Abstract

This paper gives a simple model of the pressure spectrum in wavenumber-phase velocity space for three-dimensional boundary layers. Equally as important is the definition proposed for a vector convection velocity and the method proposed for conversion of wavenumber-frequency variables into wavenumber-phase velocity variables. In three-dimensional flows one must define a phase speed that is a vector c = (c1, C3). A complete vector wave speed cannot be determined from pressure data, however, the component ck in the direction of k, ck ≡ Ω/k, can be defined and is found to be sufficient. Here k is the wave vector magnitude and α its direction. This allows the spectrum function to be well represented in the polar form Φ(k, α, ck). Moreover, for given k, a convection velocity may be defined as the point of maximum Φ(k, α, ck). Convection velocity is expressed in the form of a magnitude ck_max(k) and direction αmax(k). The three-dimensional spectrum model was produced as a generalization of Witting’s two-dimensional model. By splitting the boundary layer into two parts, the simplest generalization is formulated. Each part has its own transport velocity vector. Sample calculations are displayed as contour plots of Φ (ck, α) for fixed wavenumber k. On such plots the maximum is the convection velocity. The plots reveal an abrupt transition in the spectrum as the wavenumber increases; a result of the two layer assumption.

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