Combining a Mass Matrix formulation and a high order dissipation for the discretisation of turbulent flows

a p p o r t

d e r e c h e r c h e

9 -6 3 9 9 IS R N IN R IA /R R -- 7 0 7 9 -- F R + E N G

Thème NUM

Combining a Mass Matrix formulation and a high order dissipation for the discretisation of turbulent

flows

Anca Belme, Hilde Ouvrard

N° 7079

Octobre 2009

ThèmeNUMSystèmesnumériques

ProjetTropis

Rapportdereherhe n°7079 Otobre200923pages

Abstrat: This study fouses on analysis of a massmatrix shemewith high-ordernumerial

dissipation for thedisretisation ofthree-dimensional turbulent ows.We used amixed nite ele-

ment/nitevolumeformulationforthedisretisationofompressibleNavier-Stokesequations.VMS-

LESturbulenemodelisperformedtosimulatetheowaroundairularylinderat asubritial

Reynoldsnumber

Re

^of3900.

Key-words: numerialshemes,turbulenesimulation,ompressible

∗

INRIA,2004RoutedesLuioles,BP.93,06902Sophia-Antipolis,Frane

†

UniversityofMontpellierII,Frane

Résumé: Leprésentrapportestfoalisésurl'analysed'unshémanumériqueàmatriedemasse

et ave une dissipationnumérique d'ordreélévépour ladisrétisationdes éoulementsturbulents

tridimensionelles. Onautiliséuneformulationmixteéléments/volumesniespourladisrétisation

deséquations deNavier-Stokesompressible. UnmodèledeturbuleneVMS-LESest utilisépour

lasimulationd'unéoulementsubritiqueautour d'unylindreànombredeReynolds

Re

^3900.

Mots-lés : shémasnumériques,simulationdelaturbulene,matriedemasse

by Hughes et al. in Ref. [4℄. The VMS-LES diers fundamentallyfrom the traditional LES in

a number of ways. In this approah, one does not lter the Navier-Stokes equations but uses

instead a variational projetion. This is an important dierene beause as performed in the

traditional LES, ltering works well with periodi boundary onditions but raises mathematial

issues in wall-bounded ows. The variational projetion avoids these issues. Furthermore, the

VMS-LES method a priori separates the sales that is, before the simulation is started. And

mostimportantly, itmodels theeet of theunresolved-salesonlyin the equationsrepresenting

the smallest resolved-sales, and not in the equations for the large sales. Consequently, in the

VMS-LES,energyisextratedfromtheneresolved-salesbyasubgridsale(SGS)eddy-visosity

model, but no energy is diretly extrated from the large strutures in the ow. The proposed

modelhasbeenimplementedinanumerialsolver(AERO)fortheNavier-Stokesequationsinthe

ase of ompressible ows and perfet Newtonian gases, based on a mixed nite-element/nite-

volume sheme formulated for unstrutured grids made of tetrahedralelements. Finite elements

(P1 type) and nite volumes are used to treat the diusive and onvetive uxes, respetively.

Conerning the VMS approah, the version proposed in Ref. [5℄ for ompressible ows and for

the partiular numerial method employed in AERO has been used here. The disretisation of

the onvetive uxes for the nite volume sheme used in the software AERO exhibits a lak of

auray in theaseof irregularmesh. Inorder tooveromethis problem, weinvestigateon this

work a new mass matrix sheme with high-order numerial dissipation for the disretization of

turbulent ows. Thepresentreport isorganisedasfollows: wereviewin Setion2amassmatrix

sheme forthe1Dadvetionequationand weanalysethestabilityofthis shemefor expliitand

impliittimeadvaningusingVonNeumannanalysis. Setion3ontainsthenumerialmethodin

3Dase appliedto theompressibleNavier-Stokesequations,whih hasbeenimplementedin our

CFDsoftwareandashortdesriptionoftheVMS-LESturbulenesimulationmodel. Finallyinthe

lastsetion,onegivessomeresultsofthesimulationofturbulentowaroundairularylinder.

2 Mass Matrix Central Dierening sheme for the advetion

equation

2.1 Spatial 1D MUSCL formulation

Letusrstonsidertheone-dimensionalsalaradvetionmodel

u t + c(u) x = 0

⁽¹⁾

2.1.1 Spatialdisretisation

The nite-volume method is used forthe disretization in spae. Let

x j

^,

1 ≤ j ≤ N

^{denote the}

disretization points of the mesh. For eah disretization point, we state :

u j ≈ u(x j )

^{and we}

denetheontrolell

C j

^{astheinterval}

[x _j − ¹

2 , x _j+ ¹

2 ]

^where

x _j+ ¹

2 = ^{x j +x} ₂ ^j+1

^.

We dene the unknown vetor

U = {u j }

^aspoint approximation valuesof the funtion

u(x)

^at

eahnode

j

^ofthemesh.

Thetimeadvaningshemeiswritten :

U j,t + Ψ j (U ) = 0

wherethevetor

Ψ j (U )

^{isbuiltasfollows:}

Ψ j (U ) = 1

∆x (Φ _j+ ¹

2 − Φ _j− ¹

2 ); Φ _j+ ¹

2 = Φ(u ⁻ _j+ 1 2

, u ⁺ _j+ 1 2

);

⁽²⁾

where

u ⁻ _j± 1 2

,

u ⁺ _j± 1 2

are integration values of

u

^{at boundaries of ontrol volume}

C j

^{. The}

Φ

^{'s are}

dened as:

Φ(u, v) = cu + cv 2 − δ

2 c(v − u)

where

δ

^{isaparameterontrolingthespatial} dissipation,suhas

δ ∈ [0, 1]

^.

Thereonstruted valuesare then givenby:

u ⁻ _j+ 1 2

= u j + ¹ ₂ ∆u ⁻ _j+ 1 2

and

u ⁺ _j+ 1 2

= u j − ¹ ₂ ∆u ⁺ _j+ 1 2

(samereonstrutionfor

u ⁻ _j− 1 2

and

u ⁺ _j− 1 2

)wheretheslopes

∆u ⁻ _j+ 1 2

and

∆u ⁺ _j+ 1 2

aredenedby:

∆u ⁻ _j+ 1 2

= (1 − β)(u j−1 − u j ) + β(u j − u j−1 ) +θ ^c (−u j−1 + 3u j − 3u j+1 + u j+2 ) +θ ^d (−u j − 2 + 3u j − 1 − 3u j + u j+1 )

and

∆u ⁺ _j+ 1 2

= (1 − β)(u j+1 − u j ) + β(u j+2 − u j+1 ) +θ ^c (−u j−1 + 3u j − 3u j+1 + u j+2 ) +θ ^d (−u j + 3u j+1 − 3u j+2 + u j+3 )

It hasbeenshownthathoosing

β = ¹ ₃

^,

θ ^c = − ₁₀ ¹

^and

θ ^d = − ₁₅ ¹

^{,thisshemebeomesfth-order}

aurate. Thenumerialdissipationintroduedbythisshemeisthenmadeofsixth-orderderiva-

tivesandanbewrittenas:

D j (u) = D _j+ ¹

2 (u) − D _j− ¹

2 (u)

∆x

where

D j+ ¹ ₂ (u) = δc

60 (−u j − 2 + 5u j − 1 − 10u j + 10u j+1 − 5u j+2 + u j+3 )

⁽³⁾

2.2 Mass Matrix Sheme with entral dierening

Theusualentral-dierenesthree-pointsshemeispenalizedbyadispersionleadingerror,see[1℄.

This errorisompensatedintroduingthenite-element

P 1

^{onsistentmassmatrix. That is, the}

temporaltermofequation(1)isevaluatedbyFiniteElements. Thetimeadvaningisthuswritten

The nite-element

P 1

^{mass matrix M is dened as}

M

ij = Z

Φ ⁽ⁱ⁾ Φ ^(j)

^where

Φ ⁽ⁱ⁾

^{is the basis}

funtion forthenite-element

P1

formulation. Thatis:

Φ ⁽ⁱ⁾

^{isapieewiselinearfuntionsuhas}

Φ ⁽ⁱ⁾ (S i ) = 1

^and

Φ ⁽ⁱ⁾ (S j ) = 0

^forall

i 6= j

^,where

S j

^denotesthe

j

^{-thvertex. Misa}three-diagonal matrixequalto:

M =

^Three-diag

( 1 6 ∆x, 2

3 ∆x, 1 6 ∆x)

Oneobtainsthusforthe

j

^{-thomponentofvetor}

M U (t)

^:

(M U (t)) j = m j,j−1 U j−1,t + m j,j U j,t + m j,j+1 U j+1,t

where

m j,j−1 = 1/6∆x m j,j = 2/3∆x m j,j+1 = 1/6∆x .

Then oneanombine this disretisationof the temporal term with anumerialux forthe ap-

proximationoftheonvetifterm:

Φ _j+ ¹

2 = Φ(u j , u j+1 ) = c u j + u j+1

2

− δ 2 T _j+ ¹

2

Terms

c u j + u j+1

2

will ontributeto theentraldierenedux.

Aording to the Pasal triangle, a fth-order dierene evaluated between

j

^and

j + 1

^{an be}

written asfollows(

k > 0

^):

T j+1/2 = kc(−u j − 2 + 5u j − 1 − 10u j + 10u j+1 − 5u j+2 + u j+3 )

Onehoosestheonstant

k

^as:

k = 1

30

⁽⁵⁾

in ordertohavethesamelevelofdissipationasintheupwind ase,see(3).

Finally,startingfromexpression(2),wegetfortheoperator

Ψ j

^dened:

Ψ j (U ) = c

2∆x ( k u j − 3

+ (−6k) u j−2

+ (−1 + 15k) u j − 1

+ (−20k) u j

⁽⁶⁾

+ (1 + 15)k u j+1

+ (−6k) u j+2

k u j+3 )

2.3 Time advaning stability

2.3.1 Expliittimestepping

Let us onsider a time integration of the system

M U t = AU

^{, with A the spatial} approximation matrix and M the P1 nite element mass matrix. We an ombine the abovesheme with the

linearisedsix-stageRunge-Kuttasheme:

U ⁽⁰⁾ = U ⁿ

U ^(k) = U ⁽⁰⁾ + _{N −k+1} ^∆t M ^{− 1} Ψ U ^{(k − 1)}

, k = 1 . . . N U ⁿ⁺¹ = U ⁽⁶⁾

(7)

The stability study of the sheme is made with the lassial Fourieranalysis. Letus inlude in

equation(6)theFouriermode:

u ˆ ⁿ _j = u k e ^{ijθ k}

^where

θ k

^{isthefrequeneparameter.}

Weobtain:

m θ

dˆ u ⁿ _j

dt = − Ψ ˆ ^δ

where

m θ = ¹ ₃ (2 + cos(θ))

Ψ ˆ ^δ = _2∆x ^c (R 3 cos(3θ) + R 2 cos(2θ) + R 1 cos(θ) + R 0 + iI 1 sin(θ))ˆ u ⁿ _j

^,with:

 

 

 

 

 



R 3 = 2δk R 2 = −12δk R 1 = 30δk R 0 = −20δk I 1 = 2

denoting

λ θ

^{thelinearoperatorsuhas}

Ψ ˆ ^δ = −λ θ u ˆ ⁿ _j

^.

WeintroduetheCourantnumber

ν = ^c∆t _∆x

^{and theampliationfator}

G θ = g(z θ )

^.

z θ = ^λ _m ^{θ ∆t}

θ

,

g

^istheRK6 harateristipolynomial:

g(z) = 1 + z + z ² 2 + z ³

6 + z ⁴ 24 + z ⁵

120 + z ⁶ 720

Finaly,wehave:

 



z θ = − 2(2+cos(θ)) ^3ν (z ^R _θ + iz ^I _θ )

z _θ ^R = R 3 cos(3θ) + R 2 cos(2θ) + R 1 cos(θ) + R 0

z _θ ^I = I 1 sin(θ)

Plotting thegainfuntion

Ga(ν) = max

θ ∈ [0,π] g(z θ )

^{, wean}determinize

ν max

^{, themaximum valueof}

CFLnumberto obtainastableshemewith,that isthemaximumvalueof

ν

^suhas

|g(z θ )| ≤ 1

^.

ForaCFLsmallenough,thewholeof

z θ

^{omplexnumbersremainsinsidethe}A-Stabilityregionof theRK6timeadvaningshemeasillustratedonFIG.(1).

−4 −3 −2 −1 0 1

−4

−3

−2

−1 0 1 2 3

Real part of z

θ

Imaginary part of z θ

Figure1: Stabilityanalysis: wedepitwiththeline theboundaryoftheregionforwhih

g(z) ≤ 1

and(inside)dashed

z θ

^{omplexnumbersfor}

δ = 1

^and

ν max = 1.11

2.3.2 Impliittime stepping

The purpose ofthis setion isthe stability analysis ofthe impliit sheme.This anbealso done

withFourieranalysis. Theomputation ismadeonthesameone-dimensionalsalaronservation

law.Letususetheimpliitshemea

δ

^-shemeusingamass-matrixformulation:

T ⁿ δU ⁿ⁺¹ = ∆t ⁿ Ψ(U ⁿ )

with

δU ⁿ⁺¹ = U ⁿ⁺¹ − U ⁿ

^and

T ⁿ

^{istheimpliitmatrix.}

Inaseofarstordersheme,

T ⁿ

^{representsthefollowing}three-diagonalmatrix:

T ⁿ =

^diag

( ¹ ₆ − ν, ² ₃ + ν, ¹ ₆ )

With theFourieranalysisweobtain:

t θ = 1 + ν(1 − cos(θ)) + ¹ ₃ (2 + cos(θ)) + iν sin(θ)

Theampliationfatoristhengivenby:

G(∆t) = ^{t θ +z} _t ^θ

θ

We are must interestedon the behaviorof theampliation fator when thetime step

∆t

^tends

to

+∞

^{. The aim is to know if the builds ones shemes are} preonditionated at rst ordre with satisfatory fators of onvergene. Let us denote

lim

∆t →∞ G(∆t) = f θ (δ)

^{. We are searhing the}

Fourier'smodesfordierentshemeswithmaximisesthefuntions:

f θ (δ) = 1 − 1 4(1 − cos(θ))m θ

[(1 − cos(θ))z _θ ^R + sin(θ)z _θ ^I ] + i[ sin(θ)z ^R _θ − (1 − cos(θ))z ^I _θ

4m θ (1 − cos(θ)) ]

Thefuntion

f δ

^{aredepitedwiththreedierentvaluesof}

δ

^{inFIG.2,3and4. Weanobservethat}

ourimpliit shemeis inonditionallystable. Letus denotethat theoptimal valuethat minimise

thegainfuntion when

∆t

^goesto

∞

^is

δ = 1

^.

Figure 2: Ampliationfator,

∆t

^goesto

∞

^,ase

δ = 1

Figure3: Ampliationfator,

∆t

^goesto

∞

^,ase

δ = 0.5

Figure4: Ampliationfator,

∆t

^goesto

∞

^,ase

δ = 0.3

3 Numerial Method 3D

3.1 Introdution

InthepresenthaptertheodeAERO,usedin thepresentstudy, isdesribed. Theodepermits

to solve the Euler equations, the Navier-Stokesequations for laminar ows and to use dierent

turbulenemodelsforRANS,LESandhybridRANS/LESapproahes. Theunknownquantitiesare

thedensity,theomponentsofthemomentumandthetotalenergyperunitvolume. AEROemploys

a mixed nite-volume/nite-element formulation for the spatial disretization of the equations.

Finite-volumes are used for the onvetive uxes and nite-elements (P1) for the diusive ones.

The resultingshemeis seond order aurate in spae. Theequations an be advaned in time

withexpliitlow-storageRunge-Kuttashemes. Alsoimpliittimeadvaningispossible,basedon

alinearisedmethod thatisseondorderaurateintime.

3.2 Set of equations

In the AERO ode the Navier-Stokes equations are numerially normalised with the following

referenequantities:

ˆ

L ref = ⇒

harateristilengthoftheow

ˆ

U ref = ⇒

^veloityofthefree-streamow

ˆ

ρ ref = ⇒

^densityofthefree-streamow

ˆ

µ ref = ⇒

^{moleularvisosityofthe}free-streamow

Theowvariablesanbenormalisedwiththereferenequantitiesasfollows:

ρ ^∗ = ρ

ρ ref u ^∗ _j = u j

U ref p ^∗ = p p ref

E ^∗ = E

ρ ref U _ref ² µ ^∗ = µ µ ref

t ^∗ = t L ref

U ref

.

Thenon-dimensionalformoftheNavier-Stokesequationsforthelaminarasearereportedin the

following:

∂ρ ^∗

∂t ^∗ + ∂(ρ ^∗ u ^∗ _j )

∂x ^∗ _j = 0

∂(ρ ^∗ u ^∗ _i )

∂t ^∗ + ∂ρ ^∗ u ^∗ _i u ^∗ _j

∂x ^∗ _j = − ∂p ^∗

∂x ^∗ _i + 1 Re

∂σ _ij ^∗

∂x ^∗ _j

∂(ρ ^∗ E ^∗ )

∂t ^∗ + ∂(ρ ^∗ E ^∗ u ^∗ _j )

∂x ^∗ _j = − ∂(p ^∗ u ^∗ _j )

∂x ^∗ _j + 1 Re

∂(u ^∗ _j σ _ij ^∗ )

∂x ^∗ _i − γ ReP r

∂

∂x ^∗ _j h

µ ^∗ E ^∗ − 1

2 u ^∗ _j u ^∗ _j i

(8)

where the Reynolds number,

Re = U ref L ref /ν

^{, is based on the referenes} quantities,

U ref

^and

L ref

^{,thePrandltnumber,Pr,anbeassumedonstantforagasandequalto:}

P r = C p µ k

and

γ = C p /C v

^{is the ratio between the spei heats at onstant pressure and volume. Also}

theonstitutiveequationsfor thevisousstressesandthe stateequations may bewritten in non-

dimensionalform asfollows:

σ ^∗ _ij = − 2

3 µ ^∗ ∂u ^∗ _k

∂x ^∗ _k δ ij

+ µ ^∗ ∂u ^∗ _i

∂x ^∗ _j + ∂u ^∗ _j

∂x ^∗ _i

p ^∗ = (γ − 1)ρ ^∗ E ^∗ − 1

2 u ^∗ _j u ^∗ _j

.

⁽⁹⁾

In orderto rewrite thegoverningequations in aompat form moresuitablefor thedisrete for-

mulation,thefollowingunknownvariablesaregroupedtogetherin the

W

vetor:

W = (ρ, ρu, ρv, ρw, ρE) ^T

^.

Iftwoothervetors,

F

and

V

aredenedasfuntionof

W

,asfollows:

F =



 



ρu ρv ρw

ρu ² + p ρuv ρuw ρuv ρv ² + p ρvw ρuw ρvw ρw ² + p



 



and

V =



 

 



0 0 0

σ xx σ yx σ zx

σ xy σ yy σ zy

σ xz σ yz σ zz

uσ xx + vσ xy + wσ xz − q x uσ xy + vσ yy + wσ yz − q y uσ xz + vσ yz + wσ zz − q z



 

 



∂ W

∂t + ∂

∂x j

F j ( W ) − 1 Re

∂

∂x j

V j ( W , ∇ W ) = 0 .

⁽¹⁰⁾

It is importantto stressthat thevetors

F

and

V

are respetivelytheonvetiveuxesand the diusiveuxes.

3.3 Spatial disretization

Spatialdisretizationisbasedonamixednite-volume/nite-elementformulation. Anitevolume

upwind formulation is used for the treatment of the onvetive uxes while a lassial Galerkin

nite-elemententredapproximationisemployedforthediusiveterms.

Theomputationaldomain

Ω

^isapproximatedbyapolygonaldomain

Ω h

^{. Thispolygonaldomain}

isthendividedin

N t

tetrahedrialelements

T i

^byastandardnite-elementtriangulationproess:

Ω h =

N t

[

i=1

T i .

⁽¹¹⁾

Thesetofelements

T i

^{formsthegridusedinthe}nite-elementformulation. Thedualnite-volume gridanbebuiltstartingfromthetriangulationfollowingthemediansmethod.

Inthemediansmethodanite-volumeellisonstrutedaroundeahnode

a i

^ofthetriangulation, dividingin4sub-tetrahedraeverytetrahedronhaving

a i

^{asavertexbymeansofthemedianplanes.}

C i

^{isthe unionofthe resulting}sub-tetrahedra having

a i

^{asavertexand theyhavethe following}

property:

Ω h =

N c

[

i=1

C i .

⁽¹²⁾

where

N c

^{is thenumberofells,whihisequalto thenumberofthenodesofthe}triangulation.

3.3.1 Convetive uxes

Indiatingthebasisfuntionsforthenite-volumeformulationasfollows:

ψ ⁽ⁱ⁾ (P ) =

1

^if

P ∈ C i

0

^otherwise

theGalerkinformulationfortheonvetiveuxesisobtainedbymultiplyingtheonvetiveterms

of(10)bythebasisfuntion

ψ ⁽ⁱ⁾

^,integratingonthedomain

Ω h

^{andusingthedivergenetheorem.}

Inthiswaytheresultsare:

ZZ

Ω _h

∂F j

∂x j

ψ ⁽ⁱ⁾ x y = ZZ

C i

∂F j

∂x j Ω = Z

∂C i

F j n j σ

where

dΩ

^,

dσ

^and

n j

^{aretheelementarymeasureoftheell,ofitsboundaryandthe}

jth

^omponent

ofthenormalexternalto theell

C i

respetively.

Thetotalontributiontotheonvetiveuxesis:

X

j

Z

∂C ij

F( W , ~n) σ

where

j

^{are allthe}neighbouringnodesof

i

^,

F ( W , ~n) = F j ( W )n j

^,

∂C ij

^{istheboundarybetween}

ells

C i

^and

C j

^,and

~n

^{istheouternormaltotheell}

C i

^.

Thebasiomponentfortheapproximationoftheonvetiveuxesis theRoesheme,Ref. [15℄:

Z

∂C ij

F( W , ~n) σ ≃ Φ ^R (W i , W j , ~ν ij )

where

~ν ij = Z

∂C ij

~n σ

and

W k

^{isthesolutionvetoratthe}

k

^-thnodeofthedisretization.

IntheV6shemedeveloppedin[7℄and[6℄. Thenumerialuxes,

Φ ^R

^{,areevaluatedasfollows:}

Φ ^R (W i , W j , ~ν ij ) = F(W i , ~ν ij ) + F(W j , ~ν ij )

| {z 2 }

centred

− γ s d ^R (W i , W j , ~ν ij )

| {z }

upwinding

where

γ s ∈ [0, 1]

^{isaparameterwhihdiretlyontrolstheupwindingoftheshemeand}

d ^R (W i , W j , ~ν ij ) =    R(W i , W j , ~ν ij )    W j − W i

2 .

⁽¹³⁾

R

^{istheRoematrixandisdenedas:}

R(W i , W j , ~ν ij ) = ∂F

∂ W ( W c , ν ij )

⁽¹⁴⁾

where

W c

istheRoeaveragebetween

W i

^and

W j

^.

The lassial Roesheme is obtainedas a partiular aseby imposing

γ s = 1

^{. The auray of}

thisshemeisonly

1st

^{order. InordertoinreasetheorderofaurayoftheshemetheMUSCL}

(Monotone Upwind Shemes for Conservation Laws) reonstrution method, introdued by Van

Leer,Ref. [19℄,isemployed. ThismethodexpressestheRoeuxasafuntionof

W ij

^and

W ji

^,the

extrapolatedvaluesof

W

attheinterfaebetweentwoells

C i

^and

C j

^:

Z

∂C ij

F( W , ~n) σ ≃ Φ ^R (W ij , W ji , ~ν ij )

where

W ij

^and

W ji

^{aredenedasfollows:}

W ij = W i + 1

2 ( ∇ ~ W ) ij · ij , ~ W ji = W j + 1

2 ( ∇ ~ W ) ji · ij . ~

Toestimatethegradients

( ∇ ~ W ) ij · ij ~

^and

( ∇ ~ W ) ji · ij ~

^{,oneusedthese} expressions,Ref. [6℄:

ξ c [( ∇ ~ W ) ^U _ij · ij) ~ − 2( ∇ ~ W ) ^C _ij · ij) + ( ~ ∇ ~ W ) ^D _ij · ij)] + ~ ξ d [( ∇W ~ ) M · ij) ~ − 2( ∇W ~ ) i · ij) + ( ~ ∇W ~ ) ^D _j · ij)] ~ , ( ∇ ~ W ) ji · ji ~ = (1 − β)( ∇ ~ W ) ^C _ji · ij) + ~ β( ∇ ~ W ) ^U _ji · ij) + ~

ξ c [( ∇ ~ W ) ^U _ji · ij) ~ − 2( ∇ ~ W ) ^C _ji · ij) + ( ~ ∇ ~ W ) ^D _ji · ij)] + ~ ξ d [( ∇ ~ W ) M ^′ · ij) ~ − 2( ∇ ~ W ) i · ij) + ( ~ ∇ ~ W ) ^D _j · ij ~ )] ,

where

( ∇ ~ W ) i

^and

( ∇ ~ W ) j

^{are the nodal gradients at the nodes}

i

^and

j

respetively and are alulatedastheaverageofthegradientonthetetrahedra

T ∈ C i

^{,havingthe node}

i

^asavertex.

Forexamplefor

( ∇ ~ W ) i

^weanwrite:

( ∇ ~ W ) i = 1 V ol(C i )

X

T ∈C i

V ol(T ) 3

X

k∈T

W k ∇Φ ~ ^(i,T) .

⁽¹⁵⁾

where

Φ ^{(i,T )}

^{is the P1} nite-element basis funtion. In the 3D ase, these funtions are so dened:

Φ ⁽ⁱ⁾ (P) =

1

^if

P = S i

0

^if

P = S j

^,

i 6= j

( ∇ ~ W ) M · ij ~

^{, forthe 3D ase,is the gradientat thepoint}

M

^{in Fig. 5and itis omputed by}

interpolationofthenodalgradientvaluesatthenodesontainedinthefaeoppositetotheupwind

tetrahedron

T ij

^.

( ∇ ~ W ) M ^′ · ij ~

^{isthe gradientat thepoint}

M ^′

^{in Fig.5and itis evaluated in the}

samewayas

( ∇ ~ W ) M · ij ~

^{. Theoeients}

β

^,

ξ c

^,

ξ d

^{are parametersthat ontrolthe ombination}

offullyupwindandentredslopes. TheV6shemeisobtainedbyhoosingthemtohavethebest

aurayonartesianmeshes,Ref.[7℄:

β = 1/3, ξ c = −1/30, ξ d = −2/15 .

Thevariantoftheaboveshemealled mass-matrixwithentraldierening, sets

β

^,

ξ c

^and

ξ d

^to

zeroandinvolvesatimederivativeevaluatedwiththenite-elementonsistentmassmatrix. This

massmatrixasinthe1Daseisexpressed intermsoftheusualP1testfuntions asfollows:

M ij = R

φ ⁽ⁱ⁾ φ ^(j) dν

As in Setion 2, we an ombine this time derivative with a ux. Weget thus the following

semi-disretizationequation:

∂ W _i

∂t = ∆tM ^{− 1} X

j

Φ( W _i , W _j , ~ ν ij )

withthenumerialux denedby:

Φ(W i , W j , ~ ν ij ) = F(W i , ~ ν ij ) + F(W j , ~ ν ij )

2 − γ s

2 |R(W i , W j , ~ ν ij )| ∆W ij

where:

∆W ij = C(2( ∇W ~ ) M .~ ij − 5( ∇W ~ ) ^u _ij .~ ij + 6( ∇W ~ ) ^c _ij .~ ij − 5( ∇W ~ ) ^d _ij .~ ij + 2( ∇W ~ ) _M ^′ .~ ij) .

Followingthestudydoneforthe1Dadvetionequation(seeequation5),theonstant

C

^issetto:

C = δ

60

⁽¹⁶⁾

A variant of this sheme is also investigated and onsists in using a projeteddissipation in

ij ~

^:

|R(W i , W j , ~ ν ij )|

^beomes

R(W i , W j , ν ~ ˜ ij )

^{in whih}

ν ~ ˜ ij = ( ν ~ ij . k ^{ij ij ~ ~} k ) k ^{ij ij ~ ~} k

^{This optionhas been}

Figure5: Skethofpointsandelementsinvolvedintheomputationofgradient

investigatedfortheappliation ofaGaussianfuntion translationin setion5.1

3.3.2 Diusiveuxes

TheP1nite-elementbasisfuntion,

φ ^(i,T)

^{,restritedtothe}tetrahedron

T

^{isassumedtobeofunit}

valueonthenode

i

^{andtovanishlinearlyattheremainingvertexesof}

T

^{. TheGalerkin}formulation forthediusivetermsisobtainedbymultiplyingthediusivetermsby

φ ^{(i,T )}

^andintegratingover thedomain

Ω h

^:

Z Z

Ω h

∂V j

∂x j

φ ^{(i,T )} Ω = Z Z

T

∂V j

∂x j

φ ^{(i,T )} Ω .

⁽¹⁷⁾

Integratingbypartstheright-handsideofequation(17)weobtain:

ZZ

T

∂V j

∂x j φ ^{(i,T )} Ω = ZZ

T

∂(V j φ ^(i,T) )

∂x j Ω − ZZ

T

V j ∂φ ^(i,T)

∂x j Ω = Z

∂T

V j φ ^(i,T) n j σ − ZZ

T

V j

∂φ ^(i,T)

∂x j Ω .

⁽¹⁸⁾

X

T,i ∈ T

Z

∂T

V j φ ^{(i,T )} n j σ − Z Z

T

V j ∂φ ^{(i,T )}

∂x j

Ω

=

− X

T,i ∈ T

Z Z

T

V j ∂φ ^(i,T)

∂x j Ω + Z

Γ h =∂Ω h

φ ^{(i,T )} V j n j σ .

⁽¹⁹⁾

IntheP1formulationforthenite-elementmethod,thetestfuntions,

φ ^{(i,T )}

^{,arelinearfuntions}

on theelement

T

^{and sotheirgradient isonstant. Moreover, in the}variational formulationthe unknown variables ontained in

W

are also approximated by their projetion on the P1 basis funtion. Forthese reasonstheintegralanbeevaluateddiretly.

3.4 Boundary onditions

Firstly, therealboundary

Γ

^isapproximatedbyapolygonalboundary

Γ h

^{thatanbesplitin two}

parts:

Γ h = Γ ∞ + Γ b

⁽²⁰⁾

where the term

Γ ∞

^{represents the far-elds boundary and}

Γ b

^{represents the body surfae. The}

boundaryonditionsareset usingtheSteger-Warmingformulation([18℄)on

Γ ∞

^{andusing slipor}

no-sliponditionson

Γ b

^.

3.5 Time advaning

One the equations have been disretized in spae, the unknown of the problem is the solution

vetorat eah node of the disretization asa funtion of time,

W _h (t)

^. Consequently the spatial disretizationleadstoasetofordinarydierentialequationsintime:

d W _h

dt + Ψ( W _h ) = 0

⁽²¹⁾

where

Ψ i

^{isthetotalux,ontainingbothonvetiveanddiusiveterms,of}

W h

^throughthe

i

^-th

ellboundarydividedbythevolumeoftheell.

3.5.1 Expliittimeadvaning

Intheexpliitasea

N

^-steplow-stokageRunge-Kuttaalgorithmis usedforthedisretization of Eq.(21):

 



W ⁽⁰⁾ = W ⁽ⁿ⁾ ,

W ^(k) = W ⁽⁰⁾ + ∆t α k Ψ( W ^(k−1) ), k = 1, ... , N W ⁽ⁿ⁺¹⁾ = W ^{(N )} .

in whih the sux

h

^{has been omitted for sakeof simpliity. Dierent shemes anbeobtained}

varyingthenumberofsteps,

N

^{,andtheoeients}

α k

^.

3.5.2 Impliittime advaning

Forthe impliit time advaning sheme in AERO the following seond order aurate bakward

diereneshemeisused:

α n+1 W ⁽ⁿ⁺¹⁾ + α n W ⁽ⁿ⁾ + α (n − 1) W ^{(n − 1)} + ∆t ⁽ⁿ⁾ Ψ( W ⁽ⁿ⁺¹⁾ ) = 0

⁽²²⁾

wheretheoeients

α n

^{anbeexpressedasfollows:}

α n+1 = 1 + 2τ

1 + τ , α n = −1 − τ, α n − 1 = τ ²

1 + τ

⁽²³⁾

where

∆t ⁽ⁿ⁾

^{isthetimestepusedatthe}

n

^{-thtimeiterationand}

τ = ∆t ⁽ⁿ⁾

∆t ^{(n − 1)} .

⁽²⁴⁾

Thenonlinearsystemobtainedanbelinearisedasfollows:

α n+1 W ⁽ⁿ⁾ + α n W ⁽ⁿ⁾ + α (n−1) W ⁽ⁿ⁻¹⁾ + ∆t ⁽ⁿ⁾ Ψ( W ⁽ⁿ⁾ ) =

− h

α n+1 + δt ⁽ⁿ⁾ ∂Ψ

∂ W ( W ⁽ⁿ⁾ ) i

( W ⁽ⁿ⁺¹⁾ − W ⁽ⁿ⁾ ).

⁽²⁵⁾

Following thedefet-orretionapproah, thejaobians are evaluated using the1st order ux

sheme (fortheonvetivepart), whilethe expliituxesare omposed with 2ndorder auray.

TheresultinglinearsystemissolvedbyaShwarzmethod.

4 Variational Multisale approah for Large Eddy Simulation

AnewapproahtoLESbasedonavariationalmultisale(VMS)frameworkwasreentlyintrodued

inHughesetal. TheVMS-LESdiersfundamentallyfromthetraditionalLESinanumberofways.

Inthisnewapproah,onedoesnotltertheNavier-Stokesequationsbutusesinsteadavariational

projetion. Thisis animportant dierenebeauseasperformed in thetraditional LES,ltering

works aswellwith periodi boundaryonditions but raises mathematial issues in wall-bounded

ows. Thevariationalprojetionavoids thisissues. Furthermore,the VMS-LESmethod a priori

separatesthesalesthat is,before thesimulationisstarted. Andmostimportantly,itmodelsthe

eet of theunresolved-salesonly in theequationsrepresentingthe smallestresolved-sales,and

notintheequationsforthelargesales. Consequently,in theVMS-LES,energyisextratedfrom

the neresolved-salesby atraditionalmodelsuh asSmagorinskyeddyvisosity model, but no

energyisdiretlyextratedfromthelargestruturesinteow. Forthisreason,oneanreasonably

hopetoobtainabetterbehaviornear walls,and lessdissipationin thepreseneof largeoherent

strutures.

A less fundamental, yet noteworthy, dierene between the VMS-LES and traditional LES

methods isthattheVMS-LES approahleadstogoverningequationsthat arewrittenin termsof

theoriginal(orundeomposed)owvariablesandthemodeledeetoftheunresolved-salesonthe

intotheirlteredandutuatingpartsinthesubgrid-tensor,thenfaestheissuesofmodelingthe

subgrid-sales. In the VMS-LES approah, one does nothave to deompose into spae-averaged

and utuatingparts eahourenein theNavier-Stokesequationsofeah owvariable beause

thenal equationsareexpetedto beexpressedin termsoftheimportane asitanbeexploited

to bypassthemodeling ofsomeutuatingquantities,aswillbeillustratedhereforompressible

turbulentows.

Theinitial developement ofthe VMS-LES method foused oninompressible turbulent ows,

regular grids, and spetral disretisations where the separaton apriori of the salesis simpleto

ahieve.Forniteelementapproximations,ahierarhialbasisapproahandanalternativemethod

basedonellagglomerationwerereentlyproposedforseparatingaprioritheoarse-andne-sales.

Inmostases,theVMS-LESmethodwasappliedmainly tohomogeneousisotropiinompressible

turbulene, and reently to inompressible turbulent hannel ows, fo with it demonstrated an

improvementoverthetraditionalLESmethod.

Inthis Variational Multisale approah for Large Eddy Simulation (VMS-LES) approah the

owvariablesaredeomposed asfollows:

w i = w i

|{z}

LRS

+ w ^′ _i

|{z}

SRS

+w i SGS

(26)

where

w i

^{are the largeresolved sales (LRS),}

w ^′ _i

^{are the small resolved sales (SRS) and}

w i SGS

aretheunresolvedsales. ThisdeompositionisobtainedbyvariationalprojetionintheLRSand

SRSspaesrespetively. Inthepresentstudy,wefollowtheVMSapproahproposedinRef.[5℄for

thesimulationofompressibleturbulentowsthroughanitevolume/niteelementdisretization

onunstruturedtetrahedralgrids. If

ψ l

^arethe

N

nite-volumebasisfuntionsand

φ l

^the

N

^nite-

elementbasisfontionsassoiatedtotheusedgrid,inordertoobtaintheVMSowdeomposition

inEq. (26),thenitedimensionalspaes

V F V

^and

V F E

^,respetivelyspannedby

ψ l

^and

φ l

^,anbe

in turndeomposedasfollows[5℄:

V F V = V F V

M V _{F V} ^′ ; V F E = V F E

M V _{F E} ^′

⁽²⁷⁾

in whih

L

denotesthediret sumand

V F V

^and

V _{F V} ^′

^{arethe nitevolumespaes assoiatedto}

thelargestand smallestresolvedsales,spanned bythebasis funtions

ψ l

^and

ψ ^′ _l

^;

V F E

^and

V _{F E} ^′

arethenite elementanalogous. InRef.[5℄aprojetoroperator

P

^{in theLRSspaeisdened by}

spatialaverageonmaroellsinthefollowingway:

W = P ( W ) = X

k



 



V ol(C k ) X

jǫI k

V ol(C j ) X

jǫI k

ψ j



 



| {z }

ψ _k

W _k

(28)

fortheonvetiveterms,disretizedbynitevolumes,and:

W = P ( W ) = X

k



 



V ol(C k ) X

jǫI k

V ol(C j ) X

jǫI k

φ j



 



| {z }

φ _k

W _k

(29)

forthediusiveterms,disretizedbyniteelements. InbothEqs. (28)and(29),

I k = {j/C j ∈ C m(k) }

^,

C m(k)

^{being the maro-ell ontaining the ell}

C k

^{. The maro-ells are}

obtained by a proess known as agglomeration [9℄. The basis funtions for the SRS spae are

learlyobtainedasfollows:

ψ _l ^′ = ψ l − ψ l

^and

φ ^′ _l = φ l − φ l

^.

A keyfeature of the VMS-lesapproahis that the modeled inuene of the unresolvedsales on

largeresolvedonesissettozero,andsotheSGSmodelisaddedonlytothesmallestresolvedsales

(whih modelsthedissipativeeetoftheunresolvedsalesonsmall resolvedones). Thisleadsto

thefollowingequationsaftersemi-disretizations[5℄.

Z

C i

∂ρ

∂t Ω + Z

∂C i

ρ~ V .~nΓ = 0 Z

C i

∂ρ~ V

∂t Ω + Z

∂C i

ρ~ V ⊗ V ~nΓ + ~ Z

∂C i

p~nΓ + 1

Re Z

Ω

σ∇Φ i Ω + 1 Re

Z

Ω

τ ^′ ∇Φ ^′ _i Ω = 0 Z

C i

∂E

∂t Ω + Z

∂C i

(E + p) V .~nΓ + ~ Z

Ω

σ ~ V .∇Φ i Ω + γ

ReP r Z

Ω

∇e.∇Φ i Ω + γ ReP r t

Z

Ω

µ ^′ _t ∇e ^′ .∇Φ ^′ _i Ω = 0

(30)

where

e

^{denotes the internal energy (}

E = e + ¹ ₂ V ~ ²

^{) with}

V ~ = (u 1 , u 2 , u 3 )

^theveloity.

τ ^′

^{is the}

smallresolvedsalesSGS stressgibenby:

τ ^′ = µ ^′ _t (2S _ij ^′ − 2 3 S _kk ^′ δ ij )

with

S _ij ^′ = ¹ ₂ ( ^∂u

′ i

∂x j + ^∂u

′ j

∂x i )

^and

µ ^′ _t

^{, thesmall resolved saleseddy visosity (whih depends on the}

hosenSGSmodel).

Oneannotiethat thelaminar Navier-Stokesequations arereoveredbysubstituting

τ ^′ = 0

and

µ ^′ _t = 0

^{in Eq. (30)aboveanthattheSGSmodelisreoveredby}substituting

τ ^′ = τ

^,

µ ^′ _t = µ t

^,

e ^′ = e

^and

Φ ^′ _i = Φ i

^{in theequations,where}

τ

^and

µ t

^{denotetheusualSGS stresstensorandSGS}

eddyvisosity,respetively.

Moredetails aboutthis VMS-LESmethodologyanbefoundinRef. [5℄and[2℄.

Thegraphofagaussianisaharateristisymmetri"bellshapeurve" thatquiklyfallstowards

plus/minusinnity.

Forour2Dstudy,weonsider theadvetionequation:

∂ρ

∂t (x, y, t) + c ∂ρ

∂x i (x, y, t) = 0

usinganadvetionvetor

c = (0, 1)

^{andagaussianfuntion asinitialondition:}

ρ(x, y, z, 0) = 1 + exp ^{− 150(x+0,3)}

Theadvetionequation is solvedusing theMass Lumping V6sheme and theMass MatrixCen-

tral Dierening shemewith twooptions: projeteddissipation andnon projeteddissipationas

desribed in setion3.3. Realling thata"Mass lumping"shememeans that thetemporal term

oftheequationsistreatedbyFiniteVolumewhereasMass-Matrixshememeansatemporalterm

treatedbyFiniteElement. Weareinpartiularinterestedonthedissipationofthegaussianwhen

it istranslated usingdierentmeshes. The rstmesh is regularasshown FIG.6 andthe seond

oneisirregularwithstrongvariationsoftheloalmeshsize(seeFIG.9)

Figure6: Regularmesh

Intheregular ase, weplot respetivelyin FIG. 8and FIG. 7theGaussian translation using

theMassMatrixCentralDiereningshemeandtheMassLumpingV6sheme. Oneanobserve

that the translation is well predited. In the irregular ase, as shown FIG. 11, the translation

is well preditedwith the MassMatrix Central sheme and using theV6 sheme, onehavesome

perturbations,seeFIG.10.

Figure7: Gaussian translatedwiththeMassLumpingV6shemeonregularmesh

Figure 8: Gaussian translatedwiththeMass MatrixCentral Dierening sheme(with projeted

dissipation)onregularmesh

Figure9: Irregularmesh

Figure10: GaussiantranslatedwiththeMassLumpingV6shemeonirregularmesh

Figure11: GaussiantranslatedwiththeMassMatrixCentralDiereningsheme(withprojeted

dissipation)onirregularmesh

and atasubritial ReynoldsnumberRe

D

^{of3900 (Re}

D = u ∞ D

ν

^{)basedon ylinderdiameter}

D

andfree-steam veloity

u ^∞

^{. The}omputationaldomainasshownin FIG.12 is

−10 ≤ x/D ≤ 25

^,

−20 ≤ y/D ≤ 20

^and

−π/2 ≤ z/D ≤ π/2

^where

x

^,

y

^and

z

^{denote the}streamwise,transverseand spanwisediretion respetively. Theharateristisofthedomainarethefollowing:

L i /D = 10, L 0 /D = 25, H y /D = 20

^and

H z /D = π

Theylinderofunitdiameter isenteredon

(x, y ) = (0, 0)

^.

Theow domain isdisretized byan unstruturedtetrahedralgrid whih onsists of approxi-

matively

2.9 × 10 ⁵

^{nodes. Theaverageddistane ofthenearestpointtothe ylinderboundaryis}

0.017D

^{whihorrespondsto}

y ⁺ ≈ 3.31

^.

Forthepurposeofthesesimulations,theSteger-Warmingonditionsareimposedattheinow

and outow as well as on the upper and lower surfae (

y = ± H y

^{). In the spanwise diretion}

periodi boundaryonditionsis applied. Onthe ylindersurfaeno-slip boundary onditionsare

set.

Figure12: Computationaldomain

ToinvestigatetheinueneofnumerialshemesontheVMS-LESapproah,threesimulations

havebeenarriedoutusingtheWALEsubgrid-salemodel. Theharateristisofthesimulations

performedforthisstudyaresummarized onTAB.1. OneusestheRoe-Turkelsolveraspreondi-

tioning. Forstabilityreasons,themass-matrixshemehasbeenrunsettingthenumerialvisosity

parameter

γ

^{to its maximalvalue1. In thesamemanner, theprojeted}dissipation optionis not presentedhere.

Simulation NumerialSheme Valueofonstant

C

Dissipation

γ

^CFL

Simu1 MassMatrix+CentralDi.

1

30

^{Nonprojeted 1 20}

Simu2 MassLumping+V6

1

30

^{Nonprojeted 1 20}

Simu3 MassLumping+V6

1

30

^{Nonprojeted 0.3 20}

Simu4 MassMatrix+Centraldierening

1

15

^{Nonprojeted 1 20}

Table1: Simulationsusing VMS-LES approah withthe WALEsubgrid-sale model, onstant

C

dened insetion3.3byequation16

The CFL number has been hosen sothat avortex shedding yle is sampled in alittle less

than1000timesteps. Time-averagedvaluesandturbuleneparametersaresummarizedinTAB.2

andomparedto datafrom experimentsofNorberg,OngandWallaeandParnaudeauetal..

C d

denotesthemeandragoeient,

C _d ^′

^and

C _l ^′

respetivelytherootmeansquarevaluesofthedrag andlift,

St

^{theStrouhalnumber,}

U min

^{themean enterlinestreamwiseveloity,}

Cp back

^themean

bak-pressureoeientand

l r

^themeanreirulationlength.

St C d C _d ^′ C _l ^′ U min Cp back l r

Simulations

Simu1 0.252 0.77 0.0157 0.0367 -0.20 -0.66 1.76

Simu2 0.213 1.08 0.0490 0.4912 -0.29 -1.09 0.83

Simu3 0.215 1.01 0.0639 0.5920 -0.29 -1.02 1.08

Simu4 0.215 1.02 0.0103 0.0303 -0.29 -0.94 1.12

Experiments

[11℄ 0.215

±

^{0.05 0.99}

±

^{0.05 -0.24}

±

^{0.1 -0.88}

±

^0.05

[12℄ 0.21

±

^{0.005 1.4}

±

^0.1

[13℄ -0.34 1.51

Table2: Flowparametersforthepresentsimulations

As shown in TAB. 2, for the samelevel of numerial dissipation (Simu1, Simu2 and Simu3),

the owparametersobtained withthe mass-matrixshemeare generallyless well preditedthan

thoseobtainedwiththelassialV6sheme. Inpartiularthemeandragisunder-estimatedwith

themass-matrixsheme,whihinvolvesanover-estimationofthereirulationlength. Comparing

Simu1andSimu4,oneannotethattointrodueanumerialdissipationtwiebiggersigniantly

improvetheresults.

FIG. 13showsa proleof time-averagedand

z

^{-averagedstreamwiseveloity. Onean deter-}

minize from this plot the most dissipative sheme, that is the sheme produing themost short

reirulationlength. This assumptionis veriedonFIG. 14,showingthe pressuredistribution on

the ylinder surfae averaged in time on homogeneousz diretion. The most dissipative sheme

is indeed the one being the most far from the experimental data. From this twoplots, onean

onludethatthemass-matrixshemeisnotenoughdissipative. Thisanexplaintheenountered

robustness problem. Introduing a bigger dissipation level redue onsiderably the reirulation

length. Oneanalso note FIG. 14 thedisrepanies with theexperimental data, for

θ

^beingbe-

2 4 6 8 10

−0.5 0 0.5

x

<u> Simu1

Simu2 Simu3 Simu 4

Experiment Parnaudeau et al.

Figure13: Time-averagedstreamwiseveloityontheenterlinediretion

tween

60

^and

100

^{. Thisisprobablydueto theoarsenessoftheusedgrid.}

0 20 40 60 80 100 120 140 160 180

−1.5

−1

−0.5 0 0.5 1

Angle θ (0 at stagnation point)

<Cp>

Simu1 Simu2 Simu3 Simu 4 Experiment

Figure 14: Time-averaged and z-averaged pressure distribution on the surfae of the ylinder,

experiment: Norberg

FIG.15displaysthetotalresolvedReynoldsstress

u′u′

^{in thewakeat}

x = 1.54

^andFIG.(16)

displaysthetotalresolvedReynoldsstress

v′v′

^at

x = 1.54

^{. OneanobserveFIG.15thattheplot}

is not symmetri using the mass-matrixsheme. With alowdissipation level, the better results

are obtainedwith theV6sheme. The behaviourof theMass-Matrixsheme isimprovedusinga

biggernumerialvisosity.

−2 0 −1 0 1 2

0.05 0.1 0.15 0.2 0.25

y

<u’u’>

Figure 15: Total resolved streamwise Reynolds stress

< u′u′ >

^at

x = 1.54

^, experiments: Par- naudeau et al.;

−−

^{: Simu1;}

−

^{: Simu2;}

...

^{: Simu3;}

−.

^{: Simu4;}

+

^{: PIV1} experiment;

×

^{: PIV2}

experiment

−2 0 −1 0 1 2

0.1 0.2 0.3 0.4 0.5

y

<v’v’>

Figure 16: Total resolved streamwiseReynolds stress

< v′v′ >

^at

x = 1.54

^, experiments: Par- naudeau et al.;

−−

^{: Simu1;}

−

^{: Simu2;}

...

^{: Simu3;}

−.

^{: Simu4;}

+

^{: PIV1} experiment;

×

^{: PIV2}

experiment

fortheMass-Matrixsheme(Simu4)givesabetteragreementwiththeexperiments.

6 Conlusion

WehavepresentedinthisworkarstinvestigationofanewMassMatrixsheme,inordertoobtain

abetterauraythanforMassLumpingFVshemeonirregularmeshes.

Thisnumeriswasinstalledinaparallel odeAEROofresearhandprodution.

Therstresultsarepromising. FortheGaussiantranslation,theMass-Matrixlearlyimprovedthe

preditionusinganirregularmesh. Wehavealsoshownthattheresultsobtainedforthesimulation

oftheowaroundairular ylinderareimprovedwith theMass-Matrixshemewhenasuitable

levelofnumerialdissipationishosen.

This rstwork ontheMass-Matrixshemeneedsto be ontinued byinvestigating thedissipation

term and the behavior of this model for the simulation of turbulent ows on strongly irregular

three-dimensionalmeshes.

Aknowlegments: WethankEriLamballaisforkindlysendingusdatarelatedto[13℄. CINES

andINRIA aregratefully aknowledgedforhavingprovided theomputationalresoures.

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