Technische Universität DarmstadtFachgebiet Strömungslehre und Aerodynamik

Leiter: Prof. Dr.-Ing. habil. C. TropeaSLA

Comparative assessment of hybrid LES/RANS modelsComparative assessment of hybrid LES/RANS modelsin turbulent flows separating from smooth surfacesin turbulent flows separating from smooth surfaces

S. S. ŠarićŠarić, B. , B. KniesnerKniesner, A. , A. MehdizadehMehdizadeh, S. , S. JakirlićJakirlić, K. , K. HanjalićHanjalić*, C. *, C. TropeaTropea

Second Symposium on Hybrid RANS - LES Methods

June 17-18, Corfu, Greece, 2007

Institute for Fluid Mechanics and Aerodynamics,Institute for Fluid Mechanics and Aerodynamics,

Darmstadt University of Technology, Darmstadt, GermanyDarmstadt University of Technology, Darmstadt, Germany

**MarieMarie Curie Curie ChairChair, , UniversitaUniversita di Roma di Roma „„La SapienzaLa Sapienza““, , ItalyItaly

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OutlineOutline

Motivation/Objectives

Turbulence Modelling / Simulation strategies:

- LES, DES, Delayed DES (DDES)

- Hybrid LES-RANS (HLR)

Prediction of the turbulent flows featuring separation from smooth surfaces:

o High-Re number flow over a wall-mounted hump with separation control

o Separated flow over smoothly contoured periodic hills

•The SAS and instability sensitized RANS approach based on a

conventional RANS model (IS k-ɛ)

Conclusions/Outlook

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Motivation: currently available optionsMotivation: currently available options

RANS closures representing the mainstay of the contemporary induRANS closures representing the mainstay of the contemporary industrial strial CFDCFD are affordable, economical, applicable to arbitrarily complex geometries, multi-physics (multi-phase, reactive/combusting,…) flows,…

especially affordable in attached flow regions → nearespecially affordable in attached flow regions → near--wall regionswall regions

B U TB U T

Too much empiricism, lack of universality, difficulties in Too much empiricism, lack of universality, difficulties in predicitingpredicitingcomplex unsteady and noncomplex unsteady and non--equilibrium flows,…equilibrium flows,…

Cannot account for any spectral dynamics, and are thus especiallCannot account for any spectral dynamics, and are thus especially y limited when the flow is dominated by large, coherent eddy struclimited when the flow is dominated by large, coherent eddy structures tures with a broader spectrumwith a broader spectrum

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Motivation/ObjectivesMotivation/Objectives

LES: resolves major portion of the turbulence, captures spectral dynamics of large eddies and the physics of turbulence in general; it has been considered as the future industrial standard, but

Expensive and time consuming; uncertain for realistically high Re-number wall-bounded flows in complex geometries

LES is also uncertain in off-wall flows if the filter is not well in the inertial subrange (too coarse grid) because most SGS models cannot account for anisotropy and spectral non-equilibrium

Possible new strategy:Blending RANS and LES to utilize advantages and overcome shortcomings of each method

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FASTEST 3DAn in-hause code, based on the finite volumeFV method; collocated variable arrangement; SIMPLE algorithm, second order accuracy in both space (CDS) and time (Crank-Nicolson); block-structured, body-fitted, non-orthogonal meshes.

Numerical MethodNumerical Method

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Computational methods used: rationaleComputational methods used: rationale

LES; SGS model due to LES; SGS model due to SmagorinskySmagorinsky ((CCSS=0.1=0.1))

( )kk

i

i

SGSijji

j

i

xxU

xpUU

xtU

∂∂∂

+∂∂

−=+∂∂

+∂∂ 21 ν

ρτ ijSGS

SGSij Sντ 2−=

SCSSGS2Δ=ν

DES; grid dependent model due to the SDES; grid dependent model due to the S--A used as the SGS model A used as the SGS model ((CCDESDES=0.65=0.65))

““Delayed DES” (DDES) approach:Delayed DES” (DDES) approach:

( )zyxDES ΔΔΔ=Δ ,,max

( ) 3/1zyx ΔΔΔ=Δ

22dS~~

rκν

=22

,,

~

dUUr

jijid κ

ν=

( )Δ−−= DESd Cdfdd ,0max~ [ ]( )38tanh1 dd rf −=

22dS~~

rκν

=22

,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

= 22dS~~

rκν

=22

,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

=

( )Δ−−= DESd Cdfdd ,0max~

22,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

=

[ ]( )38tanh1 dd rf −=( )Δ−−= DESd Cdfdd ,0max~

22,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

= 22dS~~

rκν

=22

,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

=

( )Δ−−= DESd Cdfdd ,0max~

22,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

=

[ ]( )38tanh1 dd rf −=( )Δ−−= DESd Cdfdd ,0max~

22,,

~

dUUr

jijid κ

ν=22dS~

~r

κν

=

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Present hybrid LES/RANS model scheme

+y

Presently: low–Re EVM models due Chien, Launder & Sharma and Jakirlic & Hanjalic and k-ε-f-ζ models are used: ),,,,,,( Tkpwvu ε

ενν μμ

2kfCtm ==

εσνν −+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=∂∂

+∂∂

kjk

t

jjj P

xk

xxkU

tk

3,

2

2,21, εεεε

εεεσννεε P

kCf

kPC

xxxU

t kj

t

jjj −−+⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=∂∂

+∂∂

Smagorinsky / YoshizawaTpwvu ,,,, Tkpwvu SGS ,,,,,

SCSSGSm2)( Δ==νν32)( SCSSGS Δ=ε

3.0)(

.122 SC

k SSGS

Δ=

Δ=

2/3SGS

SGSkCεε( ) SGSk

j

SGSSGS

jj

SGSj

SGSSGS

Px

kxx

kUt

k ενν −+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

=∂∂

+∂

∂.21/ 2

m SGS k SGSC kν ν= = Δ

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

+∂∂

−=∂

∂+

∂∂

i

jm

j

im

jij

jii

xU

xU

xxp

xUU

tU ννν

ρ)(1 *

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

+=∂

∂+

∂∂

jt

m

jpj

j

xT

xcq

xTU

tT

PrPrνν&

incompressible, EVM

)/2~(~ 2ykDtDDtD νεεεε −=→

( )νεεεε kDDtDDtD 5.0homhom −=→

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→ continuity of continuity of ννmm is providedis provided

Matching RANS and LES at interface

ifce

νt,rans=νt,sgs

ννt,ranst,rans determined by using a 2determined by using a 2--eq. EVM: eq. EVM: DDkkransrans//DDtt; ; DDεεransrans/D/Dtt –– solved in solved in the entire solution domain; zero values of the the entire solution domain; zero values of the discretizationdiscretizationcoefficients taken in the LES region, the source terms appropriacoefficients taken in the LES region, the source terms appropriately tely manipulated manipulated

→ in such a way the boundary conditionsin such a way the boundary conditionskkransrans==kksgssgs and and εεransrans==εεsgssgs, i.e. , i.e. ννt,ranst,rans==ννt,sgst,sgs is imposedis imposed

k P P k k Uk k

A S A S⎛ ⎞+ Φ = Φ +⎜ ⎟⎝ ⎠∑ ∑ 30 300; 10 ; 10k P U sgsA S S= = = Φ

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Interface criterion, 1

reskkkk+

=mod

mod*Control parameter:

condition:

%20* ≥k move the interface away from the wall

channel

(Illustration) expressed in terms of y+:

Backward-facing step

Criteria background: fraction of resolved scales (≈80%, Pope, 2000)

Fixed interface at y+=230

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Interface criterion, 2Illustration of k*-value along the interface plane:

HLR y+int=200

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Separated flow over a wallSeparated flow over a wall--mounted 2Dmounted 2D--hump hump

936000Re == ∞

μρ cU

Rumsey, C., and Gatski, T. (2004): CFD Validation of Synthetic Jets and Turbulent Separation Control, NASA Langley RC Workshop. Williamsburg, VA, March 2004

Cord length c=0.42 m, UCord length c=0.42 m, U∝∝=34.6 m/s=34.6 m/s

Lx=6.14c (Lx=6.14c (--2.14c to 4.0c) 2.14c to 4.0c)

Ly=0.91cLy=0.91c

Lz=0.152c (LES)Lz=0.152c (LES)

Lz=0.2c (DES)Lz=0.2c (DES)

LESLES--426x145x64 (~4x10426x145x64 (~4x1066 CV’s)CV’s)

LESLES--cc-- 426x145x32 426x145x32

HLRHLR--426x145x32426x145x32

DESDES--426x145x28 (~1.7x10426x145x28 (~1.7x106 6 CV’s)CV’s)

yy++<1 , <1 , ΔΔxx++ ~80, ~80, ΔΔzz++~50 (LES) , ~50 (LES) , ΔΔzz++~150(DES)~150(DES)

RANSRANS--LES interface at yLES interface at y++=30=30--9090

Inlet B.C.s (turbulent B.L.):Inlet B.C.s (turbulent B.L.):

taken from the experiment at x=taken from the experiment at x=--2.14c2.14c

Walls: noWalls: no--slip b.c.slip b.c.

Side planes: spanwise periodicity Side planes: spanwise periodicity

Outlet plane: convective b.c.Outlet plane: convective b.c.

Greenblatt et al. (2004, 2005)

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Ratio of the filter width to the Kolmogorov length scaleKolmogorov scale assessed from the RANS computations (HJ near-wall SMC)

( ) 3/1zyx ΔΔΔΔ =

4/13

K ⎟⎟⎠

⎞⎜⎜⎝

⎛=

ενη

Assessment of the spatial resolution Assessment of the spatial resolution -- LES LES

Ideally, 1210/ K −≤ηΔ

Pope (2000)

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Pressure coefficient Pressure coefficient –– CCp p , separation eparation -- (x/c)S and reattachment and reattachment -- (x/c)R

Baseline configuration Steady suction configuration(x/c)S (x/c)R (x/c)S (x/c)R

Exp. 0.673 1.110 0.686 0.940DES 0.663 1.121 0.674 1.105LES 0.667 1.114 0.671 0.947HLR 0.669 1.183 0.677 0.949

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Mean Mean streamwisestreamwise velocity and shear stress velocity and shear stress u’vu’v’ (baseline flow)’ (baseline flow)

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DES DES vsvs LES on the same grid (baseline flow)LES on the same grid (baseline flow)

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Influence of the LES/RANS interface (baseline flow)Influence of the LES/RANS interface (baseline flow)

HLR y+int=200

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Mean U Mean U -- velocity and shear stress velocity and shear stress u’vu’v’ (steady suction flow control)’ (steady suction flow control)

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DDES DDES vsvs DES (steady suction flow control)DES (steady suction flow control)

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The LES/RANS interface (steady suction) The LES/RANS interface (steady suction)

DES ____

DDES ____ (fd=0.999)

DDES ____ (fd=0.99)

- Grid design remains the main issue in DDES

- DDES appears to be less capable than DES, as far as the hump flow with the flow control is concerned

[ ]( )38tanh1 dd rf −=

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Lx=9H

H

Ly=3.035H

Lz=4

.5H

80x100x30 = 240 000 cells (DES, HLR, DDES)

160x100x30 = 480 000 cells (SAS, IS k-ɛ)

The reference data: LES (Breuer) 13 Mio cells

-Periodicity in streamwise and spanwise directions , no-slip at the walls.

-Streamwise pressure gradient adjusted to provide the mass flow rate corresponding to Re=10 595 (based on the mean bulk velocity Ub at the hill crest and the hill height H)

Separated flow over smoothly contoured periodic hills

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Instability-Sensitive k-ε Model

Conventional, near-wall k-ε model, based on the ε∼-variable (e.g., LS model):2

,1 ,2 ,3t

j kj j j

U C P C Pt x x x k kε ε ε

ε

νε ε ε ε ενσ

⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂+ = + + − +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

% % % % %

Recall, Pε,3 term in the RSM-framework (Jakirlic and Hanjalic, 2002):

2 2 2*

,3 ,3

mod

2 2i i k i i k l i ik

l k l l k l j k j l

exact elled

u U u u U u u U UkP u Cx x x x x x x x x xε εν ν

ε⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂

= − = +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠%1442443 1444444442444444443

Recall, Pε,3 term in the EVM-framework (Rodi and Mansour, 1993):

2 2 2 2' ''

,3 ,3 ,3

mod

2 2i i i i i ik t

l k l k l k l k l k l

exact elled

u U U U U Uk kP u C Cx x x x x x x x x x xε ε εν ν ν

ε⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂∂

= − = +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠%1442443 144444444424444444443

Pε,3 term in the IS k-ε model:

( )2

5/ 22

,3 ,30.182 " 2 0.024 "t

j

kP C U S Uxε ετνν ε

ε σ

⎡ ⎤⎛ ⎞∂⎢ ⎥= + − ⎜ ⎟⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦

,1 ,2 ,31.44; 1.92; 1.0C C Cε ε ε= = = Note that the LS model contains only the first part of the Pε,3 term !

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SAS Model, Menter and Egorov (2004)

23/ 4 t

j kj j k j

k k k kU P Ct x x xμ

νσ⎛ ⎞∂ ∂ ∂ ∂

+ = − + ⎜ ⎟⎜ ⎟∂ ∂ Φ ∂ ∂⎝ ⎠2

1 2 33/ 2ˆ " t

j k tj j j

U P S U kt x k k x xφ

νζ ζ ν ζσ⎛ ⎞∂Φ ∂Φ Φ Φ ∂ ∂Φ

+ = − − + ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠1/ 4

tCμν −Φ =

Starting point: k-Φ model ( , SKL model, Menter, Egorov, 2004)

( )2

23 21 3/ 4 1/ 4 3/ 2

ˆ2 2 "j k t

j

t

j j

U P S Ut x k C f k C k

x x

μ μ μ

ε

ζ ζε ε ε ε εζ ν

ν ενσ

⎛ ⎞∂ ∂+ = − − − +⎜ ⎟⎜ ⎟∂ ∂ ⎝ ⎠

⎡ ⎤⎛ ⎞∂ ∂+ +⎢ ⎥⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

The resulting ε-equation after transformation (diff. term was not transformed):

kLΦ ≡

2

tkC fμ μνε

=Where:

Work still in progress!

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IS k-ɛ simulation of the plane channel flow

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SAS / IS k-ɛ predictions of the 2D hill flow

Isosurface of the pressure fluctuation – LES (Breuer) Vorticity magnitude coloured by pressure IS k-ɛ

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SAS / IS k-ɛ predictions of the 2D hill flow

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Conclusions/OutlookConclusions/Outlook

Different hybrid LES/RANS modeling approaches: DES, DDES, the zonal hybrid LES/RANS scheme (HLR), SAS and an Instability-Sensitized (IS) k-ε model were used to predict the flow over a wall-mounted hump at high Reynolds number and the flow over a periodic hill

The promising results obtained by the two schemes proposed by the authors - HLR and IS k-ε - with respect to the structural characteristics of the instantaneous flow field, the mean velocity field and associated integral parameters (pressure coefficient), as well as the turbulence quantities demonstrate their feasibility and applicability in a broad range of complex, wall-bounded turbulent flows.

Further development of the IS k-ε and its validation in complexflow configurations