Introduction to classical large eddy simulation (LES) of turbulent flows

1. Introduction to classical large eddy simulation (LES) of turbulent flows Andrés E. Tejada-Martínez Center for Coastal Physical Oceanography Old Dominion University

2. Outline - introduction to spatial filters • Part I: Theory • Part II: Computations - equations governing the large eddies (filtered (LES) equations) - importance of numerical discretization (i.e. the numerical solver) - subgrid-scale (SGS) models/approximations in the filtered equations • LES of isotropic turbulence and unstratified/stratified channel flows • LES of Langmuir turbulence - animations (flow over an airfoil, flow over a cavity)

3. Steps in large-eddy simulation (LES) • The Navier-Stokes equations are filtered with a low-pass filter • Filtering presents a closure problem as an unknown residual (subgrid-scale) stress which may be modeled or approximated • The modeled filtered Navier-Stokes equations governing the largest scales (or the large eddies) are numerically solved. -filtering attenuates smallest (residual) scales, while preserving largest -stress represents effect of attenuated smallest scales on largest scales

4. Large-eddy simulation large eddies resolved in LES

5. Large-eddy simulation • Residualscalesnot resolved in LES, and must be modeled • Residual scales are modeled under assumption of isotropy

6. Filtering in real space • A filtered function is defined as: • Examples of homogeneous filter kernels, , are • Filtering attenuates scales less than and splits the function as 1/2h 1/h box hat y y x-h x x+h x-h x x+h residual (small) component large component

7. Filtering in real space Scales < • Note that in general except for the sharp cutoff filter • Filtering attenuates or removes (depending on the shape of the filter • kernel) scales on the order of the filter width:

8. Filter kernels in real space • The width of G(r) may be defined with respect to the box filter as • See Turbulent Flows by S.B. Pope for functional forms of kernels

9. Filters in Fourier space: transfer function • The Fourier transform (F.T.) of a filtered function is the transfer function • of the filter multiplied by the F.T. of the un-filtered function: • For the sharp cutoff filter but in general this is not true

10. Sketch of filtered energy spectra Both filters leave low wavenumber content untouched ln(E(k)) un-filtered spectrum In the low wavenumber range filtered spectrum using cutoff filter filtered spectrum using box filter un-filtered spectrum In the high wave- number range ln(k)) • Filtering with the box filter leads to an attenuation of scales around • Filtering with the sharp cutoff filter preserves scales at less than • and completely erases scales at higher wavenumbers

11. LES and other approaches • For a turbulent flow, the Navier-Stokes (N-S) equations contain a large number of scales. • While solving these equations numerically, the grid must contain a great number of points in order to represent (resolve) all of the scales present. • In LES we filter the equations, thereby suppressing the smaller scales. With fewer scales, the filtered equations need less grid points. • In direct numerical simulation (DNS) no filtering is performed, as the simulation attempts to represent all scales down to the dissipative ones • In Reynolds-averaged N-S simulation (RANSS), we solve the ensemble-averaged N-S. Averaging suppresses all of the scales except for the largest, thus RANSS requires much fewer grid points than LES and DNS. • LES resolves many more scales than RANSS, but not as many as DNS. No. of grid points

12. Unfiltered equations

13. Unfiltered equations

14. Filtered momentum equation • Filter the momentum eqn. with an arbitrary homogenous filter of width . • Homogeneity of filter allows commutation with differentiation:

15. Filtered momentum equation • Filter the momentum eqn. with an arbitrary homogenous filter of width . • Homogeneity of filter allows commutation with differentiation:

16. Filtered momentum equation • Filter the momentum eqn. with an arbitrary homogenous filter of width . • Homogeneity of filter allows commutation with differentiation: • leads to • is an unknown stress accounting for the effect of the filtered-out small • scales on the large scales governed by the filtered equation

17. Residual (subgrid-scale (SGS)) stress • Note that in general: • Decompose the SGS stress as

18. Residual (subgrid-scale (SGS)) stress • Note that in general: • Decompose the SGS stress as

19. Residual (subgrid-scale (SGS)) stress • Note that in general: • Decompose the SGS stress as deviatoric (trace-free) component isotropic component

20. Residual (subgrid-scale (SGS)) stress • Note that in general: • Decompose the SGS stress as deviatoric (trace-free) component isotropic component • This decomposition leads to • The modified filtered pressure contains the isotropic part of the SGS stress

21. Filtered equations

22. Filtered equations SGS stress SGS stress:

23. Filtered equations SGS stress SGS density flux SGS stress: SGS density flux: (obtained in same way as the SGS stress)

24. Comments on the filtered equations • The filtered equations are numerically solved for the filtered variables • describing the large scales • The SGS stress and SGS density flux present closure problems and • must be modeled or approximated in terms of filtered variables only • In theory, the filter used to obtain the filtered equations is arbitrary • In practice, the filter is inherently assumed by the discretization (i.e. the • numerical method used to solve the filtered equations and the SGS models) • The discretization can only represent (resolve) down to scales on the order • of 1,2, or 3 times the grid cell size, h, thereby “filtering-out” smaller scales.

25. Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales ln(k))

26. Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales spectrum based on scales in inertial range ln(k))

27. Sketch of energy spectrum in LES most energetic scales usually resolved in RANSS and general ocean circulation simulations ln(E(k)) 3 spectrum -5 based on dissipative scales spectrum based on scales in inertial range resolved (large) scales sub-grid (residual) scales ln(k)) • We choose the grid size, h, to fall within the inertial range to facilitate SGS modeling

28. Role of discretization in LES Both A and B do a good job representing low wavenumber spectrum of ln(E(k)) Spectrum of Spectrum of obtained with discretization A Spectrum of obtained with discretization B ln(k)) • Discretization A behaves more like a sharp cutoff filter, while B behaves • more like a box filter • Ideally we would aim for a discretization like A

29. Smagorinsky SGS model • Recall that the SGS stress and density buoyancy flux must be • modeled or approximated

30. Smagorinsky SGS model • Recall that the SGS stress and density buoyancy flux must be • modeled or approximated Both are trace-free Smagorinsky (1967) model:

31. Smagorinsky SGS model • Recall that the SGS stress and density buoyancy flux must be • modeled or approximated Both are trace-free Smagorinsky (1967) model: Smagorinsky coefficient

32. Smagorinsky SGS model • Recall that the SGS stress and density buoyancy flux must be • modeled or approximated Both are trace-free Smagorinsky (1967) model: Smagorinsky coefficient Analogously:

33. The eddy (turbulent) viscosity • The turbulent viscosity has units . Because we are working • with the smallest resolved scales, we can set • In LES the SGS range starts at the inertial range, thus we may invoke • Kolmogorov’s 2nd hypothesis: Statistics of scales of size, say, within the inertial range have a universal form uniquely determined by the rate of energy transfer, • And we may have • In aglobal sense,the rate of energy transfer within the inertial range is • roughly equal to the SGS dissipation. Here we assume it locally:

34. Difficulties with the Smagorinsky model Smagorinsky model: • For isotropic turbulence, Lilly (1967) showed that Major difficulty: • The constant coefficient allows for a non-vanishing turbulent viscosity • at boundaries and in the presence of relaminarization • The Smagorinsky coefficient should be a function of space and time • In 1991, Germano and collaborators derived a dynamic expression • for the Smagorinsky coefficient

35. Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width

36. Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width • Consider a new filter made up from successive applications of the 1st • filter (above) and a new “test” filter. This “double” filter has width • Application of this “double” filter is denoted by a “bar-hat” in the form of

37. Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width • Consider a new filter made up from successive applications of the 1st • filter (above) and a new “test” filter. This “double” filter has width • Application of this “double” filter is denoted by a “bar-hat” in the form of • With this new filter, the filtered momentum equation becomes:

38. Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width • Consider a new filter made up from successive applications of the 1st • filter (above) and a new “test” filter. This “double” filter has width • Application of this “double” filter is denoted by a “bar-hat” in the form of • With this new filter, the filtered momentum equation becomes:

39. Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width • Consider a new filter made up from successive applications of the 1st • filter (above) and a new “test” filter. This “double” filter has width • Application of this “double” filter is denoted by a “bar-hat” in the form of • With this new filter, the filtered momentum equation becomes: • Scale invariance: Both and are in the inertial range, thus

40. Dynamic Smagorinsky model • Consider the following tensor proposed by Germano:

41. Dynamic Smagorinsky model • Consider the following tensor proposed by Germano: (resolved)

42. Dynamic Smagorinsky model • Consider the following tensor proposed by Germano: (resolved) (modeled)

43. Dynamic Smagorinsky model • Consider the following tensor proposed by Germano: (resolved) (modeled) • Minimization of the difference between these two with respect to Cs leads to: - Averaging in statistically homogenous direction(s)

44. Dynamic Smagorinsky model • Consider the following tensor proposed by Germano: (resolved) (modeled) • Minimization of the difference between these two with respect to Cs leads to: - Averaging in statistically homogenous direction(s) • Explicit application of test filter (denoted by a “hat”) is required, unlike 1st filter

45. Dynamic Smagorinsky model Sketch of spectra ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on ln(k))

46. Dynamic Smagorinsky model Sketch of spectra ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on Spectrum based on ln(k))

47. Dynamic Smagorinsky model Sketch of spectra Subtest scales ln(E(k)) Unresolved, subgrid scales Resolved scales Spectrum based on Spectrum based on ln(k)) • By applying the test filter, the Germano formulation samples the field between • the subgrid scales and the subtest scales in order to obtain the model coeff.

48. Dynamic mixed model • Recall and true SGS stress subgrid component • Inserting the former into the latter leads to + subgrid-scale terms • The subgrid-scale terms can be approximated via the Smagorinsky model • A dynamic coefficient in the Smagorinky model can be derived here as well • This mixed approach leads to a modeled SGS stress better correlated with • the true SGS stress. Both approaches lead to good correlation with true • SGS energy dissipation

49. LES methodology used in computations SGS stress SGS density flux SGS stress model: SGS density flux model: • Model coefficients in SGS models are computed dynamically as described

50. Numerical scheme used in computations • Horizontal derivatives (in x and y) are treated spectrally • Vertical (z-) derivatives are treated with 6th or 5th order implicit stencils • To prevent spurious high wavenumber content not resolvable by the grid, advection terms are: • The high order accuracy of this discretization allows for it to behave like the sharp cutoff filter - restriction to periodic boundaries in x and y - allows Dirichlet and Neumann boundaries in z 1. de-aliased in x and y 2. filtered in z with a high order implicit filter