Assessment of the Effects of Azimuthal Mode Number Perturbations upon the Implosion Processes of Fluids in Cylinders

TL;DR

This study investigates how azimuthal mode number perturbations affect fluid implosion symmetry in cylindrical systems, revealing damping of high modes and stability characteristics relevant for fusion reactor designs.

Contribution

It introduces a linearized model analyzing mode perturbations in cylindrical implosions, highlighting damping effects and stability properties of oscillatory disturbances.

Findings

High mode numbers are dampened during propagation.

Highly oscillatory perturbations exhibit bounded behavior in the far field.

The model provides insights into symmetry preservation in fluid implosions.

Abstract

Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axially-symmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analyzing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analyzing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear Klein-Gordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.