Instabilities in visco-thermodiffusive swirling flows

TL;DR

This paper develops an analytical theory for linear instabilities in swirling flows considering multiple physical effects, deriving criteria that unify and extend classical instability conditions, validated through specific flow models.

Contribution

It introduces a comprehensive analytical framework capturing visco-thermodiffusive instabilities in swirling flows, extending classical criteria to more complex, realistic conditions.

Findings

Identifies two types of instability: centrifugal and visco-diffusive McIntyre.

Derives explicit instability envelopes in parameter space.

Provides unified criteria extending classical instability theories.

Abstract

An analytical theory is presented for linear, local, short-wavelength instabilities in swirling flows, in which axial shear, differential rotation, radial thermal stratification, viscosity, and thermal diffusivity are all taken into account. A geometrical optics approach is applied to the Navier-Stokes equations, coupled with the energy equation, leading to a set of amplitude transport equations. From these, a dispersion relation is derived, capturing two distinct types of instability: a stationary centrifugal instability and an oscillatory, visco-diffusive McIntyre instability. Instability regions corresponding to different axial or azimuthal wavenumbers are found to possess envelopes in the plane of physical parameters, which are explicitly determined using the discriminants of polynomials. As these envelopes are shown to bound the union of instability regions associated with particular wavenumbers, it is concluded that the envelopes correspond to curves of critical values of physical parameters, thereby providing compact, closed-form criteria for the onset of instability. The derived analytical criteria are validated for swirling flows modelled by a cylindrical, differentially rotating annulus with axial flow induced by either a sliding inner cylinder, an axial pressure gradient, or a radial temperature gradient combined with vertical gravity. These criteria unify and extend, to viscous and thermodiffusive differentially heated swirling flows, the Rayleigh criterion for centrifugally driven instabilities, the Ludwieg-Eckhoff-Leibovich-Stewartson criterion for isothermal swirling flows, and the Goldreich-Schubert-Fricke criterion for non-isothermal azimuthal flows. Additionally, they predict oscillatory modes in swirling, differentially heated, visco-diffusive flows, thereby generalising the McIntyre instability criterion to these systems.