A zero-pressure-gradient turbulent boundary layer flowing over a transitionally rough surface (24-grit sandpaper) with$k^{+}\approx 11$and a momentum-thickness Reynolds number of approximately 2400 is studied using direct numerical simulation (DNS). Heat transfer between the isothermal rough surface and the turbulent flow with molecular Prandtl number$Pr=0.71$is simulated. The dynamic multiscale approach developed by Arayaet al.(J. Fluid Mech., vol. 670, 2011, pp. 581–605) is employed to prescribe realistic time-dependent thermal inflow boundary conditions. In general, the rough surface reduces mean and fluctuating temperature profiles with respect to the smooth surface flow when normalized by Wang & Castillo (J. Turbul., vol. 4, 2003, 006) inner/outer scaling. It is shown that the Reynolds analogy does not hold for$y^{+}<9$. In this region the value of the turbulent Prandtl number departs substantially from unity. Above this region the Reynolds analogy is only approximately valid, with the turbulent Prandtl number decreasing from 1 to 0.7 across the boundary layer for rough and smooth walls. In comparison with the smooth-wall case, the turbulent transport of heat per unit mass,$\overline{v^{\prime }v^{\prime }{\it\theta}^{\prime }}$, towards the wall is enhanced in the buffer layer, but the transport of$\overline{v^{\prime }v^{\prime }{\it\theta}^{\prime }}$away from the wall is reduced in the outer layer for the rough case; similar behaviour is found for the vertical transport of turbulent momentum per unit mass,$\overline{v^{\prime }u^{\prime }v^{\prime }}$. Above the roughness sublayer (3$k$–5$k$) it is found that most of the temperature field statistics, including higher-order moments and conditional averages, are highly similar for the smooth and rough surface flow, showing that the Townsend’s Reynolds number similarity hypothesis applies for the thermal field as well as the velocity field for the Reynolds number and$k^{+}$considered in this study.
The Behaviour of Temperature-Velocity Fluctuations and the Turbulent Prandtl Number in the Developing Thermal Turbulent Boundary Layer on the Flat Plate