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Q-LES 2007, 24-26 October, Leuven, Belgium.

Optimal Unstructured Meshing for Large Eddy Simulation

Y Addad, U Gaitonde, D LaurenceSpeaker: S. Rolfo

The University of Manchester, M60 1QD, UKSchool of Mechanical, Aerospace & Civil Engineering.

CFD group

The

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L.E.S. on unstructured grids

Unstructured FV industrial codes Geometry complexity imposes

unstructured grids (Even pipe flow requires unstr. grid. (no channel flows in industry !)

Indust. Pbs often Multiscale

L.E.S Principle = grid that captures larger

eddies + some of the energy cascade

Integral Length-scale is highly variable in any real application

Most LES today still on structured grids. PWR lower

Plenum(EDF

Code Saturne)

Why not use flexibility of unstructured FV to fix the

cell size to LES criteria LOCALLY?

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Channel Flow LES on structured gridat Re*=395 (Re*=y+ at centre )

% error on friction

Under-Resolved LES

Under-resolved LES => more dangerous than coarse RANS !

=> Q-LES very much needed now that Industry is into LES

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Empirical Guidelinesfor “hand made mesh”:

Guidelines for channel flow unstructured grids

- Much experience for channel flow, -but what about new applications (and real prediction) ?- “hand made mesh” is tedious ! - Ideal would be to feed precursor RANS results into automatic mesh generator

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Again “hand made mesh”. Can we make it automatic?

Zonal mesh adaptation to integral scale

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Kolmogorov lengthscale

Taylor microscale

Von Karman Lengthscale

Turbulent energy lengthscale

Integral lengthscale

13 4

2

2

1

1

2f

u

u

x

22 yU

yULvK

23k

Lturb

11 11 111 0

1, , ,

0, ,L x t R e r x t dr

R x t

Length scales

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Integral length scales in channel flow

Turbulent energy scale is easy RANS “model” but does not represent true (2 point correlation) integral scale for channel flow

23k

Lturb

streaks

1- x : streamwise2- y : wall normal3 – z : spanwise

Solid: stream-wise separationDashed: span-wise separation

Nb: longitudinal lenghtscales are divided by 2

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Taylor micro scales in HIT

2 2

2

2 22

2

2

22

2

'( ) '( )'( ) '( ) .....

2

' ( )'( ) '( ) ' ( ) .....

2 2

1 ....

2 ' ( )

'( )

uu

u x r u xu x r u x r

x x

r u x ru x r u x u x

x

rR

with

u x

u xx

2/121

2/3

'15

/

uL

kL

Tay

Available from RANS

Nb: Integral and Kolmororov scales can be combined to form the Taylor scale

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Taylor length scales in channel flow

Lines = “home” fine LES Symbols = only points available DNS data

(THT lab Tokyo U., N Kasagi)

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Comparison between different length-scales for the channel flow test case at Re=720.

10 Kolmogorov

Taylor spanwise

Tayor streamwise

Integral /10

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Nx Ny Nz

LES 68 to 200 46 42 to 100

DNS 256 193 192

• Re=395• Domain 2 2 • LES Ncells= 443,272• DNS Ncells = 9,486,336 (Ref: Moser et al. 1999)

Grid generation following Taylor scale

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Channel Re=395 with different grid topology

O DNS (10 Million cells)Structured Grid (0.3M)Bloc refinement 2-32h=Taylor continuous fit (0.44M)

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Bloc refinement 1-2(Benhamadouche Thesis)

Channel Re=395 with different grid topology

O DNS (10 Million cells)LES (0.5 Million cells) :- Structured Grid- Bloc refinement 2-3- Taylor/2 continuous fit

Present LES (Addad)

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Span = 1 cell to 64 cells on body)

Embedded refinement strategy

1 to 2 refinement with central differencingleads to spurious oscillations

2 to 3 refinement now systematically used

2 to 3 refinement now systematically used

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Energy conservation: Taylor-Green vortices test case.

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Energy conservation: Taylor-Green vortices test case.

Energy conservation: Taylor-Green vortices test case.

Mesh smoothing for LES see also Iaccarino & Ham, CTR briefs 05

Energy conservation: Taylor-Green vortices test case.

Error map for the U velocity component for the Cartesian mesh 60x60

Error map of U for the Cartesian mesh 60x60 + 5-8 refinement

Error map of U for the Cartesian mesh 60x60 + 1-2 refinement.

Max error where the velocity is min and the V component is max.

Max error in the middle

Energy conservation: Taylor-Green vortices test case.

Velocity components are pointing in the wrong directions.

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Conclusions

Lenghtscales from precursor RANS simulations can be used to estimate LES grid requirements.

Mesh following span-wise Taylor micro-scales in stream-wise, and span-wise directions is close to “empirical knowledge” and gives good agreement of the LES results with DNS

Less trivial test cases are necessary to define criteria 2h=f (Integral s., Taylor s., Kolmogorov)and demonstrate real benefits (jets, separated flows…)

Use of non conformal meshes can introduce spurious oscillations in the solutions. More investigations, in particular focussing on the interpolation of flow quantities at the cell faces, are studied in order to avoid the problem.

AcknowledgementsAcknowledgements This work was carried out as part of the TSEC programme KNOO and as such we are grateful to the EPSRC for funding under grant

EP/C549465/1, and to N Jarrin for Saturne code results

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Arbitrary unstructured grids

Control of cell size essential in LES

Refinements:Bloc structured (+ non-conform refin’t)or Distributed refinement?

Note: “hanging node” = 5 sided cell,no special treatment

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Colocated unstructured Finite Volumes

- Ferziger & Peric: Computational Fluid Dynamics, 3rd edt. Springer 2002.

-“Face based” data-structure => simple

- Fine for convection terms

- Approximations come from

interpolations and Taylor expansions from

cell centres to cell faces

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Interpretation:- an LES instantaneous field is NOT an instance of a filtered

DNS field (S. Pope)- but it should give the same statistics as a filtered DNS field - means eliminate all statistical bias in numerical scheme- preserve symmetries of NS rather that solve it

(as lattice Boltzmann or SPH give NS solution “statistically”)

= “phase error not so important as amplitude error”

= “position of vortex not important, but magnitude should be conserved”

- MILES ?

LES and low order schemes

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Energy conservation ? Ex. Scalar convection

1 2 1 1

1 2 2

1( ) ( ) ( )

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(( ) ( ) )2

n n n n nI I I I I I II

n nI I I

t t

t

IJ SIJm u ndS mass flux across face between cells I and J

(1 )IJ IJ I IJ J interpolation on IJ face

I contains the non-orthogonality correction

IJ interpolation weighing. If regular grid 1 2IJ

convection term for cell I is I IJ IJC m

1 2 1 2 1 2( (1 ) )n n nI I I I IJ IJ I J IJ IJ

J neighbours

C m m

cancel locally if

IJ is constant

1 2 1 2 1 2( (1 )( ) ( ))n n nJ J J J IJ IJ J I IJ IJ

I neighbours

C m m

1 2nI I

cancel 2x2 if

and

FV conserves mass & momentum,Energy can only be conserved?

1 2IJ

Conservation of convective flux of “energy” between cells I and J ?

Requirements: - centered in space and time, - regular mesh spacing, and no non-orthogonality corrections- mass flux may be explicit

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L.E.S. of H.I.T. Code_Saturne

Classic test for LES: Homogeneous Isotropic Turbulence (decay of turbulence downstream of a grid)

Viscosity = Smagorinsky(classical LES model) Viscosity = 0

E(K)= a K2

Total energy = constant

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Inviscid H.I.T test (viscosity =0, Euler eq.)

S. Berrouk, STAR-CD V4

Viscosity = 0K2 distribution as expected

STAR-CD V4: Similar to Saturne but 3 time level scheme (2nd order in space & time)