Physical-Space Decimation and Constrained Large Eddy Simulation

Shiyi Chen

College of Engineering, Peking UniversityJohns 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 TheoryKraichnan 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 denotethe Fourier modes as ( 1,2,... )ix 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

ij i j i ju u u u is the sub-grid stress (SGS).

u

i ijj

fx

1/

Local energy flux ( , ) ij ijt S r

Where is the stress from scales

and is the stress from scales

1

2ji

ijj i

uuS

x x

Local Measure of Energy Flux

221

( , )2 2t

uu u u t

r

ij i j i ju u 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 square error (Variation Procedure

i jij ij ij i j

ij ij ij

L T u u u u resolved

L T

e.g. Smagmod mod mod

e.g. Smagmod 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 ) .

1Strain rate tensor

2

rij r ij

r s ij ij

i j

ijj i

S

C S S S S

u uS

x x

CS 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

Sij s ij

simi jij sim i j

i jnlij nl

k k

msimij

mnlij

i jij i j

i jij

k

C S S

C u u u u

u uC

x x

C C

C C

S S u u u u

u uS Sx

Mixed nonlinear modelkx

Apply dynamic procedure, one can also get Dynamic Mixed model:

mod mod

1 21 2

0 , 0 ,C CC 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 Emod of a mixed model

under the constraint:

It can also been done by the energy flux εαΔ 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( ),

2k h u u u ω

Here and

Constraints on high order statistics and structures

622 2( ( , )) ( ) ( ( , ))t t

r r

or other high order constraints and etc..

Priori and Posteriori Test from Numerical Experiments

1. Priori testDNS: 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

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.

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, 6o, 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) S4 , (b) S6 , and (c) S8 .

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/LESHybrid RANS/LES: Detached Eddy Simulation: Detached Eddy Simulation

2 22 21 2 1 1 2 1

ˆ ˆ ˆ ˆ ˆ ˆ/ 1 / / /b t w w b b tD Dt C f S C f C d C f U

S-A Model

DES-Mean Velocity ProfileDES-Mean Velocity Profile

DES Buffer Layer and Transition ProblemDES Buffer Layer and Transition Problem

Lack of small scale fluctuations in the RANS area is theLack of small scale fluctuations in the RANS area is the main shortcoming of hybrid RANS/LES methodmain shortcoming of hybrid RANS/LES method

Possible Solution to the Transition ProblemPossible Solution to the Transition Problem

Hamba (2002, 2006): Overlap methodKeating et al. (2004, 2006): synthetic turbulence in the interface

Reynolds Stress Constrained Large Eddy Reynolds Stress Constrained Large Eddy Simulation (RSC-LES)Simulation (RSC-LES)

1.1. Solve LES equations in both inner and outer layers, the Solve LES equations in both inner and outer layers, the inner layer flow will have sufficient small scale fluctuations inner layer flow will have sufficient small scale fluctuations and generate a correct Reynolds Stress at the interface;and generate a correct Reynolds Stress at the interface;

2.2. Impose the Reynolds stress constraint on the inner layer Impose the Reynolds stress constraint on the inner layer LES equations such that the inner layer flow has a LES equations such that the inner layer flow has a consistent (or good) mean velocity profile; (constrained consistent (or good) mean velocity profile; (constrained variation)variation)

3.3. Coarse-Grid everywhereCoarse-Grid everywhere

LES

Reynolds Stress Constrained

Small scare turbulencein the whole space

Control of the mean velocity profile in LES by Control of the mean velocity profile in LES by imposing the Reynolds Stress Constraintimposing the Reynolds Stress Constraint

LES equationsLES equations

Performance of ensemble average of the LES equationsPerformance of ensemble average of the LES equations leads toleads to

wherewhere

RANS LES SGSij ij ijR R

2 SGSi j iji i

j i j j j

u uu up

t x x x x x

2 SGS LESiji j iji i

j i j j j j

u u Ru p u

t x x x x x x

Reynolds stress constrained SGS stress model is Reynolds stress constrained SGS stress model is adopted for the LES of inner layer flow:adopted for the LES of inner layer flow:

wherewhere

Decompose the SGS model into two parts:Decompose the SGS model into two parts:

The mean value is solved from the Reynolds The mean value is solved from the Reynolds

stress constraint:stress constraint:

(1) K-epsilon model to solve (2) Algebra eddy viscosity: Balaras & Benocci (1994) and Balaras et

al. (1996)

modijR

(3) S-A model (best model so far for separation)

For the fluctuation of SGS stress, a Smagorinsky For the fluctuation of SGS stress, a Smagorinsky

type model is adopted:type model is adopted:

The interface to separate the inner and outer layer The interface to separate the inner and outer layer is located at the beginning point of log-law region, such is located at the beginning point of log-law region, such the Reynolds stress achieves its maximum.the Reynolds stress achieves its maximum.

ResultsResults of RSC-LES of RSC-LES

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

Mean velocity profiles of RSC-LES, non-constrained LES Mean velocity profiles of RSC-LES, non-constrained LES using dynamic Smagorinsky model and DES (using dynamic Smagorinsky model and DES (ReReTT=590)=590)

Mean velocity profiles of RSC-LES, non-constrained LES Mean velocity profiles of RSC-LES, non-constrained LES using dynamic Smagorinsky model and DES (using dynamic Smagorinsky model and DES (ReReTT=1000)=1000)

Mean velocity profiles of RSC-LES, non-constrained LES Mean velocity profiles of RSC-LES, non-constrained LES using dynamic Smagorinsky model and DES (using dynamic Smagorinsky model and DES (ReReTT=1500)=1500)

Mean velocity profiles of RSC-LES, non-constrained LES Mean velocity profiles of RSC-LES, non-constrained LES using dynamic Smagorinsky model and DES (using dynamic Smagorinsky model and DES (ReReTT=2000)=2000)

Error in prediction of the skin friction coefficient:Error in prediction of the skin friction coefficient:

% Error ReReTT=590=590 ReReTT=1000=1000 ReReTT=1500=1500 ReReTT=2000=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 0.073Re2

f f Dean wallf f Dean b

f Dean b

C CError C C

C U

(friction law, Dean)

Interface of RSC-LES and DES (Interface of RSC-LES and DES (ReReTT=2000)=2000)

RSC-LES DNS(Moser)RSC-LES DNS(Moser)

Velocity fluctuations (r.m.s) of RSC-LES and DNS Velocity fluctuations (r.m.s) of RSC-LES and DNS ((ReReTT=180,395,590). Small flunctuations generated at the =180,395,590). Small flunctuations generated at the near-wall region, which is different from the DES method.near-wall region, which is different from the DES method.

Velocity fluctuations (r.m.s) and resolved shear stress:Velocity fluctuations (r.m.s) and resolved shear stress:((ReReTT=2000)=2000)

DES streamwise fluctuations in plane parallel to theDES streamwise fluctuations in plane parallel to thewall at different positions:wall at different positions:((ReReTT=2000)=2000)

y+=6y+=6 y+=200y+=200y+=38y+=38

y+=500y+=500 y+=1000y+=1000 y+=1500y+=1500

DSM-LES streamwise fluctuations in plane parallel toDSM-LES streamwise fluctuations in plane parallel tothe wall at different positions:the wall at different positions:((ReReTT=2000)=2000)

y+=6y+=6 y+=200y+=200y+=38y+=38

y+=500y+=500 y+=1000y+=1000 y+=1500y+=1500

RSC-LES streamwise fluctuations in plane parallel toRSC-LES streamwise fluctuations in plane parallel tothe wall at different positions:the wall at different positions:((ReReTT=2000)=2000)

y+=6y+=6 y+=200y+=200y+=38y+=38

y+=500y+=500 y+=1000y+=1000 y+=1500y+=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.