On the Mach- and Reynolds-Number Dependence of the Flat-Plate Turbulent Boundary Layer Wall-Pressure Spectrum

Wall-pressure measurements are performed using an array of pinhole microphones mounted into the test wall of a low noise, low turbulence flow facility. The frequency–wave-number spectra for both streamwise and spanwise directions are obtained through the spatial Fourier transformation of cross-spectral measurements in the said directions. The results of the wind-tunnel measurements reveal an interesting behavior of the low wave-number portion of the wall-pressure spectrum. Namely, this portion of the spectrum does not exhibit any Mach-number dependence for nearly incompressible flows. A low Mach-number analysis accounting for finite Reynolds-number effects is performed. This analysis shows the wall-pressure spectrum to have a leading term independent of the Mach number. The frequency dependence of the theoretical spectrum agrees well with that of the experimental spectrum. The physical mechanism for the Mach-number independence of the spectrum is attributed to a hydrodynamic dipole contribution to the wall pressure resulting from the Stokes layer induced by the turbulent eddies. The higher differential order of the viscous equations of motion allows for the existence of such Stokes layers. They have no counterpart in invisicd theories wherein no such Mach-number independence of the spectrum has been observed. A simple model problem is devised which isolates the viscous mechanism responsible for the modification of the wall pressure and the corresponding sound field. It is argued that the physical process involved is akin (albeit inversely) to that operative in the boundary-layer receptivity problem for nearly incompressible flows. Received 1 June 1998 and accepted 25 November 1998