Turbulent-Boundary-Layer Pressure Fluctuations and Wavenumber Filtering by Nonuniform Spatial Averaging

Abstract

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/U∞≫2π, where L measures streamwise element size and U∞ denotes asymptotic flow speed. The contribution Q+(ω) to Q(ω) from the convective range of high wavenumbers k1≳ω/U∝, 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 − 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 ∝ (ωR0/U∞)−2n−3. By contrast, the reduction factor for the low-wavenumber nonconvective contribution Q−(ω), which derives mainly from K≲π/R0, varies, if P (K,ω) is independent of K for most K in this range, as (ωR0/U∝)−2. The coefficient determining the magnitude of Q− relative to Q+ remains uncertain and cannot be reliably inferred from available measurements of the point cross spectrum Γ(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.