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									    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?
Test of the Constrained-SGS Model
  Forced isotropic turbulence:
   DNS: Direct Numerical Simulation. A            DSM: Dynamic Smagorinsky Model
   statistically steady isotropic turbulence
                                                  DMM: Dynamic Mixed Similarity Model
   (Re=250) data obtained by Pseudo-
   spectral method with 5123 resolution.          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




                                                   106

                                                           108.5




   (Piomelli 2002)



Near-wall treatment is key to utility of LES in practice
Hybrid RANS/LES
                             Detached Eddy Simulation




S-A Model

                                                                                                         
 Dˆ / Dt  Cb1 1  ft 2  Sˆ   Cw1 f w  Cb1 /  2  ˆ / d       ˆ  ˆ   Cb 2  ˆ  /   f t1U 2
                                                               2                                      2
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        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
          ui   ui u j    p       2ui       ij
                                                  SGS

                                       
          t     x j      xi    x j x j    x j

Performance of ensemble average of the LES equations
leads to
          ui        ui u j         p         2 ui            ij
                                                                    SGS
                                                                              Rij
                                                                                 LES

                                                                    
          t           x j         xi        x j x j         x j          x j

where


                       RANS
                     Rij     Rij   ij
                                LES    SGS
     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:

(1) K-epsilon model to solve Rijmod
(2) Algebra eddy viscosity: Balaras & Benocci (1994) and Balaras et
    al. (1996)


   where

  (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 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:

               C f  C f ,Dean                   wall                        
     Error                      100   Cf                C f ,Dean  0.073Reb1 4   (friction law, Dean)
                  C f ,Dean                     Ub 2
                                                   2




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




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




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




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




        y+=500            y+=1000            y+=1500
Multiscale Simulation of Fluid Turbulence
                 Conclusions
1. 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.

								
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