On the Wall Pressure and Shear Stress Spectra in Turbulent Boundary Layers

The spectra of the wall pressure fluctuations and the shear stress components ∂u/∂y and ∂v/∂y are determined via

numerical simulations of incompressible turbulent channel flow at Re_τ = 180. One observation of these

results is that the spectra are primarily  limited to small wavenumbers , e.g., the spectrum of ∂u/∂y is essentially

confined to k_x h ≦ 2 and k_z h ≦ 16, where h is the channel half-width.

In an attempt to better understand the results of the above simulations, we write the shear stress spectra in

terms of  the spectrum of ∂²v/∂y², the spectrum of ∂ω_y/∂y , and their cross spectrum, all at the wall.

It is then shown, in a manner that is very much akin to resolvent analysis, that the

latter three can be estimated using the (k_x,k_z) dependence of the dominant  eigenfunctions, i.e., those with the

smallest time-decay rates,  of the Orr-Sommerfeld/Squire system and their eigenvalues. 

In the case of pressure fluctuations, the spectra of the Stokes pressure, of the rapid part of the inertial

pressure, and of their sum are determined using the same methodology. The contribution of the slow part of the

inertial pressure, however,  requires further nonlinear analysis. 

The overall result is potentially a low-order model of  turbulence fluctuations at the wall.