Energy transport characteristics of converging Richtmyer–Meshkov instability

Yaowei Fu,Changping Yu,Xinliang Li

doi:10.1063/5.0022280

Abstract

In this paper, the Richtmyer–Meshkov (RM) instability in spherical and cylindrical converging geometries with a Mach number of about 1.5 is investigated by using the direct numerical simulation method. The heavy fluid is sulfur hexafluoride, and the light fluid is nitrogen. The shock wave converges from the heavy fluid into the light fluid. The main focus is on the energy transport characteristics in the mixing layer during the entire development process from early instability to late-time turbulent mixing. First, the turbulence kinetic energy transport equation is analyzed, and it is found that the production and dissipation mechanisms of the turbulence induced by the spherical and cylindrical converging RM instabilities in the mixing layer are the same. The turbulent diffusion terms are crucial in the whole development processes of the mixing layers. Before the reflected shock waves transit the interfaces, the dissipation terms can be ignored relative to other terms, and after that, the dissipation terms are close to the production terms and play an important role. The compressibility terms are approximate to the production terms and promote the production of turbulence kinetic energy in the later stage. The viscous diffusion terms can be ignored throughout the process. Then, the enstrophy transport equation is researched, and it is found that, in the mixing layers, the baroclinicity terms play a leading role in the early stage, while the vortex stretching terms play a leading role in the later stage, and the vortex stretching term of the spherical converging geometry develops faster than that of the cylindrical converging geometry. The compressibility terms are positive in the early stage, which promote the production of enstrophy. After the reflected shock waves transit the interfaces, the compressibility terms become negative, which inhibit the production of enstrophy. In addition, the results of the present direct numerical simulation also show that the density fluctuation spectra in the centers of the mixing layers of the spherical and cylindrical converging RM instabilities present the obvious −5/3 scaling law.

Highlights

When the shock wave interacts with the material interface with initial disturbance, because the direction of the pressure gradient and the direction of the density gradient are not collinear, ∇P ⋅ ∇ρ ≠ 0, baroclinic vorticity is deposited on the disturbed interface and induces the Richtmyer–Meshkov (RM) instability
After the reflected shock waves pass through the material interfaces at t = 0.04 ms, large-scale bubbles are quickly broken by scitation.org/journal/adv the reflected shock waves, and the flow fields are dominated by the small-scale structures
When the reflected shock waves pass through the interfaces, i.e., t = 0.04 ms, the vortex stretching term of the spherical converging RM instability has been far larger than the baroclinicity term

Summary

INTRODUCTION

When the shock wave interacts with the material interface with initial disturbance, because the direction of the pressure gradient and the direction of the density gradient are not collinear, ∇P ⋅ ∇ρ ≠ 0, baroclinic vorticity is deposited on the disturbed interface and induces the Richtmyer–Meshkov (RM) instability. It can be seen that previous studies on the RM instability, whether experimental, theoretical, or numerical, have mostly focused on the relatively simple case of plane shock interacting with the disturbed interface. The influence of the BP effect on the growth of disturbance amplitude of the cylindrical converging RM instability was studied experimentally by Vandenboomgaerde’s17 shock tube experiment. Based on the formal perturbation expansion and potential flow theory, Liu proposed a simple method to study the influence of nonlinear effects on the development of the cylindrical converging RM instability. The turbulence kinetic energy transport equation and enstrophy transport equation in the mixing layer of the spherical and cylindrical converging RM instabilities are analyzed and compared in detail

COMPUTATIONAL SETUP

RESULTS AND DISCUSSIONS

CONCLUSIONS