1. Dynamic subgrid-scale modeling in large- eddy simulation of turbulent flows with a stabilized finite element method Andrés E. Tejada-Martínez Thesis advisor: Kenneth E. Jansen Department of Mechanical, Aerospace, & Nuclear Engineering Rensselaer Polytechnic Institute

2. Outline Part I: Part II: - dynamic subgrid-scale modeling in large-eddy simulation - new models accounting for implicit filter characteristic - spatial filters for dynamic modeling (test filters) - physical and numerical energy dissipation - new dynamic model accounting numerical dissipation associated to the discretization (the stabilized method) of the discretization

3. Large-eddy simulation (LES) large eddies resolved in LES

4. Subgrid-scales not resolved in LES, and must be modeled Large-eddy simulation (LES) In direct numerical simulation (DNS) all scales are resolved

5. Filtering operation Consider f(x) with a wide variety of scales. A filtered function is defined as: Scales of and less are damped and f(x) is decomposed into resolved and residual components: Two homogenous filters we use are the box and hat filters. filter width y xx+hx-hyxx+hx-h 1/2h1/h boxhat

6. The filtered Navier-Stokes equations Homogenous arbitrary kernel is used. Continuity: Momentum: deviatoric subgrid-scale stress at the -level Smagorinsky model for the SGS stress: subgrid-scale stress at the -level

7. Dynamic model for Thus, we have: Application of a homogenous, secondary (test) filter,, to the once-filtered eqns. creates a second stress tensor defined as This stress is associated to, obtained from successive applications of the primary filter and test filter. Assuming scale-invariance, the deviatoric portion of can be modeled by Smagorinsky as subgrid-scale stress at the -level

8. Model continued The Germano identity between and is defined as Least squares min. between -resolved and -modeled w.r. t. to leads to a dynamic expression for for use in -resolved -modeled

9. Dynamic model with standard test filter - Averaging in statistically homogenous direction(s) filter width ratio squared standard test filter

10. Dynamic model with wide test filter wide test filter Consider replacing test filter with filter defined as the successive applications of the original test filter and a second test filter,. The residual stress generated at the -scale is The dynamic procedure leads to: filter width ratio squared

11. Comments on modeled equations The models depend on the ratios and Accurate determination of these parameters requires characterization of, and In practice is set by the choice of numerical method, thus all three filters are typically poorly characterized.

12. Discrete test filters Approximation of a filter operation using quadrature rules leads to discrete filters For example, 2-pt quadrature approximation of a box filtered function leads to: y vertexquadrature point

13. Standard and wide discrete filters We will be using the following 4 filters: filter S1 (standard with rule 1) filter W2 (wide with rule 2) filter S2 (standard with rule 2) filter W1 (wide with rule 1 )

14. Transfer functions for filter S1 on (a) triangles and (b) quads. (a) (b) Can help find test filter widths: Transfer function = Fourier transform of filter kernel = avg. radial wavenumber for a specified value of the transfer function

15. Decaying isotropic turbulence behind a bar grid isotropic far field In our LES, the larger scales are resolved while the smaller subgrid-scales are modeled bar grid

16. Decaying isotropic turbulence In decaying isotropic turbulence the mean flow is zero, motions decay in time due to a lack of kinetic energy production to balance viscous dissipation. Scales have no directional orientation. Initial conditions are obtained from experiments of Comte- Bellot and Corrsin (1963). Data exists at two non- dimensional time stations: t42 and t98. Domain is a periodic box split by 33 equidistant vertices in each direction. We use the Streamline Upwind/Petrov-Galerkin (SUPG) method w/ linear basis and a second order accurate time integrator as described in Whiting and Jansen (2001).

17. Effect of filter width ratio on energy spectra of isotropic turbulence We examine the effect of changing the filter width ratio, in on dynamic model results of decaying isotropic turbulence

18. Filter width ratio assumption based on test filter widths Recall the dynamic model coeffs.: and, where Two sets of simulations are performed on hexes: (a1) (b1). Two sets are also performed on tets: (a2) (b2) For each set, four simulations are performed: (1) std. filter S1, (2) wide filter W1, (3) std. filter S2, and (4) wide filter W2. Test filter widths and are computed based on transfer functions of the standard or wide test filter used.

19. Simulations on hexes

20. Simulations on tets

21. Dynamic filter width ratio formulation Previous results suggest Recall stresses and modeled as Consider the following identity: Least squares minimization between modeled and resolved expressions for the identity above leads to: and (1)

22. Dynamic filter width ratio formulation (DFWR2) Dividing by results in (2) Recall equation (1): Eqns. (1) and (2) can be solved for leading to its dynamic determination and thus a parameter-free model, DFWR2.

23. Comments on models The filter width ratio parameter in the classic dynamic model is not well-characterized. Results show sensitivity to filter width ratio. Its accurate determination is important. DFWR2 computes the filter width ratio dynamically without parameters. DFWR1 is derived similarly to DFWR2. DFWR1 is not parameter-free as it requires the ratio between the widths of the two test filters used. DFWR1 and DFWR2 account for implicit filtering characteristic of the numerical method.

24. DFWR on isotropic turbulence on hexes

25. Evolution of filter width ratio,

26. DFWR on isotropic turbulence on tets

27. Evolution of filter width ratio,

28. Wall-modeled turbulent channel flow Channel geometry : Reynolds # based on friction velocity, Periodicity in the x- and z-directions. Shear stress boundary condition at the walls (at is obtained via a near-wall model. 33 vertices in x, 31 in y and 33 in z. Near-wall features are not well resolved. Vertices are equidistant in each direction. Mesh is made of hexes. z y x

29. Wall modeled turbulent channel flow We compare DFWR2 (w/ S1 and W1) to 2 classic dynamic models: 1) w/ standard filter S1 and 2) w/ wide filter W1. For the classic models, test filter widths are based on the filters’ second moment and filter width ratios are taken as: No input parameters required for DFWR2. Some results are presented in wall units: 1)2)

30. Wall force Expected mean force = 0.435

31. Mean streamwise (x-) velocity Direct Numerical Simulation (DNS): Kim, Moin, and Moser (1987)

32. Dynamic model coefficient Filter width ratio predicted by DFWR2

33. Part II Physical and numerical energy dissipation

34. Stabilized FEM Formulation (SUPG) See Taylor et.al. (1998), and Hauke and Hughes (1998). For our studies: strong SUPG (numerical dissipation) weak SUPG (numerical dissipation) SUPG tensor

35. Numerical and physical dissipations Dynamic model (physical) subgrid-scale (SGS) dissipation: Numerical dissipation due to SUPG stabilization: strong SUPG and weak SUPG. Model with standard filter S1: strong (SGS) model Model with wide filter W1: weak (SGS) model Wall resolved turbulent channel simulations are performed. 65 vertices in y (normal to walls) are stretched such that there are more vertices near the walls. There are 33 equidistant vertices in x and in y. Periodicity in x and z. No-slip velocity at walls. Based on channel half width: SUPG tensor

36. Wall forces Expected mean force = 0.435

37. Dissipations and eddy viscosity

38. A modified dynamic model Model coefficients SUPG correction: Corrected SGS diss.: Corrected dynamic model:

39. Strong model w/ and w/out SUPG correction Expected mean force = 0.435

40. Strong model w/ and w/out SUPG correction

41. Weak model w/ and w/out SUPG correction Expected mean force = 0.435

42. Weak model w/ and w/out SUPG correction

43. Other statistics (for strong SUPG cases)

44. Observations SUPG correction allows the dynamic model to properly adjust to the presence of SUPG dissipation. When SUPG diss. is of the same order as SGS (dynamic model) diss., SUPG correction has a strong impact. The top performer is the weak model with SUPG correction. The worst performer is the strong model without SUPG correction. Although not shown, SUPG correction was applied to DFWR2 and helped improve results. DFWR2 with SUPG correction is of similar quality as the weak model with SUPG correction.

45. Summary and future work The models proposed here account for the implicit filter characteristic (DFWR1,2) and the dissipative nature nature (SUPG correction) of the numerical method. Two main tenets underlie the new models. Dynamic subgrid-scale models should be independent of This work has laid the foundation for improved physical subgrid-scale modeling taking into account stabilization and for improved stabilization techniques taking into account physical modeling. 1)the test filter (DFWR2) 2) a change in the numerical method brought about by stabilization (SUPG correction )

46. The Navier-Stokes equations Continuity: Momentum: viscous stress

47. Transfer functions in multi-dimensions Transfer functions can help us determine filter widths: is the average radial wavenumber for a specified iso- surface of the transfer function. Examples: Filter S1 on regularly connected (a) triangles and (b) quads. quadrature point at centroid vertex h filter support

48. Ratio between model coefficients

49. Steps in large-eddy simulation of turbulent flows The N-S equations are filtered with an arbitrary homogenous kernel. Filtering generates a subgrid-scale (SGS) or residual stress which is modeled. The modeled filtered N-S eqns. are solved numerically to represent the resolved motions. Primary filter kernel