Vorticity dynamics in isobarically closed porous channels part II: Space-reductive perturbations

In extending previous work this paper continues to address the acoustico-vortical coupling inside a porous channel of the closed-open type. The companion paper (Majdalani J., "Vorticity Dynamics in Isobarically Closed Porous Channels Part I: Standard Perturbations, " Journal of Propulsion and Power Vol. 17 No. 2) applies conventional perturbation principles to derive the temporal vorticity from the linearized vorticity transport equation. Two alternative efforts will be invested here to obtain the temporal velocity from the linearized momentum equation. These efforts rest on applying Wentzel Kramers and Brillouin (WKB) and multiple-scale expansions. The multiple-scale approach includes the innovative idea of introducing a virtually arbitrary scale that can be left unspecified during the derivation process. At the conclusion of the asymptotic analysis this unknown variable is determined. The algebraic complexity of the resulting variable justifies the reverse engineering methodology adopted in its derivation. Its complexity stems from its intrinsic function of singly representing a triple-deck structure of inner intermediate and outer length scales. This spatial scale reduction allows a conventional two-variable multiple-scale expansion to be successful. The emerging one-term formulation is conveniently short and accurate. Its simplicity enables us to obtain closed-form expressions for the velocity modulus and depth of penetration. Numerical verifications reveal that the error associated with this space-reductive perturbation solution is smaller than its precursors namely the standard perturbation solution of Paper I and the WKB solution furnished here. Most particularly the asymptotic equations are found to agree very well with independently acquired computational data. The latter are obtained from a two-dimensional Navier-Stokes solver that handles the nonlinear conservation equations.