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 (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. is the sub-grid stress (SGS).

Local energy flux Where is the stress from scales and is the stress from scales Local Measure of Energy Flux

Smagorinsky Model (eddy-viscosity model): Dynamic Models: C S is a constant.

Mixed Models: A combination of single models: Apply dynamic procedure, one can also get Dynamic Mixed model:

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.

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: Hereand

Constraints on high order statistics and structures 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 resolution. Smag DSmag

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 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 evolutionHelicity 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 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 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 Control of the mean velocity profile in LES by imposing the Reynolds Stress Constraint LES equations LES equations Performance of ensemble average of the LES equations Performance of ensemble average of the LES equations leads to leads to where where

Reynolds stress constrained SGS stress model is 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) (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 LES-DSM DES (friction law, Dean)

Interface of RSC-LES and DES ( Re T =2000)

RSC-LES DNS(Moser) 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+=6y+=200y+=38 y+=500 y+=1000y+=1500

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

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