Influence of wall thermal boundary condition on mean-flow characteristics of a Mach 2.5 fully rough turbulent boundary layer
We investigate turbulent boundary layers at Mach 2.5 over sinusoidal roughness at matched $Re_τ=784$. We considered three wall temperatures, $T_w/T_r=$ 1.0, 0.7, and 0.4. Each wall temperature was repeated with a smooth and rough surface, the latter following a three-dimensional sinusoidal profile with effective slope 0.5 and matched $k^+=79.1$. In total six direct numerical simulations were completed. Analysis of mean wall shear and heat transfer detailed the augmentation in skin friction and heat transfer coefficient between smooth and rough cases; resulting in the classical failure of the Reynolds analogy for rough walls. We also show that differences in shear stress across wall temperatures are driven by the viscous component even though the pressure component dominates. The roughness sublayer was found to be $R_{RSL}=3k-6k$, consistent with the literature. Moreover, the wall offset $d=d_Θ\approx0.5k$ for both momentum and thermal boundary layers, signifying the offset represents the half peak-to-valley height rather than the "mean roughness height." For the mean momentum boundary layer, we find the present conditions recover the incompressible roughness function $Δu_1^+$ when matched via the semi-local roughness Reynolds number $k^*$. For this reason, an equivalent sand-grain roughness of $k_s^*\approx3.7k^*$ is proposed. The existing compressible transformations hold regardless of wall temperature or roughness - contradicting recent claims otherwise. The generalized Reynolds analogy works for smooth walls but fails for rough walls; a roughness correction term is proposed and validated. Compressible mean temperature transformations fail for cold walls, especially when $(T_w-T_e)/(T_r-T_e)<0$. The thermal roughness function, $ΔΘ^+$, is reported with caution given transformation singularities.