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Hybrid LES–RANS Method Based on

an Explicit Algebraic Reynolds Stress Model

Benoit Jaffrezic, Michael Breuer and Antonio Delgado

Institute of Fluid Mechanics, LSTM Erlangen

University of Erlangen–Nurnberg

bjaffrez/breuer@lstm.uni-erlangen.de

DESider 2007Corfu, June 17, 2007

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Outline

• Motivation

• Hybrid LES–RANS Method

– Objectives and requirements– RANS/LES models

∗ Linear eddy–viscosity model∗ Extension to anisotropy by an explicit algebraic Reynolds stress model

– Interface definition

• Numerical Method

• Test Cases

• Results

– Plane Channel Flow

– Interface Behavior

– Periodic Hill Flow

• Conclusions and Outlook

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Motivation for Hybrid LES–RANS Methods

Objective: Prediction of complex turbulent flows at high Re using LES

• within acceptable simulation times and reasonable accuracy

RANS LES • Airfoil–LES at Re = 107 (Spalart)• Grid points: 1011 ; Time steps: 5× 106

• Floating–Point–Operations: 1020

=⇒ possible in 4 decades (?)

Requirements of many aerodynamic applications:

I. Computation of attached turbulent thin boundary layers up to separation

II. Computation of flows past separation line including Reynolds stresses

Idea of LES–RANS coupling:

RANS: requirement I fulfilled, but not requirement II

LES : requirement II fulfilled, but requirement I too expensive

splitting into attached eddies (RANS) and detached eddies (LES)

Hybrid methods with RANS and LES regions

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Hybrid LES–RANS Approach

Objectives and Requirements

x/h

y/h

LES zone

RANS zonesLES-RANS Interface

PSfrag replacements

0 0

1

2

2

3

4 6 8

Interface: Non–Zonal Approach

• Automatic switching between LES and RANS

• Gradual transition between LES and RANS (no explicit boundaries)

• No grid influence and no interface predefinition

RANS/LES Models: Primary Model Requirements for Non–Zonal Approach

• Similar models for both modes to facilitate the blending

• Cheap models avoiding large computational effort

• RANS model especially designed for near–wall region

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RANS/LES Models

One–equation model for both LES and RANS modes

• RANS Option 1: LEVM (Linear Eddy–Viscosity Model = L)=⇒ Near–wall one–eq. RANS model by Rodi et al. (1993)

based on transport eq. for kmod

• RANS Option 2: NLEVM (Non–Linear Eddy–Viscosity Model = nL)=⇒ Extension of the LEVM by an Explicit Algebraic Reynolds Stress Model

by Wallin and Johansson (2000)

• LES=⇒ One–eq. SGS model based on transport eq. for ksgs by Schumann (1975)

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Linear Eddy–Viscosity Model (LEVM)

• Unique one–equation model for RANS and LES

∂kmod

∂t+ Uj

∂kmod

∂xj=

∂

∂xj

[

(ν +νtσk

)∂kmod

∂xj

]

− (u′i u′j)mod

∂Ui

∂xj− ε

RANS 1

• One–eq. model by Rodi et al. (1993)(kmod → kRANS) with (v′2)1/2 as veloci-ty scale, Durbin (1991)

• Turbulent eddy viscosity:

νt = (v′2)1/2 lµ,v

v′2/kRANS = f(y∗) for y∗ ≤ 60

• Dissipation: ε = (v′2)1/2kRANS/lε,v

• Length scale: lµ,v and lε,v analyticallydetermined (l ∼ y)

LES

• SGS model by Schumann (1975)(kmod → kSGS)

• Subgrid–scale eddy viscosity:

νt = Cµ k1/2SGS ∆

• Dissipation: ε = Cd k3/2SGS/∆

• Length scale: ∆ = (∆x ·∆y ·∆z)1/3

defined by filter width

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NLEVM: Extension of LEVM by EARSM (1)

Motivation: Take Reynolds stress anisotropy into account: aij =u′iu

′j

k− 2

3δij

LEVM: u′iu

′jmod

=2

3δij kmod − 2 νt Sij

LEVM + EARSM: u′iu

′jmod

=2

3δij kmod − 2 νt Sij + a

(ex)ij kmod

︸ ︷︷ ︸

aij kmod

EARSM of Wallin & Johansson (2000):(Full RSM → weak equilibrium assumption + near–wall treatment by Durbin)

a(ex)ij = function(S2, Ω2, Sn ·Ωm, fdamp, . . .) with:

S =Sij

1/τ=

τ

2

(∂Ui

∂xj+

∂Uj

∂xi

)

normalized strain tensor

Ω =Ωij

1/τ=

τ

2

(∂Ui

∂xj− ∂Uj

∂xi

)

normalized rotation tensor

τRANS = max

(k

ε, Cτ

√ν

ε

)

time scale

[

LES: τLES =∆√k

]

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NLEVM: Extension of LEVM by EARSM (2)

LEVM by Rodi et al. + EARSM of Wallin & Johansson =⇒ RANS 2

∂kmod

∂t+ Uj

∂kmod

∂xj=

∂

∂xj

[

(ν +νtσk

)∂kmod

∂xj

]

− (u′i u′j)mod

∂Ui

∂xj︸ ︷︷ ︸

Pk

− ε

• Production term in NLEVM (→ Use extra anisotropy)

Pk =(

−2 νt Sij + a(ex)ij kmod

)∂Ui∂xj

(in LEVM: Pk = −2 νt Sij∂Ui∂xj

)

• Turbulent eddy viscosity based on EARSM: νt = Ceffµ k τRANS

• Extra anisotropy term from EARSM: a(ex)ij = f(S2, Ω2, Sn ·Ωm, fdamp, . . .)

• Dissipation from LEVM: ε = (v′2)1/2kRANS/lε,v

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LES–RANS Interface

Objectives

• Non–Zonal Approach

• Use of a physical (turbulent) parameter in the interface definition

• Interface dynamically determined

• Applicable for separated flow

x/h

y/h

LES zone

RANS zonesLES-RANS Interface

PSfrag replacements

0 0

1

2

2

3

4 6 8

Use of the instantaneous turbulent kinetic energy k (=⇒ kmod)

LES–RANS switching criterion y∗ = k1/2 · y/ν kmod influence: low kmod =⇒ thick RANS region; high kmod =⇒ thin RANS region

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LES–RANS Switching Criteria (1)

• First interface criterion

y∗ ≤ Cswitch,y∗ =⇒ RANS modey∗ > Cswitch,y∗ =⇒ LES mode

with y∗ = k1/2mod · y/ν and Cswitch,y∗ = 60 (Validity region of RANS 1)

LES–RANS Interface+

kmod contour

x/h

y/h

0 2 4 6 8

0

1

2

3

RANS islands

dynamically computed interface

RANS islands in LES region

kmod closely follows kcrit defined as kcrit =(Cswitch ν

y

)2

representing the LES–RANS interface 0

0.5

1

1.5

2

2.5

3

0 0.002 0.004 0.006 0.008 0.01

PSfrag

replacements

y/h

kcritkmod

k

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LES–RANS Switching Criteria (2)

• Second interface criterion

Combine the 1st criterion with a sharp interface

y∗ ≤ Cswitch,y∗ =⇒ RANS modey∗ > Cswitch,y∗ =⇒ LES mode

with y∗ = k1/2mod · y/ν and Cswitch,y∗ = 60

+ Sharp Interface Treatment

NO CONVERSION

CONVERSION

WALL

WALL

LES

LES LES

LES

LES

LES

LES

RANS

RANS

WALL

WALL

LES

LES LES

LES

LES

LES

LES RANS

RANS

RANS

LES

LES–RANS Interface + kmod contour

x/h

y/h

0 2 4 6 8

0

1

2

3

RANS islands have no real influence on the statistical results =⇒ Computations perfor-med with the 1st criterion (cheapest formulation)

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LES–RANS Method

Res. Mod.

Mod.

LES

URANS

RANS

Mod.Res.

Mod.: Modeled scalesRes.: Resolved scales

k = wave number

Energy spectrumPSfrag replacements

logE(k)

kclog k

RANS mode operates as an Unsteady RANS (UR-ANS)=⇒ Presence of resolved scales in RANS mode

Total contributions (ktot, (u′iu′j)tot, ...) are the

sum of the respective modeled and resolved fields=⇒ ktot = kmod + kres

=⇒ (u′iu′j)tot = (u′iu

′j)mod + (u′iu

′j)res

k

0

1

2

3

4

5

6

0.1 1 10 100

PSfrag replacements

logE(k)

kclog k

y+

ktotkreskmod

DNS

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Numerical Method: LESOCC

• LESOCC (Large Eddy Simulation On Curvilinear Coordinates)

• Navier–Stokes solver (incompressible fluid)

• 3–D finite volume approach– Curvilinear body–fitted coordinate system– Non–staggered (cell–centered) grid arrangement– Block–structured grids

• Spatial discretization– Viscous fluxes: central differences O(∆x2)– Convective fluxes: five different schemes, central diff. O(∆x2), CDS–2

• Temporal discretization– Predictor step (moment. eqns.): low–storage Runge–Kutta scheme, O(∆t2)– Corrector step (pressure correction equation): SIP solver (ILU)

• Pressure–velocity coupling: Momentum interpolation of Rhie & Chow

• High–performance computing techniques

– Highly vectorized– Parallelized by domain decomposition and explicit message passing– Vector–parallel computers and SMP–clusters (Hitachi SR8000–F1, SGI, ...)

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Test Cases

Test Case 1: Plane Channel Flow

L = 2dy

x L = 2 pi d

L = pi dz

x

y

z

d

• Reference data provided by Moser et al.(1999); DNS at Reτ = 590

• 2 grid resolutions:

– Grid A: 128× 128× 128 CVs∆x+ = O(30), ∆z+ = O(15)y+1stpt

= 0.68

– Grid B: 64× 64× 64 CVs∆x+ = O(60), ∆z+ = O(30)y+1stpt

= 1.46

Test Case 2: Periodic Hill FlowRe = 10,595

L x Lz

yL

h

xy

0 2 4 6 80

1

2

3

x/h = 6.x/h = 2.

Distribution of k

Zoom

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Periodic Hill Flow at Reb = 10,595

Grids

• Reference Solution: WR–LESHighly Resolved LES (no wall model); Dynamic Smagorinsky model12.4× 106 control volumeswall–normal resolution (1stCV height): ∆ycrest/h = 0.002

• Present Test Griddesigned according to DES requirements160× 100× 60 ≈ 1.0× 106 control volumeswall–normal resolution (1stCV height): ∆ycrest/h = 0.005

x

y

0 2 4 6 80

1

2

3

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Channel Flow Test Case / Adjusted RANS Model

Result of the Near–Wall LEVM Modification: −→ Lm

kmodeled + kresolved = ktotal

0

1

2

3

4

5

6

0.1 1 10 100

LLm

DNS

PSfrag replacements

y+

kmod

kresktot

0

1

2

3

4

5

6

0.1 1 10 100

LLm

DNS

PSfrag replacements

y+

kmod

kres

ktot

0

1

2

3

4

5

6

0.1 1 10 100

LLm

DNS

PSfrag replacements

y+

kmod

kres

ktot

• Version Lm: modified near–wall model using adjusted model const. for high–Re

• Resolved field almost unchanged

• Modeled field adjusted

Better agreement between the hybrid LES–RANS technique and the DNS data

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Channel Flow at Reτ = 590

Grid A Grid B

Grid B

Grid B

0

5

10

15

20

25

1 10 100

PSfrag replacements

y+

U+

u′u′rmstot

v′v′rmstot

u′v′tot

LnL

DNS

0

5

10

15

20

25

1 10 100

PSfrag replacements

y+

U+

u′u′rmstot

v′v′rmstot

u′v′tot

LnL

DNS

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 10 100

PSfrag replacements

y+

U+

u′u′rmstot

v′v′rmstot

u′v′tot

LnL

DNS

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1

0

1 10 100

PSfrag replacements

y+

U+

u′u′rmstot

v′v′rmstot

u′v′tot

LnL

DNS

• L: Linear hybrid methodwith 1st interface criterion

• nL: Non–Linear hybridmethod with 1st interfacecriterion

• Best predictions by non–linear hybrid method

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Interface Behavior: First criterion

LES–RANS Interface + kmod contour

x/h

y/h

0 2 40

0.5

1

1.5

RANS islands

Interface dynamically determined =⇒ evolves at each time step as well as spatially

Criterion based on inst. kmod =⇒ weak RANS–island influence on a statistical viewpoint

• Inst. separation point, kmod =⇒ thin RANS region, priority to LES mode

• Downstream to separation kmod =⇒ thicker RANS region, priority to RANS mode

y∗ recognizes the zones of low and high turbulence intensity through kmod, e.g.,localization of the inst. separation point

Method adapts itself to the turbulent flow features =⇒ priority to RANS or LES

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Results: Streamlines for Hill Flow Test Case

Wall–resolved LES (WR–LES)xsep/h = 0.190, xreatt/h = 4.694

x

y

0 2 4 6 80

1

2

3

Hybrid Version Lxsep/h = 0.254, xreatt/h = 4.751

x

y

0 2 4 6 80

1

2

3

DES Spalart–Allmaras modelxsep/h = 0.173, xreatt/h = 5.197

x

y

0 2 4 6 80

1

2

3

Hybrid Version nLxsep/h = 0.231, xreatt/h = 4.701

x

y0 2 4 6 80

1

2

3

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Mean Velocity Profiles at x/h = 0.05 to 8

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12

PSfrag

replacements

y/h

U/Ub

L

nL

DES

WR−LES

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u′u′tot Profiles at x/h = 0.05 to 8

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8 9

PSfrag

replacements

y/h

u′u′tot/U2b

L

nL

DES

WR−LES

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Further Examples of Improvements by NLEVM

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 1 2 3 4 5 6 7 8 9

PSfrag

replacements

y/h

x/h

τw

v′v

′tot /U2b

LnL

DESWR−LES

x/h = 0.05

0

0.5

1

1.5

2

2.5

3

0 0.01 0.02 0.03 0.04

PSfrag

replacements

y/h

x/hτw

v′v′tot/U2b

L

nL

DES

WR−LES

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Conclusions and Outlook

Hybrid LES–RANS Technique:

• Non–Zonal LES–RANS approach

• Two unique one–equation models for the velocity scale

– LEVM −→ v′2–formulation for the near–wall RANS region

– LEVM + EARSM −→ anisotropy of Reynolds stresses

• Encouraging results for plane channel and periodic hill flow

• Use of the EARSM enhances the results

• More detailed tests of the interface region required

• Investigations / adjustments of EARSM for more complex flows

• Future test cases (challenging for RANS) required to evaluate the potential ofthis hybrid LES–RANS technique

– Ahmed body, 3–D hill flow, stalled airfoil flow