Effects of steepness on turbulent heat transfer over sinusoidal rough surfaces

We conducted a direct numerical simulation (DNS) study to investigate the impact of surface undulation steepness on rough wall turbulent heat transfer. The flow geometry was turbulent open-channel flow over three-dimensional sinusoidal rough surfaces. To examine the effects of steepness, we systematically varied the streamwise and spanwise wavelengths of the sinusoidal roughness while keeping the roughness height constant. The friction Reynolds number ranged from 180 to 600, and we considered a passive scalar with the fluid Prandtl number was 0.7, assuming air flow conditions. In the fully rough regime, the velocity roughness function is expressed as a function of the inner-scaled equivalent sand grain roughness $k_s^{+}$ independent of steepness, whereas the steeper surfaces with shorter wavelengths result in larger temperature roughness functions at the same $k_s^{+}$ value. Analysis of the physical mechanisms that increases the roughness function shows that the pressure drag primarily contributes to the increase in the velocity roughness function, while the temperature roughness function is mainly augmented by the roughness-induced wall heat transfer term, correlating with the steepness of the surface undulations. It is also suggested that the effective slope, which quantifies the steepness of rough surfaces, could improve the predictive accuracy of existing correlations for the temperature roughness function.