2.1. Method of eliminating reverberating waves

Figure 1(a) illustrates the initial configuration of the A/B/A-type heavy fluid layer with sinusoidal interfaces on both sides. Here, $a_1^0$ and $a_2^0$ denote the initial amplitudes of the first and second interfaces, respectively; $L_0$ denotes the initial fluid-layer thickness; $\lambda$ denotes the initial wavelength of the fluid-layer interface; and $\rho _A$ and $\rho _B$ denote initial densities for fluid A and fluid B, respectively. Notably, the initial amplitudes of both perturbed interfaces satisfy the small-amplitude hypothesis in this work, i.e. $ka_1^0 \ll 1$ and $ka_2^0 \ll 1$, where $k$ ($= 2{\rm \pi} /\lambda$) is the wavenumber. For the small-amplitude scenario, the method of eliminating reverberating waves, derived based on one-dimensional (1-D) shock dynamics theories (Han & Yin Reference Han and Yin1993), is considered to be applicable.

Figure 1. Schematics of a heavy fluid layer ($\rho _B > \rho _A$) for the shock compression in the absence of reverberating waves. (a) The moment ($t < 0$) before the arrival of the incident shock IS. (b) The moment ($t = t^+$) when the transmitted shock TS$_1$ just leaves the first interface. (c) The moment ($t = t^\ast$) when the shock TS$_1$ reaches the middle position of the initial second interface. Here, II$_1$ and II$_2$ denote the initial first and second interfaces, respectively; SI$_1$ and SI$_2$ denote the shocked first and second interfaces, respectively; $a_1^0$ and $a_2^0$ denote initial amplitudes of the first and second interfaces, respectively; $a_1^+$ denotes the amplitude of the first interface at $t^+$; $a_1^\ast$ and $a_2^\ast$ denote the amplitudes of the first and second interfaces at $t^\ast$, respectively; $L_0$ denotes the initial layer thickness; $L^\ast$ denotes the layer thickness at $t^\ast$; $\lambda$ denotes the interface wavelength; and $\rho _A$ and $\rho _B$ ($\rho _A^\ast$ and $\rho _B^\ast$) denote the pre-shock (post-shock) densities for fluid A and fluid B, respectively.

As shown in figure 1(a), when the incident shock IS propagates from fluid A to fluid B, total transmission (i.e. no reflected wave) occurs at the first interface, provided that the following condition (Han & Yin Reference Han and Yin1993) is satisfied:

(2.1)\begin{equation} c_B/c_A= \kappa, \end{equation}

with

(2.2)\begin{equation} \kappa= \frac{\gamma _{B}\left[ 1+ \dfrac{\gamma _{B}+1}{2 \gamma _{B}}(p_{2}/p_{1}-1)\right] ^{1/2}}{\gamma _{A}\left[ 1+ \dfrac{\gamma _{A}+1}{2 \gamma_{A}}(p_{2}/p_{1}-1)\right] ^{1/2}}, \end{equation}

where $c_B/c_A$ is the ratio of sound speed in fluid B to that in fluid A; $\gamma _A$ and $\gamma _B$ are the adiabatic indices for fluid A and fluid B, respectively; and $p_2/p_1$ is the ratio of pressure behind IS to that in front of IS, which can be expressed as

(2.3)\begin{equation} p_{2}/p_{1}=1+\frac{2\gamma_A}{\gamma_A+1}(M_s^2-1), \end{equation}

with $M_s$ denoting the incident shock Mach number. By substituting $c_A=\sqrt {\gamma _A p_A/\rho _A}$ and $c_B=\sqrt {\gamma _Bp_B/\rho _B}$ into (2.1) and simplifying, we obtain

(2.4)\begin{equation} 1+\frac{2}{p_{2}/p_{1}-1}=\frac{\rho_B-\rho_A}{\gamma_A \rho_A-\gamma_B\rho_B}. \end{equation}

Substituting (2.3) into (2.4) yields

(2.5)\begin{equation} R_{0}=\frac{\rho_{B}}{\rho_{A}}= \frac{\gamma _{A}(\gamma _{A}+1)M_{s}^{2}}{\gamma _{B}- \gamma _{A}+ \gamma _{A}(\gamma _{B}+1)M_{s}^{2}}, \end{equation}

where $R_0$ is the initial density ratio. Equation (2.5) gives the relationship among $R_0$, $M_s$, $\rho _{A}$, $\rho _{B}$, $\gamma _{A}$ and $\gamma _{B}$ under the total transmission condition.

When the IS passes through the first interface, the transmitted shock TS$_1$ is generated, as depicted in figure 1(b). Due to the total transmission of IS at the first interface, the value of $p_2/p_1$ remains equal for both IS and TS$_1$. As a result, the gas parameters on both sides of the second interface also satisfy (2.4), that is, TS$_1$ is also totally transmissive at the second interface. Here, it can be concluded that, for an A/B/A-type heavy fluid layer, if the shock is totally transmissive at the first interface, it is also totally transmissive at the second interface. The total transmission at both fluid-layer interfaces causes the post-shock flow velocity to remain unvaried. As a consequence, the shock-induced jump velocities for the first and second interfaces are equal. In the remainder of this paper, the parameter $\Delta U$ will be used to represent the jump velocity for both the first and second interfaces under the total transmission condition.

2.2. Theory of individual freeze-out for the first and second interfaces

The freeze-out theory for heavy fluid layers focuses on the case where the first and second interfaces are in phase, as illustrated in figure 1(a). This choice is limited by the fact that freeze-out only theoretically emerges for in-phase cases (Mikaelian Reference Mikaelian1995). To construct the freeze-out theory, we need to know the parameters (including the density ratio, the fluid-layer thickness and the interface amplitudes) at the initiation of interface coupling. For the shock-accelerated fluid layer, the interface coupling is initiated after the shock compression. Figure 1 shows schematics of the shock-compression process for the heavy fluid layer in the absence of reverberating waves. The time when the incident shock IS just contacts the first interface is defined as $t = 0$. When the IS completely passes through the first interface ($t = t^+$), as shown in figure 1(b), the density of fluid A changes from $\rho _A$ to $\rho _A^\ast$, and the first interface amplitude is compressed from $a_1^0$ to $a_1^+$. When the transmitted shock TS$_1$ reaches the second interface centre ($t = t^*$), as shown in figure 1(c), the fluid-layer thickness is compressed from $L_0$ to the minimum value of $L^\ast$, the amplitude of the first interface changes to $a^*_1$ and the amplitude of the second interface is compressed from $a^0_2$ to $a^*_2$. In this work, the interface coupling is considered to be initiated at $t = t^*$. Then, the post-shock quantities at $t = t^\ast$, including the density ratio $R^\ast$, the layer thickness $L^\ast$ and the interface amplitudes $a^*_1$ and $a^*_2$, required for the establishment of the freeze-out theory, will be determined.

By solving the 1-D Riemann problem of the shock–interface interaction, the post-shock densities $\rho _A^\ast$ and $\rho _B^\ast$ can be obtained and thus $R^\ast$ can be written as

(2.6)\begin{equation} R^\ast=\rho_B^\ast{/}\rho_A^\ast. \end{equation}

The duration of the whole compression process of the fluid layer can be denoted as $L_0/V_{ts1}$, where $V_{ts1}$ represents the velocity of TS$_1$. Throughout this period, the downstream movement of the first interface can be quantified as $\Delta U$($L_0/V_{ts1}$). As a result, $L^*$ can be written as

(2.7)\begin{equation} L^\ast=L_0-\Delta U(L_0/V_{ts1})=L_0(1-\Delta U/V_{ts1}). \end{equation}

Notably, under the total transmission condition, $L^\ast$ remains unvaried since the jump velocities for both interfaces are equal.

The development of the first interface's amplitude during the transition from $a_1^0$ to $a_1^\ast$ can be divided into two stages: (I), 0 $< t \leqslant t^+$, the shock-compression stage; (II), $t^+ < t \leqslant t^*$, the independent perturbation growth stage. In stage I, the amplitude ($a_1^+$) after shock compression can be expressed as

(2.8)\begin{equation} a_1^+=a_1^0(1-\Delta U / V_{is}), \end{equation}

where $V_{is}$ is the velocity of IS. In stage II, the amplitude growth for the first interface initially enters a startup period. According to the work of Lombardini & Pullin (Reference Lombardini and Pullin2009), the amplitude growth rate of the first interface in the startup period can be expressed as

(2.9)\begin{equation} \frac{{\rm d}a_1}{{\rm d}t}=\frac{C_{1}\varGamma}{\Delta UA^{+}} \dot a^{Rich}_{1}\frac{t}{\tau}. \end{equation}

Here, $\tau$ is the startup time, which can be found using

(2.10)\begin{equation} \tau = \frac{1}{2k}\left(\frac{1-A^{+}}{V_{f}+\Delta U}+\frac{1+A^{+}}{V_{ts1}-\Delta U}\right), \end{equation}

where $A^+ = (\rho _B^*-\rho _A^*)/(\rho _B^*+\rho _A^*)$ is the post-shock Atwood number, and $V_{f}$ is the velocity of the reflected wave from the interface. In the total transmission condition, $V_{f}$ is considered to be equal to the local sound velocity. The parameter $\varGamma$ is the circulation imposed by IS on the first interface (Samtaney & Zabusky Reference Samtaney and Zabusky1994), which can be expressed as

(2.11)\begin{equation} \varGamma=\frac{c_{A}}{M_s}\left[ \frac{\gamma_B(1-\phi(p_3/p_1,\gamma_B))}{\gamma_A(\gamma_B-1)R_{0}}-\frac{1-\phi(p_3/p_2,\gamma_A)\phi(p_2/p_1,\gamma_A)}{\gamma_A-1}\right], \end{equation}

with

(2.12)\begin{equation} \phi({\xi},\eta) \equiv \xi\frac{(\eta-1)\xi+\eta+1}{(\eta+1)\xi+\eta-1}, \end{equation}

where $p_3$ is the pressure behind TS$_1$. Here, $C_{1} = [(1-A^{+})(V_{f}+\Delta U)-(1+A^{+})(-V_{ts1}+\Delta U)]/(V_{f}+V_{ts1})$ and $\dot a^{Rich}_{1}$ is the perturbation growth rate obtained from the model of Richtmyer (Reference Richtmyer1960) as

(2.13)\begin{equation} \dot{a}^{Rich}_1=ka_{1}^{+}\Delta U A^+. \end{equation}

After the startup period, the amplitude growth for the first interface enters a linear period. The amplitude growth rate in this period can be given by (2.13).

Notably, at $t = t^\ast$, the amplitude growth of the first interface can be categorized into two scenarios: (i) for $t^\ast -t^+ \leq \tau$, the amplitude growth is still in the startup period; (ii) for $t^\ast -t^+ > \tau$, the amplitude growth has entered the linear period. By integrating the time term in (2.9) and (2.13), the amplitude $a_{1}^{*}$ in the two scenarios can be expressed as

(2.14)\begin{equation} a_{1}^{*}= \left\{ \begin{array}{@{}ll@{}} \displaystyle a_{1}^{+}+\dfrac{C_{1}\varGamma\dot a^{Rich}_{1}}{2\Delta U\tau A^{+}}(t^{*}-t^{+})^2, & t^{*}-t^{+} \leq \tau; \\ \displaystyle a_{1}^{+}+\dfrac{C_{1}\varGamma\dot a^{Rich}_{1}}{2\Delta UA^{+}}\tau+\dot a^{Rich}_{1}(t^{*}-t^{+}-\tau), & t^{*}-t^{+} > \tau. \end{array} \right. \end{equation}

The amplitude $a_{2}^{*}$ can be expressed as

(2.15)\begin{equation} a_2^*=\frac{a_2^0+a_2^+}{2}, \end{equation}

where $a_2^+$ is the amplitude for the second interface at the time when TS$_1$ completely passes through it. The $a_2^+$ can be expressed as

(2.16)\begin{equation} a_2^+=a_2^0(1-\Delta U/V_{ts1}). \end{equation}

In the previous studies involving a heavy/light single-mode case (Meyer & Blewett Reference Meyer and Blewett1972; Holmes et al. Reference Holmes, Dimonte, Fryxell, Gittings, Grove, Schneider, Sharp, Velikovich, Weaver and Zhang1999; Guo et al. Reference Guo, Cong, Si and Luo2022a), $a_2^\ast$, rather than $a_2^+$, is employed as the post-shock amplitude for theoretical modelling since $a_2^\ast$ performs better in predicting the amplitude growth rate for the heavy/light configuration. In the current work, $a_2^\ast$ is also selected as the post-shock amplitude of the second interface to establish the freeze-out theory.

By replacing the pre-shock quantities $R_0$, $L_0$, $a_1^0$ and $a_2^0$ in the model of Mikaelian (Reference Mikaelian1995) with the post-shock quantities $R^\ast$, $L^\ast$, $a_1^\ast$ and $a_2^\ast$, we can obtain the following expression to predict the freeze-out for the first interface:

(2.17)\begin{equation} a_{1}^{*}= a_{2}^{*}\sin \theta^\prime, \end{equation}

and the following expression to predict the freeze-out for the second interface:

(2.18)\begin{equation} a_{2}^{*}= a _{1}^{*}\sin \theta^\prime. \end{equation}

In (2.17) and (2.18), $\theta ^\prime$ is the coupling angel with consideration of the post-shock quantities $R^\ast$ and $L^\ast$ and

(2.19)\begin{equation} \sin \theta^\prime =\frac{2W}{1+W^2}, \end{equation}

with

(2.20)\begin{align} W =1+ \sinh(kL^{{\ast}})\tanh\left(\frac{kL^\ast}{2}\right)+ \frac{\sinh(kL^\ast)}{R^\ast} \left\{1+\left[1+{R^\ast}^{2}+2R^\ast\coth(kL^\ast)\right]^{1/2}\right\}. \end{align}

When employing equation (2.17) to forecast the first interface's freeze-out, we find that the calculated value of $a_1^\ast$ is too large, resulting in a significant amplitude growth rate of the first interface. As a result, the coupling effect imposed by the second interface fails to freeze the amplitude growth of the first interface. Similarly, (2.18) also yields an overestimation of $a_2^\ast$, leading to its inability to predict the freeze-out for the second interface. The failure of (2.17) and (2.18) indicates that considering only the modification of post-shock quantities for the original model of Mikaelian (Reference Mikaelian1995) is insufficient.

It should be noted that, in the original model of Mikaelian (Reference Mikaelian1995), $R_0$ and $L_0$ are independent parameters. However, in the total transmission condition, $R_0$ is determined by (2.5), establishing a relationship between $R_0$ and the adiabatic indices ($\gamma _A$ and $\gamma _B$). Any change in $R_0$ requires $\gamma _A$ or $\gamma _B$ to change accordingly to ensure total transmission. Nevertheless, based on the shock dynamics theory (Han & Yin Reference Han and Yin1993), variations in $\gamma _A$ or $\gamma _B$ can affect the values of $\Delta U$ and $V_{ts1}$, thus influencing the value of $L^\ast$ (see (2.7)). Consequently, under the total transmission condition, changes in $R_0$ can lead to variations in $L^\ast$ so that $R^\ast$ and $L^\ast$ are not independent. It is believed that directly substituting the independent parameters $L_0$ and $R_0$ with the dependent parameters $L^\ast$ and $R^\ast$ into (2.19) causes it to be incapable of accurately quantifying the interface coupling strength required for freeze-out, thus leading to the failure of (2.17) and (2.18). To account for additional effects arising from the correlation between $R^\ast$ and $L^\ast$, an empirical parameter $\beta$ is introduced in (2.19), resulting in a modified form of (2.19):

(2.21)\begin{equation} \sin \theta^\ast=\frac{2\beta W}{1+W^2}, \end{equation}

where $\theta ^\ast$ is the modified coupling angle.

Then, we describe the determination process of the parameter $\beta$. As mentioned above, (2.17) and (2.18) yield an overestimation for $a_1^\ast$ and $a_2^\ast$, respectively, resulting in their inability to predict the individual freeze-out for the first and second interfaces. As a consequence, to ensure the effectiveness of the freeze-out theory, it is necessary to introduce a parameter less than 1 to multiply $\sin \theta ^\prime$ in (2.17) to reduce the resulting value of $a_1^\ast$, and in (2.18) to reduce the resulting value of $a_2^\ast$. In other words, the parameter $\beta$ introduced should be smaller than 1. Naturally, $\beta$ is expected to be a constant, representing the simplest approach for the modification. To determine the optimal value for $\beta$, we explore some values within the range of 0 to 1. For each attempted $\beta$ value, numerical simulations are conducted to confirm whether freeze-out occurs. Using cases I$_1$3 and I$_2$6$^m$ (details provided in tables 1 and 2 in §§ 4.2 and 4.3) as examples, figure 2(a,b) shows the dimensionless amplitude growth of the first and second interfaces predicted by the freeze-out theory with three $\beta$ values ($\beta = 1$, 0.4 and 2/3). It is evident that, for $\beta$ values that are either too large ($\beta = 1$) or too small ($\beta = 0.4$), the amplitude growth of both interfaces remains unfrozen. However, when $\beta = 2/3$, the amplitude growth of the first and second interfaces is effectively arrested. Importantly, this $\beta$ value ($=2/3$) is found to be effective for the freeze-out theory in predicting freeze-out for the first and second interfaces over a broad range of initial conditions, as observed in figures 8 and 11 in §§ 4.2 and 4.3. Consequently, $\beta = 2/3$ is recommended in (2.21).

Figure 2. The dimensionless amplitude growth for the first (a) and second (b) interfaces predicted by the freeze-out theory with different $\beta$ values. Cases I$_1$3 and I$_2$6$^m$ are designed for examining the freeze-out for the first and second interfaces, respectively, with their initial conditions detailed in tables 1 and 2 in §§ 4.2 and 4.3.

Table 1. Initial conditions of cases I$_1$1–I$_1$14 designed for testing the validity of (2.22) in predicting freeze-out for the first interface. Case S1 is a single-mode case, which is utilized for comparison. In cases I$_1$1–I$_1$3, fluid A is Xe. In cases I$_4$ and I$_5$, fluid A is Kr and Ar, respectively. In cases I$_1$1–I$_1$4, fluid B is a mixture of C$_4$F$_{10}$ and C$_5$H$_{12}$, and the volume fractions of C$_4$F$_{10}$ and C$_5$H$_{12}$ in cases I$_1$1–I$_1$4 are 74.39 % and 25.61 %, 67.30 % and 32.70 %, 64.31 % and 35.69 %, 24.58 % and 75.42 %. In case I$_1$5, fluid B is a mixture of C$_4$H$_{10}$ and C$_3$H$_{8}$, and the volume fractions of C$_4$H$_{10}$ and C$_3$H$_{8}$ are 59.80 % and 40.20 %. In cases I$_1$6–I$_1$14, fluid B is SF$_6$, and fluid A is the artificial gas whose $\rho _A$ and $\gamma _A$ are determined by (2.5) to satisfy the total transmission condition. For cases I$_1$1–I$_1$3, the selection of $L_0$ results in the same $L^\ast$, and similarly, for cases I$_1$7, I$_1$11-I$_1$12. The purpose of maintaining the same $L^\ast$ is to emphasize the effect of the Mach number. The units of $L_0$, $\lambda$, $a_1^0$ and $a_2^0$ are mm, the unit of $\Delta U$ is m s$^{-1}$ and the unit of $\rho _A$ and $\rho _B$ is kg m$^{-3}$.

Table 2. Initial conditions for cases I$_2$1–I$_2$3, I$_2$4$^m$–I$_2$15$^m$ and S2. Cases I$_2$1–I$_2$3 are designed for examining the influence of whether the second interface completes phase reversal on the predictions of (2.23). Cases I$_2$4$^m$–I$_2$15$^m$ are designed for testing the validity of (2.23) and (2.27) in predicting freeze-out for the second interface. The superscript ‘$m$’ denotes that $L_0$ of the cases is calculated using (2.27). Case S2 is a single-mode case, which is utilized for comparison. In cases I$_2$4$^m$–I$_2$6$^m$, fluid A is Xe. In cases I$_2$7$^m$ and I$_2$8$^m$, fluid A is Kr and Ar, respectively. In cases I$_2$4$^m$–I$_2$7$^m$, fluid B is a mixture of C$_4$F$_{10}$ and C$_5$H$_{12}$, and the volume fractions of C$_4$F$_{10}$ and C$_5$H$_{12}$ in cases I$_2$4$^m$–I$_2$7$^m$ are 74.39 % and 25.61 %, 67.30 % and 32.70 %, 64.31 % and 35.69 %, 24.58 % and 75.42 %. In case I$_2$8$^m$, fluid B is a mixture of C$_4$H$_{10}$ and C$_3$H$_{8}$, and the volume fractions of C$_4$H$_{10}$ and C$_3$H$_{8}$ are 59.80 % and 40.20 %. In cases I$_2$1–I$_2$3 and I$_2$9$^m$–I$_2$15$^m$, fluid B is SF$_6$, and fluid A is the artificial gas whose $\rho _A$ and $\gamma _A$ are determined by (2.5) to satisfy the total transmission condition. The units of $L_0$ ($L_0^m$), $\lambda$, $a_1^0$ and $a_2^0$ are mm, the unit of $\Delta U$ is m s$^{-1}$ and the unit of $\rho _A$ and $\rho _B$ is kg m$^{-3}$.

Notably, $\sin \theta ^\ast$ varies from 0 to 2/3. As the value of $\sin \theta ^\ast$ becomes larger, the interface coupling strength is higher. To illustrate the dependence of $\sin \theta ^\ast$ on initial conditions, figure 3 presents the variations of $\sin \theta ^\ast$ vs $kL_0$ and $R_0$, with consideration of the influence of $M_s$. As shown in figure 3(a), as $kL_0$ decreases, $\sin \theta ^\ast$ increases and when $kL_0$ approaches zero, $\sin \theta ^\ast$ converges to 2/3. As shown in figure 3(b), as $R_0$ increases, $\sin \theta ^\ast$ increases but saturates rapidly. Particularly, at a relatively large $kL_0$ such as $kL_0 = 3.14$, increasing $R_0$ alone cannot make $\sin \theta ^\ast$ converge to 2/3. As a result, enhancing of the interface coupling strength is more effectively achieved by reducing $kL_0$ rather than increasing $R_0$. In addition, as $M_s$ increases, $\sin \theta ^\ast$ also increases. This phenomenon occurs because a stronger shock impact leads to a greater compression in both interface thicknesses and fluid densities, resulting in an increase in both $kL^\ast$ and $R^\ast$ and, consequently, an increase in $\sin \theta ^\ast$.

Figure 3. Variations of $\sin \theta ^\ast$ as a function of $kL_0$ (a) and $R_0$ (b). In (a), the values of $R_0$ corresponding to each $M_s$ are 2, 3 and 4, respectively. In (b), the values of $M_s$ corresponding to each $kL_0$ are 1.2, 1.6 and 2.0, respectively. The lines for $kL_0^m$ are obtained based on (2.27), as described in § 2.3.

Finally, with consideration of the quantity $\sin \theta ^\ast$, the model for predicting the freeze-out for the first interface can be expressed as

(2.22)\begin{equation} a_{1}^{*}= a_{2}^{*}\sin \theta^*, \end{equation}

and the model for predicting the freeze-out for the second interface can be expressed as

(2.23)\begin{equation} a_{2}^{*}= a _{1}^{*}\sin \theta^*. \end{equation}

Equations (2.22) and (2.23) quantify the amplitudes of the first and second interfaces at the onset of interface coupling, as well as the interface coupling strength required for freeze-out.

Based on the variations of the first and second interfaces’ amplitudes during the shock compression stage, the initial amplitudes ($a_{1}^{0}$ and $a_{2}^{0}$) of the first and second interfaces required for freeze-out can be determined. For instance, we can first set a value for $a_{1}^{0}$, and then, the $a_{2}^{0}$ required for freeze-out for the first interface can be determined by combining (2.22), (2.8), (2.14), (2.15) and (2.16); and the $a_{2}^{0}$ required for freeze-out for the second interface can be determined by combining (2.23), (2.8), (2.14), (2.15) and (2.16).

2.3. Supplementary condition for the second interface's freeze-out

After the shock impact, phase reversal generally arises at the second interface of the heavy fluid layer (Meyer & Blewett Reference Meyer and Blewett1972). Previous fluid-layer studies have indicated that, once phase reversal of the second interface is accomplished, the spike continually grows downstream so that freeze-out cannot be achieved by the interface coupling (Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020; Cong et al. Reference Cong, Guo, Si and Luo2022). Consequently, the freeze-out for the second interface needs to be achieved before phase reversal. This indicates that (2.23) is not sufficient to achieve the freeze-out for the second interface.

After the shock passes through the fluid layer ($t > t^\ast$), the coupling effect is considered to propagate at the local sound velocity ($c_B^\ast$) within the fluid layer. The value of $c_B^\ast$ can be obtained by the shock dynamics theory (Han & Yin Reference Han and Yin1993). As the first interface evolves, its induced coupling effects continuously transmit to the second interface. If the first interface imposes sufficient coupling effects on the second interface, the second interface's phase reversal could be prevented (Liang & Luo Reference Liang and Luo2023). The coupling transfer time ($t^{ct}$) between the first and second interfaces is expressed as

(2.24)\begin{equation} t^{ct} = L^\ast{/}c^\ast_B. \end{equation}

Note that, as the initial layer thickness increases, the $t^{ct}$ becomes larger. Consequently, over the same time interval, the total amount of coupling effects imposed on the second interface decreases with increasing initial layer thickness. As a result, there is a maximum initial layer thickness ($L_0^m$) at which the total coupling effects imposed on the second interface just prohibit the second interface's phase reversal.

Starting from $t^\ast$, the phase-reversal time ($t^{pr}$) for the second interface is given by

(2.25)\begin{equation} t^{pr} = \frac{a^0_2}{V_{ts1}}+\frac{a_2^+}{\dot{a}^{Rich}_2}, \end{equation}

where $\dot {a}^{Rich}_2$ ($= ka_2^+\Delta UA^+$) is the amplitude growth rate in the phase reversal stage (Zhai et al. Reference Zhai, Dong, Si and Luo2016). To prevent phase reversal at the second interface, $t^{ct}$ should be smaller than $t^{pr}$. This leads to the relation

(2.26)\begin{equation} t^{ct}=\delta t^{pr}, \end{equation}

where $\delta \in (0, 1)$ is a proportional factor. Combining (2.26) with (2.7), the maximum initial layer thickness $L_0^m$ is expressed as

(2.27)\begin{equation} L_{0}^m=\frac{\delta c_{B}^\ast}{1-\Delta U/V_{ts1}}\left(\frac{1}{k\Delta UA^+}+\frac{a_2^0}{V_{ts1}}\right). \end{equation}

When $L_0 < L_0^m$, the phase reversal at the second interface can be prevented so that the freeze-out for the second interface is feasible.

In (2.27), only the factor $\delta$ is unknown. To determine $\delta$, we perform numerical simulations with various initial conditions. Specifically, eight numerical cases, I$_2$4$^m$–I$_2$6$^m$ and I$_2$9$^m$–I$_2$13$^m$, each with unique initial conditions detailed in table 2 in § 4.3, are considered. For each case, simulations with different initial layer thicknesses are conducted to specify the numerical maximum initial layer thickness ($L_{0}^{m,num}$) that prevents the phase reversal at the second interface. Then, by substituting the $L_{0}^{m,num}$ values into (2.27), eight numerical values for $\delta$ are obtained. It is shown that each case yields a distinct numerical value for $\delta$, indicating that a unified $\delta$ cannot be determined. In other words, $\delta$ is influenced by the initial conditions, much like $t^{ct}$ and $t^{pr}$. Here, $\sin \theta ^\ast$, a non-dimensional parameter dependent on initial conditions, is employed to specify $\delta$. Figure 4 shows the $\delta$–$\sin \theta ^\ast$ plot. It is observed that, as $\sin \theta ^\ast$ increases, $\delta$ also increases. We find that a quadratic function, $\delta = 9\sin ^2\theta ^\ast /8$, can well characterize the relationship between $\delta$ and $\sin \theta ^\ast$. As a result, $\delta = 9\sin ^2\theta ^\ast /8$ is recommended for defining $\delta$. Notably, $\sin \theta ^\ast$ varies from 0 to 2/3 so that $\delta$ varies from 0 to 1/2. Consequently, as illustrated in (2.26), the maximum value of the coupling transfer time $t^{ct}$ must be less than 1/2 of the phase reversal time $t^{pr}$ to prevent the second interface's phase reversal.

Figure 4. The variations of $\delta$ with $\sin \theta ^\ast$. Symbols denote the numerical values of $\delta$, obtained from simulations of cases I$_2$4$^m$–I$_2$6$^m$ and I$_2$9$^m$–I$_2$13$^m$. The initial conditions of the eight numerical cases are presented in table 2. The red line indicates the quadratic function, $\delta = 9\sin ^2\theta ^\ast /8$.

The theoretical lines of the dimensionless form of $L_0^m$, denoted as $kL_0^m$, are presented in figure 3(a). The value for the dimensionless initial layer thickness ($kL_0$) on the left-hand side of the lines of $kL_0^m$ enables the freeze-out for the second interface. In short, to achieve freeze-out for the second interface, it is essential not only for the amplitudes of the first and second interfaces to satisfy (2.23), but also for the initial layer thickness to remain smaller than $L_0^m$ (i.e. $L_0 < L_0^m$).

3.1. Numerical set-up and code validation

The initial configuration of the A/B/A-type heavy fluid layer, adopted in the simulations, is depicted in figure 1(a). The detailed settings for the initial parameters including $a_1^0$, $a_2^0$, $M_s$, $L_0$, $\rho _A$, $\rho _B$ and $\lambda$ will be described in §§ 4 and 5. The computational domain's length is set to be at least ten times half the wavelength $\lambda$ (in this work, three different wavelengths with values of 120, 240 and 480 mm are considered) to avoid interference of reflected waves from the right boundary with the evolving interface during the designated simulation time. To optimize computational efficiency, calculations are performed only for half the computational domain along the $y$-direction. In the simulations, the nodes with a volume fraction of fluid B between 5 % and 95 % are considered as the interface. The amplitude of the evolving first (second) interface, denoted as $a_1$ ($a_2$), is defined as half of the distance between the bubble tip and the spike tip at the nodes with a fluid B volume fraction of 50 %. In the simulations, the temperature is 295 K and the pressure is 101.3 kPa.

The interactions between a shock and the heavy fluid layer are described by two-dimensional (2-D) compressible multi-component Euler equations. The optimized six-point weighted compact nonlinear scheme (WCNS) with minimum dispersion and adaptive dissipation (Zhou et al. Reference Zhou, Ding, Huang and Luo2023b) is used. The optimized WCNS scheme is extended to multi-species flows by incorporating the double-flux algorithm of Abgrall & Karni (Reference Abgrall and Karni2001), and has shown a good capability to simulate RMI problems (Zhou et al. Reference Zhou, Ding, Cheng and Luo2023a). For time integration, a third-order total variation diminishing Runge–Kutta method is adopted. Further details about the numerical scheme can be found in the previous work (Zhou et al. Reference Zhou, Ding, Huang and Luo2023b).

The experimental data from a shocked air/SF$_6$/air fluid layer in the previous work (Li et al. Reference Li, Cao, Wang, Zhai and Luo2023) are utilized for code validation. The simulation employs identical initial conditions to those used in the experiment. The incident shock Mach number is 1.19. The first interface of the fluid layer is a flat one, while the second interface is a single-mode case with an initial amplitude and wavelength of 4 and 60 mm, respectively. The volume fraction of SF$_6$ within the layer is 82.6 %, and the surrounding air outside the layer is pure. Figure 5(a) shows the comparison of interface amplitude growths between the experimental data and our simulation results. It is found that the experimental and numerical results achieve a good agreement.

Figure 5. Code validation based on amplitude growths of heavy-fluid-layer interfaces under the conditions of lower (a) and higher (b) Mach numbers. Symbols represent experimental results obtained from the air/SF$_6$/air fluid-layer case in the previous work (Li et al. Reference Li, Cao, Wang, Zhai and Luo2023). Lines represent numerical results based on four kinds of mesh sizes.

To evaluate grid convergence, four different mesh sizes (0.4 mm, 0.2 mm, 0.15 mm and 0.1 mm) are adopted to simulate an air/SF$_6$/air fluid layer impacted by a shock with ${M_s = 2.0}$. Figure 5(b) shows the temporal amplitude variations for the first and second interfaces for each mesh size at $M_s = 2.0$. Additionally, figure 5(a) includes the amplitude growth data of the lower Mach number case, considering these four mesh sizes. It is evident that the amplitude growth data exhibit convergence as the mesh size decreases from 0.4 to 0.1 mm, both for higher and lower Mach number cases. Particularly, the numerical results for cases with mesh sizes of 0.2 mm, 0.15 mm and 0.1 mm are in close agreement. To strike a balance between computational cost and accuracy, a mesh size of 0.15 mm is utilized for cases with $\lambda = 120$ mm, while a mesh size of 0.2 mm is utilized for cases with ${\lambda = 240}$ mm or 480 mm.

3.2. Total transmission validation

The total transmission of a shock at small-amplitude fluid-layer interfaces is assessed. The initial conditions for the examination are listed as follows: $M_s$ is set to 2.0 to investigate the total transmission at a relatively high Mach number; SF$_6$ is chosen as the heavy fluid with $\rho _B$ and $\gamma _B$ of 6.08 kg m$^{-3}$ and 1.10, respectively; $R_0$ is set to 3, causing $\rho _A$ to be 2.03 kg m$^{-3}$; $\gamma _A$ is determined to be 4.72 using (2.5); and $a_{1}^0$ and $a_{2}^0$ are 0.36 and 1 mm, respectively, both satisfying the small-amplitude hypothesis. Notably, the values of $a_1^0$ and $a_2^0$ satisfy (2.22) so that the perturbation growth of the first interface would be frozen. Since the reverberating waves within the layer arise due to reflection from the second interface, only the total transmission at the second interface is examined.

Figure 6 shows the numerical schlieren images and corresponding pressure fields of a shocked SF$_6$ fluid layer for the examination case. As shown in figure 6, the incident shock IS passes through the first interface, giving rise to the transmitted shock TS$_1$ (155 $\mathrm {\mu }$s). Subsequently, TS$_1$ impacts the second interface, leading to the formation of a transmitted shock TS$_2$ (245 $\mathrm {\mu }$s). Notably, a reflected region RR$_1$ is formed when TS$_1$ traverses the second interface. The generation of RR$_1$ region is primarily ascribed to the diffuse initial interface across four grids in this work. Due to the slight diffusion of the interface, the complete satisfaction of the total transmission condition is not achieved, resulting in the generation of reflected waves. Furthermore, the derivation of the total transmission condition is based on 1-D assumptions so that the 2-D small-amplitude perturbation may still give rise to reflected waves. In the partially enlarged view of the pressure plot at 245 $\mathrm {\mu }$s, as shown in figure 6, the pressure first decreases and then increases when traversing the RR$_1$ region from left to right. This indicates that the RR$_1$ region consists of rarefaction and compression waves. These waves in the RR$_1$ region successively strike the first interface, resulting in an RR$_2$ region consisting of reflected compression and rarefaction waves (320 $\mathrm {\mu }$s). In this examination case, the ratio of the pressure ahead of the RR$_1$ (RR$_2$) region to that behind the RR$_1$ (RR$_2$) region is 1.0098 (0.9987). We also determine the two pressure ratios for all cases considered in this work (see tables 1–3 for details of the cases), and the pressure ratio involving the RR$_1$ (RR$_2$) region varies from 1.0041 (0.9865) to 1.0148 (0.9995) for different cases. This indicates that the reverberating waves resulting from the initial 2-D diffuse interfaces are weak for all cases so that the shock can be considered totally transmissive at the second interface.

Figure 6. Numerical schlieren images and pressure fields for a shocked heavy-fluid-layer case, designed for examining the total transmission. Note that two partially enlarged views of the pressure fields are provided at the bottom. Here, RR$_1$ and RR$_2$ denote the regions consisting of rarefaction and compression waves; II$_1$ and II$_2$ denote the initial first and second interfaces, respectively; SI$_1$ and SI$_2$ denote the shocked first and second interfaces, respectively, and similarly hereinafter. The black dashed lines in the pressure plots denote a half-wavelength interface, which are intentionally plotted to enhance understanding of the interface evolution. Numbers in the numerical schlieren images denote time with unit of $\mathrm {\mu }$s.

Table 3. Initial conditions of cases designed for testing freeze-out for double interfaces and for fluid-layer width growth. The units of $L_0$, $\lambda$, $a_1^0$ and $a_2^0$ are mm, the unit of $\Delta U$ is m s$^{-1}$ and the unit of $\rho _A$ and $\rho _B$ is kg m$^{-3}$.

Notably, the maximum values of $ka_1^0$ or $ka_2^0$ for all cases in this work are no more than 0.1. These initial amplitude values satisfy the small-amplitude requirements of $ka_1^0 \ll 1$ and $ka_2^0 \ll 1$, because the values of $ka^0_{1,2}$ are an order of magnitude smaller than unity. Such a small amplitude can significantly reduce the influence of dimensionality on the total transmission condition. Furthermore, the reverberating waves resulting from the small-amplitude interfaces are very weak. As a result, $ka^0_{1,2} = 0.1$ may serve as a threshold for the total transmission of a shock at the perturbed fluid-layer interface.

4.1. Initial condition settings for parametric studies

Equations (2.22), (2.23) and (2.27) are all functions of $\sin \theta ^\ast$, which is dependent on the incident shock Mach number ($M_s$), the initial density ratio ($R_0$), the initial fluid-layer thickness ($L_0$) and the interface wavelength ($\lambda$). As a result, the individual freeze-out for the first and second interfaces under the conditions of various $M_s$, $R_0$, $L_0$ and $\lambda$ will be examined.

In the previous freeze-out studies with lower Mach numbers ($M_s < 1.4$) (Chen et al. Reference Chen, Wang, Zhai and Luo2023a,Reference Chen, Xing, Wang, Zhai and Luob; Liang & Luo Reference Liang and Luo2023), after the incident shock passes through the interface, both the transmitted shock and reflected wave rapidly propagate away from the interface. As a result, after a very short time, there are no large changes in pressure in the vicinity of the interface so that compressibility effects are weak in the lower Mach number cases (Richtmyer Reference Richtmyer1960). As the Mach number increases, the interface velocity shows a significant rise. For instance, in the cases of $M_s = 2.0$ considered in this work, the velocity ($\Delta U$) of the first interface is approximately 80 % of that ($V_{ts1}$) of the transmitted shock TS$_1$. The velocity ratio ($\Delta U/V_{ts1} \approx 0.8$) at $M_s = 2.0$ is much higher than the corresponding ratio ($\Delta U/V_{ts1} \approx 0.4$) at $M_s = 1.2$. As a result, in relatively high Mach number cases, the transmitted shock recedes slowly from the interface (Glendinning et al. Reference Glendinning2003; Motl et al. Reference Motl, Oakley, Ranjan, Weber, Anderson and Bonazza2009). This results in changes in pressure behind the transmitted shock influencing the interface perturbation growth for a relatively long time; thus, compressibility effects become significant in higher Mach number cases (Sadot et al. Reference Sadot, Rikanati, Oron, Ben-Dor and Shvarts2003; Guo et al. Reference Guo, Zhai, Ding, Si and Luo2020; Zhang et al. Reference Zhang, Zhou, Ding and Luo2022). The presence of compressibility effects adds complexity to the coupling between the two interfaces, possibly influencing the realization of freeze-out. Furthermore, compressibility effects may cause the two models of (2.22) and (2.23) to be invalid in predicting freeze-out, because these two models are initially derived by treating the shock as an instantaneous acceleration on incompressible fluids (Mikaelian Reference Mikaelian1995). In the present study, to examine the applicability and effectiveness of the models (2.22) and (2.23) under conditions of relatively high Mach numbers, the $M_s$ is consistently set at 2.0, except for cases specifically designed for freeze-out examinations at lower Mach numbers ($M_s = 1.2$ and 1.6).

We first employ real gases to form different $R_0$. After examining 56 real gases (see Appendix A for details of the gases), it is found that, under the total transmission condition, the largest $R_0$ is achieved when xenon (Xe) is used as fluid A and a mixture of decafluorobutane (C$_4$F$_{10}$) and neopentane (C$_5$H$_{12}$) is used as fluid B. Specifically, at $M_s = 1.2$, 1.6 and 2.0, the largest $R_0$ under the total transmission condition is determined to be 1.49, 1.40 and 1.36, respectively. Additionally, we consider the following two cases to investigate the potential effect of gas species on the validity of our freeze-out theory: krypton (Kr) is employed as fluid A and a mixture of C$_4$F$_{10}$ and C$_5$H$_{12}$ is employed as fluid B; argon (Ar) is employed as fluid A and a mixture of butane (C$_4$H$_{10}$) and propane (C$_3$H$_{8}$) is employed as fluid B. At $M_s = 2.0$, the $R_0$ value for these two cases under the total transmission condition is 1.35 and 1.31, respectively. Further details of the real-gas cases can be found in tables 1 and 2 (denoted as cases I$_1$1–I$_1$5 and I$_2$4$^m$–I$_2$8$^m$). The discussion about the total transmission under real-gas conditions is presented in Appendix A.

Notably, the largest $R_0$ formed by real gases is no more than 1.49 under the total transmission condition, as $M_s$ varies from 1.2 to 2.0. This density ratio is relatively small, so that it is insufficient for validating the effectiveness of our freeze-out theory from the perspective of $R_0$. To address this limitation, artificial gases are utilized, providing a convenient means to adjust $R_0$. In this work, using artificial gases, we form $R_0$ values with a range of 2–4. The upper limit for $R_0$ is based on the fact that, beyond $R_0 = 4$, the change in $\sin \theta ^\ast$ is minimal, resulting in a limited variation in interface coupling strength (see figure 3b). As illustrated in tables 1 and 2, for cases I$_1$6–I$_1$14, cases I$_2$1–I$_2$3 and cases I$_2$9$^m$–I$_2$15$^m$, the artificial gas is utilized as fluid A, and the real gas SF$_6$ is utilized as fluid B. The properties including $\rho _A$ and $\gamma _A$ for the artificial gases can be determined by (2.5).

The choice of $L_0$ involved in parametric studies is made to first ensure that the layer thickness ($L^\ast$) at the coupling onset time ($t^\ast$ in this work) remains under 1/3 of the interface wavelength. This criterion is based on the finding that the interface coupling becomes non-negligible when the layer thickness at the onset time is less than 1/3 of the interface wavelength (Taylor Reference Taylor1950). Particularly, in freeze-out test cases for the second interface, the choice of $L_0$ is further limited by $L_0^m$, as described in § 2.3.

In most cases, $\lambda$ is fixed at 120 mm, which is the same as the transverse size of the shock tube used in our previous work (Cong et al. Reference Cong, Guo, Si and Luo2022; Guo et al. Reference Guo, Cong, Si and Luo2022a,Reference Guo, Si, Zhai and Luob). To explore potential effects arising from variations in interface wavelengths on freeze-out, we also include two additional wavelengths of 240 and 480 mm. These values, generally unexplored in previous shock-tube studies, offer insights into the impact of different interface wavelengths on the freeze-out phenomenon.

Notably, (2.22) and (2.23) are derived based on the small-amplitude hypothesis (i.e. $ka_1^0 \ll 1$ and $ka_2^0 \ll 1$). As a result, the initial amplitudes of all cases in this work are set in accordance with this condition. When freeze-out for the first interface is examined, $a_2^0$ is first specified, and then the corresponding $a_1^0$ is determined by combining (2.15), (2.22), (2.14) and (2.8). When freeze-out for the second interface is examined, $a_1^0$ is first specified, and then the corresponding $a_2^0$ is determined by combining (2.8), (2.14), (2.23) and (2.15). For all cases, $ka_1^0$ has a range of 0.0084–0.0628, and $ka_2^0$ has a range of 0.0162–0.0628.

4.2. Freeze-out for the first interface

Cases I$_1$1–I$_1$3 and I$_1$11–I$_1$12, cases I$_1$6–I$_1$8, cases I$_1$1–I$_1$5 and I$_1$9–I$_1$10 and cases I$_1$13–I$_1$14, with their initial conditions summarized in table 1, are designed to examine the validity of (2.22) in predicting the freeze-out for the first interface from the perspectives of $M_s$, $L_0$, $R_0$ and $\lambda$, respectively. The inclusion of different gases in these cases also offers an opportunity to assess the validity of (2.22) from the perspective of the gas species. Additionally, case S1, as a single-mode scenario, is included for comparison. As illustrated in table 1, all values of $a_1^0$ determined by our freeze-out theory are smaller than the values of $a_2^0$.

Case I$_1$7 is taken as an example to illustrate the interface evolution observed in cases I$_1$1–I$_1$12, all featuring an identical wavelength of 120 mm. Figure 7(a) presents the numerical schlieren images of interface evolution for case I$_1$7. Notably, numbers in images of figure 7(a) denote the dimensionless time ($k\Delta Ut$). The fluid layer in case I$_1$7 first experiences a compression process resulting from the IS impact. Then, phase reversal of the second interface occurs after the impact of TS$_1$ ($k\Delta Ut = 1.9\unicode{x2013}6.3$). Note that, at $k\Delta Ut = 6.3$, the first interface's morphology remains almost unchanged compared with its post-shock state ($k\Delta Ut = 1.9$). This phenomenon occurs because the second interface has a higher initial amplitude and thus a faster amplitude growth rate (Richtmyer Reference Richtmyer1960; Meyer & Blewett Reference Meyer and Blewett1972) in comparison with the first interface. As a result, after phase reversal, the upstream growth of the second interface's bubble effectively squeezes the bubble of the first interface, hindering its evolution. As time elapses, the second interface constantly develops so that it imposes a stronger squeeze on the first interface. In a late stage ($k\Delta Ut = 13.9$), the coupling effect from the second interface even causes the bubble tip of the first interface to slightly protrude upstream, further impeding the first interface's evolution.

Figure 7. Numerical schlieren images of interface evolution for cases I$_1$7 (a) and I$_1$14 (b). Here, ‘Bu’ and ‘Sp’ denote bubbles and spikes, respectively, and ‘$x_{B1}$’ and ‘$x_{S1}$’ denote the tips of bubbles and spikes for the first interface, respectively. Numbers denote the dimensionless time $k\Delta Ut$.

Case I$_1$14 is taken as an example to illustrate the interface evolution observed in cases I$_1$13–I$_1$14, featuring larger interface wavelengths of 240 and 480 mm, respectively. Figure 7(b) presents the interface evolution for case I$_1$14. It is observed that, at a late stage ($k\Delta Ut = 13.9$), the bubble tip in case I$_1$14 becomes nearly flat but remains downstream of the spike tip, which is different from the observation in case I$_1$7. The distinction arises primarily from the larger initial amplitude and greater initial layer thickness in case I$_1$14. When subjected to the coupling effect of the second interface, the larger initial amplitude requires more time for the bubble tip of the first interface to move upstream of the spike tip. Moreover, the greater layer thickness allows for a longer transfer time of coupling effect from the second interface, thus delaying movement of the bubble tip at the first interface. As a result, the bubble tip of the first interface in case I$_1$14 is more prone to remain downstream of the spike tip compared with case I$_1$7.

The dimensionless amplitude growth of the first interface for cases I$_1$1–I$_1$14 is shown in figure 8, in which the time is scaled as $k\Delta Ut$ and the amplitude is scaled as $ka_1$. The amplitude $a_1$ for the evolving first interface is defined as $(x_{B1}-x_{S1})/2$, where $x_{B1}$ and $x_{S1}$ denote positions of the bubble tip and the spike tip for the first interface, respectively (see figure 7). It is observed that the amplitude data for cases I$_1$13–I$_1$14 do not collapse with those for cases I$_1$1–I$_1$12. This is primarily because the initial amplitudes for cases I$_1$13-I$_1$14 are larger than those for cases I$_1$1–I$_1$12, resulting in consistently higher evolving amplitude data for these two cases. Notably, the amplitude data for cases with higher $L_0$, $R_0$ and $M_s$ (such as cases I$_1$7–I$_1$8, I$_1$10, I$_1$13–I$_1$14) exhibit oscillations, which are attributed to the reflected rarefaction and compression waves from the 2-D diffuse interface, as detailed in § 3.2. The effect of the rarefaction waves leads to an increase in amplitude for the first interface, while compression waves contribute to a decrease, resulting in the oscillatory amplitude growth. As $L_0$ decreases, the period of the oscillation shortens; and as $R_0$ and $M_s$ decrease, the strength of the reverberating waves diminishes. Consequently, the oscillation is not evident for the cases with smaller values of $L_0$, $R_0$ and $M_s$.

Figure 8. Temporal variations of the first interface's amplitudes for cases I$_1$1–I$_1$14 in a dimensionless form. Case S1 is a single-mode case included for comparison.

Overall, the amplitude growths of the first interface for cases I$_1$1–I$_1$14 are roughly frozen, as illustrated in figure 8. This indicates that the model (2.22) is robust in predicting freeze-out for the first interface over a relatively wide range of initial conditions. Furthermore, (2.22) shows insensitivity to variations in $M_s$, $R_0$, $L_0$, $\lambda$ and gas species.

4.3. Freeze-out for the second interface

First, the influence of whether the second interface accomplishes phase reversal on the prediction of (2.23) is examined. Case I$_2$1, with $L_0 = 20$ mm, is designed as an incomplete phase-reversal scenario, and cases I$_2$2 and I$_2$3, with $L_0 = 35$ and 60 mm, are designed as completed phase-reversal scenarios. The selection of $L_0$ values for cases I$_2$1–I$_2$3 is based on the value of $L_0^m$, which is calculated as 28.7 mm using (2.27). Further details of the initial conditions for cases I$_2$1–I$_2$3 are presented in table 2. The numerical schlieren images of the interface evolution for cases I$_2$1–I$_2$3 are shown in figure 9. For case I$_2$1 with a smaller $L_0$, as shown in figure 9(a), the bubble tip of the first interface constantly squeezes that of the second interface and even occupies the position where the bubble tip of the second interface would be, thus preventing phase reversal of the second interface. Conversely, for cases I$_2$2 and I$_2$3 with a larger $L_0$, as shown in figure 9(b,c), phase reversal of the second interface is completed.

Figure 9. Numerical schlieren images of the shocked heavy fluid layer for cases I$_2$1 (a), I$_2$2 (b) and I$_2$3 (c). Here, ‘$x_{B2}$’ and ‘$x_{S2}$’ denote the tips of bubbles and spikes for the second interface, respectively. Numbers denote the dimensionless time $k\Delta U(t-t^\ast$).

The dimensionless amplitude growth of the second interface for cases I$_2$1–I$_2$3 is presented in figure 10, in which the time is scaled as $k\Delta U(t-t^\ast )$ and the amplitude is scaled as $ka_2$. The amplitude $a_2$ for the evolving second interface is defined as $(x_{B2}-x_{S2})/2$, where $x_{B2}$ and $x_{S2}$ denote positions of the bubble tip and the spike tip for the second interface, respectively (see figure 9). It is evident that the amplitude growth of the second interface for case I$_2$1 is frozen, while the two phase-reversal cases display continuous negative perturbation growth. This indicates that although (2.23) defines the relationship between $a_1^0$ and $a_2^0$, freeze-out for the second interface cannot be achieved once phase reversal is accomplished.

Figure 10. Time variations of the second interface's amplitudes for cases I$_2$1–I$_2$3 in a dimensionless form. Case S2 is a single-mode case included for comparison.

Second, the validity of (2.23) in predicting freeze-out for the second interface at $L_0 = L_0^m$ is examined. Cases I$_2$4$^m$–I$_2$6$^m$ and I$_2$12$^m$–I$_2$13$^m$, cases I$_2$4$^m$–I$_2$11$^m$ and cases I$_2$14$^m$–I$_2$15$^m$, with the initial conditions presented in table 2, are designed to examine the second interface's freeze-out from the perspectives of $M_s$, $R_0$ and $\lambda$, respectively. Figure 11 shows the dimensionless amplitude growths of the second interface in cases I$_2$4$^m$–I$_2$15$^m$. Given that cases I$_2$9$^m$–I$_2$15$^m$ have larger initial amplitudes in comparison with cases I$_2$4$^m$–I$_2$8$^m$, their amplitude growth data are correspondingly higher. Overall, the second interface's amplitude growths are roughly frozen for all of these cases, indicating that both (2.23) and (2.27) are necessary to achieve freeze-out for the second interface. Additionally, apart from the limitation of $L_0^m$, the model (2.23) is insensitive to variations in $M_s$, $R_0$, $\lambda$ and gas species.

Figure 11. Time variations of the second interface's amplitudes for cases I$_2$4$^m$–I$_2$15$^m$ in a dimensionless form.

In short, the smaller value of $a_2^0$ determined by our freeze-out theory compared with the value of $a_1^0$ (see table 2) results in a lower-amplitude growth rate for the second interface. As a result, the bubble of the first interface more effectively squeezes the counterpart of the second interface, preventing the second interface's phase reversal when $L_0 < L_0^m$ and thereby leading to freeze-out for the second interface.