On the Reynolds analogy for high-speed rough-wall flows: implications for wall modelling

TL;DR

This work addresses the challenge of predicting heat transfer in high-speed compressible flows over rough walls by testing the generalized Reynolds analogy (GRA) with DNS data for prism-shaped roughness at $M_ opinfty=2$ and $M_ opinfty=4$. It demonstrates that roughness breaks near-wall momentum–energy coupling, but outer-layer similarity is restored and the GRA remains valid when wall-flux terms are incorporated, with outer profiles collapsible via the $u$-transformation $u_ ext{VD}$ that accounts for density variations through the van1951 model. Building on this, the authors formulate a GRA-based wall model that couples heat-transfer predictions to a drag-prediction framework for prism roughness using an exponential-sheltering layer and a density-aware logarithmic law, with a matching location guiding the integration. A priori tests against the DNS data show strong accuracy for wall shear stress and heat flux, particularly at $M_ opinfty=2$ (errors within a few percent), and robust heat-transfer predictions at $M_ opinfty=4$—outperforming traditional correlations such as Hill’s. The study offers a physically grounded pathway to WMLES of rough-wall, high-speed flows with limited tunable parameters and highlights directions for extending the approach to other roughness geometries and flow conditions.

Abstract

We study the validity of the generalized Reynolds analogy (GRA) in compressible turbulent boundary layers over prism-shaped roughness by mining direct numerical simulation data of Mach 2 and Mach 4 compressible turbulent boundary layers with adiabatic and cooled surfaces. Although the direct influence of roughness strongly disrupts the near-wall coupling between momentum and energy, we show that this breakdown is confined to the roughness sublayer. Above this layer, the enthalpy and velocity fields recover a smooth-wall-like similarity, and the GRA becomes asymptotically valid by naturally accounting for roughness-enhanced wall shear stress and heat flux. Building on these results, we propose a GRA-based wall model for predicting heat transfer over rough surfaces, which is coupled with a drag-predictive physics-based method developed for prism-shaped roughness by means of compressibility transformations.

Figures (3)

• Figure 1: Profiles of the total enthalpy to velocity ratio $(\bar{H}_g-\bar{H}_w)/\bar{u}$ compared to their respective velocity constant scale $U_w$, as per Eq. \ref{['eq zhang enthalpy']}, as function of $y/\delta_{99}$ Here, $U_w=-Pr ( q_w/\tau_w)$, and for adiabatic cases $U_w=0$. The markers indicate the roughness height $k/\delta_{99}$ for each case.
• Figure 2: Comparison between the mean velocity profile $u^+$ as function of $y^+$ before (a) and after (b) applying the transformation of van1951turbulent. Rough-wall cases are shifted in the wall-normal direction by the virtual origin $d$. Grey lines represent the log law $u^+=(1/\kappa) \ ln(y^+)+5.2$. The subsonic case M03 from cogo2025a, which has the same roughness pattern, is included for reference.
• Figure 3: Mean velocity (a,c) and temperature (b,d) profiles as function of the wall normal coordinate $y^+$ and $y/\delta_{99}$, respectively. Panels (a,b) report cases at $M_\infty=2$, while Panels (c,d) at $M_\infty=4$. Each figure shows two wall temperature conditions: adiabatic ($\varTheta\approx1$) and cold wall ($\varTheta=0.25$). For the velocity profiles, adiabatic cases (M2A and M4A) are manually shifted upwards by $\varDelta u^+=5$ (left axis) in order to distinguish them from cold wall cases (right axis). The matching location for the model is located at $y^+=300$.