Low-wavenumber wall pressure fluctuations in turbulent flows within concentric annular ducts

Compressible direct numerical simulations of turbulent channel flows in concentric annular ducts of height $2\delta$ are performed to study the low-wavenumber wall pressure fluctuations (WPF) over cylindrical walls at a bulk Mach number $M_b = 0.4$ and bulk Reynolds number $Re_b=3000$. The radius of the inner cylinder $R$ is varied between $0.2\delta$, $\delta$, $2\delta$ and $\infty$. As $R$ decreases, the one-point power spectral density of the WPF decreases at intermediate but increases at high frequencies. When $R$ decreases, the 1D (streamwise) wavenumber-frequency spectrum of the WPF decreases at high wavenumbers. At low wavenumbers, however, as $R$ reduces to $0.2\delta$ the 1D wavenumber-frequency spectrum exhibits multiple spectral peaks whose strengths increase with frequency. Examination of the 2D wavenumber-frequency spectra shows that these represent acoustic duct modes that closely match theoretical predictions. The acoustic modes of higher radial orders exhibit increasingly high amplitude on the inner than on the outer walls. The low-wavenumber components of the $0$th-order (azimuthal) 2D wavenumber-frequency spectrum are of great importance in practice, and their magnitude increases as $R$ reduces; this increase is increasingly pronounced at higher frequencies. Analytical modelling and numerical validation show that this increase appears to arise from the ``geometric'' effects connected with the Green's function, and they are generated mainly by radial and azimuthal disturbances. Disturbances closer to the wall are shown to be increasingly important in WPF generation as $R$ reduces, which highlights a potential in WPF control using wall treatment on thin cylinders.