Analytical scalings of the linear Richtmyer-Meshkov instability when a rarefaction is reflected

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.