Reynolds Stress Constrained Multiscale Large Eddy Simulation for Wall-Bounded Turbulence Shiyi Chen Yipeng Shi, Zuoli Xiao, Suyang Pei, Jianchun Wang, Yantao Yang State Key Laboratory of Turbulence and Complex Systems College of Engineering, Peking University and Johns Hopkins University

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 inertia range, scalar flux, intermittency exponents, Reynolds stress structures… Through constrained variation principle.. such as decimation theory, renormalized perturbation theory… physical space?

Comparison of PDF of SGS dissipation at grid scale (a posteriori) Test of the Constrained-SGS Model 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 the steady state energy spectra. Comparison of PDF of SGS dissipation at grid scale (a posteriori)

Large Eddy Simulation resolution challenge at high Re Near-wall treatment is key to utility of LES in practice 106 108.5 (Piomelli 2002)

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

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) 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

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

K-epsilon model to solve 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)

The interface to separate the inner and outer layer 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 ReT =180 ~ 590

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

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

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

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

Error in prediction of the skin friction coefficient: (friction law, Dean) % Error ReT=590 ReT=1000 ReT=1500 ReT=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

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

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

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

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

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

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

Multiscale Simulation of Fluid Turbulence

Conclusions RSC-LES uses the same grids resolution as DES; 2. The inner layer flow solved by RSC-LES possesses sufficient small scale fluctuations; 3. The transition of the mean velocity profile obtained by the RSC-LES from the inner layer to the outer layer at the interface is smooth; 4. RSC-LES is a simple method and may improve DES, and the forcing scheme… 5. We have used time averaging scheme for non-uniform systems, RSC-LES works nicely.