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

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 ($\delta v_i^{\infty }$) 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 (${\psi }_i$) grows as ${\psi }_{\infty }+\delta v_i^{\infty }t$, where ${\psi }_{\infty }$ 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)] for a single shock moving into one fluid.

Figure 1

A planar incident shock (not shown) has compressed fluid $b$ from density ${\rho }_{b0}$ to ${\rho }_{b1}$ with pressure $p_1$. At $t=0$, the incident shock arrives at the corrugated contact surface ($x=0$). At $t>0+$ a corrugated reflected shock moves to the right with velocity $D_r+{\dot{\psi }}_r\left(t\right)$. The transmitted shock moves into fluid $a$, to the left, with velocity $D_t+{\dot{\psi }}_t\left(t\right)$. Tangential velocities $\delta v_{ya}$ and $\delta v_{yb}$ evolve at both sides of the contact surface. At the same time, the material surface ripple grows with normal velocity $U+\delta v_i$.

Figure 2

Dependence of the bulk vorticity parameters $F_a$ and $F_b$ on the incident shock Mach number ($M_i$) for the given pair of gases. The vorticity parameters are normalized with their values at high compression ($F_a^{\infty }=-0.1174, F_b^{\infty }=0.0157$). The weak shock scaling is indicated with a dashed line.

Figure 3

Spatial profiles of the asymptotic normal velocity (a), tangential velocity (b), and a density map of the vorticity field (c) on which the streamlines have been superposed. The fluid parameters are shown inside the plot.

Figure 4

(a) Temporal evolution of the contact surface ripple velocity for ${\gamma }_a=1.0935, {\gamma }_b=7/5, R_0=5.25$, and $M_i=2.86$. The horizontal dashed line shows the asymptotic velocity $\delta v_i^{\infty }/\left(k{D}_i{\psi }_0\right)$. (b) Temporal evolution of the contact surface ripple corrugation $[{\psi }_i\left({\tau }_d\right)-{\psi }_{\infty }]/{\psi }_0$ as a function of the dimensionless time: ${\tau }_d=k\delta v_i^{\infty }t$. Different choices of $\gamma , R_0$, and $M_i$ are shown together. The markers correspond to different experiments in [36] and [37]: (1) magenta star: $\text{He}/{\text{SF}}_6,R_0=39,M_i=1.95$; (2) green circle: $\text{He}/{\text{SF}}_6,R_0=39,M_i=1.41$, (3) blue triangle: $\text{air}/{\text{SF}}_6,R_0=5.23,M_i=1.45$, (4) orange diamond: $\text{He}/{\text{SF}}_6,R_0=39,M_i=1.09353$, (5) red square: $(\text{He}+\text{Ar})/\text{Ar},R_0=1.817,M_i=1.90$.

Figure 5

Dependence of the relative difference between the exact growth rate and the irrotational approximation as a function of the incident shock Mach number.

Figure 6

Dependence of the relative difference between the exact growth rate and the irrotational approximation as a function of the preshock density ratio.

Figure 7

Dependence of the relative difference between the exact growth rate and the irrotational approximation as a function of the isentropic exponent ${\gamma }_a$.

Figure 8

Dependence of the relative difference between the exact growth rate and the irrotational approximation as a function of the isentropic exponent ${\gamma }_b$.

Figure 9

Functional dependence of the velocity ratios $|u_i/v_{ia}|$ and $u_i/\left|v_{ib}\right|$ as a function of the incident shock Mach number $M_i$.

Figure 10

Functional dependence of the velocity ratios $|u_i/v_{ia}|$ and $u_i/\left|v_{ib}\right|$ as a function of $R_0$.

Figure 11

Functional dependence of the velocity ratios $|u_i/v_{ia}|$ and $u_i/\left|v_{ib}\right|$ as a function of ${\gamma }_a$.

Figure 12

Functional dependence of the velocity ratios $|u_i/v_{ia}|$ and $u_i/\left|v_{ib}\right|$ as a function of ${\gamma }_b$.

Figure 13

Comparison between the exact value of $u_i$ (obtained with at least five iterations) and the approximate formula Eq. (42) (without iterations) for (a) ${\gamma }_a=5/3$ and (b) ${\gamma }_a=1.1$.

Figure 14

Comparison between the exact normal velocity $u_i$ as a function of the shock strength with the different terms of the expansion shown in Eq. (45). Curve (a) represents the first, linear term: $a_1(M_i-1)$, curve (b) includes up to the second power: $a_1(M_i-1)+a_2{(M_i-1)}^2$, curve (c) up to the third: $a_1(M_i-1)+a_2{(M_i-1)}^2+a_3{(M_i-1)}^3$, curve (d) up to the fourth term: $a_1(M_i-1)+a_2{(M_i-1)}^2+a_3{(M_i-1)}^3+a_4{(M_i-1)}^4$, and (e) is the exact value.

Figure 15

Comparison between the exact freeze-out contour curve (obtained according to formalism developed in [17]) and the Taylor expansion shown in Eq. (53). The solid curves give the exact values of $(M_i,R_0)$ at which freeze-out is expected for that choice of isentropic exponents. The dashed curves represent the approximate estimates according to the Taylor expansion of Eq. (53).

Figure 16

Comparison between the exact normal velocity $u_i$ and Eq. (55) valid at large values of $M_i$.

Figure 17

Comparison between the exact normal velocity $u_i$ and the Taylor expansion shown in Eq. (56) as a function of the variable $R_0-R_0^m$. Each dashed line refers to a particular order of the expansion: blue circles represent the linear term, the green square includes up to the quadratic term, the orange diamond includes up to the cubic term, and the red triangle up to the quartic term.

Figure 18

Comparison between the exact normal velocity $u_i$ and Eq. (57), valid at large values of $R_0$.

Figure 19

Comparison between case 5 of [36] and our theoretical model. Curve (a) is calculated with Eq. (35) using $u_i^{\left[5\right]}$ in Eq. (34); (b) uses $u_i^{\left[0\right]}$; (c) is given by Eq. (52); (d) uses $u_{\mathrm{irrot}}$ given by Eq. (41); and (e) uses Eq. (51). The red dashed line (f) is given by Eq. (37).

Figure 20

(a) Temporal evolution of the contact surface ripple for cases 1 and 2 of [36]. The solid lines are calculated with Eq. (35) using $u_i^{\left[0\right]}$ in Eq. (34). The dashed lines are given by Eq. (37). (b) Same as (a) for cases 6, 7, and 8 of [36].

Figure 21

(a) Plot of $k[{\psi }_i\left(t\right)-{\psi }_{\infty }]$ as a function of $k\delta v_i^{\infty }t$ for case 1 of [36]. (b) Same as (a) for cases 5, 6, 7, and 8 of [36]. The dashed line represents the asymptotic behavior predicted by Eq. (37).

Figure 22

Plot of $k[{\psi }_i\left(t\right)-{\psi }_{\infty }]$ as a function of $k\delta v_i^{\infty }t$ for cases 1 and 2 of [37]. The solid lines are calculated with Eq. (35) using $u_i^{\left[0\right]}$.