On Viscous and Compressible Cyclonic Motions in Confined Hemispherical Chambers

Abstract

This dissertation presents a theoretical investigation into the mechanics of steady-state, swirl-driven flows confined within an impermeable hemispherical chamber. A hierarchy of three distinct analytical models is developed to systematically isolate and characterize the roles of geometric confinement, weak compressibility, and viscosity. First, a generalized incompressible, inviscid baseline model is derived from the Bragg--Hawthorne equation (BHE). An eigenfunction expansion technique yields a discrete spectrum of solution modes, and the analysis characterizes the evolution of the meridional streamline topology with increasing radial and polar mode numbers. Second, the inviscid framework is extended to incorporate first-order compressibility effects using a Rayleigh--Janzen expansion in the squared injection Mach number. This analysis yields closed-form expressions for the dilatational shifts of several flowfield features, including the mantle surface where the axial velocity reverses; this study also identifies a composite parameter that captures the effects of compressibility. Third, viscous phenomena near solid boundaries are resolved using a multiple-scales asymptotic framework, providing a model for the boundary-layer structure. Collectively, these theoretical results provide a more complete physical understanding of hemispherical vortex flows and establish a set of exact solutions for the verification of computational methods.