On an Acoustic Oscillation Energy for Shear Flows

Abstract

The pressure perturbation and its derivative for an acoustic wave in a uniformly moving medium are canonical variables whose Hamiltonean is the acoustic energy. In the present paper is introduced a generalized acoustic Hamiltonean or oscillation energy that: (i) is conserved in a uniform flow, when it reduces to the acoustic energy; (ii) is not conserved in an homentropic unidirectional shear flow, in which case it has minimum at the critical layer where the Doppler shifted frequency vanishes. These properties are illustrated in the case of an unidirectional shear flow with an hyperbolic tangent velocity profile, representing a boundary layer over a flat rigid or impedance wall. The exact solution in the boundary layer can be matched to plane waves in the free stream, and reflection and transmission coefficients are defined for a incident wave of unit amplitude. If there is no critical layer in the boundary layer, a single powers series solutions covers the whole flow region from the free stream up to the wall; it is shown that for a rigid wall the modulus of the reflection coefficient is unity, but the argument is not zero, i.e. all sound is reflected with a phase shift. Also it is shown that, in this case, of a boundary layer without critical layer over a rigid wall, the phase of the transmission coefficient is one-half the phase the phase of the reflection coefficient, i.e. sound takes the same time to cross the boundary layer both ways, bouncing from the wall in between. In more general cases, viz. impedance wall and/or boundary layer with a critical layer, the modulus and phase of the reflection and transmission coefficients are calculated as functions of the angle of incidence, free stream Mach number and boundary layer thickness measured on a wavelength scale. When a critical layer is present in the boundary layer, the two pairs of solutions around it and the free stream must be matched, in order to obtain the acoustic field across the whole boundary layer, and calculate the reflection and transmission coefficients. The special case of critical layer at the free stream is also discussed. The cases discussed include boundary layers with and without a critical layer, subsonic and supersonic free streams with one zone of silence and one or two propagation zone(s), and rigid, inductive, reactive and mixed impedance walls. The generalized acoustic Hamiltonean or oscillation energy is also used to obtain a simple approximation for the modulus of the transmission factor, which is reasonably accurate not far from normal incidence.