Modeling Challenges and Approaches in LES for Physically Complex Flows

J. Andrzej DomaradzkiPeter DiamessisXiaolong Yang

Department of Aerospace and Mechanical EngineeringUniversity of Southern California

Los Angeles

Financial support: NSF and ONR

Introduction

Complexity sources in LES modeling: • GEOMETRY• PHYSICS: governing equations have additional

terms

Classical LES equations for a constant density, incompressible flow:

2

2

1( )

i ii j ij

j i j j

u p uu u

t x x x x

“Complex” Physics• Compressibility• Rotation and Stratification (Stable/Unstable)

2

2

1( )

i ij i ij

j i j j

u p uu u

t x x x x

2 j ijl lu ig T

Additional terms in the momentum equation are linear and do not require modeling

2

2( )j j

j j j

T Tu T

t x x x

ij i j i ju u u u

j j ju T u T

• Temperature Equation

• Subgrid Scale Stresses

Same form as for a flow without Coriolis force

Same form as for a passive scalar

• Can/should traditional models be used?

Incompressible MHD equations

2

2

1( )

1 1( )

( ) ( )

i ij i ij

j i j j

b extj i ij j i

j j j

u p uu u

t x x x x

b b B bx x x

2

2( ) exti i

j i j i j ij j j

b bb u u b B u

t x x x

Acquis Défis

Turbulence homogène

Pas de cascade !

Anisotropie:

Spectre d’énergie :

Début des DNS et LES

Transferts angulaires

Quid de ?

Quid des petites échelles ?

Quid du 2ème scalaire ?

LES spécifique à la MHD ?

Ecoulement de Hartmann

Régime Q2D intense

Casc. inverse d’énergie

Le tourbillon Q2D MHD

LES et RANS

Fonctions de paroi

Rôle des couches de Ha

Promoteurs de turbulence ?

Ecoulements complexes RIEN !

Profils de vitesse en M

Entrée et sortie de l’aimant

Géométries complexes

(divergents, coudes, etc…)

Turbulence MHD (Rm << 1) – from R. Moreau

l//l

tJ

12

E k 3t 2

u//

u

Rotating Turbulence• Rotating turbulence: refers to flows observed in a frame of

reference rotating with a solid body angular velocity .

• Rotating flows are distinguished from ‘non-rotating’ flows by the presence of the Coriolis force (turbomachinery, geophysical flows).

• Rossby number: for turbulent rotating flow:

• Qualitative Observations:

* energy decay is reduced compared with non-rotating turbulence * the inertial spectrum is steeper than the Kolmogoroff k-5/3 form * for initially isotropic flow Reynolds stress remains isotropic but

length scales become anisotropic

/Ro U L /(2 )Ro K

Modeling Difficulties

Implications for SGS models

• For dynamic model*S SC C

Approximate velocity models (nonlinear, deconvolution, estimation) avoid these difficulties and satisfy transformation properties automatically

but *S SC C makes the model too dissipative for rotating turbulence

and the model is inconsistent with thetransformation properties

Truncated Navier-Stokes Equations (TNS)• Variation of the Velocity Estimation Model (VEM)

(Domaradzki and Saiki (1997))

• Based on two observations:

- the dynamics of small scales are strongly determined by the large, energy carrying eddies

- the contribution of small scales to the dynamics of large scales (k<kc) comes mostly from scales within kc<k<2kc

• Implemented for low Reynolds number rotating turbulence by Domaradzki and Horiuti (2001) to avoid difficulties with rotational transformation properties for classical SGS models (Horiuti (2001))

Truncated N-S dynamics (spectral space)

E(k)

k2kckc

Unresolved scales

Large physical scales (on “coarse” mesh): computed by N-S eqns.

Estimated scales (on “fine” mesh): Artificial energy accumulation due to absence of (natural or eddy) viscosity.

Filter small-scales at fixed interval and replenish using estimation model

TNS=Sequence of DNS runs with TNS=Sequence of DNS runs with periodic processing of high modesperiodic processing of high modes

Multiscale modeling

E(k)

k2kckc

Unresolved scales

Large scales computed from (inviscid) TNS eqs.

Estimated small scales computed from a separate dissipative equation forced by the inviscid solution.

Scales periodically replaced by estimated scales

Similar to Dubrulle, Laval, Nazarenko, Kevlahan (2001)

Properties• N-S equations are solved

- SGS stresses are not needed - Transformation properties (Galilean, rotating frame) always satisfied - Commutation errors are avoided

• Applicable to strongly anisotropic flows (VLES)

• Straightforward inclusion of additional effects (convection, compressibility, stable stratification, rotation)

• Requires determination of the filtering interval (based on a small eddy turnover time or a limiter on the small energy growth)

TNS for rotating turbulence

• TNS with VEP applied to simulate low/high Re number turbulence with/without rotation

• DNS data of Horiuti (2001) • Mesh size is 2563 for DNS and 643 for TNS• The initial condition for TNS is obtained by truncating the

full 2563 DNS field to 323 grid• Low Re: • High Re:

max0.0014, 1.5t 16

max2.5 10 , 20t

Energy Spectrum, low Re

0 10

Energy decay, high Re

-1.2

High Re: spectral slope predictions

• n=2: Zhou (1995); Baroud et al. (2002).

• n=11/5=2.2: Zeman (1994).

• n=7/3=2.33: Bershadskii, Kit, Tsinober (1993).

• n=3: Smith and Waleffe (1999); Cambon et al. (2003).

nk

Energy spectrum, high Re

-3-3

-2-2

Anisotropy Indicators

• Length scales

• Reynolds stress tensor and anisotropy tensor

(=0 for isotropic turbulence)

0,

( ) ( )

( ) ( )

u x u x rn drL

u x u x

2

ˆ( ) ( ) Re ( )

/ / 3

ij i j ij

ij ij ij

R u x u x U k dk

b R q

Integral length scales

0

5,10

1

50

100

Anisotropy Tensor

• Directional and polarization anisotropy tensor

E(k) is the total energy for all modes in a wavenumber shell

22

2

( )( ) /4

Re /

e zij ij ij

eij ij

zij i j

b b b

E kb e k P dk qk

b ZN N dk q

| |k k

Directional anisotropy tensor

0

1

5

100

Summary of Observations

• Spectral slope n=-2 at earlier times (t<5) and n=-3 at later times (t>15)

• Anisotropy indicators largest for

- times t>5

- moderate rotation rates • Anisotropy indicators small for and

0

Spectral Exponent Hypothesis Approximately isotropic state characterized by n=-2

Strongly anisotropic state characterized by n=-3

Two different views of LES

• Classical view:

- governing LES equations are derived from Navier-Stokes eqs. and are are different from them

- unknown SGS stress is modeled using physical principles

- there exists a unique best solution to the SGS modeling problem

Additional “complex” physics often requires substantial changes in models developed for simpler flows.

• Competing view:

- governing LES equations are simply Navier-Stokes eqs.

- LES modeling problem is of numerical nature: how to accurately solve Navier-Stokes eqs. on coarse grids

- there may be many solutions to the problem, e.g. regularization of the equations or the solutions, using numerical dissipation in place of physical dissipation (MILES/ILES), etc.

Potential Advantage: if the equations are known there are no modeling problems!

Disadvantages: ILES is not robust because there is no guarantee that the implicit dissipation is equal to the physical dissipation

D = 10 m

U = 10 m/s

N = 0.003 /s

Re = 108

F = 500

D = 10 km

U = 10 m/s

N = 10-4 /s

Re = 1010

F = 10

Guadalupe island

Turbulent wakes in stably stratified fluids

3 4Re 10 ,10

21, 200

UD

UF

ND

20H

D

Experiments: Spedding et al. (1996, 1997,2001,2002).

Numerical Method: Computational Domain and Flow Configuration

• Periodic in horizontal directions: Fourier discretization.• Bottom: Solid Wall. Top: Free Surface.

Divide into spectral subdomains (elements). Legendre polynomial discretization.

Wake of a towed sphere

U

Numerical Method: Spectral Multidomain Discretization

• Partition domain into M subdomains with:– Height Hk and order polynomial approximation Nk.

– Non-uniform local Gauss-Lobatto grid (No stretching coefficients !).

Well-resolved wake core, subsurface

Ambient regionnot over-resolved

Numerical Techniques Dealing with Under-Resolution to Maintain Spectral Accuracy

and Stability • Spectral Filtering.• Strong Adaptive Interfacial Averaging.• Spectral Penalty Methods (J. Hesthaven – SIAM J. Sci.

Comp. Trilogy)

Attempting to satisfy eqs. with limited resolution arbitrarily close to boundaries leads to catastrophic instabilities

Solution: Implement BC in a weak form by collocating equation at boundary with a penalty term

iu RHS BCt

Truncated Navier-Stokes Dynamics

Flow Parameters and Runs Performed

• Domain size: 16Dx16Dx12D -Timestep t~0.03 D/U.

• Initialization procedure that of Dommermuth et al. (JFM 2002) = Relaxation.

Initial velocity data that of Spedding at Nt=3.

Re=UD/ Fr=2U/ND Pr= Resolution Nk

5x103 1 128x128x165 32

5x103 4 1 -””- -””-

2x104 1 128x128x249 44

2x104 4 1 -””- -””-

Flow Structure: Isosurfaces of |ω| at Fr=

Re=5KRe=5K Re=20KRe=20K

Flow Structure: Isosurfaces of ωz at Fr=4

Re=5KRe=5K Re=20KRe=20K

Vertical Vorticity, z at Horizontal Centerplane(Nt=56, x/D=112)

Fr=4Fr=4

Fr= Fr=

Re=5KRe=5K Re=20KRe=20K

Fr and Re Universality of Wake Power Laws: Mean centerline velocity

Fr and Re Universality of Wake Power Laws: Wake Horizontal Lengthscale

, Fr=4

Conclusions

• A range of subgrid scales adjacent to the resolved range dominates dynamics of the resolved eddies

• These subgrid scales can be estimated in terms of the resolved scales (estimation model)

• Dynamics of the resolved eddies is approximated by Truncated Navier-Stokes equations for resolved and estimated scales

• The method consists of a sequence of underresolved DNS and a periodic processing of the solution

Conclusions• TNS approach captures well temporal evolution of wake mean

velocity profile, length scales and vorticity field structure.

• For Reynolds numbers considered both TNS and stability filtering produced essentially the same results (supports ILES?)

• For decaying isotropic turbulence the inertial range spectrum is maintained during flow evolution

• For decaying rotating turbulence good comparison with DNS data is obtained at low Re

• At high Re decreased kinetic energy decay rates are observed for increasing rotation rate and the asymptotic spectrum proportional to 3k

Can LES with complex physics be best addressed by minimizing explicit modeling that affects the form and properties of the governing equations !?