Estimation of possible excitation frequencies for shallow rectangular cavities.

Abstract

The physical processes that sustain discrete-frequency oscillations of cavities adjacent to compressible flow are modeled analytically, yielding a formula which predicts excitation frequencies as a function of Mach number and cavity geometry. These physical processes are similar to those used by Powell in describing the mechanism underlying the production of edge tones. The empirically determined constants appearing in Rossiter's formula for excitation frequencies are computed from the model. It appears that instability of the shear layer as well as interaction between the shear layer and the cavities' trailing edge is required to sustain discrete-frequency oscillation. It is suggested that the simultaneous excitation of two or more discrete frequencies (which are not harmonic), as have been observed in practice, correspond to the simultaneous participation of two or more vortex sheet displacement modes. The model analyzed yields qualitatively correct acoustic mode shapes in the cavity. Theoretical results include the calculation of possible excitation frequencies over the range 0.8 ^ M ^ 3 for rectangular cavities and show their dependence on cavity depth. Analytic results are in general agreement with experimental data. Nomenclature A = magnitude of harmonic force acting on vortex sheet at \x\ ^ e, y = 0 a = sound speed B = constant in mass-flux equation [Eq. (27)] C(u) = Fresnal integral D = cavity depth / = frequency H(u) = Hankel function i = (-l)" 2 K = complex wave number in x direction and Fourier transform variable L = cavity length M = u/ a , Mach number m = mass flux at cavity trailing edge n, m = summation indices in x, y direction, respectively P = perturbation pressure R = image-correction sum r = thermal recovery factor rj, 7 = source and observation point position vectors, respectively S = -i(tt-KU} 51 = fL/U+, exterior Strouhal number 5 2 = coL/a_, interior Strouhal number S(u) = Fresnal integral t = time U= convection velocity (note U = 0 inside the cavity) x, y = coordinate system (see Fig. 1) y = ratio of specific heats A = dispersion relation e = small physical length Y\ = shear-layer displacement K = fractional vortex convection speed A a = wavelength of acoustic wave A B = wavelength of shed vortices

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