatial schemearek2=0.andk4=0.032.

Fig.1showstheL_∞ normoftheerrorwithrespectto the ex-act solution,corresponding topureadvectionofthe initialvortex atconstantvelocity.Firstly, wenoticethataconvergenceorderof aboutthreeisobtainedforthesingle-blockcomputation,whichis higherthantheorderofthepresenttimeintegrationschemeand closetotheorderofthespatialscheme,showingthatthetime er-rorismuchsmallerthanthespatialerror.ThesimplifiedIRS treat-ment at inter-block boundaries reduces the convergence order if only twolayers ofghost cellsare used.Increasingthenumberof ghostcellsallowsrestoringtheoriginalaccuracyand,forgh=4or higher,thesolutionissuperposedtothesingle-blockone.

In Fig. 2 we report the iso-contours of the fluctuating pres-sure field (on the grid 200 × 200) by using an increasing num-ber ofghostcells.A highermaximum CFLnumber(equal to5)is used here,toemphasize thedifferencesamongthevariouscases. In the same figure, we alsoreport the resultsobtained withthe second-order implicit Gear scheme andthe same maximum CFL, aswell astheexactsolution.Increasingthenumberofghost-cells fromgh=2togh=5decreasesthespuriousnoisegeneratedwhen crossingtheinternalborder.Thankstotheinnersubiterations,the Gearschemegenerateslowererrorsattheinterfaceusingonlytwo layersofghostcells(asrequiredbythespatialscheme).However, it introducessignificant dispersionanddissipation errorsatinner meshpoints,leadingtoconsiderabledeformationandsmearingof thetransportedvortex.TheL_∞ errorsforanincreasingnumberof ghost cellsare plottedinFig.3forthegrid 200× 200andthree valuesoftheCFL number.Itisworth notingthattheeffectofthe ghost cellsontheerrorbecomessignificantastheCFLnumberis increasedupto10.

Forthissimple2Dproblemwedidnotobserveanyappreciable impact ofthenumberofghost cellsinuseon thecomputational cost of the IRS4 scheme(differences of theorder of 7% between gh=2andgh=6).InallcasestheIRScalculationsallowreducing theoverall CPUcost bya factorlargerthan2withrespecttothe Gearscheme.

3.2.TurbinecascadeVKILS-59

ForpreliminaryassessmentoftheIRS schemeonamore chal-lenging flow configuration, and in view of the LES of a turbine flow,the IRS4isthen appliedto a2D unsteadyflow inaturbine cascade. Morespecifically, we chose the VKI LS-59 transonic tur-binerotor cascade, previously considered in severalexperimental andnumerical studies [30–34]. To allow comparisons withother resultsintheliterature,theflowismodeledthroughtheunsteady RANS equations,supplementedby the Wilcoxk₋

ω

[1]modelof turbulence.TheVKILS-59isahigh-loadedrotorbladewithathick, roundedtrailingedge originally designedfornear-sonicexitflow conditions. This rotor blade has been extensively tested in vari-ous European wind tunnels [30]. Experiments are available in a widerange ofconditions,andSchlierenphotographs clearly indi-cate the existence of vortex shedding downstream of the blade trailing edge, which is responsible foran appreciable fraction of profile losses. The flow conditions considered for thisstudy cor-respondto an outlet isentropic Mach number equal to 1 and to a Reynolds number (based on the chord andexit conditions) of 7.44× 105_{.Theinletangleis30}◦_{.Thischoiceismotivatedbythe} fact that mostnumerical computations available inthe literature havebeen madefor theseconditions,and experimental distribu-tionsoftheisentropicMachnumberatthewallarealsoavailable forclosebyconditions.Indeed,theconfigurationisaffectedbyboth geometricalandoperationaluncertainties, sincetheblade geome-triesused inthe various testsreportedin [30] exhibit slight dif-ferencesthataffecttheshocklocationandintensity,andthe out-letisentropicMachandReynoldsnumberalsoexhibitsmall differ-encesinthevarioustests. Thecomputationaldomain,constituted ofasinglebladepassage,isdiscretizedbyasingle-blockC-gridof 384 × 32cells,withafirstcell heightleadingto



y+≈ 2.Non reflectiveboundary conditionsare applied attheinlet andoutlet boundaries,andperiodicconditionsareimposedatthelowerand upperboundariesofthedomain.Theunsteadysimulationsare ini-tializedwitha(partiallyconverged)steadyRANSfieldobtained us-ingtheimplicitbackwardEulerschemeavailableintheDynHoLab code[27]andalocaltimestep.

Allthe resultspresentedin thisSection were obtainedby ap-plying the third-order spatial scheme with artificial dissipation coefficients k2=0.5 and k4=0.032. The IRS4 was applied using

θ

=0.01assmoothingparameterandaconstanttimestepleading toamaximumCFL≈ 7,whichcorrespondstoapproximately5000 time-stepspervortexsheddingperiod.Forthepresentsingle-block grid, the effect of the number of ghost cellsis restricted to the treatmentoftheconnectionlinebehindthetrailingedge. Numer-icaltestsconductedby varyingthe numberofghost cellsdidnot revealanysignificantinfluenceontheaccuracyandcomputational cost ofthe simulations.As a consequence onlythe results corre-spondingtogh=2arereportedinthefollowingofthissection.

Forthisslow unsteady flowproblem (only the meanflow in-stabilityinthewakeiscapturedbytheunsteadyRANSsolver),the explicitRK schemeleadsto verysevere constraintson the maxi-mumallowabletimestep,andwasdiscarded.Forcomparisonwith theIRS4, wecarried out twosimulations usingthe Gearscheme. Forthefirstsimulation,we usethesamephysicaltimestepasin theIRS case.Insuch conditions,6to10 subiterationsareneeded tosatisfytheprescribedtoleranceontheresidual.Thesecondone usesatime steptentimeslarger(i.e.CFLmax≈ 70andabout500 timestepsperperiod).Inthiscase,approximately20subiterations pertimesteparenecessarytoconvergetheinnerloop.Inallcases, thesimulations were run over an integrationtime corresponding to20sheddingperiods.

InFig.4,snapshotsofthedensitygradientcorrespondingtothe threesimulationsdescribedpreviouslyarecomparedtoaSchlieren pictureobtained experimentally by Kiock etal.[30]

(correspond-Fig. 2. Vortex advection problem: Isocontours of the fluctuating pressure ( p = 100 Pa, min = −150 Pa, max = 150 Pa) on the grid 200 × 200 and CFL = 5. Positive and negative values are represented with solid and dashed lines, respectively.

ingtoslightlydifferentexitconditions).The flowischaracterized byshockwavesdepartingfromthebladetrailingedgeandby vor-texsheddinginthewake.Inallcases,aninstabilityofthewakeis observed.However,thesimulationbasedontheGearschemeand 500time stepsperperioddoesnotresolveaccuratelythevortices shedin the wake.Fig. 5 showsthe distributionof the isentropic Machnumberalongtheblade,averagedover10sheddingperiods. The present results are compared to two series of experimental datafromKiocketal.[30]andtothenumericalresultsofMichel etal. [34] (based on the Spalart-Allmaras turbulence model and usingaResidual-Based-Compactschemeforthespatial

discretiza-tion and a second-order implicit time discretization). This quan-tity of interest is little affected by the time integration scheme, andalltheresultsareingoodagreementwiththenumericaldata and matchreasonably well the experimental measurements. The largest differencesare observed at the upperside in the vicinity ofx/c=0.6,whichcorrespondstothereflectionoftheimpinging shockandisaparticularlysensitivezone.Fig.6showstheFourier transform of the time-dependent tangential force acting on the blade.Awell-definedpeakcorrespondstothesheddingfrequency. The corresponding Strouhal number (based on the trailing edge thickness and exit velocity) is about 0.21 for the computations

Fig. 3. L ∞ norm of the error E as a function of the number of ghost cells gh on the

grid 200 × 200.

Fig. 5. VKI LS-59 cascade: time-averaged wall distribution of the isentropic Mach number for various time integration schemes. Comparison with experimental

[30]and numerical results [34] .

Fig. 4. VKI LS-59 cascade: snapshots of the density gradient computed with the IRS4 scheme at CF L max ≈ 7 (a), and with the Gear scheme at CF L max ≈ 7 (b), and CF L max ≈ 70

Fig. 6. VKI LS-59 cascade: Fourier spectra of the tangential force on the blade for various time integration schemes, as a function of the Strouhal number.

Table 1

VKI LS-59 cascade: CPU cost [s] for various time integra- tion scheme. The cost corresponds to the time integration over two vortex shedding cycles. The computational costs were measured on an IBM x3750M4 supercomputer (Intel Sandy Bridge Processors). Single-processor calculation.

Scheme IRS4

GEAR GEAR

CFL 7 CFL 70 Number of iterations 10,000 10,000 1000 CPU Time (s) 10,260 15,582 4190 CPU cost per iteration 1.026 1.558 4.19

withthesmallest time step,and about0.20forthe computation withCFLmax≈ 70.Inallcases,thelatterisingoodagreementwith previousnumericalresults,e.g.[34],andinreasonableagreement withthe rangeoffrequency[0.2,0.4] observedexperimentally for asimilarcascadeconfiguration[35],althoughthelargernumerical errorsintroduced bythe second-order implicittime schemewith alargetimestepleadtoaslightlylowervalue.

With the presentsetting, the cost per physical time step and permeshcelloftheIRS4coupledwithRK6is1.2timeslowerthan thecost of the Gearscheme withthe sameCFLmax, seeTable 1. WhentheCFLmaxisincreased,thecomputationalcostperiteration ofthe second-orderschemeisnearly 3timeslargerthantheIRS. Inthiscase,however,amuchlargertimestepisallowed(although withsomelossofaccuracy)andthustheoverallCPUcostislower. Thisisan expectedresult,sincethesecond-order explicitscheme withNewton-Raphson subiterations is particularly well suited to slowunsteadyproblems.

4. Large-eddysimulationoftheVKILS-89turbinecascade

In thissection, the IRS4 scheme is applied to the LES of the flow around the VKI LS-89 planar turbine cascade instrumented atthe von Kármán Institute by Arts etal. [16]. The chord blade C is 67.647 mm long with a pitch-to-chord ratio of 0.85 and a staggerangle

χ

=55◦.The flowangleatturbineinlet isequal to 0◦. For this configuration, several sets of experimental data are available, characterized by various inlet turbulent intensities and

pressure ratios. Hereafter, we select flow conditions correspond-ing totheexperiment calledMUR129. Thiscorresponds toan in-lettotalpressureP0=1.87× 105Pa,outletisentropicMachnumber

M_is,2=0_.840 and outlet Reynolds number Re2=106. An isother-mal wall conditionis used, withwall temperatureTw=298K. In

thisexperiment,theinlet free-streamturbulenceintensityisvery low,Tu₌1%.Sinceourmaingoalistoassessnumericalmethods ratherthantocloselyreproducetheexperimentalconfiguration,no inlet turbulencewasprescribed inthe presentnumerical simula-tions. This has the advantage of simplifying the numerical setup andreducing the number ofparameters susceptible to affectthe computedsolutions.Forthecomputedconditions,theflowfieldis subsonic,andnaturalboundarylayer transitionoccursattherear ofthebladeuppersurface.

ThecomputationaldomainusedforthepresentLESisdisplayed in Fig.7(a). The mesh is a H-type structured mesh composed of 850cellsinthestreamwise direction,180inthe pitchwise direc-tionand200inthespanwise direction.Thetotal numberofcells forthebladepassageisequalto30.6 × 106_{.Thebladeupperand} lower surfacesare discretized by 550cellseach. Aclose-up view ofthe computational mesh(everyfour pointsare represented) is providedinFig.7(b).Thecorrespondingdistributionsof



y+,



x+ and



z+ are showninFig.8.Thefriction velocity,used for com-putingthewallcoordinates,isbasedonthelocalvaluesofthewall shearstress. The averagefirst layersize is2.5μm, corresponding to



y+≈ 2. Theaverage resolutionsinthe streamwiseand span-wise directionsare



x+≈ 100and



z+≈ 25.Thesevalues corre-spond to a coarse LES, but are similar to those of Collado et al.

[6].The LES isinitialized with a preliminary2D laminar calcula-tionextrudedinthespanwisedirection.Asinusoidalperturbation oftheconservativevariableswithanamplitudeof10%isappliedin thespanwisedirectiononlyontheinitialsolution,tospeed-upthe initial transient towarda fully 3D field. The simulations are first runoverabouttenflow-throughtimestoevacuatetheinitial tran-sient. Aflow-through time iscalculated as thetime requiredfor aparticle droppedatthe bladeleadingedgeto reachthetrailing edge, when traveling ata constant velocity approximated as the arithmeticaverageofthevelocityatthepassageinlet(x=0inour referenceframewithoriginatthebladeleadingedge)andthe ve-locityatthepassageoutlet(x=Ccos

χ

).Afterwards,thestatistics arecollectedoverthefivesubsequentflow-throughtimes.

Theartificialdissipationcoefficientsofthespatialschemewere takenequaltok2=0.andk4=0.064forthisseriesofcalculations. For simulations based on the IRS4 time scheme, the dimen-sional time-step is set equal to 3× 10−8 _{seconds, which} corre-spondsto amaximumCFL numberofapproximately7.The value of thesmoothing parameter is setto

θ

=0.01 andtwo layers of ghostcellsareused.Theresultsarecomparedtothoseofthe ex-plicit scheme andthe second-order implicit scheme. Due to the highcomputational cost of thesimulation, thelatter wascarried outwithamaximumnumberofNewtonsubiterationsequalto10.

Fig.9presentsa typicalfieldofthetime-averagedMach num-ber, obtained with the IRS4 scheme. The flow field is smooth andsubsonic everywhere.An overviewofthe instantaneous flow field isgiveninFig.10, showinganiso-surface oftheQ-criterion (Q=103_{) colored by the velocity magnitude (the domain} span-wise length was reproduced three times for an easier visualiza-tion), as well as the distribution of the density gradient in the backgroundplane.Aclose-upviewonthetrailingedgeisdisplayed inFig.10(b).Althoughtheaverageflowissubsonic,instantaneous weakshocksareobservedinthetrailingedgeregion.Itisalso pos-sibletoobservetheinstabilitygrowthandtransitioninthe bound-arylayeratthesuctionside,withtheformationofstructures rem-iniscentofhairpins,andtheonsetofturbulence.Finally,thewake ischaracterized by coherentvortex sheddings.Note that,starting fromadistanceofabout1/3 ofthebladechord fromthetrailing

0.10 C 0.85 C 0.7 C 1.5 C

(a)

(b)

Fig. 7. Computational domain for the LES of the LS-89 cascade (a) and close-up view of the grid (b) (every four points are represented).

edge, the mesh doesno longer provide a sufficient resolutionof the turbulent structures in the wake.However, thishas little in-fluenceonthepredictionoftheflowclosetothebladewall.Flow snapshots fortheRK6 andGearschemes arequalitatively similar andarenotreported.

Fig. 11(a)provides the time averagedwall distribution ofthe isentropic Mach number, calculated with various schemes (IRS4, RK6, and Gear). The time step selected for the RK6 simulation corresponds to amaximum CFL ofabout1,whereas fortheGear schemeweusedthesametime stepasfortheIRS4computation. This quantity isweakly sensitive to the time integration scheme

−1.0

−0.5

0.0

0.5

1.0

Curvilinear abscissa (m)

0

1

2

3

4

5

Δ

y

+

Intrados

Extrados

(a)

−1.0

−0.5

0.0

0.5

1.0

Curvilinear abscissa (m)

0

50

100

150

200

250

Δ

x

+

Intrados

Extrados

(b)

−1.0

−0.5

0.0

0.5

1.0

Curvilinear abscissa (m)

0

10

20

30

40

50

Δ

z

+

Intrados

Extrados

(c)

Fig. 8. VKI LS-89 cascade: Time- and span-averaged distributions of y+ (a), _x+

Fig. 9. VKI LS-89 cascade: time-averaged isocontours of the Mach number (IRS4).

andall theresultsarein excellentaccordancewiththe computa-tionofColladoetal.[6],correspondingtothesameflowconditions andusingacomputationalgridwithasimilarresolutiontotheone usedinthepresentcomputations.Theweakinfluenceofthetime schemeshowsthatspatialresolutionismoreinfluentialthantime accuracyforthiscase.SincenoexperimentalisentropicMach num-berdataareavailable forcaseMUR129,weconsiderexperimental results[16]forslightlydifferentflow conditions(MUR43), charac-terizedbythesameinletturbulenceintensityandoutletMachand ReynoldsnumbersasMUR129,butadifferenttotalinletpressure P0=1.435× 105Pa.

In figure 11(b), we report the time-averaged convective heat transfercoefficientatthewall,definedas:

H = q w T 0− Tw

(14)

where qw is the time-averaged wall heat flux, T0 the total free streamtemperatureandTw thewall temperature.The results

ob-tainedwithvarioustimeschemesarecomparedtothe experimen-taldataofArtsetal.[16]forcaseMUR129andto thenumerical resultsofColladoetal.[6]andSeguietal.[9].Thesolutionof Col-ladoetal.[6]wasobtainedbyusingaspatialschemesimilartothe presentoneandaH–O–Hgridwitharesolutionsimilartotheone consideredinthisstudy;timeintegrationwascarriedout usinga second-orderdualtime steppingmethodandthetimestep corre-spondstoaCFLofabout25.ThesolutionofSeguietal.[9]isbased insteadonathird-orderaccurateexplicitfiniteelementsolverand ahybrid unstructuredgrid ofabout60_{× 10}6 _{elements. Thegrid} wasadaptedtoobtain averyfinenearwallspacingofabout5,6 and6wallunitsinthewall-normal,streamwiseandspanwise di-rections,respectively.Severalconsiderationsareinorder.First, de-spitetherathercoarsegridresolution,all thepresentsimulations comparefairly to the experimental andnumerical data fromthe literature.Specifically,thepresentLEScapturesreasonablywellthe increaseofHattheuppersurfacebladenearacurvilinearabscissa of0.07,whichisduetotheboundarylayertransition.Ontheother hand,the presentcoarse simulations tend to underestimateH in themiddleoftheuppersurfaceandintherearpartofthelower surface.Thismaybe duetothecoarsegridresolutioninsuch

re-Fig. 10. VKI LS-89 cascade: iso-contour of the Q-criterion ( Q = 10 3 ) colored with

the velocity norm (a) and close up view near the trailing edge (b).

gion,characterizedbyanextremelyfineboundarylayer.Alsonote thatthereferenceexperimentsandsimulationsuseanon-zero tur-bulenceintensityattheinlet.Second,all thetime schemesinuse providesimilarresults,exceptforminordifferencesinthe transi-tion region atthe suction side. Inparticular, the IRS4 provides a solution in closeagreement with theexplicit scheme by using a timestepseventimeslarger.

Asensitivitystudyto theIRS parameter

θ

andthenumberof ghost cellsgh showsthat thesolution is essentially independent ofthechoiceoftheseparameters(seeFig.12).Thisindicatesonce againthat thesolution quality isdominatedby spatialresolution forthiscase.Furthermore,duetomesh clustering,highvaluesof theCFLnumberarereachedonlyintheclosevicinityofthewall, andthe localCFL is closeto unityelsewhere, reducing the influ-enceofIRSonthesolutionaccuracy.

Forfurtherinvestigationoftheeffectoftimeintegrationerrors, acomparisonoftheresolved turbulentkineticenergyfieldsis re-portedin Fig. 13.This quantity dependson the resolved velocity fluctuations andis thus more sensitive to numerical errors than meanflowquantities.BoththeIRS4andGearschemesprovide iso-contoursofthekineticenergyinexcellentagreementwiththe ref-erenceexplicitsolution,showingthatturbulentstructuresarewell resolvedintime.

Curvilinear Abscissa Is ent ropi c M ac h num be r -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 0.2 0.4 0.6 0.8 Explicit CFL 1 GEAR CFL 7 IRS4 CFL 7 Experiment Collado et al.

(a)

Curvilinear Abscissa He at T ra n sf er -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 200 400 600 800 1000 Explicit CFL 1 GEAR CFL 7 IRS4 CFL 7 Experiment Collado et al. Segui et al.

(b)

Fig. 11. VKI LS-89 cascade: time-averaged and span-wise averaged wall distribu- tions of the isentropic Mach number (a) and of the convective heat transfer coeffi- cient (b). Various time integration schemes.

We conclude thissection withsome considerations aboutthe computational cost of the simulations. In Table 2 we report the overall CPU time corresponding to the calculation of flow statis-tics (five flow-though times) forvarious time schemes and num-bers of ghost cells. Due to the severe constraints on the maxi-mum allowable time step, the RK6 requires about one order of magnitudemoretimestepstocovertheintegrationtimeinterval. The IRS4 scheme, on the other hand, allows increasing the time step (whilepreservinga comparableaccuracy) andhasa compu-tational cost per iteration only35% higher iftwo layers of ghost cellsareused.As aconsequence,theoverallCPU timeis reduced by morethanafactor5withrespecttotheexplicitcomputation. This represents an extremely substantial improvement for costly LES.Thecomputational gainislesserifthenumberofghost cells is increased. However,even whenusing 4ghost cells,the overall

Curvilinear Abscissa H eat Tr an sf er -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 200 400 600 800 1000 Theta 0.005 Theta 0.01 Theta 0.02 Experiment

(a)

Curvilinear Abscissa He at T ra n sf er -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0 200 400 600 800 1000 gh = 2 gh = 3 gh = 4

(b)

Fig. 12. LS-89 cascade: sensitivity of the heat transfer coefficient to the IRS param- eter θ(a) and the number of ghost cells gh (b).

Table 2

LES of the VKI LS-89 cascade: computational cost for various time integration scheme. The cost is evaluated for the time interval corresponding to the compu- tation of flow statistics (five flow-through times). The computational costs were measured on an IBM x3750M4 supercomputer (Intel Sandy Bridge Processors) and 250 processors were used in all cases.

Scheme gh Number ofiterations CPU time/Processor(s) Time periteration and processor RK6 2 450,000 933,750 2.075

IRS4 2 60,000 168,000 2.800 IRS4 3 60,000 219,000 3.651 IRS4 4 60,000 245,160 4.086 GEAR 2 60,000 348,600 5.810

Explicit CFL 1 IRS4 CFL 7

(a)

Explicit CFL 7 GEAR CFL 7

(b)

Fig. 13. VKI LS-89 cascade: close-up view of the turbulent kinetic energy isocon- tours in the trailing edge region for various time integration schemes.

computational cost is almost four times lower than the explicit scheme.These costs are given for the present calculations using 250blocksofapproximately493_{pointsandmaybedependenton} thelevel of parallelization. The CPU cost forthe Gear scheme is 1.5÷2timesgreaterthanIRS4(dependingonthenumberofghost cellsused),althoughitstillrepresentsaconsiderableimprovement (aboutafactor3)withrespecttoRK6.

The precedingresults showthe interestof usingtime implicit schemes for the LES of turbomachinery flows, the IRS4 scheme providing the best compromise between cost and accuracy. The computational gain is only weakly dependent on the number of ghost cellsinuse, which,on the other hand, haveno significant influenceonthesolutionaccuracyforthepresentcase.

5. Conclusions

A high order implicit time integration scheme based on the combinationofahigh-accurateimplicitresidualsmoothing opera-torwithanoptimizedRunge–Kuttaschemewasextendedto

struc-tured curvilinear grids by means of a finite volume formulation andapplied tothenumericalsimulationofunsteadyflowsin tur-bomachinery, the focus being put on large eddy simulations of turbulentflows throughturbinebladecascades.Theproposed ap-proach,calledIRS4, isdesignedtoenlargethestabilitydomainof theunderlyingRunge–Kuttaschemesignificantlywithoutincurring incostlymatrixinversionsandNewton-Raphsonsubiterationsand while preservingan accuracy similar to the explicitscheme over a rangeofCFL numbers ࣠10. TheIRS4 schemerequires the in-versionofascalarpentadiagonalmatrixpermeshdirection,which can be done veryefficiently. The schemeisparallelized by intro-ducing layers ofghost cellsatthe interface betweenneighboring meshblocksandby simplifyingthesmoothingoperators atblock borders,withaminimumof2layers.Numericaltestsforavortex advectionproblemshowthatverylowerrorlevelscanbeachieved byusing4to5ghostcellsinsteadoftwo.

Preliminaryvalidationsoftheschemewerecarriedoutforthe simulationoftransonicturbinerotorcascade,namely,theVKI LS-59 cascade, by solving theunsteady RANS equations.The flow is characterizedbyameanflowinstabilityinthewakeregionleading totheformationofavonKármánvortexstreet.AlthoughtheIRS4 isnot especially tailoredforcapturingthiskindoflow-frequency phenomenon,theoverall computationalcost isofthesameorder of that of a second-order implicit scheme(Gear scheme) with a timesteptentimeslarger.Thisisduetothecostoftheinner New-ton subiterations used to solve thefully implicitscheme at each physicaltimestep.

Finally, the IRS4 wasapplied to the LES of the high-pressure LS-89turbinecascade.Thesimulationscould becarriedoutusing amaximumCFL ofapproximately7(i.e.about7timeslargerthan the explicit scheme) and a comparable accuracy to the underly-ingexplicitscheme.Thisleadstoareductionofthecomputational costbyafactor5forthesameaccuracy,usingtwolayersofghost cells. This factor reduces to about 4 when the number of ghost celllayersisincreasedtofour.Notethat,forthepresent computa-tions, the solutionwasfound to be little sensitiveto the param-eters chosen for the IRS4 scheme and dominated by the spatial resolution.Specifically,varyingthenumberofghostcelllayersdid notaffecttheresults.TheIRS4solutionwasalsocomparedtothat obtainedusingtheGearschemeandthesamemaximumCFL.No appreciable differences between the numerical solutions are ob-served.However, theGearscheme isabout1.7timesmorecostly thantheIRS4,althoughitstill allowsagainofafactor3with re-specttotheexplicitscheme.Inallcases,thepresentnumerical re-sultswerefoundtocomparefairlywithexperimentaland numeri-caldatafromtheliterature,despitetheuseofarathercoarsegrid. Inconclusion,theIRS4wassuccessfullyextendedandassessed forcomplexturbulentflowsinturbomachineryconfigurationsand was demonstrated to be a promising numerical technique to speed-uplargeeddysimulationsofcomplexwallboundedflows.

Acknowledgments

This work was granted access to the HPC resources of IDRIS National Scientific Computing Centre, under the allocation num-ber A0052A07332. Thisstudywasalsofunded byDGA (Direction Généraledel’Armement)underPhDcontract2016600043.

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