Multiple asymptotic solutions for axially travelling waves in porous channels

Abstract

Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

Direct numerical simulation and biglobal stability investigations of the gaseous motion in solid rocket motors

“…As shown by Majdalani, Vyas & Flandro (2002, 2009, the effects of time-dependent propellant grain regression on the mean flow may be safely dismissed in the context of linear stability and vorticoacoustic wave analysis. The characteristic frequencies of these unsteady effects are sufficiently low compared with other flow disturbances to justify ignoring them.…”

“…In this study, only a brief overview of the relevant theories will be presented. Substantially more detailed descriptions can be found in work by Chedevergne et al (2006) and Majdalani (2009).…”

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

“…This may be owed to its association with several studies involving hydrodynamic instability [23][24][25][26][27][28], acoustic instability [29][30][31][32][33][34][35], wave propagation [36][37][38][39], particle-mean flow interactions [40], and rocket performance measurements [41][42][43]. The Taylor-Culick solution was originally verified to be an adequate representation of the expected flowfield in SRMs both numerically by Sabnis et al [44] and experimentally by Dunlap et al [45,46], thereby confirming its character in a nonreactive chamber environment.…”

Internal Flows Driven by Wall-Normal Injection

“…The instantaneous flow field plays a key role in describing acoustic instability, particle-mean flow interactions, erosive burning, nozzle erosion, and thrust performance. The traditional modus operandi is to decompose the instantaneous motion into a steady average flow and an amalgam of unsteady wave contributions (Chedevergne et al, 2007;Culick, 2006;Majdalani, 2009). In this context, the mean flow represents the bulk motion of the gases and can be approximated by the steady-state solution for a porous tube or channel with wall-normal injection.…”