1. Convergence Studies of Turbulent Channel Flows Using a Stabilized Finite Element Method Andrés E. Tejada-Martínez Department of Civil & Environmental Engineering and Florida Advanced Numerical Simulation Institute (FANSI) University of South Florida Alisa Trofimova, Kenneth E. Jansen and Richard T. Lahey Jr. Department of Mechanical, Aerospace & Nuclear Engineering and Scientific Computation Research Center Rensselaer Polytechnic Institute

2. Motivation A number of turbulent channel flow computations with stabilized methods have recently appeared Analysis of these simulations has been limited to mean velocity, RMS of velocity fluctuations and Reynolds shear stress An investigation of additional statistics often computed in finite-difference and spectral simulations of this problem is still lacking - Jansen, CMAME, 2000 - Tejada-Martínez and Jansen, CMAME, 2005 - Bazilevs et al., ICES Report, 2007 - Rispoli et al., Comp. & Fluids, 2007 - Akkerman et al., ICES Report, 2007 - RMS of pressure fluctuations - Two-point velocity autocorrelations - One-dimensional energy spectra - Balances of budget terms transport of TKE

3. Part I Part II - weak form - stabilized finite element formulation: - variational equation for resolved kinetic energy - impact of stabilization on mean velocity and single-point statistics - single/two-point statistics at different resolutions in flow at Outline - impact of stabilization on the smaller resolved scales - problem setup Streamline upwind / Petrov-Galerkin (SUPG) method

4. See Taylor et.al., CMAME, 1998 and Hauke and Hughes, CMAME (1998) is not well-defined and should be chosen carefully Stabilized FE formulation (SUPG)

5. Resolved kinetic energy Dissipation due to advection stabilization Dissipation due to continuity stabilization - For turbulent channel flows: Tejada-Martínez and Jansen, CMAME, 2005 - As mesh size and time step are decreased it is desirable to decrease as well so that remains constant and does not adversely effect convergence results

6. Problem setup - Turbulent channel flow driven by constant body force F such that F Periodicity in (streamwise direction) Periodicity in (spanwise direction) No-slip in (wall-normal direction) Boundary Conditions:

7. Mesh and time step parameters N 1 xN 2 xN 3 Δt c1 c1 coarse mesh 32x65x320.05 8 medium mesh 64x129x640.025 4 fine mesh 256x193x1920.006 4 - All meshes are comprised of tri-linear hexahedral elements - Grid points are clustered near walls - Results compared to spectral DNS of Moser et al., Phys. Fluids, 1999 - DNS of Moser et al. was on a 256x193x192 mesh, same as our fine mesh case

8. Effect of mesh resolution on mean velocity - Under-prediction of mean velocity on coarse mesh is also observed in finite difference schemes with no LES subgrid-scale (SGS) model

9. Mean velocity in finite difference methods Results of Morinishi and Vasilyev, Phys. Fluids, 2001 Flow is at on 32x64x32 grid

10. Effect of mesh resolution on RMS of velocity - Over-prediction of u-rms on coarse mesh is also observed in finite difference schemes with no LES subgrid-scale (SGS) model

11. U-rms in finite difference methods Results of Morinishi and Vasilyev, Phys. Fluids, 2001 Flow is at on 32x64x32 grid

12. Effect of mesh resolution on shear stresses - Peak Reynolds shear stress is not greatly affected by mesh resolution

13. Effect of mesh resolution on RMS of pressure

14. Effect of mesh resolution on spanwise spectra - High wavenumber damping due to SUPG stabilization occurs on scales between 1 and 2 times the grid cell size in streamwise direction

15. Effect of mesh resolution on spanwise spectra - High wavenumber damping in spanwise spectra is less than in streamwise spectra. Consistent with streamline upwinding in SUPG

16. Balance of Budget Terms in TKE Transport Flow on 256x193x192 mesh - Small residual is due to TKE dissipation by SUPG stabilization - TKE dissipation by SUPG stabilization is not accessible in our computation

17. Impact of stabilization on mean velocity - coarse mesh flow at - Increased effect of SUPG stabilization acts as a surrogate LES-like subgrid-scale model leading to improved mean velocity on coarse mesh

18. Impact of stabilization on RMS of velocity - coarse mesh flow at - Increased effect of SUPG stabilization leads to poorer prediction of RMS of velocity

19. Impact of stabilization on RMS of pressure - coarse mesh flow at - Increased effect of SUPG stabilization leads to improved prediction in the near-wall region. RMS of pressure in core is insensitive to strength of SUPG

20. Impact of stabilization on streamwise spectra - coarse mesh flow at - Increased effect of SUPG stabilization leads to slightly more high wavenumber damping

21. Impact of stabilization on spanwise spectra - coarse mesh flow at - Spanwise spectra is almost insensitive to increased strength of SUPG dissipation as would be expected

22. Summary - On DNS grids, stabilized FEM results proved to be in excellent agreement with results of Moser et al. (1999) obtained with a spectral method - Stabilized FEM faithfully represented wide range of turbulent scales without introducing excessive energy dissipation - Relative to spectral method, damping by the stabilized FEM was relegated to small scales on the order of 1 to 2 times the streamwise grid cell size - Small scale damping was more pronounced in streamwise direction - On coarse mesh, increase of stabilization effect can improve mean velocity and near-wall RMS of pressure but can lead to poorer RMS of velocity