Resolution of Pressure in Turbulence

The finite size of a transducer-sensing element limits its space resolution of a pressure field associated with a local turbulent flow. Such pressure fields are translated at a speed comparable to the characteristic velocity of the flow. Consequently, a lack of resolution in space causes an apparent inability to resolve in time. This problem—an example of the mapping of a random function of several variables by a linear operator—is examined here. With the help of a formalism which has been previously discussed and of some recent experimental information about the spatial structure of turbulent pressure fields in boundary layers, the mapping or distortion of statistical quantities associated with the second-order moments of the pressure field is given. The attenuation of the frequency spectral density and of the cross-spectral density is given explicitly in table form and in asymptotic form. The numerical results indicate that the attenuation caused by the finite size of transducers is generally more severe than previous computations had suggested, mainly because the lateral correlation of pressure is highly frequency-dependent, a typical turbulent pressure-wave component being inclined to the stream direction at roughly 45 degrees. The results are applied to an evaluation of contemporary measurements of turbulent pressure fields in shear flows. It is shown that the transducer size used introduces undesirable large errors in these measurements, which lead to doubts even about the magnitude of the intensity of turbulent pressure fluctuations. Asymptotic formulas for the attenuation of large transducers are given which yield estimates of the degree to which a flush-mounted sonar receiver immersed in a boundary layer is able to reject the background noise provided by turbulent pressure fluctuations.