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 (PKU&JHU)
Suyang Pei (PKU)
Jianchun Wang (PKU)
Zhenghua Xia (PKU&JHU)
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…
Decimation Theory
Kraichnan 1975, Kraichnan and Chen 1989
Constraints:
(Intermittency Constraint)
(Energy flux constraint:
Direct

Interaction

Approximation)
Let us do Fourier

Transform of the Navier

Stokes Equation and denote
the Fourier modes as
( 1,2,...)
i
x i N
(S < N)
Lead to factor
(Small Scale
)
(Large Scale)
Large Eddy Simulation (LES)
After filtering the Navier

Stokes equation, we have the equation for the
filtered velocity
One needs to model the SGS term using the resolved motion .
u
,
1
In this way, we decompose the velocity
into the sum of a resolved
component and a subgridscale (SGS) com
ponent .
is called SGS stress
i i
i j i
j i j j
ij i j i j
u u
p
u u f
t x x x x
u u u u
u
u u
.
ij i j i j
uu u u
is the sub

grid stress (SGS).
u
(,) ( )
i i
u u x t G x x dx
i ij
j
f
x
1/
Local energy flux
(,)
ij ij
t S
r
Where is the stress from scales
and is the stress from scales
1
2
j
i
ij
j i
u
u
S
x x
Local Measure of Energy Flux
2
2
1
(,)
2 2
t
u
u u u t
r
ij i j i j
uu u u
mod mod mod
Germano identity: = is the stress at 2.
Let be the resolved model stress,
A dynamic procedure is to minimize the s
quare error (Variation Procedure
i j
ij ij ij i j
ij ij ij
L T u u u u resolved
L T
e.g. Smag
mod mod mod
e.g. Smag
mod mod mod
):
= = (,);
or the meansquare error:
= = ( )
ij ij ij ij s s
ij ij ij ij s s
L L L L C C t
L L L L C C t
x
Smagorinsky Model
(eddy

viscosity model):
Dynamic Models:
2 2 1/2
2
with , and (2 ).
1
Strain rate tensor
2
r
ij r ij
r s ij ij
i j
ij
j i
S
C S S S S
u u
S
x x
C
S
is a constant.
is the SGS stress at scale
is the SGS stressat scale 2.
ij i j i j
ij i j i j
u u u u
T u u u u
2
Mixed Models:
A combination of single models:
2
2
1 2
1 2
2
2 2
Smagorinsky model
Similarity model
Nonlinear model
Mixed similarity model
S
ij s ij
sim
i j
ij sim i j
i j
nl
ij nl
k k
msim
ij
mnl
ij
i j
ij i j
i j
ij
k
C S S
C u u u u
u u
C
x x
C C
C C
S S u u u u
u u
S S
x
Mixed nonlinear model
k
x
Apply dynamic procedure, one can also get Dynamic Mixed model:
mod mod
1 2
1 2
0 , 0,
C C
C C
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
E
mod
of a mixed model
under the constraint:
It can also been done by the energy flux ε
αΔ
through scale
αΔ.. If the system does
not have a good inertial range scaling, the extended self

similarity version has
been used.
(,)
t
r
2
(,)
t
r
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:
1
( ),
2
k h
u u u
ω
Here
and
Constraints on high order statistics and structures
6
2
2 2
( (,)) ( ) ( (,))
t t
r r
or other high order constraints and etc..
Priori
and
Posteriori
Test from
Numerical Experiments
1. Priori test
DNS: A statistically steady
isotropic turbulence (Re
=270
)
obtained by Pseudospectral
method with 512
3
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
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 512
3
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.
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%.
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.
Simulations start from a
statistically steady state
turbulence field, and then
freely decay.
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)
S
4
, (b)
S
6
, and (c)
S
8
.
A. Statistically steady nonhelical turbulence
Freely Decaying Isotropic Turbulence:
Comparison of the SGS energy dissipations as a function of
simulation time for freely decaying isotropic turbulence (
a priori
).
Simulations start from a Gaussian random field with an initial
energy spectrum:
Initial large eddy turn

over time:
Statistically steady helical turbulence
Free decaying helical turbulence
Energy spectra evolution
Helicity spectra evolution
Decay of mean kinetic energy and mean helicity
Reynolds Stress Constrained
Multiscale Large Eddy Simulation
for Wall

Bounded Turbulence
Hybrid RANS/LES
: Detached Eddy Simulation
2 2
2 2
1 2 1 1 2 1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
/1///
b t w w b b t
D Dt C f S C f C d C f U
S

A Model
DES

Mean Velocity Profile
DES Buffer Layer and Transition Problem
Lack of small scale fluctuations in the RANS area is the
main shortcoming of hybrid RANS/LES method
Possible Solution to the Transition Problem
Hamba (2002, 2006): Overlap method
Keating et al. (2004, 2006): synthetic turbulence in the interface
Reynolds Stress Constrained Large Eddy
Simulation (RSC

LES)
1.
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;
2.
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)
3.
Coarse

Grid everywhere
LES
Reynolds Stress Constrained
Small scare turbulence
in the whole space
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
RANS LES SGS
ij ij ij
R R
2
SGS
i j ij
i i
j i j j j
u u
u u
p
t x x x x x
2
SGS
LES
ij
i j
ij
i i
j i j j j j
u u
R
u p u
t x x x x x x
Reynolds stress constrained SGS stress model is
adopted for the LES of inner layer flow:
where
Decompose the SGS model into two parts:
The mean value is solved from the Reynolds
stress constraint:
(1)
K

epsilon model to solve
(2)
Algebra eddy viscosity: Balaras & Benocci (1994) and Balaras et
al. (1996)
mod
ij
R
(3) S

A model (best model so far for separation)
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.
Results
of RSC

LES
Mean velocity profiles of RSC

LES of turbulent
channel flow at different Re
T
=180 ~ 590
Mean velocity profiles of RSC

LES, non

constrained
LES using dynamic Smagorinsky model and DES
(
Re
T
=590)
Mean velocity profiles of RSC

LES, non

constrained
LES using dynamic Smagorinsky model and DES
(
Re
T
=1000)
Mean velocity profiles of RSC

LES, non

constrained
LES using dynamic Smagorinsky model and DES
(
Re
T
=1500)
Mean velocity profiles of RSC

LES, non

constrained
LES using dynamic Smagorinsky model and DES
(
Re
T
=2000)
Error in prediction of the skin friction coefficient:
%
Error
Re
T
=590
Re
T
=1000
Re
T
=1500
Re
T
=2000
LES

RSC
1.6
2.5
3.3
0.3
LES

DSM
15.5
21.3
30.2
35.9
DES
19.7
17.0
13.5
14.1
,
1 4
,
2
,
100 0.073Re
2
f f Dean
wall
f f Dean b
f Dean
b
C C
Error C C
C
U
(friction law, Dean)
Interface of RSC

LES and DES (
Re
T
=2000)
RSC

LES DNS(Moser)
Velocity fluctuations (r.m.s) of RSC

LES and DNS
(
Re
T
=180,395,590). Small flunctuations generated at the
near

wall region, which is different from the DES method.
Velocity fluctuations (r.m.s) and resolved shear
stress:(
Re
T
=2000)
DES streamwise fluctuations in plane parallel to the
wall at different positions:
(
Re
T
=2000)
y+=6
y+=200
y+=38
y+=500
y+=1000
y+=1500
DSM

LES streamwise fluctuations in plane parallel to
the wall at different positions:
(
Re
T
=2000)
y+=6
y+=200
y+=38
y+=500
y+=1000
y+=1500
RSC

LES streamwise fluctuations in plane parallel to
the wall at different positions:
(
Re
T
=2000)
y+=6
y+=200
y+=38
y+=500
y+=1000
y+=1500
Multiscale Simulation of Fluid Turbulence
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.
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