Vorticity Dynamics in Isobarically Closed Porous Channels Part II: Space-Reductive Perturbations

Abstract

In extending previous work thispaper continues to addresstheacoustico-vortical coupling insidea porouschanneloftheclosed-opentype.Thecompanionpaper (Majdalani,J.,“ VorticityDynamicsinIsobaricallyClosedPorous Channels Part I: Standard Perturbations,” Journal of Propulsion and Power , Vol. 17, No. 2 ) applies conventional perturbationprinciples toderivethetemporalvorticity from thelinearizedvorticity transportequation.Two alternative efforts will be invested here to obtain the temporal velocity from the linearized momentum equation. These effortsreston applying Wentzel,Kramers, and Brillouin (WKB)and multiple-scaleexpansions.Themultiple-scale approach includes the innovative idea of introducing a virtually arbitrary scale that can be left unspecie ed during the derivation process. At the conclusion of the asymptotic analysis, this unknown variable is determined. The algebraiccomplexity oftheresulting variablejustie esthereverseengineering methodology adoptedinitsderivation. Its complexity stems from itsintrinsicfunction ofsingly representing a triple-deck structureofinner,intermediate, and outerlength scales. This spatial scale reduction allows a conventional two-variablemultiple-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 verie cations reveal that the error associated with this space-reductive perturbation solution is smaller than its precursors, namely, the standard perturbation solution ofPaper Iand the WKB solution furnished here. Mostparticularly, theasymptotic 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.