Knowledge of the wavevector/frequency spectrum of wall pressure, P(K,ω) [K = (k1,k3)], for a normal turbulent boundary layer has been largely confined to properties depending on the mean-convective ridge (k1=ω/uc). Recent theoretical work yields the wavevector dependence of P(K,ω), for flow at low Mach numbers, also in the acoustic wave number domain where K ≤ ω/c, except for undetermined functions of ωδ/U∞. In the nonconvective but incompressive domain of wavevectors (important in underwater acoustics), apart from the proportionality to K2 where ω/c ≤ K ≤ δ-1, the scaling, dependence and magnitude of P(K,ω) remain to be established.
This domain is approached here by theoretical modeling of the velocity-derivative sources of pressure. The expression for the pressure spectrum derived from the pertinent Poisson equation is cast so that source models may be formulated as spectra in frequency and three-component wavevector, and inhomogeneity normal to the wall treated via dependence of source strength, correlation scales, and mean convection velocity on geometric mean wall distance. A model for the frequency dependence is formulated on the notion of fluctuating local convection. Convection of a frozen wave pattern of the turbulent velocity-product field generates a disturbance in this velocity product, and hence in wall pressure, at frequency ω even if the streamwise wavenumber component of the convected pattern is much smaller than the minimum mean convective wavenumber, ω/U∞. Such generation occurs by virtue of wavenumber components normal to the wall on the order of the ratio of frequency to probable normal convection velocities. The effective rms normal convection velocity is argued to be of the order of the local rms normal turbulence velocity. (This local-convection model for pressure differs essentially from one based jointly on assumption of a space-time quasinormal velocity distribution and application of the local-convection model to two-component velocity spectra.) The model yields the source wavevector/frequency spectrum in terms of the pure wavevector spectrum. A wavevector spectrum constructed to accord with Kronauer-Morrison wave structure yields, in the nonconvective domain where ω-uck1/v*K > > 1 (v* = friction velocity) but K δ > > 1:
P(K, ω) = a'B(k 1/K)o2v*7K(ω-u ck1)-4
for Ω < < 1, where a' is a constant, B(k1/K) an uncertain anisotropy factor, uc a convection velocity, and Ω = 5(ω-uck1)v/v* a viscous-sublayer parameter; for Ω > > 1, an exponential cutoff is predicted. This sharp cutoff is characteristic of the local-convection model with a normal distribution of convection velocity. An alternative source wavevector spectrum yields form (A) with an additional factor v* K(ω-uck1)-1. A recent wind-tunnel measurement is interpreted to provide an upper limit on a' in either case.
Application of the model to the mean-convective domain suggests isotropy of the pertinent Kronauer-Morrison wave strength and hence an angular dependence of P(K,ω) as ctcm(k1/K)2 where ct, cm derive respectively from pure-turbulence and mean-shear source terms and ctcm is comparable with or somewhat less than unity.
Chase, D. M., "Wavevector/Frequency Spectrum of Turbulent-Boundary-Layer Pressure" (1971). Symposia on Turbulence in Liquids. 80.
Symposium on Turbulence in Liquids (1971: Oct. 4-6, Rolla, MO)
Chemical and Biochemical Engineering
Article - Conference proceedings
© 1972 University of Missouri--Rolla, All rights reserved.