Modeling Challenges and Approaches in LES for Physically Complex Flows

1. Modeling Challenges and Approaches in LES for Physically Complex Flows J. Andrzej Domaradzki Peter Diamessis Xiaolong Yang Department of Aerospace and Mechanical Engineering University of Southern California Los Angeles Financial support: NSF and ONR

2. Introduction Classical LES equations for a constant density, incompressible flow: Complexity sources in LES modeling: • GEOMETRY • PHYSICS: governing equations have additional terms

3. “Complex” Physics • Compressibility • Rotation and Stratification (Stable/Unstable) Additional terms in the momentum equation are linear and do not require modeling

4. Temperature Equation • Subgrid Scale Stresses Same form as for a flow without Coriolis force Same form as for a passive scalar • Can/should traditional models be used?

5. Incompressible MHD equations

6. Turbulence MHD (Rm << 1) – from R. Moreau

7. Rotating Turbulence • Rotating turbulence: refers to flows observed in a frame of reference rotating with a solid body angular velocity . • Rotating flows are distinguished from ‘non-rotating’ flows by the presence of the Coriolis force (turbomachinery, geophysical flows). • Rossby number: for turbulent rotating flow: • Qualitative Observations: * energy decay is reduced compared with non-rotating turbulence * the inertial spectrum is steeper than the Kolmogoroff k-5/3 form * for initially isotropic flow Reynolds stress remains isotropic but length scales become anisotropic

10. Modeling Difficulties

11. Implications for SGS models but makes the model too dissipative for rotating turbulence • For dynamic model and the model is inconsistent with the transformation properties Approximate velocity models (nonlinear, deconvolution, estimation) avoid these difficulties and satisfy transformation properties automatically

12. Truncated Navier-Stokes Equations (TNS) • Variation of the Velocity Estimation Model (VEM) (Domaradzki and Saiki (1997)) • Based on two observations: - the dynamics of small scales are strongly determined by the large, energy carrying eddies - the contribution of small scales to the dynamics of large scales (k<kc) comes mostly from scales within kc<k<2kc • Implemented for low Reynolds number rotating turbulence by Domaradzki and Horiuti (2001) to avoid difficulties with rotational transformation properties for classical SGS models (Horiuti (2001))

13. Truncated N-S dynamics (spectral space) Estimated scales (on “fine” mesh): Artificial energy accumulation due to absence of (natural or eddy) viscosity. E(k) Large physical scales (on “coarse” mesh): computed by N-S eqns. Unresolved scales kc 2kc k Filter small-scales at fixed interval and replenish using estimation model TNS=Sequence of DNS runs with periodic processing of high modes

14. Multiscale modeling Scales periodically replaced by estimated scales E(k) Large scales computed from (inviscid) TNS eqs. Unresolved scales kc 2kc k Estimated small scales computed from a separate dissipative equation forced by the inviscid solution. Similar to Dubrulle, Laval, Nazarenko, Kevlahan (2001)

15. N-S equations are solved - SGS stresses are not needed - Transformation properties (Galilean, rotating frame) always satisfied - Commutation errors are avoided Applicable to strongly anisotropic flows (VLES) Straightforward inclusion of additional effects (convection, compressibility, stable stratification, rotation) Requires determination of the filtering interval (based on a small eddy turnover time or a limiter on the small energy growth) Properties

16. TNS for rotating turbulence • TNS with VEP applied to simulate low/high Re number turbulence with/without rotation • DNS data of Horiuti (2001) • Mesh size is 2563for DNS and 643 for TNS • The initial condition for TNS is obtained by truncating the full 2563DNS field to 323 grid • Low Re: • High Re:

17. Energy Spectrum, low Re

18. Energy decay, high Re -1.2

19. High Re: spectral slope predictions • n=2: Zhou (1995); Baroud et al. (2002). • n=11/5=2.2: Zeman (1994). • n=7/3=2.33: Bershadskii, Kit, Tsinober (1993). • n=3: Smith and Waleffe (1999); Cambon et al. (2003).

20. Energy spectrum, high Re -2 -3

21. Anisotropy Indicators • Length scales • Reynolds stress tensor and anisotropy tensor (=0 for isotropic turbulence)

22. Integral length scales 5,10 50 1 100 0

23. Anisotropy Tensor • Directional and polarization anisotropy tensor E(k) is the total energy for all modes in a wavenumber shell

24. Directional anisotropy tensor 5 100 1 0

25. Summary of Observations • Spectral slope n=-2 at earlier times (t<5) and n=-3 at later times (t>15) • Anisotropy indicators largest for - times t>5 - moderate rotation rates • Anisotropy indicators small for and • Spectral Exponent Hypothesis • Approximately isotropic state characterized by n=-2 • Strongly anisotropic state characterized by n=-3

26. Two different views of LES • Classical view: - governing LES equations are derived from Navier-Stokes eqs. and are are different from them - unknown SGS stress is modeled using physical principles - there exists a unique best solution to the SGS modeling problem Additional “complex” physics often requires substantial changes in models developed for simpler flows.

27. Competing view: - governing LES equations are simply Navier-Stokes eqs. - LES modeling problem is of numerical nature: how to accurately solve Navier-Stokes eqs. on coarse grids - there may be many solutions to the problem, e.g. regularization of the equations or the solutions, using numerical dissipation in place of physical dissipation (MILES/ILES), etc. Disadvantages: ILES is not robust because there is no guarantee that the implicit dissipation is equal to the physical dissipation Potential Advantage: if the equations are known there are no modeling problems!

28. D = 10 m U = 10 m/s N = 0.003 /s D = 10 km U = 10 m/s N = 10-4 /s Re = 108 F = 500 Re = 1010 F = 10 Turbulent wakes in stably stratified fluids Guadalupe island

29. Experiments: Spedding et al. (1996, 1997,2001,2002).

30. Numerical Method: Computational Domain and Flow Configuration • Periodic in horizontal directions: Fourier discretization. • Bottom: Solid Wall. Top: Free Surface. Divide into spectral subdomains (elements). Legendre polynomial discretization. U Wake of a towed sphere

31. Numerical Method: Spectral Multidomain Discretization Well-resolved wake core, subsurface Ambient region not over-resolved • Partition domain into M subdomains with: • Height Hk and order polynomial approximation Nk. • Non-uniform local Gauss-Lobatto grid (No stretching coefficients !).

32. Numerical Techniques Dealing with Under-Resolution to Maintain Spectral Accuracy and Stability • Spectral Filtering. • Strong Adaptive Interfacial Averaging. • Spectral Penalty Methods (J. Hesthaven – SIAM J. Sci. Comp. Trilogy) • Attempting to satisfy eqs. with limited resolution arbitrarily close to boundaries leads to catastrophic instabilities • Solution: Implement BC in a weak form by collocating equation at boundary with a penalty term

33. Truncated Navier-Stokes Dynamics

34. Flow Parameters and Runs Performed • Domain size: 16Dx16Dx12D -Timestep Dt~0.03 D/U. • Initialization procedure that of Dommermuth et al. (JFM 2002) = Relaxation. Initial velocity data that of Spedding at Nt=3.

35. Flow Structure: Isosurfaces of |ω| at Fr= Re=5K Re=20K

36. Flow Structure: Isosurfaces of ωz at Fr=4 Re=5K Re=20K

37. Vertical Vorticity, wz at Horizontal Centerplane(Nt=56, x/D=112) Re=5K Re=20K Fr=4 Fr=

38. Fr and Re Universality of Wake Power Laws: Mean centerline velocity

39. Fr and Re Universality of Wake Power Laws: Wake Horizontal Lengthscale , Fr=4

40. Conclusions • A range of subgrid scales adjacent to the resolved range dominates dynamics of the resolved eddies • These subgrid scales can be estimated in terms of the resolved scales (estimation model) • Dynamics of the resolved eddies is approximated by Truncated Navier-Stokes equations for resolved and estimated scales • The method consists of a sequence of underresolved DNS and a periodic processing of the solution

41. Conclusions • TNS approach captures well temporal evolution of wake mean velocity profile, length scales and vorticity field structure. • For Reynolds numbers considered both TNS and stability filtering produced essentially the same results (supports ILES?) • For decaying isotropic turbulence the inertial range spectrum is maintained during flow evolution • For decaying rotating turbulence good comparison with DNS data is obtained at low Re • At high Re decreased kinetic energy decay rates are observed for increasing rotation rate and the asymptotic spectrum proportional to Can LES with complex physics be best addressed by minimizing explicit modeling that affects the form and properties of the governing equations !?