A source-term interpretation of turbulent rough wall-pressure spectra

Abstract: Wall-pressure fluctuations beneath a turbulent boundary layer drive the flow-induced noise and structural loading of rough surfaces. On a smooth wall their variance grows with the logarithm of the Reynolds number, a growth carried, in a source-term reading, by the nonlinear turbulence--turbulence part of the pressure source acting across the logarithmic layer. We ask what sets the same growth once the wall is fully rough. Standard rough-wall phenomenology answers it: above the roughness the mean flow keeps its smooth-wall logarithmic form, so the active source range is cut off at the roughness height rather than the viscous length, and the variance grows on the logarithmic span between the roughness height and the layer thickness in place of the Reynolds number. Splitting the pressure source into its two physical parts then distinguishes separate contributions to the energy: a roughness-local one from the mean-shear source, fixed at high frequency on the roughness scale, and an energetic one from the nonlinear source at the outer scale, which alone carries the growth. Calibrated against rough-wall cases, the model collapses the spectral shape onto this energetic peak and holds it Reynolds-independent at fixed geometry. The canopy contribution decays with distance from the wall and contributes a finite offset. The growth coefficient is predicted to be the smooth-wall one. The present range is too narrow to discriminate that rate from twice or half it, so the contribution is a parameter-free prediction and the identification of a suitable roughness-size sweep that would test it.