A body-forcing model for heat transfer in rough-wall turbulent channel flow

A body-forcing strategy is proposed to model the effects of surface roughness on heat transfer in a turbulent flow. The model introduces a source term into the temperature transport equation that depends on a characteristic velocity scale, the local wall-relative temperature and the roughness density. We test it in direct numerical simulation (DNS) of a turbulent open channel at a constant Prandtl number of 0.71, and through extensive comparison with roughness-resolved DNS data, we show that the instantaneous local friction velocity is the appropriate velocity scale. Despite its simple formulation, it accurately captures the salient physics of rough-wall heat transfer, such as the breakdown of the Reynolds analogy, the characteristic plateau of the temperature roughness function and the fully rough behaviour of the ratio of roughness lengths of momentum and heat transfer. An appropriate decomposition of the total heat flux reveals that the scaling of the conductive and body-forcing Stanton numbers is similar to that of the exposed and sheltered Stanton numbers from fully resolved roughness simulations, respectively. This suggests that the primary effect of roughness on heat transfer can be effectively captured so long as the surface heat flux is modelled correctly in a statistical sense, even if the precise contribution from individual roughness elements is not accurately resolved. Finally, the body-forcing model can be used over regular no-slip smooth walls, bypassing the need to resolve the exact details of the roughness geometry.