Characterisation of rough-wall drag in compressible turbulent boundary layers

TL;DR

This study investigates how roughness drag in compressible turbulent boundary layers can be characterized across different Mach and Reynolds numbers, revealing empirical scalings that improve the understanding of roughness effects in such flows.

Contribution

It introduces empirical scalings and correction factors to better relate roughness parameters from incompressible to compressible turbulent boundary layers.

Findings

$ riangle U^+$ is insensitive to velocity transformation.

Mach-number-dependent shift observed in the fully rough regime.

Correction factor $ ext{sqrt}(1/F_c)$ improves data consistency.

Abstract

In compressible turbulent boundary layers (TBLs), roughness drag is typically characterised by first applying a velocity transformation to account for compressibility, after which the momentum deficit $\Delta U^+$ (Hama, 1954) and the equivalent sand-grain roughness $k_s$ are inferred. In practice, $k_s$ is often obtained from measurements at a single Mach number $M$ and Reynolds number $Re$, effectively forcing the roughness into the $\Delta U^+$--$\log(k_s)$ relation of Nikuradse (1933). This raises a key question: if a rough surface has a known $k_s$ in incompressible flow, under what conditions can this value be used in compressible flows? This question is explored using data obtained through a series of experiments of TBLs on rough walls (P60- and P24-grit sandpapers) over $0.3 \leq M \leq 2.9$ and $7427 \leq Re_{\tau} \leq 30292$, including independent variation of $Re_{\tau}$ at $M=2$. Results show that $\Delta U^+$ is largely insensitive to the velocity transformation, but the fully rough regime exhibits a Mach-number-dependent shift in the logarithmic relation. Three empirical scalings are examined: an equivalent incompressible $k_s$, a viscosity-scaled roughness $k_{*} = k/\nu_\infty^+$ with $\nu_\infty^+ = \nu_\infty/\nu_w$, and a correction factor $\sqrt{1/F_c}$ where $F_c$ depends on $T_\infty/T_w$. The last provides the most consistent improvement across datasets, although all corrections remain empirical and rely on smooth-wall compressibility transformations. This paves the way for future work to develop custom transformation for a rough-wall TBL that can account for roughness properties and other parameters including wall conditions.