A theory is described for propagation of vortical waves across alternate rigid and compliant panels. The structure in the fluid side at the junction of panels is a highly vortical narrow viscous structure which is idealized as a wave driver. The wave driver is modelled as a ‘half source cum half sink’. The incoming wave terminates into this structure and the outgoing wave emanates from it. The model is described by half Fourier–Laplace transforms respectively for the upstream and downstream sides of the junction. The cases below cutoff and above cutoff frequencies are studied. The theory completely reproduces the direct numerical simulation results of Davies & Carpenter (J. Fluid Mech., vol. 335, 1997, p. 361). Particularly, the jumps across the junction in the kinetic energy integral, the vorticity integral and other related quantities as obtained in the work of Davies & Carpenter are completely reproduced. Also, some important new concepts emerge, notable amongst which is the concept of the pseudo group velocity.
The generic problem considered is the propagation of vortical waves across junctions between one wave-bearing medium and another. It is assumed that the eigensolutions are known for the corresponding spatially homogeneous problems. The task is how to determine the amplitudes of the reflected and transmitted waves given the amplitude of the incident wave. In general, there may be more than one incident, reflected or transmitted wave. It is shown how this sort of problem may be solved in terms of the homogeneous eigensolutions by drawing an analogy between the junction and a wave-driver. The particular illustrative problem studied is that of a Tollmien-Schlichting wave, propagating along a rigid-walled channel flow, that is incident on a section of the channel where the walls consist of compliant panels. It is shown how the wave system over the compliant panels and the amplitude of the Tollmien-Schlichting wave leaving the compliant section may be determined in terms of the incident wave. The technique developed for this problem is considered to be generic.
The problem of wave propagation in a fluid flow across the junction between the rigid and the compliantpanelsin a channelhas been studied.In vorticalTollmien-Schlichting-type waves, thejump conditions are obtainedby the half-Fouriertransforms defined on both the sides of thejunction along withthe adjointmethod. The methoddevelopedis fairlygenericand is applicableto similarproblems. A comparison of the results obtained in the present study with those obtained from direct numerical simulations' shows good agreement.
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