Fully rough wall-pressure variance growth rate

Determine whether the inner-scaled wall-pressure variance beneath a fully rough turbulent boundary layer grows with the geometric range parameter \(\eta_s=\ln(\delta/k_s)\) at the predicted smooth-wall coefficient \(A_{Q,\eta}^{\mathrm{base}}\approx 2.3\), or instead at a rate differing by a factor of two.

Background

The paper predicts that, once the wall is fully rough, the nonlinear pressure source acts over the logarithmic wall-normal interval from the roughness height $k_s$ to the boundary-layer thickness $\delta$. Consequently, the integrated inner-scaled wall-pressure variance should grow as $A_{Q,\eta}^{\mathrm{base}}\ln(\delta/k_s)$, with the coefficient inherited from the smooth-wall value of approximately 2.3. Roughness-dependent effects are predicted to contribute only a non-growing offset.

The fourteen Virginia Tech cases span only a narrow range in $\eta_s$, and the fitted free slope has a broad uncertainty interval. A controlled sweep of $\delta^+$ at fixed $k_s^+$, with pressure gradient controlled, is needed to establish the growth rate directly.