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In this article, a long rectangular channel is considered with two transpiring walls that are a small distance apart. The channel’s head end is hermetically closed while the aft end is either open (isobaric) or acoustically closed (choked). A mean flow enters uniformly across the permeable walls, turns, and exits from the downstream end. The slightest unsteadiness in flow velocity is inevitable and occurs at random frequencies. Small pressure disturbances are thus produced. Those waves whose oscillations match the enclosure’s natural frequencies are promoted. Inception of small pressure perturbations alters the flow character and leads to a temporal field that we wish to analyze. The mean flow is of the Berman type and can be obtained from the Navier-Stokes equations over different ranges of the cross-flow Reynolds number. The unsteady component can be formulated from the linearized momentum equation. This has been carried out in numerous studies and has routinely given rise to a singular, boundary-value, double-perturbation problem in the cross-flow direction. The current study focuses on the resulting second-order differential equation that prescribes the rotational wave motion in the transverse direction. This equation exhibits unique features that define a general class of ODEs. Due to the problem’s oscillatory behavior, two general asymptotic formulations are derived, for an arbitrary mean-flow profile, using WKB and multiple-scales expansions. The fundamental asymptotic solutions reveal the same similarity parameter that controls the rotational wave character. The multiple-scales solution unravels the problem’s characteristic length scale following a unique, nonlinear variable transformation. The latter is derived rigorously from the problem’s solvability condition. The advantage of using a multiple-scales procedure lies in the ease of construction, accuracy, and added physical insight stemming from its leading-order term. For verification purposes, a specific mean-flow solution is used for which an exact solution can be derived. Comparisons between asymptotic and exact predictions are gratifying, showing an excellent agreement over a wide range of physical parameters.

Two Forms of the Rotational Solution for Wave Propagation Inside Porous Channels