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Transient RANS and Hybrid RANS/LESTransient RANS and Hybrid RANS/LES

by by K. K. HanjaliHanjalić

Title 1/4

Department of Multi-scale Physics, Delft University of TechnologyDelft, The Netherlands

Transient -RANS based VLESRationale, justification, validation and limitations Application to thermal R-B convection at extreme Ra’s(“ultra-turbulent” regime)Examples of practical relevance:

Diurnal dynamics over a mezzo-scale town valleyDiurnal wind over Arctic ice sheet

Hybrid RANS/LES (HRL)Rationale and a priori tests; zonal and seamless coupling; interface issuesOne-, two- and multi-equation RANS modelsExamples of application of HRL in attached and separated flows

Plane channel at high ReHill flow

CONTENT: CONTENT:

Limitations of LES and needs for combined (hybrid) LES/RANS

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Motivation: Predictions of complex wallMotivation: Predictions of complex wall--bounded turbulent flows bounded turbulent flows and heat transfer at very high Reynolds and and heat transfer at very high Reynolds and Rayleigh Rayleigh numbers numbers

The mainstay of the contemporary industrial CFD are the RANS turbulence closures: affordable, economical,…but:

too much empiricism, lack of universality, difficulties in predictingcomplex unsteady and nonequilibrium flows, ..

LES: less empirical, captures better the turbulence physics, considered as the future industrial standard,…but:

expensive and time consuming, especially for high Re and Ra number wall-bounded flows in complex geometries:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

∂∂

+∂∂

−= ij

j

i

ji

ii

xU

xxP

FDtUD

τνρ1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

∂∂

+= ijTjp xT

xcq

DtTD

θτσν

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

−==i

j

j

itijjiij x

UxUkuu νδτ

32

itT

tii x

Tuτ∂∂

−==σνθθ

Time- or ensemble-averaged (RANS) or filtered (LES) momentum and energy Equations:

Common practice in RANS approach: Linear Eddy Viscosity/Diffusivity models:

3

LES of wall-bounded flows require high resolution grid in all directions for resolving near-wall processes (∆x+ O(50), ∆y+ O(1), ∆z+ O(20))

Options for very high Re and Ra numbers:• Hybrid LES/RANS (Balaras, Davidson, Spalart, Hamba, Piomelli,…• RANS-based VLES

For resolving viscous near-wall boundary layer: No of grid cells ∝ Reτ1.8

as compared to Reτ0.4 for outer layer (Chapman, 1979).

Hence, for high Reynolds and Raleigh numbers LES still too expensive

Grids issues for LES and RANS for wallGrids issues for LES and RANS for wall--bounded turbulent flows bounded turbulent flows

For R-B conv.: ∆/H ≈ O(Pr2/NuRa)1/4 ⇒ Total No of grid cells ∝ Ra!

In contrast, for near-wall RANS N∝ ln Reτ , for R-B N∝ Ra1/3

Hybrid LES-RANS (HRL) strategies (including DES)Substantial part of turbulence is modelled by RANSSignificantly smaller number of cells (large aspect ratio)Criteria for location the RANS-LES interface:

Decided by user or Controlled by cell dimensions –comparison between length scale and typical mesh size(critical in some separating flows)

Wall functionsNo universal characterStandard log-law adequate in simple wall-attached flowsInadequate for separated flows

Wall functions versus nearWall functions versus near--wall RANS (Hybrid) wall RANS (Hybrid)

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Key questions regarding HRL Key questions regarding HRL

Where to locate the interface?

Which matching conditions are to be used at the interface?

How does a RANS model react to unsteadiness (“receptivity”)?

Will the dynamics be rightly returned?

What is the impact of RANS layer on the LES region?

Will the modelled contribution correctly compensate the reduction in the resolved contribution?

Which models are suitable?

( )

( )

22

1 1 2

13 6 6

31 1 23 3 6

1 1 3

62 22 22 2

1 ,

1, , , 1 , ,1

, ,

b w w bj j j

wt w

w

w

D c S c f cDt x x xd

cf f f f gc f g c

g r c r r r S S fSd d

υ υ υυ υ

υ

ν ν ν νν ν νσ

ν χ χν ν χν χ χ

ν ν

κ κ

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

⎡ ⎤+= = = = − = ⎢ ⎥+ + +⎣ ⎦

= + − = = +

• One-equation transport model (Spalart-Allmaras, 1993, Nikitin et al. 2000) used as RANS model in the near-wall region, and as an ssg model for LES in the outer region

DES approach of Spalart et al.

• Switching from RANS to LES: and d is the distance from the nearest wall

min( , ), max( , , )DESd d C where x y z∆ ∆ ∆ ∆ ∆= =

5

DES: RANS/LES interface location in a channel flow

Grid: 96x64x64

Grid: 64x64x32

DES: RANS/LES interface location in a channel flow

Grid: 64x64x32

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Prescribed RANS layer by reference to the distance to the wall, or separate RANS and SGS models (ideally same type of model)

A-priori test – overlap of the two domainsA-posteriori test – two separate domains

A priori test of the response of a RANS model to external LES perturbations

Rationale:• Identifying / quantifying the response of the RANS layer to LES

Methodology• LES provides information to RANS• RANS does not provide information to LES• LES is solved down to the wall

Case Description• Periodic channel flow• Reb = 10935 – DNS of Moser, Kim and Mansour (1999)• Computational domain: 2πh x 2h x πh• Grid: 96 x 64 x 64 with • Interface location: • SGS model: Smagorinsky

A priori study

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A-priori test resultsInstantaneous streamwise velocity profiles for the a-priori RANS and equivalent LES;

LES RANS

Time history for the velocity U (y+ = 30) and Uτ for a-priori RANS and equivalent LES

AA--priori priori test of Wall Function approach for LEStest of Wall Function approach for LESWall-normal variations of the correlations for the a-priori RANS and LES, (Temmerman, Leschziner & Hanjalic, 2002)

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Time-averaged kinetic energy and eddy viscosity for the reference DNS, the a-priori RANS and the equivalent LES

A-priori test results

A posteriori study: coupled RANS and LES

Prescribedy+

int

LES

RANS

LES→RANS data transfer

°° •

••

°°

•

RANS and LES regions are solved using the same solverCoupling strategy:• Switch from RANS to LES at an imposed location and blending

of RANS and LES viscosity on the LES-RANS interface;

• Switch from RANS to LES controlled by wall distance d and cell side ∆

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• Grid: Nx × Ny × Nz= 64 x 32 x 64 with y+(1) = 0.5• Interface locations: y+

int=65 (Reτ=590); y+int=135 (Reτ=2000);

Case Description: Fully developed channel flow

• Reb = 10935 (Reτ=590) (DNS by Moser, Kim and Mansour (1999) • Reb = 40000 (Reτ=2000) (Experiments by Wei & Willmarth, 1998• Computational domain: 2πh x 2h x πh

Hybrid RANS/LES and Coarse LES

Fine-resolved LES• Reτ=590 : Nx × Ny × Nz= 64 x 64 x 128 and 96x64x64, y+(1) = 0.5• Reτ=2000: Nx × Ny × Nz= 512x128x128, y+(1) = 0.75

Channel Flow – Results, Reτ=2000(L. Temmerman)

Time-averaged velocity and shear stress profiles for the LES computations.

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HRL Modelling Practice: HRL Modelling Practice: OneOne--eqneqn. RANS. RANS((L.L.Temmerman, 2001))

RANS model: one-equation transport model for turbulence energy (Wolfshtein, 1969);SGS model: One-equation transport model for SGS energy (Yoshizawa and Horiuti, 1985)Assumption: RANS and LES grids are identical at the interface;Target:• Viscosity:

• Velocity:

• Modelled energy: .

LESRANS UU intint =

LESt

RANSt int,int, νν =

LESRANS kk intmod,intmod, =

res resSGS LES t RANSν ν ν ν+ = +

' ' ' '( / 3) iji j k k ijresLES

ij ij

u u u u S

S S

δν

−=

SGS tν ν=

2

tkC fµ µνε

=

( )( )( )

2

22

/

/

SGSf kC

f k

µµ

µ

ε ν

ε=

A two-layer hybrid scheme: Matching criteriaMatching criteria: continuity of total eddy viscosity at the interface

with overbar denoting filtered, and <> some local smoothing.

Resolved stresses continuous across the interface ⇒

5.0klCt µµν =One-eqn model: k-ε model:

5.0

int,RANS

SGS

klC

µµ

ν=

Cµ at the interface:

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AdjustmentAdjustment of Cof Cµµ

( ) ( )( ),int

int int

1 exp0.09 0.09

1 exp

yC C

yµ µ

− − ∆⎡ ⎤⎣ ⎦= + −− − ∆⎡ ⎤⎣ ⎦

,int

int int

0.09 for 2727

0.09 1 exp( ( ( 34)) / )0.09

1 exp( ( 34) /

for 27

)y

yC y

C y y yC

y y y

µ

µµ

+

++

+

+

= ≤

⎡ ⎤⎡ ⎤− − − − =

⎧⎪⎪⎨⎪ >

∆⎣ ⎦ ⎣ ⎦= +⎡ ⎤− − − =⎩ ⎦⎪ ∆⎣

Variant 1

Variant 2

Time-averaged velocity profiles for the hybrid RANS-LES computations.

Channel Flow – Results, Reτ=2000One-equation RANS, (L. Temmerman)

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HRL: Channel Flow, Reτ=2000, One-equation RANS, (L. Temmerman)

Interface issues in Two-equation RANS modelInterface B.C. for 2-eqn RANS, kint and εint - options for k:

resLESint kk = 250 )UU(.kk iiSGSint −==

where iU

iU- test-filtered velocity- filtered velocity

c: Scale similaritya:

b: Isotropic spectrum distrib.2

32382

3⎟⎠⎞

⎜⎝⎛

∆== − SGS/

/S

SGSintv

CCkk πκ

Interface B.C. for 2-eqn RANS, kint and εint - options for εint :

n

/

int y.k52

23

=ε Or, from least-square error between the total viscosity on both sides of interface.

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Channel Flow – Results, Reτ=2000Two-equation RANS (M. Hadziabdic)

Streamwise vorticity, Reτ=590

fine-resolved LES (96x64x64)

coarse LES (64x64x32)

hybrid RANS/LES (64x64x32)

∆ z+

RANS/LES interface

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Zonal k-v2-f RANS/LES: Equations(Hadziabdic & Hanjalic 2003)

• RANS model:

RANS region

Buffer region

LES region

• LES model: Dynamic Smagorinsky model

2k fε υ− − −

( ) εανν ⋅−+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+∂∂

=∂

∂+

∂∂ P

xk

xxkU

tk

jt

jj

j

( ) 3151

801 /LES

.tot

lRANSLES

RANS ZYX.L,kCL,LL,max ∆⋅∆⋅∆⋅==⎟⎟

⎠

⎞⎜⎜⎝

⎛=

εα

1≤α

1>α

5.11 ≤< α

modrestot kkk +=

Hybrid RANS (k-v2-f) / LES (dynamic): Velocity profiles

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Hybrid RANS (k-v2-f) / LES (dynamic): Shear stress and kinetic energy

Reτ=2000 Reτ=20000

Hill Flow Hill Flow –– Case DescriptionCase DescriptionPeriodic channel flow with constriction at both endsRe number based on channel height and bulk velocity is 21560Data from highly resolved LES computations (5 x 106 nodes) by Temmerman et al (2003)Domain size: (h=hill height)Grid details:• Discretisation for HRL• Near-wall resolution:• Spanwise and streamwise resolution:

hhh 5.4036.39 ××

112 64 56× ×1)1( ≈+

cyzx ∆=∆

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Averaged streamlines for the reference simulation, LES, DES and RANS-LEScases.

(x/h)sep. = 0.22(x/h)reat. = 4.72

(x/h)sep. = 0.21(x/h)reat. = 5.30

(x/h)sep. = 0.23(x/h)reat. = 4.64

(x/h)sep. = 0.23(x/h)reat. = 5.76

Hill Flow - Results

Streamwise velocity profiles at x/h = 2.0.

Hill Flow - Results

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Resolved motion in the URANS region is as strong as the resolved motion in the equivalent LES region.Hence, in the RANS region, both resolved and modelled contributions to the motion are substantial.The sum of both contributions is too high, hinting at the need of an ad hoc modification to reduce the total motion.

Some observations

Channel FlowChannel Flow: Encouraging results.The response to the parameters change is small.Response to the location of the interface: in proportion of modelled motion;

Hill FlowHill Flow: agreement with the reference data reasonable;Compared to the channel case, Cµ has a similar behaviour.Difficult to draw definitive conclusions because of the low Re.

New hybrid RANS-LES (HLR) method allowing:• Freedom in locating the interface;• Dynamic adjustment of the RANS model to ensure

continuity across the interface.For identical grids, the HRL results are significantly better than those obtained with LES for the same (coarse) mesh.Application to a recirculating flow:• Results are non-conclusive due to low Reynolds number;• The hybrid RANS-LES approach overestimates the

recirculation zone length.Fundamental inconsistency in on the LES side next to RANS

(unrealistic streaks structure, insufficient stress); Needs for further adjustment (smoothing, extra forcing, artificial backscatter, …) irrespective of RANS model

Concluding Remarks on HRLConcluding Remarks on HRL

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Schematics of TRANS –VLES rationaleSemi-deterministic Modelling (SDM), (Ha Minh et al.)

TT--RANS Niche: HighRANS Niche: High--Ra challenge in thermal RB convectionRa challenge in thermal RB convection

λv /H ∝ Ra-1/7

Nu∝ Ra1/3 for Ra<1012 (Pr O(1))

Nu∝ Ra1/2 for Ra→∞

λθ/H ∝ Ra-1/3

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( ) ( )refii

refij

j

i

jj

ij

i TTgx

PP1xU

xxU

Ut

U−β+

∂−∂

ρ−⎟

⎟⎠

⎞⎜⎜⎝

⎛τ−

∂∂

ν∂∂

=∂∂

+∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛τ−

∂

∂ν∂∂

=∂

∂+

∂

∂θj

jjjj x

TPrxx

TU

tT

⎟⎟⎠

⎞⎜⎜⎝

⎛τ−

∂

∂ν∂∂

=∂

∂+

∂

∂cj

jjjj x

CScxx

CU

tC

T-RANS EQUATIONS AND SUBSCALE MODELS:

ε−++= kkk GPDDt

kD

YGPPDDt

D21 −+++=

εεεεε

θθθ ε−+=θ

PDDt

D 2

+final closure: 3eqn. model

0)/( =− iuDtD ϕiffDassuming weak equilibrium

Subscale ASM/AFM/ACM

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

∂∂

+∂∂

−= 2θηβξττε

τ θφθ ij

ij

jiji g

xU

xTk

C

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂ξτ+

∂

∂τ

ε−=τ φ

j

icj

jijci x

UxCk

C

jiiji

j

j

itij ugkCk

32

xU

xU

θβε

+δ+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂+

∂∂

ν−=τ

Verification: Long-term averaged temperature profiles and heat flux in R-B convection for different Ra numbers; DNS and TRANS

(Kenjereš and Hanjalić, 1999-2003)

Wall scaling with heat-flux-based buoyancy velocity

φψ+=−~~^ΨΦΦΨΨΦ

^

.constρcqθwWT

zTα

p

w~~

==−−∂∂

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MON1(z/D=0.5,x/L=0.5, y/L=0.5) MON2(z/D=0.01,x/L=0.5,y/L=0.5)

T-RANSLES

Time spectra of <U>, <V>, <W> and <T> signals at characteristic monitoring points, Ra=109

Mean vertical profiles of temperature for different Ra

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HighHigh--Ra number challenge in thermal RB convection Ra number challenge in thermal RB convection

λv /H ∝ Ra-1/7

Nu∝ Ra1/3 for Ra<1012 (Pr O(1))

Nu∝ Ra1/2 for Ra→∞

λθ/H ∝ Ra-1/3

Ra1/3

Ra1/2

Ra-1/3

Ra-1/7

Ra-2/9

INITIAL STRATIFICATIONS

∆Τ=2

Residential Industrial

TEMPERATURE CONCENTRATION

z

y

z/H=1/3

z

y

CASE (I): weak CASE (II): strong

∆T=4

z/H=2/3

T=T(x,y,z,τ)

C=C(x,y,z,τ)

∆Τ=1

1600m

800m

T-RANS of pollutant dispersion in a town valley

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INITIAL STRATIFICATIONCASE (I): weak CASE (II): strong

Instantaneous trajectories in vertical plane over hilly terrain

Q0>0

Q0<0

TIME

Passive pollutant dispersion visualized by concentration isosurface

Evolution of the pollutant front (C=0.05 Cmax): strong stratification

T-RANS of pollutant dispersion in a town valley

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380 km 45km

3 km

Diurnal winds over Arctic ice sheet

Solution domain:380x45x3 km

Mesh:180x40x40

Ra~1010 , Pr~ 1

Assumed near-ground temperature

V. van Huijen, S. Kenjeres and K.Hanjalic

Some instantaneous streamline patterns over an ice sheet

Wind velocity profiles: comparison with measurements

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Some Conclusions on T-RANS/VLES:

There will be an increasing research on merging RANS and LESstrategies for very high Re and Ra numbers and complex flows

Both RANS and LES will long be in use, each in its niche, but industrycannot count on LES for large-scale problems in the foreseeable future

T-RANS can be used to predict natural convection at very high Ra and in complex domains, which are inaccessible to LES, DNS or other methods.

T-RANS based VLES captures well main flow features in flows dominatedby large-scale (pseudo)deterministic structures

References:

1. Balaras, E., Benocci, C., Piomelli, U., Two-layer approximate boundary conditions for large-eddy simulations, AIAA Journal 34 (1996), 1111-1119.

2. Cabot, W., Moin, P., Approximate wall boundary conditions in the large eddy simulation of high Reynolds number flow, Flow, Turbulence and Combustion, 63 (1999), 269-291.

3. Hanjalic, K., Hadziabdic, M., Temmerman, L. and Leschziner M., Merging RANS and LES strategies: zonal or seamless coupling, Invited lecture DLES V, Munchen Aug. 27-29, 2003 (to appear in R. Friedrich, B. Geurs and O. Metais, (eds) Durect and Large-Eddy Simulations V, Kluwer Acad. Publ. 2004

4. L.Temmerman, M.A.Leschziner, K.Hanjalic, A priori studies of a near-wall RANS model within a hybrid LES/RANS scheme , 5th Internacional Symposium on Engineering Turbulence Modelling and Measurements, Mallorca, Spain, 16-18 September, 2002

5. Spalart, P.R., Jou, W-H., Strelets, M., Allmaras, S.R., Comments on the feasibility of LES for wings and on the hybrid RANS/LES approach, in Advances in DNS/LES, 1st AFOSR Int. Conf. On DNS/LES (Greden Press) (1997).

6. Spalart P.R. and Allmaras, S.R., A one-equation turbulence model for aerodynamic flows. AIAA Paper 92-0439. (1992).

7. Temmerman, L., Leschziner, M., Mellen, C. and Froehlich J., Investigation of subgrid-scale models and wall-function approximations in Large Eddy Simulation of separated flow in a channel with streamwise periodic constrictions, Int. J. Heat Fluid Flow (to appear).