Turbulent wall pressure and wall shear fluctuations calculated from the Orr-Sommerfeld equation with nonlinear forcing terms

Abstract

The wavenumber‐frequency spectral densities of turbulent wall pressure fluctuations are investigated over a rigid flat plate. Nonlinear Reynolds stress terms of the inhomogeneous Orr‐Sommerfeld equation are regarded as a known forcing function. The forcing function is modeled after Bark’s hydrodynamic bursting formulation. The inhomogeneous Orr‐Sommerfeld equation is solved by the method of Eckhaus in terms of discrete homogeneous solutions. The method of Eckhaus is then extended and proved for the continuous Orr‐Sommerfeld eigenfunctions. Turbulent wall pressure fluctuations in terms of wavenumber‐frequency spectral densities are numerically computed and compared to the experimental results of Martin as well as to his transformation of Blake’s data fitted to a modified Corcos model. The wavenumber‐frequency spectral densities numerically computed from the discrete eigenfunctions compared well with Martin’s transformations on the convective ridge, but the continuous eigenfunctions made insignificant contributions there. However, it is shown that the continuous eigenfunction contributions compare well with the low‐wavenumber, high‐frequency wavenumber‐frequency spectral density measurements of Martin.