Abstract
Direct numerical simulations of two-dimensional (2D) and three-dimensional (3D), single-mode and multi-mode, incompressible immiscible Rayleigh–Taylor (RT) instabilities are performed using a phase-field approach and high-order finite-difference schemes. Various combinations of Atwood number, Reynolds number, surface tension, and initial perturbation amplitude are investigated. It is found that at high Reynolds numbers, the surface tension, if significant, could prevent the formation of Kelvin–Helmholtz type instabilities within the bubble region. A relationship is proposed for the vertical distance of the bubble and spike vs the Atwood number. The spike and bubble reaccelerate after reaching a temporary plateau due to the reduction of the friction drag as a result of the formation of the spike vortices and also the formation of a momentum jet traveling upward within the bubble region. The interface for a 3D single-mode instability grows exponentially; however, a higher Reynolds number and/or a lower Atwood number could result in a noticeably larger surface area after the initial growth. It is also shown that a 3D multi-mode RT instability initially displays an exponential interface growth rate similar to single-mode RT instabilities. Due to the collapse and merging of individual single-mode instabilities, the interface area for a multi-mode RT instability is strongly dependent to the mesh resolution after the exponential growth rate. However, the ratio of kinetic energy over released potential energy exhibits an almost steady state after the initial exponential growth, with values around 0.4, independently of the mesh resolution.
Highlights
The Rayleigh–Taylor (RT) instability develops at the perturbed interface of two fluids of different densities subjected to relative acceleration.[1–3] It is named after Rayleigh[4] and Taylor[5] who were the first to study such flow patterns using classical linear stability theories
A phase-field approach based on the Allen–Cahn formulation in conjunction with a continuous surface force (CSF) surface tension model was implemented to perform the simulations
At low and medium Atwood numbers, unless the surface tension effect is relatively significant, RT instabilities could be safely modeled without considering the surface tension force
Summary
The Rayleigh–Taylor (RT) instability develops at the perturbed interface of two fluids of different densities subjected to relative acceleration.[1–3] It is named after Rayleigh[4] and Taylor[5] who were the first to study (separately) such flow patterns using classical linear stability theories. Liang et al.[19] reported spike and bubble reaccelerations for a relatively wide Reynolds number range (10 Re 5000) based on computational studies of 3D single-mode immiscible RT instabilities with At 1⁄4 0.15. Livescu et al.[35] used direct numerical simulation (DNS) to study the evolution of 3D multi-mode RT instabilities with At 1⁄4 0:5; 0:75, and 0.9 after the gravity is set to zero or reversed They reported significant changes to some turbulence quantities when the acceleration was altered. The present study aims to contribute to the field of the incompressible immiscible RT instabilities by performing a quantitative comparative study of the formation and evolution of 2D and 3D single-mode RT instabilities with a wide range of combinations of the Atwood number (0:2 At 0:75), Reynolds number (300 Re 10 000), surface tension force, and initial perturbation amplitude. The density of the heavy fluid is chosen as the reference density q~r 1⁄4 q~h , and both fluids are assumed to have a similar viscosity (l~r 1⁄4 l~h 1⁄4 l~l ) in the present study
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