Generalized Scaling Technique for the Solution of the Vortical Wave Eigenfunction Equation

Abstract

The description of the unsteady wave motion in an acoustic chamber is relevant to a multitude of aerospace applications such as rockets, gas turbines, scramjets, and preburners. Several related problems arise in the development and propagation of vorticoacoustic boundary layers when considering the effects of arbitrary injection profiles on acoustic wave motions. The reverse problems correspond to arbitrary suction profiles associated with chambers that are intended for filtration or particle separation. Two wave solutions can be developed for such flow regimes: The first being a relatively straightforward potential flow motion for the compressible, irrotational acoustic wave, while the other leads to a complex eigenfunction solution for the rotational, incompressible vortical wave. Unlike the acoustic wave equation, the eigenfunction equation does not readily lend itself to straightforward analysis. Its exact solutions are tractable only for simple base flow profiles and geometries. Moreover, the inherently stiff nature of the eigenfunction equation makes a numeric solution to the resulting problem a non-trivial task. Previous work has tackled the solution of this class of problems by solving equations that are derived for specific geometric configurations and injection conditions. The resulting approximations have relied heavily on multiple-scales and WKB approximations, as these have the ability to accommodate both linear and nonlinear scales. The present work provides a generalized formulation to the rotational wave eigenfunction equation assuming arbitrary coordinate systems and mean flow profiles. The ensuing solutions are obtained using an extension of multiple-scales theory, namely, the Generalized Scaling Technique (GST). The GST method extends the applicability of multiple scales by determining the problem's underlying scales through analysis rather than conjecture. The problem is also approximated using the WKB expansion method. In doing so, we not only confirm our GST results, but also supply sufficient evidence to re-establish the WKB method as a special case of the GST framework. Finally, our solution is validated numerically using a spectral Chebyshev collocation approach. Nomenclature a = chamber radius or half-height s a = speed of sound Re = injection Reynolds number,

Transverse Vortico-Acoustic Waves in the Presence of Strong Mean Flow Shear Layers

“…Comparisons are made to the GST formulation by Batterson and Majdalani,43 which reproduces the WKB solutions obtained by Majdalani and Flandro 24 and Zgheib and Majdalani 44 for the axisymmetric, cylindrically-shaped solid rocket motor configuration. Experimental measurements taken by Brown et al 29 serve to validate both the previous analytical solutions and the general numerical formulations shown here.…”