Combined Rayleigh–Taylor and Kelvin–Helmholtz instabilities on an annular liquid sheet

Abstract

This paper describes the three-dimensional destabilization characteristics of an annular liquid sheet when subjected to the combined action of Rayleigh–Taylor (RT) and Kelvin–Helmholtz (KH) instability mechanisms. The stability characteristics are studied using temporal linear stability analysis and by assuming that the fluids are incompressible, immiscible and inviscid. Surface tension is also taken into account at both the interfaces. Linearized equations governing the growth of instability amplitude have been derived. These equations involve time-varying coefficients and have been analysed using two approaches – direct numerical time integration and frozen-flow approximation. From the direct numerical time integration, we show that the time-varying coefficients evolve on a slow time scale in comparison with the amplitude growth. Therefore, we justify the use of the frozen-flow approximation and derive a closed-form dispersion relation from the appropriate governing equations and boundary conditions. The effect of flow conditions and fluid properties is investigated by introducing dimensionless numbers such as Bond number ($Bo$), inner and outer Weber numbers ($We_{i}$, $We_{o}$) and inner and outer density ratios ($Q_{i}$, $Q_{o}$). We show that four instability modes are possible – Taylor, sinuous, flute and helical. It is observed that the choice of instability mode is influenced by a combination of both $Bo$ as well as $We_{i}$ and $We_{o}$. However, the instability length scale calculated from the most unstable wavenumbers is primarily a function of $Bo$. We show a regime map in the $Bo,We_{i},We_{o}$ parameter space to identify regions where the system is susceptible to three-dimensional helical modes. Finally, we show an optimal partitioning of a given total energy ($\unicode[STIX]{x1D701}$) into acceleration-induced and shear-induced instability mechanisms in order to achieve a minimum instability length scale (${\mathcal{L}}_{m}^{\ast }$). We show that it is beneficial to introduce at least 90 % of the total energy into acceleration induced RT instability mechanism. In addition, we show that when the RT mechanism is invoked to destabilize an annular liquid sheet, ${\mathcal{L}}_{m}^{\ast }\sim \unicode[STIX]{x1D701}^{-3/5}$.

Axisymmetric Plumes in Viscous Fluids

“…Since this necessarily involves an extra spatial dimension, solution methods for such problems involve an order of magnitude extra computing power, although the underlying spectral method is not greatly more complicated than that developed here. Such computations would require a substantially more powerful machine than used here, and perhaps also a different programming environment, and it is significant that most investigations of fully three-dimensional flows of this sort are therefore limited to linearized analyses, such as the stability calculation for helical waves on a fluid jet, presented recently by Vadivukkarasan and Panchagnula [34]. We have, in fact, developed spectral approaches that can account for such three-dimensional time-dependent fluid geometry, and these will be discussed in a future article.…”

Single- and double-helix vortex breakdown as two dominant global modes in turbulent swirling jet flow

“…In contrast to round jets, stability analyses for annular swirling jets are less common. Instability-related studies focus mainly on two-phase flows where a liquid annular jet emerges in a gas environment (Vadivukkarasan & Panchagnula 2017;Matas, Delon & Cartellier 2018) or on reacting flows in combustor geometries (Terhaar, Oberleithner & Paschereit 2015). To the authors' knowledge, no studies exist for single-phase non-reacting annular swirling flows.…”

Breakup Morphology and Mechanisms of Liquid Atomization

“…However, under some conditions of prefilming atomization, the droplet size increases with the increase of gas velocity, and then decreases with the increase of gas velocity. So, the classical KH-RT atomization model [92][93][94] is modified and extended to the prefilming air-blast atomization [95].…”