A zonal and regime-based generalised Reynolds analogy for compressible Couette–Poiseuille flow

Abstract

To address the limitation of the generalised Reynolds analogy (GRA) in handling flows with a spatial mismatch between velocity and temperature extrema, we propose a zonal and regime-based GRA which integrates a zonal decomposition approach based on the extrema of velocity and temperature profiles with a regime-based approach that accounts for different temperature–velocity (T–V) relations. The new GRA is verified using compressible turbulent Couette–Poiseuille (C–P) flow, which occurs between two plane plates driven by the combination of relative moving walls and the application of a pressure gradient. Direct numerical simulations (DNS) are implemented at ${\textit{Re}}_0 = 4000$, $\textit{Ma}_0 = 0.8$ and $1.5$. Two flow regimes are recognised: one is the Couette regime (C regime), featuring opposite-direction wall frictions on the bottom and top walls, and the other is the Poiseuille regime (P regime), characterised by same-direction wall frictions. For C-regime flow, the temperature maximum point and the minimum magnitude point of the velocity gradient divide the entire channel into three zones, with each zone modelled via canonical GRA. For P-regime flow, the velocity maximum point presents a strong singularity for canonical GRA. We propose a new set of T–V relations with non-uniform distribution of the effective Prandtl number (${\textit{Pr}}_e$) rather than the typical constant-${\textit{Pr}}_e$ assumption. Comparisons with DNS results indicate that the new T–V relation improves the prediction of temperature profile in compressible C–P turbulence, particularly for the two P-regime flows with higher $\textit{Ma}_0$, where the original GRA model shows clear deviations from the DNS.