Formula for growth rate of mixing width applied to Richtmyer-Meshkov instability

The mixing zone width and its growth rate are of great significance in the study of the Richtmyer-Meshkov instability (RMI). In this paper, a formula for the growth rate of the mixing width is proposed for analysis of the RMI-induced mixing process. A new definition of the mixing width h˙, based on the mass fraction φ, is used to derive the formula of the growth rate of the mixing width, h˙. In the derivation, the velocity field and the diffusion term are concisely introduced into the formula by using the mass equation and mass fraction equation. This formula is used together with two-dimensional (2D) and three-dimensional (3D) numerical data to quantitatively study the effects of compressibility and the diffusion process on the development of the RMI. The results based on our simulations show the following. After a shock, the magnitudes of the contributions of compressibility and diffusion to h˙ increase initially, and in the middle stage of the RMI, they appear to attain a maximum value, around 10%; however, under some conditions (e.g., absolute value of Atwood number ~0.9), this value can be more than 10%. The results indicate that compressibility and the diffusion process become important in the later stages of the RMI and the neglect of these physical processes is not always suitable. This study shows that the derived formula is not only an approach for modeling of the mixing zone width but also a quantitative tool for the study of an RMI-induced mixing process. This formula is expected to be useful in the analysis of turbulent mixing in the later stages of the RMI process.