Compressible turbulent boundary layers over two-dimensional square-rib roughness

TL;DR

This study uses direct numerical simulations to analyze how surface roughness and wall cooling affect compressible turbulent boundary layers, proposing new methods to improve velocity and temperature predictions.

Contribution

It introduces a fitting-based optimization for the virtual origin, a modified rough-wall GRA, and a refined strong Reynolds analogy for better modeling of roughness and cooling effects.

Findings

The Griffin–Fu–Moin transformation outperforms classical methods in outer-layer velocity similarity.

The classical Reynolds analogy breaks down due to thermal and momentum disparities over rough surfaces.

The modified rough-wall GRA accurately predicts temperature-velocity relations despite near-wall heterogeneity.

Abstract

Direct numerical simulations are performed to investigate the combined effects of surface roughness and wall heat transfer on spatially developing compressible turbulent boundary layers at $Ma=2.5$. The roughness consists of transverse square bars with $\lambda_x/k=8$ and $k^+ \approx 35$, under adiabatic and wall-cooling ($T_w/T_r = 0.5$) conditions. Dynamically, the conventional zero-moment method fails to yield a consistent zero-plane displacement for the present cavity-type roughness. Instead, a fitting-based optimization procedure is proposed to determine the kinematic virtual origin, which successfully restores the logarithmic behavior. Based on this displacement, Griffin--Fu--Moin (GFM) transformation outperforms the classical van Driest transformation in recovering outer-layer similarity for the velocity defect. Thermodynamically, the physical disparity between momentum form drag and the absence of a corresponding heat transfer mechanism disrupts the classical Reynolds analogy. The effective turbulent Prandtl number ($Pr_e$) deviates severely from unity within the roughness sublayer, leading to the breakdown of the classical Generalized Reynolds Analogy (GRA). To address this, a modified rough-wall GRA (rGRA) is formulated by introducing an equivalent slip-plane or reference-point boundary conditions, which accurately reconstructs the temperature-velocity relationship by bypassing the near-wall thermal heterogeneity. Finally, the refined strong Reynolds analogy (RSRA) is shown to maintain predictive accuracy for fluctuation intensities in the outer layer despite near-wall modulation by roughness and cooling.