Physical-Space Decimation and Constrained Large Eddy Simulation

1. 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)

2. 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…

3. Decimation Theory Kraichnan 1975, Kraichnan and Chen 1989 Let us do Fourier-Transform of the Navier-Stokes Equation and denote the Fourier modes as (S < N) (Small Scale) (Large Scale) Constraints: Lead to factor (Intermittency Constraint) (Energy flux constraint: Direct-Interaction-Approximation)

4. Large Eddy Simulation (LES) After filtering the Navier-Stokes equation, we have the equation for the filtered velocity is the sub-grid stress (SGS). One needs to model the SGS term using the resolved motion .

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

6. Smagorinsky Model (eddy-viscosity model): CS is a constant. Dynamic Models:

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

8. Constrained Subgrid-Stress Model (C-SGS) Assumption: the model coefficients are scale-invariant in the inertialrange, or close to inertial range. The proposed model is to minimize the square error Emodof a mixed model under the constraint: It can also been done by the energy flux εαΔthrough scale αΔ.. If the system does not have a good inertial range scaling, the extended self-similarity version has been used.

9. 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: Here and

10. Constraints on high order statistics and structures or other high order constraints and etc..

11. Priori and Posteriori Test from Numerical Experiments 1. Priori test DNS: 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

12. 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 the steady state energy spectra. Comparison of PDF of SGS dissipation at grid scale (a posteriori)

13. 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%.

14. Simulations start from a statistically steady state turbulence field, and then freely decay. Energy spectra for decaying isotropic turbulence (a posteriori), at t = 0, 6o, and 12 o, whereois the initial large eddy turn-over time scale.

15. 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.

17. A. Statistically steady nonhelical turbulence

18. Freely Decaying Isotropic Turbulence: Simulations start from a Gaussian random field with an initial energy spectrum: Initial large eddy turn-over time: Comparison of the SGS energy dissipations as a function of simulation time for freely decaying isotropic turbulence (a priori).

19. Statistically steady helical turbulence

20. Free decaying helical turbulence Energy spectra evolution Helicity spectra evolution

21. Decay of mean kinetic energy and mean helicity

22. Reynolds Stress Constrained Multiscale Large Eddy Simulation for Wall-Bounded Turbulence

23. Hybrid RANS/LES: Detached Eddy Simulation S-A Model

24. DES-Mean Velocity Profile

25. DES Buffer Layer and Transition Problem Lack of small scale fluctuations in the RANS area is the main shortcoming of hybrid RANS/LES method

26. Possible Solution to the Transition Problem Hamba (2002, 2006): Overlap method Keating et al. (2004, 2006): synthetic turbulence in the interface

28. 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

29. 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

30. 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)

31. 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.

32. Results of RSC-LES Mean velocity profiles of RSC-LES of turbulent channel flow at different ReT =180 ~ 590

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

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

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

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

37. Error in prediction of the skin friction coefficient: (friction law, Dean)

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

39. 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)

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

41. 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

42. 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

43. 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

44. Multiscale Simulation of Fluid Turbulence

45. 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.