We conducted direct numerical simulations (DNSs) of turbulent flow over
three-dimensional sinusoidal roughness in a channel. A passive scalar is
present in the flow with Prandtl number Pr=0.7, to study heat transfer by
forced convection over this rough surface. The minimal channel is used to
circumvent the high cost of simulating high Reynolds number flows, which
enables a range of rough surfaces to be efficiently simulated. The near-wall
temperature profile in the minimal channel agrees well with that of the
conventional full-span channel, indicating it can be readily used for
heat-transfer studies at a much reduced cost compared to conventional DNS. As
the roughness Reynolds number, k+, is increased, the Hama roughness
function, ΔU+, increases in the transitionally rough regime before
tending towards the fully rough asymptote of κm−1log(k+)+C, where
C is a constant that depends on the particular roughness geometry and
κm≈0.4 is the von K\'arm\'an constant. In this fully rough
regime, the skin-friction coefficient is constant with bulk Reynolds number,
Reb. Meanwhile, the temperature difference between smooth- and rough-wall
flows, ΔΘ+, appears to tend towards a constant value,
ΔΘFR+. This corresponds to the Stanton number (the temperature
analogue of the skin-friction coefficient) monotonically decreasing with Reb
in the fully rough regime. Using shifted logarithmic velocity and temperature
profiles, the heat transfer law as described by the Stanton number in the fully
rough regime can be derived once both the equivalent sand-grain roughness
ks/k and the temperature difference ΔΘFR+ are known. In
meteorology, this corresponds to the ratio of momentum and heat transfer
roughness lengths, z0m/z0h, being linearly proportional to z0m+,
the momentum roughness length [continued]...Comment: Accepted (In press) in the Journal of Fluid Mechanic