modeling challenges and approaches in les for physically complex flows j. andrzej domaradzki peter...
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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 !?