Smoothed particle hydrodynamics simulation of converging Richtmyer-Meshkov instability

Abstract

The Smoothed Particle Hydrodynamics (SPH) method based on the Harten-Lax-van Leer Riemann solver is improved to study converging Richtmyer-Meshkov instability (RMI). A new density summation algorithm is proposed, which greatly suppresses the pressure oscillation at the material interface. The one-dimensional Sod problem is first simulated for code verification. Then, the SPH program is extended to two dimensions to simulate the converging RMI at a square air/SF6 interface, and the numerical results compare well with the experimental ones [Si et al., "Experimental investigation of cylindrical converging shock waves interacting with a polygonal heavy gas cylinder,"J. Fluid Mech. 784, 225-251 (2015)]. Nonlinear mode coupling and pressure disturbance are found to act evidently, causing a very fast growth spike. Performing a Fourier analysis of the interface profiles, amplitude growths of the first three harmonics are obtained. The first harmonic presents an increasing growth rate at early stages due to geometric convergence. The second harmonic experiences a long period of linear growth due to the counteraction between geometric convergence and nonlinearity, whereas the third harmonic saturates very early for stronger nonlinearity. For all three harmonics, the perturbation growth rate reduces evidently at the late stage due to the Rayleigh-Taylor stabilization caused by interface deceleration. It is found that the instability growth at early stages depends heavily on the incident shock strength, while the late-stage asymptotic growth rate is nearly constant, regardless of shock strength. It is also found that intensifying the incident shock is an effective way to produce extreme thermodynamic state at the geometric center even though it causes a faster instability growth.