Estimation of the Wavevector-Frequency Spectrum of Turbulent Boundary Layer Wall Pressure by Multiple Linear Regression

Abstract

This paper examines the wavevector-frequency spectrum of the turbulent boundary layer wall pressure in the incompressive, inviscid domain in the intermediate and high frequencies range, i.e, ωδ*/U ∞ >> 0.5. It is shown that the wavevector-frequency spectrum can be normalized by a factor so that it becomes simply a function of nondimensional Strouhal wavenumber Uc k1 /ω and Uc k3 /ω, where Uc is the convective flow velocity, and k1 and k3 are the wavenumbers in the plane of the wall along the streamwise and the crossflow directions, respectively. The normalization factor is the point pressure frequency spectrum times (Uc /ω)2 . It follows that the normalized wavevector-frequency spectrum can be scaled with respect to the Strouhal wavenumbers Uc k1 /ω and Uc k3 /ω. The rationale of using a linear regression model for estimating the normalized wavevector-frequency spectrum with a set of measured response data from a wavevector filter is presented. The contention is that the actual spectrum can be obtained by the multiplication of a trial spectrum with a correction spectrum. The correction spectrum is approximated by a polynomial in Uc k1 /ω with a set of coefficients to be determined. The multiple linear regression model relates the response of a measuring system to these coefficients which are determined by least square minimization of a set of measured response data. The advantages of the regression approach are that it relaxes the requirements of the wavevector filter’s ability to discriminate against the spectral elements outside the wavenumber bandwidth of the filter, and this approach is capable of better estimating the entire wavevector spectrum as compared to the existing methods which are limited to measurements of the low-wavenumber spectra. Some preliminary numerical results are presented.