Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification

Abstract

A vortex placed at a density interface winds it into an ever-tighter spiral. We show that this results in a combination of a centrifugal Rayleigh–Taylor (CRT) instability and a spiral Kelvin–Helmholtz (SKH) type of instability. The SKH instability arises because the density interface is not exactly circular, and dominates at large times. Our analytical study of an inviscid idealized problem illustrates the origin and nature of the instabilities. In particular, the SKH is shown to grow slightly faster than exponentially. The predicted form lends itself for checking by a large computation. From a viscous stability analysis using a finite-cored vortex, it is found that the dominant azimuthal wavenumber is smaller for lower Reynolds number. At higher Reynolds numbers, disturbances subject to the combined CRT and SKH instabilities grow rapidly, on the inertial time scale, while the flow stabilizes at low Reynolds numbers. Our direct numerical simulations are in good agreement with these studies in the initial stages, after which nonlinearities take over. At Atwood numbers of 0.1 or more, and a Reynolds number of 6000 or greater, both stability analysis and simulations show a rapid destabilization. The result is an erosion of the core, and breakdown into a turbulence-like state. In studies at low Atwood numbers, the effect of density on the inertial terms is often ignored, and the density field behaves like a passive scalar in the absence of gravity. The present study shows that such treatment is unjustified in the vicinity of a vortex, even for small changes in density when the density stratification is across a thin layer. The study would have relevance to any high-Péclet-number flow where a vortex is in the vicinity of a density-stratified interface.

Stability of a vortex in radial density stratification: role of wave interactions

“…This led Fabre et al (2006) to conclude that there are no quasi-modes for a Gaussian vortex in the 2D limit. We extend the viscous stability study to other values of n, with the governing equations and method described in Dixit & Govindarajan (2010). The results at Re = 10 5 shown in figure 15 indicate that there are significant vorticity perturbations inside the core for n > 2, suggesting again that quasi-modes exist for steeper profiles.…”

“…These are the standard equations for an incompressible fluid where density varies across the flow (see e.g. (Birman, Martin & Meiburg 2005) and (Dixit & Govindarajan 2010)). We use a unit Schmidt number, so Re = P e = Ξ/ν.…”

“…For a Gaussian vortex however, shear and vorticity gradient co-exist everywhere, and this clouds the interaction between the two. In our recent work on a more generic density stratification (Dixit & Govindarajan 2010), we obtained one instance of a Rankine vortex surrounded by lighter fluid immediately outside the core, where a configuration which is unstable in terms of density stratification is rendered neutrally stable. A desire to understand this result motivated the present study.…”

Dynamics of circular arrangements of vorticity in two dimensions

“…where the operator D ≡ d/dr, angular velocity Ω = U θ /r, and base flow vorticity Z = U θ + U θ /r. Primes denote derivative with respect to r. The linear stability equations (12,13,14) are the same as equation (3.1) in [32], written for constant density. The boundary conditions depend on the mean profiles, which obey U θ ∼ r as r → 0, U θ ∼ 1/r as r → ∞.…”

Radial Injector Mixing Effects on Detonation Zone Position in Rotating Detonation Engine

“…However, this paper utilizes the superposition of potential flow models in the development of the DNS solutions [58]. Potential flow model assumptions assume irrotational flow, thus ignoring dominant features of the late stage vorticity field [59].…”