Testing an analytic model for Richtmyer–Meshkov turbulent mixing widths

Abstract

We discuss a model for the evolution of the turbulent mixing width h(t) after a shock or a reshock passes through the interface between two fluids of densities ρ A and ρ B inducing a velocity jump V . In this model, the initial growth rate is independent of the surface finish or initial mixing width h 0 , but its duration t * is directly proportional to it:, α and θ are dimensionless, A-dependent parameters measured in past Rayleigh-Taylor experiments, and β is a new dimensionless parameter we introduce via t * = (h 0 / V )β. The mixing width h and its derivativeḣ remain continuous at t = t * since h * = h 0 + 2α A V t * andḣ * = 2α A V . We evaluate β ∼ 6 at A ≈ 0.7 from air/SF 6 experiments and propose that the transition at t = t * signals isotropication of turbulence. We apply this model to the recent experiments of Jacobs et al. (Shock Waves 23:407-413, 2013) on shock and reshock, and discuss briefly the third wave causing an unstable acceleration of the interface. We also consider the experiments of Weber et al. (Phys Fluids 24:074105, 2012) and argue that their smaller growth rates reflect density gradient stabilization.

Turbulent mixing induced by Richtmyer-Meshkov instability

“…Mikaelian's model [1] was used to extract the growth exponent θ from the experimental measurements of mixing layer width. Although the experiments are technically compressible, the weak shock strengths used here allow for the incompressible estimation for t * to be used.…”

“…Mikaelian's model [1] was used to extract the growth exponent θ where a wide range of values was found, indicating that the initial density distribution has a strong effect on this parameter. It is interesting to note that numerical simulations [5] yield θ values in relatively tight range 0.26 ≤ θ ≤ 0.3 with a narrow band very short wavelength initial perturbation.…”

“…Mikelian's model [1] separates RMI instability process initiated with a random initial perturbation into two stages. In the first stage it is assumed that the mixing layer width, h grows linear in time for t < t * where…”

The Growth of Richtmyer-Meshkov Instability under Multiple Impingements

“…where the value of * 1 t is 0.519 and has a difference with the Mikaelian model [16] which gives a linear growth velocity of 19.98 m/s (blue line in figure), also with the extended model [21] [22]. Then after the following interaction of the reflected rarefaction wave with the interface, the TMZ also evolves as a negative exponential law but with a different factor * 2 0.875 t = .…”

“…Assuming no dependence on the initial conditions and neglecting the molecular dissipation effects, Mikealian [16] developed a simple model by replacing the acceleration for RTI growth rate with an impulsive acceleration as 0.28 a uA = ∆  following any shock or reshock, which was also validated by some investigations [17]- [19]. This model was also extended to t * , for which it takes a ∼ t for t ≤ t * and a ∼ t θ for t ≥ t * [20] [21]. All in all, these theoretical models are developed under some assumptions and limitations, and only give an approximate predict of the RMI growth.…”