Scaling of rough-wall turbulence by the roughness height and steepness

Abstract

A roughness scaling behaviour is tested by performing the direct numerical simulation (DNS) of a turbulent channel flow over three-dimensional sinusoidal rough walls. By systematically varying the roughness height k(+) and the roughness steepness S, the results for three groups of cases are considered and compared with those for flat-wall turbulence. The results show that the mean velocity and Reynolds stresses are highly dependent on both k(+) and S. To describe these specific relationships, we define a coupling scale k(+) S. With this coupling scale, all the simulated data for the roughness function (Delta U+), the ratio of the pressure drag to the total wall resistance (gamma(p)), the normalized bulk mean velocity (U-b(+)) and the peak of the streamwise turbulent velocity fluctuations ((u(p)'(+)) over bar) collapse onto single curves, which shows that there is a strong direct correlation between them, i.e. Delta U+, gamma(p), U-b(+), (u(p)'(+)) over bar proportional to f(k(+) S). Furthermore, a model for the prediction of wall resistance based on the roughness function can be established by defining a drag increasing ratio (DI). Accordingly, the wall resistance coefficient C-f can be estimated directly from k(+)S of a given rough surface. These results suggest that this coupling scale provides a useful alternative to the equivalent sand grain roughness k(s).