lecture4 cfd

LES turbulence closuresHybrid RANSE/LES formulations

ApplicationsConclusions and perspectives on LES and hybrid RANSE/LES modeling

Turbulence and its physical modeling 3/3

Michel VISONNEAU

LHEEA - CNRS UMR 6598Ecole Centrale de Nantes, Nantes, FRANCE

CFD in ship hydrodynamics - Kul.24-Z courseNovember - December, 2013

ECN-CNRS Turbulence and its physical modeling 3/3



Outline of lecture 3

Large Eddy Simulation strategy

Hybrid RANSE-LES formulations

Illustrations on 3D turbulent flows




Large Eddy Simulation strategy (LES)




Foundations of the LES strategy 1/2

A turbulent flow contain a large range of different scales. The largescale motions are much more energetic than the small scales and itwould be wise to devise a simulation method treating more accuratelythe large eddies than the small ones.LES is such a method : Three-dimensional, unsteady and quiteexpensive (but much less than Direct Numerical Simulation !)




Foundations of the LES strategy 2/2

How can we define the quantities to be computed ? By filtering !

ui(x) =∫

G(x ,x ′)ui(x ′)dx ′ (1)

where G(x,x’) is a localized function called filtering kernel.Each filter may be associated with a length scale ∆, which means thateddies larger than ∆ will be computed, smaller ones will be filteredand should be modeled.




Filtered Incompressible Navier-Stokes equations

∂ρui

∂t+

∂ρuiuj

∂xj=− ∂p

∂xi+

∂

∂xj

[µ(

∂ui

∂xj+

∂uj

∂xi

)](2a)

∂ρui

∂xi= 0 (2b)

By introducing the subgrid Reynolds stress by : τSij =−ρ(uiuj −ui ui)

∂ρui

∂t+

∂ρuiuj

∂xj=− ∂p

∂xi+

∂

∂xj

[µ(

∂ui

∂xj+

∂uj

∂xi

)+ τ

Sij

](3)

which really looks like the Unsteady RANSE.




Kinetic energy transport equation 1/2

By averaging the instantaneous kinetic energy, E(x , t) = 12 U ·U, one

obtains E = 12 U ·U, which can be decomposed into E = Ef + kr where

Ef = 12 U ·U, filtered kinetic energy kr = 1

2 τRii .

The conservation equation of Ef is obtained by multiplying themomentum conservation equation by Uj , which leads to :

∂Ef

∂t+ Uj

∂Ef

∂xj− ∂

∂xi[U j(2νSij − τij

r − pρ

δij)] =−εf −Pr (4)

with the following definitions : εf = 2νSijSij , for the viscous dissipationand Pr =−τr

ijSij for the rate of production.




Kinetic energy transport equation 2/2

Estimate of normalized quantities vs the filter width ∆in the viscous sub-layer for high Re (from Pope)




Smagorinski-like models

FormulationThe most popular model was proposed by Smagorinski in 1963. It is alinear eddy-viscosity based model:

τSij −

13

τSkk δij = µt

(∂ui

∂xj+

∂uj

∂xi

)(5)

where µt is an eddy-viscosity which is defined by :

µt = C2Sρ∆2

(SijSij

)1/2and CS ≈ 0.2




Remarks

Of course, CS should not be uniform in reality and there are manyproposals to introduce specific formulas based, for instance, onVan-Driest damping close to the wall!!

CS = CS0

(1−e−n+/A+

)2(6)

where n+ = nuτ/ν is the normal distance to the wall, Uτ =√

τW/ρ isthe shear velocity and A+ = 25.

Such a non-local expression is not in agreement with the fundations ofa SGS model !




Dynamic models - Formulation from Germano et al. (1990)

An idea of a self-consistent procedure 1/2

Let us introduce two filtering procedures, the grid filter g and the testfilter t which is characterized by a wider length scale

τij = (uiuj)g−ug

i ugj (7)

Filtering again with the test filter provides a different subgrid scalestress

Tij = (uiuj)gt −ugt

i ugtj (8)

The tensor Lij = Tij − τtij = (ug

i ugj )t −ugt

i ugtj describes the resolved

turbulent stresses. The so-called Germano identity may be used tocompute dynamically the local values of CS by applying theSmagorinski model to Tij and τij .





An idea of a self-consistent procedure 2/2

Lij −13

Lkk δij =−2CSMij (9)

where:

Mij =(∆t)2 |Sgt |Sgt

ij − (∆g)2(|Sg|Sg

ij

)t(10)

CS may be computed as :

CS =−12

LklSgkl

MmnSmn(11)

or as proposed by Lilly (1991)

CS =−12

LklMkl

MmnMmn(12)





Subgrid tensor Expression Zone of usage

τij uiuj −uiuj l < ∆

Tij uiuj − ui uj l < ∆

Lij Tij − τij ∆ < l < ∆

Table: Sub grid tensors for the dynamic LES models





ReferencesGermano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991), "ADynamic Subgrid-Scale Eddy Viscosity Model", Physics of FluidsA, Vol. 3, No. 7, pp. 1760-1765.

Lilly, D. K. (1991), "A Proposed Modification of the GermanoSubgrid-Scale Closure Method", Physics of Fluids A, Vol. 4, No.3, pp. 633-635.




Dynamic models - Advantages

In shear flows, the dynamic model is able to reduce automaticallythe Smagorinski model parameter CS ,

It also decreases it in a correct manner close to the wall,

It compensates the use of inadequate length scale for anisotropicgrids by changing the value of the parameter




Dynamic models - WALE (Wall-Adapting LocalEddy-viscosity) subgrid model from Nicoud & Ducros (1999)1/4

The subgrid model should be based on a flow invariant representativeof turbulence, which may be the velocity gradient ( gij ). Smagorinskihas selected the second invariant of the symmetric part of this tensor,which leads to several limitations:

There is no contribution of the rotation rate,

Close to the wall, this invariant is O(1), which leads to theunphysical behaviour of the turbulent viscority νt ∼ O(1) close tothe wall.





Nicoud & Ducros (1999) propose to introduce the symmetric part ofthe velocity gradient squared:

sdij =

12

(g2ij + g2

ji )−13

δijg2kk

, which can be expressed in terms of deformation and rotation tensors.

sdij s

dij =

16

(S2S2 + Ω2Ω2) +23

S2Ω2 + 2IVSΩ (13)

with IVSΩ = Sik SkjΩjlΩli

With the use of such an invariant, the model is able to detect theturbulent fluctuations by using the rate of strain and the rate of rotation.





In case of a pure shear stress, this invariant will cancel, which is inagreement with the fact that shear stress zones contribute less toturbulence than convergent or rotational flows. In the case, forinstance, of a laminar Poiseuille flow, the turbulent diffusion will benegligible contrary to the Smagorinski’s model based on SijSij (large inpure shear stress), unable to reproduce the transition from laminar toturbulent regimes in such a configuration.





Close to the wall (y ' 0), one can also see that the WALE formulationsd

ij sdij tends to zero like y2, which provides a good near-wall behaviour

of the turbulent viscosity :

νt = (Cw ∆)2 (sdij s

dij )

3/2

(SijSij)5/2 + (sdij s

dij )

5/4(14)

where Cw is a true constant here.




Dynamic models - Problems

This procedure produces highly varying parameter in space andtime,

It may produce negative values of the tubulent eddy viscositywhich may lead to numerical instabilities,

This may require additional time and space averaging !

Finding a good and robust model for the subgrid scale is still a subjectof current research !




Zonal coupling methodsContinous transitional coupling methods

Hybrid LES/RANSE formulations : motivations

Although the rapid growth of the computational performance, thecomputation of massive 3D flows by LES and, a-fortiori, by DNS is still

out of reach.The simulation of flows around complete realistic airplanes is

absolutely inconceivable, because of the lack of computationalmemory and power. However, the RANS modelling approach may

determine mean fields for industrial applications at low-cost and withan acceptable accuracy if the flow is attached. It would be ideal to use

only LES in zones where a RANS modelling fails, i.e in recirculatingregions which are generally ruled by strong detached vortical

structures.

How can we build such an “HYBRID LES” strategy ?





Optimistic estimate of the year of accomplishment of variouscomputations around a 3D wing (from Spalart (2002)

Method Modelisation part Needed number of points Year

2DURANS High 105 19803DURANS High 107 1995

DES High 108 2000LES Weak 1011.5 2045DNS No 1016 2080





Hybrid LES/RANSE formulations : coupling strategies

How can we couple the RANS and LES regions ? Two approachescan be considered :

To define a-priori the boundaries dividing each region associatedto each respective turbulence model (RANSE or LES),

To design a continous transition law between each turbulencemodel depending on flow simulation parameters withoutspecifying a-priori the respective zones.

Remark: for both coupling strategies, there is an intermediate region“gray zone”, in which occurs the transition between each formulation,where the model used is neither RANSE nor LES and therefore has alimited reliability.





Hybrid LES/RANSE formulations : zonal coupling strategies

Main motivation : to decide which method will be used in whichdomain i.e more control on the computational process.One can impose to use LES in the boundary layer, a strategywhich will not be possible with DES-like modellings.However, one has to model the exchange of information betweenRANS and LES domains, a task which may seeminsurmountable...

Difficulties related with zonal couplingECN-CNRS Turbulence and its physical modeling 3/3





If Γ is the interface between the RANS domain DRANS and the LESdomain DLES , NS the Navier-Stokes Equations and MT the closureequations, the following relations have to be fulfilled :

Sur DRANS ∪Γ : NSRANS(X DRANSRANS ,X DRANS

Γ ) = 0

MTRANS(X DRANSRANS ,X DRANS

Γ ) = 0(15)

Sur DLES ∪Γ : NSLES(X DLESLES ,X DLES

Γ ) = 0

MTLES(X DLESLES ,X DLES

Γ ) = 0(16)

Where X is a flow or turbulence variable. Moreover, one should verifythat the Reynolds averaging applied to the instantaneous LESvariables does converge towards zero :

< XLES >−XRANS v 0 (17)






Two specific operators have to be built:

a Coarsening operator to build RANSE fields from LES,

an Enrichment operator to build LES from RANSE

See Quémére or Davidsson for more details.





Hybrid LES/RANSE formulations : continous transitionalcoupling strategies

This approach is based on the use of unique transport equationsvalid for the whole domain but incorporating “commutators” oftendepending on the grid fineness.

These commutators create a continuous transition from an LESmodel (when the cell size is small enough) to RANSE modelwhen the cell size becomes too coarse to capture turbulentvortical structures.

The most famous model in that category is DES (Detached EddySimulation) proposed by Spalart although there are otheralternatives briefly described here.





Other hybrid formulations

VLES (Very Large Eddy Simulation) proposed by Speziale, whichconsists in attenuating the Reynolds Stress tensor by comparingthe cell size ∆ to the Kolmogorov scale Lk ,LNS (Limited Numerical Scale) proposed by Batten whichcompares the value of the turbulent viscosity computed with theSmagorinski model to its RANSE counterpart. Therefore, theRANSE value is not systematically used when the grid is coarse,but a combination of LES and RANSE can be used, insteadSAS (Scale Adapted Simulation) is based on a modification of thek−ω SST model to make it sensitive to unsteady fluctuationsdetected by

Lvk−SAS =

√√√√√∣∣∣∣∣∣

∂Ui∂xi

∂Ui∂xi

∂2Ui∂x2

m

∂2Ui∂x2

n

∣∣∣∣∣∣ (18)





Detached-Eddy Simulation DES - Return to theSpalart-Allmaras model

This is a model based on the solution of one transport equation for ν.Turbulent viscosity is defined by a related variable, ν, and a wallfunction, fv1, ie : νt = νfv1

Far from the boundary layer, fv1 is equal to 1 and νt = ν. The transportequation for the turbulent viscosity si therefore :

∂ρν

∂t+

∂

∂xj

(ρνUj

)= cb1(1− fv1)ρSν

+1σ

[∂

∂xj

(ρ(ν + ν)

∂ν

∂xj

)+ cb2ρ

∂ν

∂xj

∂ν

∂xj

]−[cw1fw −

cb1

κ2 ft2]ρ

[ν

d

]2+ ft1ρ∆U2

(19)





Detached-Eddy Simulation DES - The DES-SA model 1/2

Idea: to use RANS in the neighborhood of walls and only use LES indetached regions. Spalart et al. (1997) proposed to modify thetransport equation of SA by replacing in the destruction term, thedistance d by d .This DES-SA model reads :

∂ν

∂t+

∂νuj

∂xj︸︷︷︸convection

= cb1Sν︸︷︷︸production

− cw1fw (ν

d)2︸︷︷︸

destruction

+1σ

[∂

∂xj((ν + ν)

∂ν

∂xj+ cb2

∂2ν

∂xj2 ]︸︷︷︸

diffusion(20)





Detached-Eddy Simulation DES - The DES-SA model 2/2

The length scale is defined by :d = min(d ,CDES∆) with ∆ = max(∆x ,∆y ,∆z) with CDES = 0.65.

when d ∆ , one is back with the classical Spalart-Allmarasmodel close to the wall where one finds very elongated cells andwhere ∆x ,∆z d ∝ ∆y .

When d ∆, ∆ becomes the length scale for the DES-SAmodel.





Detached-Eddy Simulation DES - The DES-SST model 1/2

Menter & Kuntz(2004) have proposed an adaptation of this concept tothe k−ω SST model.The destruction term E =−β∗ρωk is replaced by E =−β∗ρωkFDES

where the function FDES is defined by:

FDES = max(Lt

CDES∆,1) (21)

where Lt =√

kβ∗ω .

Where the mesh is fine enough, FDES is larger than 1, which increasesthe dissipation term in the turbulent kinetic transport equation, reducesthe turbulent viscosity which makes the simulation more unsteady. A

larger part of the turbulence spectrum is therefore computed instead ofbeing modelled.





Detached-Eddy Simulation DES - The DES-SST model 2/2

However, for some flows, FDES switches too early to the LES mode(e.g. inside the boundary layer):

Menter & Kuntz have introduced a modification of DES-SST (thefunction FSST ) in order to protect the boundary layer from anon-desired switch to LES:

FDES = max(Lt

CDES∆(1−FSST ),1) (22)

with FSST = 0,F1,F2 being the usual functions of k−ω SST model.ECN-CNRS Turbulence and its physical modeling 3/3




Detached-Eddy Simulation DES - The DDES (Delayed DES)model 1/3

To protect the boundary layer from a premature switch to LES,Menter& Kuntz have previously introduced functions depending from√

k/ω and distance to the wall.

Several meshes for the boundary layer. Top: mesh adapted to DES,left : ambiguous mesh, right : mesh adapted to LES






To introduce a length scale, Spalart proposes to use :

rd ≡νt + ν√

Ui,jUi,jκ2d2

(23)

a parameter which is going to be equal to 1 in the Log zone anddecreases to zero at the edage of the boundary layer. this term rd willbe used in the function fd ≡ 1− tanh([8rd ]3) which is designed to tendtowards 1 in LES zones where rd 1 and towards 0 elsewhere, beingmoreover insensitive to values of rd higher than 1 close to the wall.






The above-mentioned procedure applied to DES-SA leads to a newdefinition of the DES length scale:

d ≡ d− fd max(0,d−CDES∆) (24)

If fd is equal to zero, one gets the RANS model (d = d). If fd tendstowards 1, then one returns to the now classical DES model where(d = min(d ,CDES)).To switch from DES to DDES, one just have to multiply by fd the termwhich defines the difference between RANS and LES in equation 24.Note: This is not a minor modification of DES since, once thismodification is applied, the length scale d will not only depend on thegrid but also on the turbulent viscosity.




Applications

Three-dimensional hill - Comparisons with Simpson’s experiments(2005)




Three-dimensional hill - Comparisons with Simpson’sexperiments (2005)

Complex flow which is highly 3D through combined streamwise andcrosswise velocity gradients :

Re = 130000, computations with ISIS-CFD with a DES-SA modelon a coarse mesh comprised of 500 000 points,

Separation over a curved smooth surface (not imposed bygeometry)

Acceleration over the top of the hill and deceleration on theleeward side

Test case studied during the last 11th ERCOFTAC/IAHRWorkshop (2005).





Mean streamlines Instantaneous streamlines





Statistic SST closure Experiments DES-SA closure





X/h

Cp

-2 0 2 4

-1

-0.5

0

0.5

Expkw-SSTkw-EASMDES-SA

Cp profile at the spanwise direction x/h = 3.69





<Vx>

Y/h

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 ExpDESRANS

<Vz>

Y/h

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 ExpDESRANS

U velocity component W velocity component





<uu>

Y/h

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 ExpDESRANS

<ww>

Y/h

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6ExpDESRANS

uu normal stress ww normal stress




Applications

Automotive flows - The Willy body(see the self-contained presentation on automotive flows)




Conclusions 1/2

LES gives an answer to a fundamental drawback: RANS willnever be able to extract information from the dynamics of largeenergy containing eddies,

For many applications, an accurate computation of unsteadinessis crucial (aero-acoustics, fluid-stucture interaction,...) and RANSis not a well-defined unsteady approach when unsteadiness isnot imposed by the boundary conditions„

The computational time is still a fundamental problem which willnot be solved easily in the coming years (note that, most often,LES grids are coarser than RANS grids, which should not be thecase!).




Conclusions 2/2

LES-like approach is by far the best one when the flow isdominated by vortex-shedding phenomena, large recirculatingzones and can potentially model transition from laminar toturbulent regimes,

A huge effort has to be invested to make LES as reliable asRANS from a numerical point of view,

Well established Verification and Validation procedures areurgently needed to impose in the long term this promisingmodeling strategy.


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