# Physical-Space Decimation and Constrained Large Eddy Simulation

Urban and Civil

Nov 16, 2013 (4 years and 7 months ago)

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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 (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)

(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 subgrid-scale (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 mean-square 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
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

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.