Vorticoacoustic waves in axisymmetric porous tubes with arbitrary suction

This work focuses on developing asymptotic approximations for the axially standing waves in porous tubes with arbitrary levels of wall suction. The analysis starts by linearly decomposing the Navier–Stokes equations into steady and time-dependent components. The steady part is solved using Berman's similarity transformation, which gives rise to an irrotational mean flow profile, or by solving the Bragg–Hawthorne equation to arrive at an inviscid rotational solution. Using the Helmholtz theorem, the unsteady field is further decomposed into a compressible, irrotational acoustic part and a solenoidal, rotational vortical part. The two fields remain coupled only through the wall boundary condition. Procedurally, the rotational field gives rise to a doubly perturbed eigenfunction that can be treated using multiple-scales, Wentzel–Kramers–Brillouin (WKB), and generalized-scales method (GSM). The ensuing perturbation analysis is undertaken assuming a Strouhal number of the same order as the crossflow Reynolds number, R. The analysis begins with a multiple-scales expansion in 1/R assuming a linear coordinate transformation that captures the near-wall viscous effects, thus helping to unravel the inner scaling structure. The character of the solution is found to be consistent with that of a weakly under-damped wave that is strongly influenced by the mean flow. WKB and GSM solutions are also constructed to the extent of providing additional means to verify the results while capturing the problem's scaling nonlinearities. These asymptotic solutions are formulated and characterized for two mean flow profiles and compared against exact and numerical solutions. Finally, the Stokes layer thickness and Richardson annular effect are evaluated and discussed.