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

Abstract

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.

Nonlinear interaction between bulk point vortices and an unstable interface with nonuniform velocity shear such as Richtmyer–Meshkov instability

“…A strong incident shock leaves large bulk vortices behind the transmitted and reflected rippled shocks, and the large bulk vortices suppress the linear growth of RMI because the flow near the interface generated by the bulk vortices is opposite to the flow caused by the interface vorticity. 34,36 This phenomenon motivates our present work. However, as the shocks separate from the interface, the perturbations of the pressure and density behind the rippled shocks will decay in time.…”

“…34 For our initial conditions (11), where the normalized sheet strength c has a maximum (positive) value at x ¼ Àp=2 and a minimum (negative) value at x ¼ p=2, the positions of the bulk vortices approximately correspond to x $ 6p=2 and y $ p; 3p; Á Á Á, and Àp=5; À4p=5; Á Á Á, in which the positive and negative vortices appear alternately in the y direction. 34,35 Here, for simplicity, we set the initial distribution of four and eight point vortices close to the vortex sheet so that two (four) point vortices are placed over the vortex sheet and the other two (four) vortices under the sheet for the case of four (eight) vortices. The configuration of point vortices and their signs are related to the linear RMI discussed above.…”

“…Also, due to the conservation of tangential velocity at the rippled shock fronts, transverse velocity perturbations are generated inside the compressed fluids, which account for vorticity generation in the bulks. 10,28,33,34,36 In the present work, we assume that the bulk vortices are given by point vortices for simplicity. The spatial distribution of the bulk vortices was clearly visualized, for instance, in Fig.…”

“…[22][23][24][25][26][27] Also, due to the conservation of tangential velocity at the rippled shock fronts, transverse velocity perturbations are generated inside the compressed fluids, which account for vorticity generation in the bulks. 23,25,[28][29][30][31][32][33][34] Cobos-Campos and Wouchuk 34,35 reported that bulk vortices with sufficiently small vorticity stabilize the interfacial instability, at least in the (compressible) linear stage. 25,[30][31][32][33][34] When the incident shock is strong and/or the amplitude of initial perturbations is large, it is known that a lot of vortices are left behind with the transmitted shock.…”

“…23,25,[28][29][30][31][32][33][34] Cobos-Campos and Wouchuk 34,35 reported that bulk vortices with sufficiently small vorticity stabilize the interfacial instability, at least in the (compressible) linear stage. 25,[30][31][32][33][34] When the incident shock is strong and/or the amplitude of initial perturbations is large, it is known that a lot of vortices are left behind with the transmitted shock. 23,26,36 These bulk vortices can interact with the interface and they affect the vorticity distribution on the interface.…”