Turbulence model and numerical scheme assessment for buffet computations

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Turbulence model and numerical scheme assessment for buffet computations

Eric Goncalvès da Silva, Robert Houdeville

To cite this version:

Eric Goncalvès da Silva, Robert Houdeville. Turbulence model and numerical scheme assessment for buffet computations. International Journal for Numerical Methods in Fluids, Wiley, 2004, 46 (11), pp.1127-1152. �10.1002/fld.777�. �hal-00530301�

Assessment for Buet Computations

Eri GONCALVES

LEGI

BP 53,38041 Grenoble edex 9,Frane

Eri.Gonalveshmg.inpg.fr

Robert HOUDEVILLE

ONERA-Toulouse, Departmentof Aerodynamis and Energetis,

BP 4025, Toulouse, 31055edex, Frane

Robert.Houdevilleoneert.fr

Abstrat

The preditionof shok-indued osillationsovertransoni rigid airfoilsisimportant for a

betterunderstandingofthebuetingphenomenon. TheunsteadyresolutionoftheNavier-

Stokesequations isperformedwithvarioustransport-equation turbulenemodelsinwhih

orretionsareaddedfornon-equilibriumows. Thelakofnumerialeienyduetothe

CFL stabilityonditionisirumventedbytheuseofawalllawapproah andadualtime

steppingmethod. Moreover,variousnumerialshemesareusedtotryandbeindependent

of thenumerial disretization.

ComparisonsaremadewiththeexperimentalresultsobtainedforthesuperritialRA16SC1

airfoil. TheyshowtheinterestinusingtheSSTorretionorrealizabilityonditionstoget

orretpreditions ofthefrequeny,amplitude and pressureutuations overtheairfoil.

Keywords

Buet, Unsteady Flows, Turbulene Modelling,Numerial Sheme

x, y, z ^{loalwall frame (boundarylayer)} U_τ ^{fritionveloity}

C_f ^{shearstress oeient} C_p ^{pressureoeient}

F_c, F_d ^{onvetive and diusiveux densities} P_k ^{turbulent kineti energy prodution} M∞ ^{innite Mahnumber}

Re_c ^{Reynoldsnumberbased onthemean hord} Ti ^stagnationtemperature

u, v, w ^{veloityomponentsinthe loalwall frame} q ^{total heatux,}q^v+q^t

k ^{turbulent kineti energy} P ^{statipressure}

P_r, P_rt ^{Prandtl numbers}

T ^meanstatitemperature

α ^{angleof attak} ε dissipationrate

κ ^{von Karmanonstant} ω ^speidissipation

µ, µ_t ^{moleular andeddy visosity}

ρ ^density

τ ^{total stresstensor,} τ^v+τ^t

w ^{wall value}

+ ^{wall sale}

1 ^{adjaent ell withrespetto the wall} v

visous

t

turbulent

The numerial simulation of unsteadyturbulent ows around airfoilsis motivatedbythe

need to better understand omplex owphenomena appearing inaeronauti appliations

suh as ows overairraft wings. Thepresent work fouses on the transonibuet. This

aerodynamiphenomenonresultsinaself-sustainedperiodimotionoftheshokwaveover

the surfae of the airfoil, due to the development of instabilities ausedby theboundary

layerseparationandtheshokwaveinteration. Theshok-induedosillations(SIO)over

rigid airfoils in transoni regime have been lassied by Tijdeman for fored instabilities

using a moving trailing edge ap [1℄. A detailed desription of the physial features of

SIOisgivenbyLee [2℄. Thisproblemisofprimaryimportane foraeronautiappliations

as it an lead to the bueting phenomenon through the mehanial response of the wing

struture. The large amplitude periodi variation of lift assoiated withbuet limitsthe

ruisingspeedofommerial airraftandseverelydegradesthemaneuverabilityofombat

airraft. Aurate preditionsofsuh ow phenomenais ofsignianttehnologial inter-

estandtheirsimulationremainsahallengingproblemduetotheomplexphysisinvolved.

Today, despite the fast improvement of omputer performanes, the unsteady resolu-

tion of the Navier-Stokes equations remains a diult problem. Three-dimensional time-

dependent omputations obtained with large eddy simulations (LES) and espeiallywith

diretsimulations (DNS) arenot yetpratial for this kind ofappliations beause ofthe

highdemands inomputerresoures. Inthisstudy,theReynoldsdeompositionwasused

withanaveragedstatistialproessingresulting intheRANSequationsfor themeanow

quantities. Thisapproah leads to alowfrequeny separation between modeledand om-

puted sales. It is well known that these equations an be legitimely used for ows in

whih thetime sale ofthe meanowunsteadiness ismuh larger than theharateristi

timesaleofthe turbulene. Thisistheasewiththetransonibuetinwhihtheshok-

indued osillation frequenyis around100Hz.

TheReynoldsdeompositionintroduesadditionalunknownquantitiesliketheReynolds

stress tensor and requires a turbulene model to lose theequation system. Various tur-

bulene losures an be found in the literature of unsteady numerial simulation ows

assoiatedwithbuet,osillatingairfoilsordynamistall. Modelsaremoreor lesssophis-

models [5, 6, 7, 8, 9℄, to EARSM [8℄, RSM [9℄ and non-linear models [9℄. Regarding the

shok loation for steady ows, algebrai models annot give preditions withan aept-

ablelevelofauray. Thestandard eddy-visositymodelsbasedonthelinearBoussinesq

relation are known to be aited by numerous weaknesses, inluding seriously exessive

generationofturbuleneatimpingement zones, aninabilityto apturetheboundarylayer

separation andaviolationofrealizabilityat largerates ofstrain. Moreover,thesesmodels

are formulated following thespetral energy of Kolmogorov withan equilibrium assump-

tion of turbulene and they are alibrated for steady ows. However, for unsteady ows,

the preseneof oherent strutures an break this equilibriumand lead to a dierent en-

ergydistribution. Anobservedonsequeneistheover-produtionofeddy-visosity,whih

limits the unsteadiness development and modies the ow topology. The present study

investigated some orretionsfor standard linearmodels suhastheshear stree transport

(SST) Menter orretion and the use of realizability onstraints. A rst study was on-

duted,onsistingofnumerialsimulationoftransonibuetoverairfoils[10 ℄withtheSST

orretion. It were shown the great inuene of this limiter for two-equation models and

goodresults wereobtained. Otherwaysoflimitingtheeddy-visosityortheprodutionof

turbulenekinetienergyanbeused,suhasadereasethevalueoftheC_µ^oeient[12℄

or theintrodution of thevortiityintheprodution term [13℄but they werenot tested.

Another important aspet onerns the numerial methods and the omputer ost.

Indeed,unsteadyRANSomputationswithturbulenemodelsremainexpensive. Expliit

methods solved the equations using a global timestep omputed as the minimum of the

loal time step assoiated with eah grid ell. The CFL stability riterion drastially

redues the method eieny for ne meshes for whih the dimensionless mesh size at

the wall must be of unity order, in wall units. To overome this diulty, a wall law

approah isusedto relaxthe meshrenement nearthewall[11 ℄. Moreover,omputations

areperformedwithaneientimpliitmethodallowingsomelargetimestepsandwiththe

dualtime-steppingapproahallowingtheuseofaelelerationtehniquessuhasmultigrid

algorithm andloaltimestep. Finally,the paperpresentsanumerial sheme omparison

to study the inuene of the sheme on these unsteady omputations and to try and be

independent of thespatial disretization.

Thenumerial simulationswerearriedoutusinganimpliitCFDodesolvingtheunou-

pled RANS/turbulent systems for multi-domain strutured meshes. This solver is based

on aell-entered nite-volume disretization.

2.1 Governing equations

The ompressible RANS equations oupled with a two-equation turbulene model in in-

tegral form are written for a ell of volume Ω ^{limited by a surfae} Σ ^{and with an outer}

normaln^{. Theseequations an be expressedas:} d

dt Z

Ω

w dΩ + I

Σ

F_c.n dΣ− I

Σ

F_d.n dΣ = Z

Ω

S dΩ ⁽¹⁾

w=





















 ρ ρV ρE

ρk ρΨ























; F_c =























ρV ρV ⊗V +pI

(ρE+p)V ρkV ρΨV























; F_d=























0

τ^v+τ^t (τ^v+τ^t).V −q^v−q^t

(µ+µ_t/σ_k)^gradk (µ+µ_t/σΨ)^gradΨ























where w ^{denotes the} onservative variables, F_c ^and F_d ^{the onvetive and diusive ux}

densities and S the soure terms whih onern only the transport equations. Ψ ^{is the}

length sale determiningvariable.

The exatexpressionof the eddyvisosityµ_t ^{andthesoure termsdependsontheturbu-}

lene model, aswell astheonstants σ_k ^and σΨ^.

The total stress tensor τ ^{is evaluated following the Stokeshypothesisand the Boussinesq}

assumption. Thetotalheatuxvetorq^{isobtainedfromtheFourierlawwiththeonstant}

Prandtl numberhypothesis.

τ = τ^v+τ^t= (µ+µ_t) 1

2(^gradV + (^gradV)^t)−2

3(^divV)I

+ 2

3kI ⁽²⁾ q = q^v+q^t=−

µ P_r + µ_t

P_rt

C_p^gradT ⁽³⁾

For the mean ow, the spae-entered Jameson sheme [14 ℄ was used. It was stabilized

by a salar artiial dissipation onsisting of a blend of 2

nd

and 4

th

dierenes. For the

turbulene transport equations, the upwind Roe sheme [15 ℄ was used to obtain a more

robust method. The seond-order auray was obtained by introduing a ux-limited

dissipation [16 ℄. The Harten'sentropy orretionwasused.

Time integration was performed through a matrix-free impliit method [17, 18℄. The

impliit method onsists in solving a system of equations arising from the linearization

of a fully impliit sheme, at eah time step. The main feature of this method is that

thestorage of the Jaobian matrix is ompletely eliminated, whih leads to a low-storage

algorithm. The visous ux Jaobian matries are replaed by their spetral radii. The

onvetive ux are written withthe Roe sheme instead of the Jameson sheme beause

of thedissipationterm, theuseof aninonsistentlinearization having no onsequene for

steady omputations. The Jaobian matries whih appear from the linearization of the

entered uxes areapproximated with thenumerial uxes and thenumerial dissipation

matriesare replaedbytheir spetralradii.

Conerning theturbulenetransportequations,thediusive uxJaobian matrixarealso

replaedbytheir spetralradii. Thesoure termneeds aspeialtreatment [19 ℄. Onlythe

negativepartofthesouretermJaobian matrixisonsideredandreplaedbyitsspetral

radius.

The impliittime-integration proedure leadsto a systemwhihan besolveddiretly or

iteratively. Thediretinversion an be memory intensive and omputationally expensive.

Therefore, an impliit relaxation proedure is preferred and the point Jaobi relaxation

algorithm washosen.

For steady state omputations, onvergene aeleration was obtained using a loal

timestep and the fullapproximation storage(FAS)multigrid methodproposed byJame-

son[20 , 21℄. Foring funtionsaredened ontheoarsergrids and added totheresiduals

usedforthesteppingsheme. Theorretionsomputedoneahoarsegridaretransferred

bak to the ner one by trilinear interpolations. Theturbulent equations areonly solved

onthenegridandthe omputededdyvisosityµt ^istransferred totheoarsegrids. The

Forunsteadyomputations,thedualtimesteppingmethod,proposedbyJameson[21℄,

was used to takle the lak of numerial eieny of the global time stepping approah.

The derivative withrespetto thephysial timeis disretized bya seond-order formula.

Making the sheme impliit with respet to the dual time provides fast onvergene to

the time-aurate solution. Between eah time step, the solution is advaned in a dual

time and aeleration strategies developed for steady problems an be used to speed up

the onvergene intitious time. The initialization of the derivative withrespetto the

physial time wasperformedwitharst-order formula.

2.3 Far eld onditions

At the outer edge ofthe omputational domain,a non-reeting onditionis used witha

vortiityorretion inorderto simulateauniform inniteow. Itisdeduedfromtheow

eld induedby asingle vortex,thestrength ofwhih isgivenbytheairfoil lift[22 ℄.

2.4 Turbulene Models

Various popular two-equation turbulene models were used in the present study : the

Smith k−l ^{model [23 , 24 ℄, the Wilox} k−ω ^{model [25℄, the Menter SST} k−ω ^model

[26 , 27℄, the high Reynoldsversion ofthe Jones-Launder k−ε^{model [28 ℄,theKok}k−ω

model [29℄and also theone-equationSpalart-Allmaras model [30,31℄.

Asthedisretizationshemedoesnotinsurethepositivityoftheturbulentonservative

variables,limiterswereusedto avoidnegativek^orΨ^{values. Theselimiters weresetequal}

to theorresponding imposedboundary values inthe far eld.

SST orretion

The Menter orretion is basedon the empirial Bradshaw'sassumption whih bindsthe

shear stress to the turbulent kineti energy for two-dimensional boundary layer. This

orretion was extendedforthe k−ε^{modeland the} k−l^model.

The nonequilibrium orretion of Smith [32℄, developed for the k−l ^{model, onsists in}

modifying the omputationof the eddy visositybyintroduinga funtion σ ^: µ_t=σµ_teq ; σ= α−0.25α^1/2+ 0.875

α^3/2+ 0.625 ; α= min P_keq,0

ε ⁽⁴⁾

where the subsript eq ^{denotes the} equilibrium value. The non-equilibrium funtion was hosen to limit the eddy-visosity when prodution is greater than dissipation and to in-

rease thevisosityabove theequilibriummodelvalueintheontraryase.

Durbin orretion - link with realizability

Based on the realizability priniple (the variane of the utuating veloity omponents

shouldbepositive andtheross-orrelationsboundedbytheShwartzinequality), amini-

mal orretionwasderivedfor two-equation turbulenemodelsandwasshownto urethe

stagnation-pointanomaly[33℄. Theonditiontoensurerealizabilityinathree-dimensional

owis:

C_µ≤ 1 s√

3 ; s= k

εS ; S² = 2S_ijS_ij −2

3S_kk^{2 (5)}

A weakly non-linear model was thus obtained [35℄ with a C_µ ^{oeient funtion of the}

dimensionless meanstrainrate :

C_µ= min

C_µ^o, c s√

3

with c≤1 ⁽⁶⁾

where C_µ^{o is settothe onstant value}0.09^{. Durbin xedthevalueoftheonstant}c ^to0.5

for good results inimpinging jets [34 ℄. Then, the following relation wasobtained for the

k−ε^model:

µ_t=ρC_µk²

ε ; C_µ= min

C_µ^o,0.3 s

(7)

And for the k−ω ^model:

µ_t=ρC_µk

ω ; C_µ= min

1, 0.3 C_µ^os

(8)

ItshouldbenotedthatthisorretionissimilartotheSSTformulabyreplaingΩ^with S^{. Yet, the Durbin orretion is} established with mathematial onepts and is available

for three-dimensional ows whereas the SST orretion is based on an empirial two-

dimensionalhypothesis. Thismodelhasbeensuessfullytestedonshokwave/boundary

layer interations withthe Wilox k−ω ^{model[35 ℄.}

The Kokmodel hasbeen builtin orderto resolve thedependeneon freestreamvalues of

ω^{. Theturbulene transport equations ofthemodelare given by:}

∂ρk

∂t + ^div[ρkV−(µ+σ_kµ_t)^gradk] = P_k−β^∗ρkω

∂ρω

∂t + ^div[ρωV−(µ+σ_ωµ_t)^gradω] = P_ω−βρω² +σ_dρ

ω ^gradk.^gradω

Kok obtained additionalonstraintsfor theonstants:

σ_ω−σ_k+σ_d>0 σ_k−σ_d>0

The hoie of Kokwas:

σ_ω = 0.5 ; σ_k= 2/3 ; σ_d= 0.5

Theonstant values were hanged, following all onstraints, to showthesensitivityof

the modelto the ross-diusion term gradk.^gradω ^intheω ^{equation for these unsteady}

omputations.

test 1: σω = 0.5 ; σ_k = 2/3 ; σ_d = 0.65

test 2: σ_ω = 0.5 ; σ_k = 1 ; σ_d = 0.85

2.5 Wall law approah

At the wall, a no-slip ondition was used oupled to a wall law treatment. It onsistsin

imposingthe diusiveuxdensities, requiredfor theintegration proess,inadjaent ells

to a wall. Theshear stress τ ^{and theheatux}q ^{areobtained from an analytialveloity}

prole :

u⁺ = y^{+ if} y⁺<11.13

u⁺ = 1

κlny⁺+ 5.25 ^if y⁺>11.13 u⁺ = u/U_τ ; y⁺= yU_τ

ν_w

(9)

In equation(9), u^{representsthevanDriest [36,37 ℄} transformed veloity forompressible ows.

Conerningtransport-equationturbulenemodels,k^wassetto0^{atthewallanditspro-}

dutionwasimposedaordingtotheformulationproposedbyViegasandRubesin[38 ,39℄.

Theseond variablewasdeduedfromananalytial relationandwasimposedinadjaent

ells to a wall. The harateristi length sale of the Chen model [40℄ was used for the

dissipation rate ε^{and the spei} dissipation ω^{. For theSmith model, a standard linear}

law forthe lengthl ^wasused.

For the one-equation Spalart-Allmaras model, the transported quantity was imposed in

adjaent ellstoawallbyusingthe losurerelations ofthemodel,theveloityproleand

a mixing-length formulation for the eddy-visosity. Moredetails onerning thewall law

approah aregiven in[11℄.

For unsteady boundary layers, the existene of a wall law was assumed valid at eah

instant. As shown in [41 ℄, the veloity phase shift is nearly onstant in the logarithmi

region and equal to the shift of the wall shear stress phase. This is true for a Strouhal

numberup to 10.

Whenusingthewalllawapproahwiththemultigridalgorithm,thewalllawboundary

ondition wasappliedon the negridand theno-sliponditionwasapplied on theoarse

grids.

3 Numerial results

3.1 Experimental onditions

The experimental study was onduted inthe S3MA ONERA wind tunnel [42 ℄ withthe

RA16SC1 airfoil. It is a superritial airfoil with a relative thikness equal to 16% and

a hord length equal to 180^{mm. The RMS pressure utuations were measured from} 36 ^{Kulite transduers installed in the airfoil. The ow onditions were :} M∞ = 0.732^, T_i = 283K^, Re_c = 4.2 10^{6 and the angle of attak varied from} 0 ^to 4.5^{◦. Transition was}

xed nearthe leading edgeat x/c= 7.5^{%on both sidesof theairfoil.}

For the omputations, experimental orretions were used. The Mah number was de-

reased by0.09 ^{and the angleofattakwasdereasedby}1^{◦ at allinidenes withrespet}

to experiment. ThegridwasaC-typetopology. Itontained 321×81 ^{nodes,241 ofwhih}

were on the airfoil(f. gure1,2). They^{+ values ofthe oarse mesh,at theenter ofthe}

rst ell,arepresentedingure3 for asteady omputation atα= 4^◦.

Thenumerial parameters usedfor the omputationswere :

- thedimensionless time step, ∆t^∗ = ∆ta_i c = 0.2

where c ^{isthe hord ofthe airfoiland} a_i ^{the stagnation soundveloity}

- grid levels for themultigrid method,2

- sub-iterationsof thedualtime steppingmethod,75 upto 100

By inreasing the number of sub-iterations, it was heked that the same solution was

ahieved.

- theCFL number, 200

- Jaobi iterationsfor the impliit stage,14

- the artiial dissipation of the Jameson sheme introdues two oeients, one for the

seond-dierene term: χ2 = 0.5 ^{and one for the} fourth-dierene term: χ4 = 0.016^{. For}

theseond gridlevel, theoeient χ4 ^wasxedat 0.032

- theoeient of the Harten'sorretion, 0.05

Computations started from a uniform ow-eld using a loal time step and one grid

level. After 50 iterations, the dual time stepping method was used with the mulgrid

algorithm and osillationsdevelopwitha growing amplitude.

3.3 Comparison of turbulene models

The frequeny f ^{and theamplitude of the lift oeient} ∆C_L ^{arereportedin table 1 for}

all turbulenemodels and forthree angles ofattakα= 3,4^and 5^◦,orresponding tothe buet onset, established phenomenon and buet exit, i.e. the return to a steady state,

respetively.

The apaity of turbulene models to restitute the natural unsteadiness of the ow