PhysicalSpace Decimation and Constrained Large Eddy Simulation

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Physical-Space Decimation and Constrained Large
Eddy Simulation

• Shiyi Chen
• College of Engineering, Peking University
• Johns Hopkins University

Collaborator Yi-peng Shi (PKU)
Zuoli Xiao (PKUJHU)
Suyang Pei (PKU)
Jianchun Wang (PKU)
Zhenghua Xia (PKUJHU)
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Question How can one directly use fundamental
physics learnt from our research on turbulence
for modeling and simulation? Conservation of
energy, helicity, constant energy flux in the
inertial range, scalar flux, intermittency
exponents, Reynolds stress, Statistics of
structures Through constrained variation
principle.. Physical space decimation theory
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Decimation Theory Kraichnan 1975, Kraichnan and
Chen 1989
Let us do Fourier-Transform of the Navier-Stokes
Equation and denote the Fourier modes as
(S lt N)
(Small Scale)
(Large Scale)
Constraints
Lead to factor
(Intermittency Constraint)
(Energy flux constraint Direct-Interaction-Approx
imation)
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Large Eddy Simulation (LES)
After filtering the Navier-Stokes equation, we
have the equation for the filtered velocity
is the sub-grid stress (SGS).
One needs to model the SGS term using the
resolved motion .
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Local Measure of Energy Flux
Local energy flux
Where is the stress from
scales and is the
stress from scales
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Smagorinsky Model (eddy-viscosity model)
CS is a constant.
Dynamic Models
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Mixed Models
A combination of single models
Apply dynamic procedure, one can also get Dynamic
Mixed model
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Constrained Subgrid-Stress Model (C-SGS)
Assumption the model coefficients are
scale-invariant in the inertial range, or close
to inertial range.
The proposed model is to minimize the square
error Emod of a mixed model under the constraint
It can also been done by the energy flux ea?
through scale a?.. If the system does not have a
good inertial range scaling, the extended
self-similarity version has been used.
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Energy and Helicity Flux Constraints
Consider energy and helicity dissipations, we add
the following two constraints

is determined by using the method of Lagrange
multipliers

Here
and
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Constraints on high order statistics and
structures
or other high order constraints and etc..
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Priori and Posteriori Test from Numerical
Experiments
1. Priori test
DNS A statistically steady isotropic turbulence
(Re?270) obtained by Pseudospectral method with
5123 resolution.
Smag 0.357 0.345 0.299 0.410
0.376 0.340 DSmag 0.360 0.348
0.301 0.413 0.378 0.350
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Test of the C-SGS Model (Posteriori test)

• Forced isotropic turbulence

DNS Direct Numerical Simulation. A statistically
steady isotropic turbulence (Re?250) data
obtained by Pseudo-spectral method with 5123
resolution.
DSM Dynamic Smagorinsky Model DMM Dynamic Mixed
Similarity Model CDMM Constrained Dynamic Mixed
Model
Comparison of PDF of SGS dissipation at grid
scale (a posteriori)
Comparison of the steady state energy spectra.
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PDF of SGS stress (component ?12) as a priori, SM
and DSM show a low correlation of 35, DMM and
CDMM show a correlation of 70.
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Simulations start from a statistically steady
state turbulence field, and then freely decay.
Energy spectra for decaying isotropic turbulence
(a posteriori), at t 0, 6?o, and 12 ?o, where
?o is the initial large eddy turn-over time scale.
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• Prediction of high-order moments of velocity
  increment

High-order moments of longitudinal velocity
increment as a function of separation distance r,
where ? is the LES grid scale. (a) S4 , (b) S6 ,
and (c) S8 .
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A. Statistically steady nonhelical turbulence
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• Freely Decaying Isotropic Turbulence

Simulations start from a Gaussian random field
with an initial energy spectrum Initial large
eddy turn-over time
Comparison of the SGS energy dissipations as a
function of simulation time for freely decaying
isotropic turbulence (a priori).
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Statistically steady helical turbulence
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Free decaying helical turbulence
Energy spectra evolution
Helicity spectra evolution
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Decay of mean kinetic energy and mean helicity
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Reynolds Stress Constrained Multiscale Large
Eddy Simulation for Wall-Bounded Turbulence
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Hybrid RANS/LES Detached Eddy Simulation
S-A Model
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DES-Mean Velocity Profile
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DES Buffer Layer and Transition Problem
Lack of small scale fluctuations in the RANS area
is the main shortcoming of hybrid RANS/LES
method
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Possible Solution to the Transition Problem
Hamba (2002, 2006) Overlap method Keating et al.
(2004, 2006) synthetic turbulence in the
interface
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Reynolds Stress Constrained Large Eddy Simulation
(RSC-LES)

• Solve LES equations in both inner and outer
  layers, the inner layer flow will have sufficient
  small scale fluctuations and generate a correct
  Reynolds Stress at the interface
• Impose the Reynolds stress constraint on the
  inner layer LES equations such that the inner
  layer flow has a consistent (or good) mean
  velocity profile (constrained variation)
• Coarse-Grid everywhere

LES
Small scare turbulence in the whole space
Reynolds Stress Constrained
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Control of the mean velocity profile in LES
by imposing the Reynolds Stress Constraint
LES equations
Performance of ensemble average of the LES
equations leads to
where
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Reynolds stress constrained SGS stress
model is adopted for the LES of inner layer
flow
Decompose the SGS model into two
parts The mean value is solved
from the Reynolds stress constraint

• K-epsilon model to solve
• Algebra eddy viscosity Balaras Benocci (1994)
  and Balaras et al. (1996)

where
(3) S-A model (best model so far for separation)
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For the fluctuation of SGS stress, a
Smagorinsky type model is adopted
The interface to separate the inner and outer
layer is located at the beginning point of
log-law region, such the Reynolds stress achieves
its maximum.
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Results of RSC-LES
Mean velocity profiles of RSC-LES of turbulent
channel flow at different ReT 180 590
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Mean velocity profiles of RSC-LES,
non-constrained LES using dynamic Smagorinsky
model and DES (ReT590)
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Mean velocity profiles of RSC-LES,
non-constrained LES using dynamic Smagorinsky
model and DES (ReT1000)
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Mean velocity profiles of RSC-LES,
non-constrained LES using dynamic Smagorinsky
model and DES (ReT1500)
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Mean velocity profiles of RSC-LES,
non-constrained LES using dynamic Smagorinsky
model and DES (ReT2000)
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Error in prediction of the skin friction
coefficient
(friction law, Dean)
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Interface of RSC-LES and DES (ReT2000)
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Velocity fluctuations (r.m.s) of RSC-LES and DNS
(ReT180,395,590). Small flunctuations generated
at the near-wall region, which is different from
the DES method.
RSC-LES
DNS(Moser)
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Velocity fluctuations (r.m.s) and resolved shear
stress(ReT2000)
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DES streamwise fluctuations in plane parallel to
the wall at different positions(ReT2000)
y6
y200
y38
y1000
y1500
y500
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DSM-LES streamwise fluctuations in plane
parallel to the wall at different
positions(ReT2000)
y6
y200
y38
y1000
y1500
y500
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RSC-LES streamwise fluctuations in plane
parallel to the wall at different
positions(ReT2000)
y6
y200
y38
y1500
y500
y1000
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Multiscale Simulation of Fluid Turbulence
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Conclusions

• As a priori, the addition of the constraints not
  only improves the correlation between the SGS
  model stress and the true (DNS) stress, but
  predicts the dissipation (or the fluxes) more
  accurately.
• As a posteriori in both the forced and decaying
  isotropic turbulence, the constrained models show
  better approximations for the energy and helicity
  spectra and their time dependences.
• Reynold-Stress Constrained LES is a simple method
  and improves DES, and the forcing scheme, for
  wall-bounded turbulent flows.
• One may impose different constraints to capture
  the underlying physics for different flow
  phenomenon, such as intermittency, which is
  important for combustion, and magnetic helicity,
  which could play an important role for
  magnetohydrodynamic turbulence, compressibility
  and etc.