Wall-modelled large-eddy simulation of turbulent channel flow with unstable stratification

Abstract

Unstable thermal stratification modifies near-wall momentum and heat transport, causing the mean velocity profile to depart from the classical logarithmic law and complicating wall modelling for turbulent mixed convection. We develop a buoyancy-modified logarithmic-quadratic wall model for incompressible Poiseuille--Rayleigh--Bénard flow. The model combines an approximately linear relation between the near-wall mean temperature and mean streamwise velocity with a thermally modified mean-gradient representation inspired by mixing-length scaling. A priori assessments using wall quantities from the direct numerical simulation (DNS) database show that the calibrated wall law reconstructs near-wall velocity and wall-function-equivalent eddy-viscosity profiles. We implement the wall model in wall-modelled large-eddy simulations (WMLES) at friction Reynolds numbers up to $Re_τ\approx6000$ and Rayleigh numbers up to $Ra=10^{10}$. For cases with DNS reference profiles at $Ra=10^8$ and $10^9$, the maximum pointwise absolute relative errors are $3.6\%$ for the mean velocity and $1.9\%$ for the mean temperature. For cases with available DNS global-transport data, the maximum relative deviations in the Nusselt number $Nu$ and skin-friction coefficient $C_f$ are $11.7\%$ and $15.9\%$, respectively. The WMLES reduces the mesh count by factors of approximately $195$--$542$ relative to the corresponding DNS meshes. For the $Ra=10^{10}$ and $Ri_b=0.1$ case, extrapolation of reference DNS resolution strategies gives a mesh count of order $10^{11}$, approximately three orders of magnitude larger than the present WMLES mesh count. We also examine how the balance between shear and buoyancy reorganises flow structure, and we identify signatures consistent with the coexistence of streamwise-elongated motions resembling very-large-scale motions and buoyancy-associated streamwise rolls.