Wall pressure spectra calculations for equilibrium boundary layers

Abstract

Assuming information about the mean velocity and vertical turbulent velocity, it is possible to calculate the flow direction wavenumber spectrum of pressure fluctuations ϕ(k₁ δ)/τ₀²δ. The law of the wall plus Cole's wake function represented the mean velocity profiles. A scale-anisotropic model of R₂₂ was used and the component intensity û₂ was assumed to vary across the boundary layer in constant proportionality to the Reynolds stress. Calculated zero-pressure-gradient spectra rise as k₁^1.5 at low wavenumbers. Curves for various Reynolds numbers are closely similar, and diverge only slightly around the peak in the spectrum. A high wavenumber spectrum ϕk₁v/u_*. u_*/τ₀²v is independent of Reynolds number. The calculations reveal an overlap region in which ϕ ∼ k₁⁻¹. Imposing an equilibrium pressure gradient increases the spectrum at the low and mid wavenumbers, but has no effect in the overlap region. The spectrum peak for II = 6 is a factor 10² higher than for the zero-pressure-gradient layer. It is proposed that the convective velocity U_c(k₁) has an overlap region. The overlap law is found to be \[ \frac{U_c}{u_{*}} = -\frac{1}{\kappa}\ln k_1\delta +\frac{1}{\kappa}\ln\frac{u_{*}\delta}{\nu}+A, \] where K and A are the same constants as in the mean velocity expression. Comparison with experiments shows very good agreement. A rough convective ‘wake’ function is formulated for the low-wavenumber range. Wavenumber spectra are converted to frequency spectra, and compared with experiments. Data from a zero pressure gradient and an adverse pressure gradient II = 3 show reasonable agreement with the calculations.