The resolution of turbulent pressures at the wall of a boundary layer

Abstract

This paper is a discussion of the resolution by a piezo-electric transducer of the local pressure fluctuations at the wall of a turbulent boundary layer. The first part applies to transducers whose sensing area is large in comparison with the boundary layer thickness, and the second part to transducers whose sensing area is either of the order of, or smaller than, the boundary layer thickness. For large transducers, the main result, derived for a sensing surface of arbitrary shape and with an arbitrary spatial sensitivity distribution, is that under circumstances which are perhaps unrealistically stringent, the attenuation of a given frequency component will increase as the cube of the transducer face linear dimension. In general, it should be expected to increase only as the square of the dimension. In the second part, the probable dependence of the resolution on frequency and on transducer size is indicated by using both recent experimental evidence and physical reasoning bearing on the nature of the pressure cross-spectral density. Both approaches suggest that in a co-ordinate system translated downstream at an appropriate velocity, as the streamwise wave number increases, the spectral density in wave number-frequency space is associated with proportionately increasing spanwise wave numbers and almost proportionately increasing frequencies. Experimental evidence suggests departure from this finding if the fixed axis longitudinal wave length is very large (of the order of the boundary layer thickness) in which case both the characteristic time in convected axes and the characteristic lateral coherence length appear to reach asymptotic finite values. For shorter waves, the strong relationship between the two space scales and the convective time scale necessarily implies that the resolution of the frequency spectral density by a finite transducer continues to deterioriate as the frequency increases. It is concluded that the attenuation predictions previously published by the author are not likely to be seriously in error.