When a planar shock hits a corrugated contact surface between two fluids, hydrodynamic perturbations are generated in both fluids that result in asymptotic normal and tangential velocity perturbations in the linear stage, the so called Richtmyer-Meshkov instability. In this work, explicit and exact analytical expansions of the asymptotic normal velocity (δv_{i}^{∞}) are presented for the general case in which a shock is reflected back. The expansions are derived from the conservation equations and take into account the whole perturbation history between the transmitted and reflected fronts. The important physical limits of weak and strong shocks and the high/low preshock density ratio at the contact surface are shown. An approximate expression for the normal velocity, valid even for high compression regimes, is given. A comparison with recent experimental data is done. The contact surface ripple growth is studied during the linear phase showing good agreement between theory and experiments done in a wide range of incident shock Mach numbers and preshock density ratios, for the cases in which the initial ripple amplitude is small enough. In particular, it is shown that in the linear asymptotic phase, the contact surface ripple (ψ_{i}) grows as ψ_{∞}+δv_{i}^{∞}t, where ψ_{∞} is an asymptotic ordinate different from the postshock ripple amplitude at t=0+. This work is a continuation of the calculations of F. Cobos Campos and J. G. Wouchuk, [Phys. Rev. E 90, 053007 (2014)PLEEE81539-375510.1103/PhysRevE.90.053007] for a single shock moving into one fluid.

The Richtmyer-Meshkov instability for the case of a reflected rarefaction is studied in detail following the growth of the contact surface in the linear regime and providing explicit analytical expressions for the asymptotic velocities in different physical limits. This work is a continuation of the similar problem when a shock is reflected [Phys. Rev. E 93, 053111 (2016)1539-375510.1103/PhysRevE.93.053111]. Explicit analytical expressions for the asymptotic normal velocity of the rippled surface (δv_{i}^{∞}) are shown. The known analytical solution of the perturbations growing inside the rarefaction fan is coupled to the pressure perturbations between the transmitted shock front and the rarefaction trailing edge. The surface ripple growth (ψ_{i}) is followed from t=0+ up to the asymptotic stage inside the linear regime. As in the shock reflected case, an asymptotic behavior of the form ψ_{i}(t)≅ψ_{∞}+δv_{i}^{∞}t is observed, where ψ_{∞} is an asymptotic ordinate to the origin. Approximate expressions for the asymptotic velocities are given for arbitrary values of the shock Mach number. The asymptotic velocity field is calculated at both sides of the contact surface. The kinetic energy content of the velocity field is explicitly calculated. It is seen that a significant part of the motion occurs inside a fluid layer very near the material surface in good qualitative agreement with recent simulations. The important physical limits of weak and strong shocks and high and low preshock density ratio are also discussed and exact Taylor expansions are given. The results of the linear theory are compared to simulations and experimental work [R. L. Holmes et al., J. Fluid Mech. 389, 55 (1999)JFLSA70022-112010.1017/S0022112099004838; C. Mariani et al., Phys. Rev. Lett. 100, 254503 (2008)PRLTAO0031-900710.1103/PhysRevLett.100.254503]. The theoretical predictions of δv_{i}^{∞} and ψ_{∞} show good agreement with the experimental and numerical reported values.

An analytical model to study the perturbation flow that evolves between a rippled piston and a shock is presented. Two boundary conditions are considered: rigid and free surface. Any time a corrugated shock is launched inside a fluid, pressure, velocity, density, and vorticity perturbations are generated downstream. As the shock separates, the pressure field decays in time and a quiescent velocity field emerges in the space in front of the piston. Depending on the boundary conditions imposed at the driving piston, either tangential or normal velocity perturbations evolve asymptotically on its surface. The goal of this work is to present explicit analytical formulas to calculate the asymptotic velocities at the piston. This is done in the important physical limits of weak and strong shocks. An approximate formula for any shock strength is also discussed for both boundary conditions.

As firstly predicted by D'yakov and Kontorovich (DK), an initially disturbed shock front may exhibit different asymptotic behaviours depending on the slope of the Rankine-Hugoniot curve.Adiabatic and isolated planar shocks traveling steadily through ideal gases are stable, in the sense that any perturbation on the shock front decays in time with the power t −3/2 (or t −1/2 in the strongshock limit). While some gases whose equation of state cannot be modelled as a perfect gas, as those governed by van der Waals forces, may induce constant-amplitude oscillations to the shock front in the long-time regime, fully unstable behaviours are seldom to occur due to the unlikely conditions that the equation of state must meet. In this work, it has been found that unstable conditions are might be found when the gas undergoes an endothermic or exothermic transformation behind the shock. In particular, it is reported that constant-amplitude oscillations can occur when the amount of energy release is positively-correlated to the shock strength and, if this correlation is sufficiently strong, the shock turns to be fully unstable. The opposite highly-damped oscillating regime may occur in negatively-correlated configurations. The mathematical description then adds two independent parameter to the regular adiabatic index γ and shock Mach number M 1 , namely: the total energy added/removed and its sensitivity with the shock strength. The formulation in terms of endothermic or exothermic effects is extended, but not restricted, to include effects associated to ionization, dissociation, thermal radiation, and thermonuclear transformations, so long as the time associated to these effects is much shorter time than the acoustic time.

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.