Direct numerical simulation and statistical analysis of stress-driven turbulent Couette flow with a free-slip boundary

Abstract

The effects of free-slip boundary on shear turbulence are studied numerically using the direct numerical simulation (DNS) approach. The flow considered in this study is a stress-driven turbulent Couette flow between two flat boundaries. The top boundary has an imposed shear stress in the streamwise direction and a free-slip condition for the streamwise velocity fluctuation and the spanwise velocity, and the bottom boundary satisfies the no-slip condition. This type of flow has a mean flow pattern similar to the turbulent plane Couette flow between a stationary flat plate and a moving flat plate but exhibits considerable differences in turbulence statistics due to the effects of the free-slip boundary. Statistical analysis based on the DNS data and theoretical derivation based on Taylor series expansion show that near the free-slip surface the turbulence variances of the three velocity components vary as quadratic functions of the vertical distance from the boundary while the Reynolds shear stress exhibits a linear behavior, which are very different from the counterparts near the no-slip boundary. The free-slip surface condition also leads to zero horizontal vorticities at the surface but allows nonzero vertical vorticity in the meantime, leading to considerable differences in the near-boundary statistics of vorticities and coherent vortex structures. Comparison of three DNS runs with different grid resolutions shows that smoothly resolving the more energetic turbulent flow structures near the free-slip boundary requires a higher horizontal grid resolution than that used for resolving the structures near the no-slip boundary.