Linear theory of Richtmyer–Meshkov like flows

Abstract

The hydrodynamic flow generated by rippled shocks and rarefactions (Richtmyer-Meshkov like flows) is presented. When a corrugated shock travels inside an homogeneous fluid, it leaves pressure, density and velocity perturbations in the compressed fluid. The velocity perturbations generated in the composed fluid are inherently rotational. Vorticity is an important quantity in order to determine the asymptotic rate of growth in the linear stage. The size of the strongest vortices generated by the rippled shocks is analyzed as a function of the shock Mach number for different boundary conditions downstream. Comparison to experiments and simulations is provided for the RMI in the shock and rarefaction reflected cases and the validity of the growth law ψ δ + ∞ ∞ v t i is emphasized.

Kinetic energy of the rotational flow behind an isolated rippled shock wave

“…It is useful to study the behavior of the the characteristic times τ a , τ c , and τ e in figure 2, which locate the first relative minima and maxima of the shock pressure perturbation. The scalings of the the first three zero crossings of ps , represented by τ b , τ d , and τ f are discussed in [58]. The products (D−U)τ a , etc will be seen later to well approximate the characteristic lengths of the kinetic energy distribution.…”

“…The approach used to calculate the times τ a , τ c , and τ e is the same as that used by Euler [61] to obtain the first zeros of the ordinary Bessel functions. This approach has been successfully applied to calculate the characteristic times τ b , τ d , and τ f , corresponding to the first three zeros of the function ˜( ) p r s s [58]. In our case here, we are interested in the first three zeros of its derivative p r d d s s .…”

“…The fact that vorticity is generated behind rippled shock fronts and that the corresponding rotational velocity field contributes significantly to the energy budget of the perturbed flow has been well known since the publication of old shock-turbulence interaction studies [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. Its importance in RMI flows was realized later [9,10,[51][52][53][54][55][56][57][58]. Due to the rotational character of the velocity fields generated behind corrugated shock fronts, there is a significant amount of kinetic energy stored inside the vortical structures, which, so far, has not been quantified for arbitrary shock strength and/or arbitrary values of fluid compressibility.…”

“…In order to satisfy the conservation of mass, normal and tangential linear momentum, and energy across the rippled shock surface, the shock surface will show damped shape oscillations with a characteristic time t c . The product º l D t v c would then determine the longitudinal size of the vortices generated downstream, as analyzed in [58]. We consider an isolated shock, for which the interaction with the piston that generated it can be neglected.…”

Successive Picket Drive for Mitigating the Ablative Richtmyer–Meshkov Instability

“…[30] Vorticity is considered as important quantity to determine the RMI growth rate in the ear stage. When the shock is weaker, the stored vorticity intensity is lower, [31] and then the RMI growth decreases. Hence, the 𝜂 cl a term grows slowly due to the relatively weak shock.…”