Weakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries

Abstract

The relationship between the weakly nonlinear (WN) solutions of the Rayleigh-Taylor instability in spherical geometry [Zhang et al., Phys. Plasmas 24, 062703 (2017)] and those in planar geometry [Wang et al., Phys. Plasmas 19, 112706 (2012)] is analyzed. In the high-mode perturbation limit (Pn(cos θ), n≫1), it is found that at the equator, the contributions of mode P2n along with its neighboring modes, mode P3n along with its neighboring modes, and mode Pn at the third order along with its neighboring modes are equal to those of the second harmonic, the third harmonic, and the third-order feedback to the fundamental mode, respectively, in the planar case with a perturbation of the same wave vector and amplitude as those at the equator. The trends of WN results in spherical geometry towards the corresponding planar counterparts are found, and the convergence behaviors of the neighboring modes of Pn, P2n, and P3n are analyzed. Moreover, the spectra generated from the high-mode perturbations in the WN regime are provided. For low-mode perturbations, it is found that the fundamental modes saturate at larger amplitudes than the planar result. The geometry effect makes the bubbles at or near the equator grow faster than the bubbles in planar geometry in the WN regime.

Plasma Waves and Rayleigh–Taylor Instability: Theory and Application

“…Zhou and Cabot utilized three big datasets with a goal of determining degree of departure, using direct numerical simulation with different Atwood numbers at moderate Reynolds number, of homogeneous and isotopic turbulence [100] Luttwak used staggered Mesh Godunov scheme to study RTI [104]. Zhang et al compare the RTI in planar geometry and spherical geometry using the third-order WN solutions for RTI in their research [105]. Zhang et al using a model established the multi-mode incompressible RTI for spherical geometry in 2D and derived the solutions within the second order.…”

Interface coupling effects of weakly nonlinear Rayleigh–Taylor instability with double interfaces*

“…[4] Generally, it will happen when a heavy fluid is accelerated by a light one, if some perturbations are on the interface between two fluid layers. Focusing on this kind of RTI, many studies have been done in its linear, [1,2] nonlinear, [5,6] and turbulence mixing [7,8] regimes. However, in some practical applications, the RTI with multiple material interfaces is more attractive.…”

Simulation of the Weakly Nonlinear Rayleigh-Taylor Instability in Spherical Geometry*

“…5, for moderate Atwood numbers, NSA decreases with 𝑛. NSA is larger than the corresponding planar NSA when 𝑛 is small, and is lower than the planar NSA when 𝑛 is large. [14] However, one shall expect the spherical NSA to approach the planar NSA when 𝑛 is sufficiently large. Here we will show this is indeed the case.…”