Direct numerical simulation of turbulent flow over a wavy wall

The Navier–Stokes equations have been solved, by a pseudospectral method, for pressure-driven flows between a no-slip wavy wall and a slip flat wall. Periodic boundary conditions were used in the streamwise and spanwise directions. The physical domain is mapped into a computational domain that is a rectangular parallelepiped using a nonorthogonal transformation. The pseudospectral solution procedure employed in previous studies, for example, Lam and Banerjee [Phys. Fluids A 4, 306 (1992)], eliminated the pressure and solved for the wall–normal velocity and vorticity. The other velocity components were calculated using the definition of vorticity, and the continuity equation. This procedure leads to oscillations in the pressure field when solutions were attempted in the mapped computational domain. To overcome the problem, the procedure had to be modified and the pressure solved for directly using a fractional time step technique. For the cases examined here, these modifications resulted in spectral accuracy being maintained. Flow over sinusoidal wave trains has been simulated and the results compare well with available experiments. The simulations show significant effects of the wavy boundary on the mean flow and the turbulence statistics. The mean velocity profile differs substantially from the profile for the flat-wall case, particularly in the buffer region where the fluid is under the influence of both the wavy wall and the slip boundary. The velocity fluctuations in the streamwise direction decrease in the buffer region. This effect becomes more pronounced when the wave amplitude increases. Most of the redistribution of energy, from the streamwise direction to the spanwise and wall–normal directions, occurs in a thin layer close to the boundary, downstream of the wave troughs. The energy primarily redistributes into spanwise fluctuations. High shear stress regions form downstream of the wave troughs, and streaky structures and quasi-streamwise vortices are also seen to initiate in these regions. The length of the streaks, and the extent of the quasi-streamwise vortices, scale with wave length for the two cases investigated.