Dependence on roughness Reynolds number in the fully rough regime

Determine whether the fully rough wall-pressure spectrum and its variance growth are independent of the roughness Reynolds number \(k_s^+\) when \(\delta/k_s\) is held fixed.

Background

The source-term framework predicts that, in the fully rough regime, the viscous length is buried within the roughness sublayer and $k_s^+$ loses independent dynamical significance. The active nonlinear-source range should therefore depend on $\delta$ and $k_s$ through $\eta_s=\ln(\delta/k_s)$, rather than through $k_s^+$.

The two speed families in the Virginia Tech campaign provide a partial comparison at different $k_s^+$ values and overlapping $\eta_s$, but the data cannot separate genuine residual roughness-Reynolds-number effects from non-equilibrium development, calibration bias, and experimental scatter. A matched-$\delta/k_s$ sweep over roughness sizes or speeds is required.