TURBULENT-BOUNDARY-LAYER PRESSURE FLUCTUATIONS AND WAVENUMBER FILTERING BY NONUNIFORM SPATIAL AVERAGING

The area averaged pressure spectrum Q(ω) due to a turbulent boundary layer (TBL) measured by transducers with spatially varying response is treated by the wavenumber approach in the high frequency, domain ωL/Uââ2π, where L measures streamwise element size and Uâ denotes asymptotic flow speed. The contribution Q+(ω) to Q(ω) from the convective range of high wavenumbers k1âω/Uâ, in which the wavenumber spectrum P (K,ω) [where K = (k1,k3)] of TBL pressure has a high ridge, is reduced by a response that varies smoothly over the element and across its periphery. For a circularly symmetric element of radius R0 having a response function and its first n â 1 spatial derivatives continuous and vanishing at the periphery, Q+(ω) is shown, by use of asymptotic Fourier transforms and similarity variables for the TBL pressure, to be reduced relative to the point spectrum by a factor for the â (ωR0/Uâ)â2nâ3. By contrast, the reduction factor for the low wavenumber nonconvective contribution Qâ(ω), which derives mainly from Kâ2π/R0, varies, if P (K,ω) is independent of K for most K in this range, as (ωR0/Uâ)â2. The coefficient determining the magnitude of Qâ relative to Q+ remains uncertain and cannot be reliably inferred from available measurements of the point cross spectrum ⋔(ζ,ω). Rectangular elements and effects of shape and orientation are also considered. Various properties of TBL pressure depending mainly on P (K,ω) in the convective range are computed from an approximate P (K,ω) formed to be space/time isotropic in a convected frame and with similarity form characteristic of the universal inner layer. © 1969, Acoustical Society of America. All rights reserved.