Multiple-Scales Solution to the Acoustic Boundary Layer in Solid Rocket Motors

Abstract

The acoustic boundary-layer structure is investigated in a cylindrical tube where steady sidewall injection is imposed upon an oscillatory e ow. Culick’ s steady, rotational, and inviscid solution is assumed for the mean e ow. The time-dependent velocity is obtained by superimposing the acoustic (compressible, inviscid, irrotational ) and the vortical (incompressible, viscous, rotational ) velocity vectors. A multiplescales perturbation technique that utilizes proper scaling coordinates is applied to the axial momentum equation by retaining the viscous terms and ignoring the axial convection of vorticity. A closed-form expression for the time-dependent axial velocity is derived that agrees well with the corresponding numerical solution, cold-e ow experimental data, and Flandro’ s near-wall analytic expression. A similarity parameter that controls the thickness of the rotational region is identie ed. The role of the Strouhal number in controlling the wavelength of rotational waves is established. An accurate assessment of the amplitude and phase relation between unsteady velocity and pressure components is obtained. Increasing viscosity is found to reduce the depth of penetration of the rotational region.