4.1. Experimental observation

Schlieren images of the shock-induced quasi-1-D LMH fluid-layer evolution are shown in figure 3(a–c) for $2h_0=10.0$, 30.0 and 50.0 mm, respectively. The moment when the IS impacts the average position of the first initial interface (II$_1$) is defined as $t=0$. Taking the L50-1D case as an example, the wave patterns and interface motions are discussed in detail. After the IS impacts the II$_1$, the reflected shock (RF$_1$) and the transmitted shock (TS$_1$) are immediately generated, and the shocked first interface (SI$_1$) begins to move forwards (46 $\mathrm {\mu }$s). Then, the TS$_1$ impacts the second initial interface (II$_2$), and a transmitted shock (TS$_2$) is refracted outside the fluid layer and followed by the shocked second interface (SI$_2$) (262 $\mathrm {\mu }$s). Meanwhile, a weak shock wave (RF$_2$) is reflected inside the fluid layer since the SI$_2$ is a fast/slow interface relative to the motion of the TS$_1$. Theoretically, after the RF$_2$ impacts the SI$_1$, rarefaction waves (RW) should be reflected inside the fluid layer since the SI$_1$ is a slow/fast interface relative to the motion of the RF$_2$. However, due to the RW's limited strength, the density variation induced by the RW is restricted, and it is challenging to distinguish the RW. Finally, all waves are refracted outside the fluid layer, and the interfaces at the two sides of the fluid layer move at the same speed. Moreover, it is noted that the two interfaces coalesce to one at a late time when the initial layer is very thin (1288 $\mathrm {\mu }$s in the L10-1D case).

Figure 3.

Schlieren images of the shock-induced quasi-1-D LMH fluid-layer evolution in cases (a) L10-1D, (b) L30-1D and (c) L50-1D, where SI$_1$ (SI$_2$) denotes the shocked first (second) interface, RF$_1$ (TS$_1$) denotes the reflected (transmitted) shock after the IS impacts II$_1$ and RF$_2$ (TS$_2$) denotes the reflected (transmitted) shock after the TS$_1$ impacts II$_2$. Numbers indicate time in $\mathrm {\mu }$s.

The displacements ($y_{{SI}_n}$) and velocities ($v_{{SI}_n}$) of the two interfaces are measured from experiments ($n=1$ for the first interface and 2 for the second interface), and are shown in figures 4(a) and 4(b), respectively. The pixel size of the Schlieren images introduces the experimental measurement uncertainty and is 0.4 mm in this work. The size of the error bars equals the size of the symbols in the figures. Time is scaled as $tv_{{TS_1}}/(2h_0)$, interface displacement is scaled as $(y_{{SI}_n}-y_{01})/(2h_0)$, with $y_{01}$ denoting the initial position of the first interface, and interface velocity is scaled as $v_{{SI}_n}/\Delta v_{1}$. The values of $v_{{TS_1}}$ and $\Delta v_{1}$ in all cases are calculated according to the 1-D gas dynamics theory (Drake Reference Drake2018), as listed in table 2. The dimensionless displacements and velocities of the first (second) interface converge in all $h_0$ cases, as shown in figure 4(a,b), which indicates that the dimensionless movements of the two interfaces are independent of the initial layer thickness.

Figure 4.

The dimensionless displacements (a) and velocities (b) of the first interface (solid symbols) and the second interface (hollow symbols). Black and orange solid lines represent the 1-D theory predictions for the first interface motions in Stages I ($t_{1}^{*}>t>0$) and II ($t>t_{1}^{*}$), respectively. Black and purple dashed lines represent the 1-D theory predictions for the second interface motions in Stages I ($t_{2}^{*}>t>t_{2}^{0}$) and III ($t>t_{2}^{+}$), respectively.

Table 2.

Physical parameters of the waves and interfaces of an LMH fluid layer in the laboratory reference, where $v_{{TS_1}}$ denotes the speed of the TS$_1$, $v_{{RF}_2}$ denotes the speed of the RF$_2$, $v_{{RWH}}$ denotes the speed of the RWH, $v_{{RWT}}$ denotes the speed of the RWT, $\Delta v_{1}$ ($\Delta v_{2}$) denotes the speed of the SI$_1$ (SI$_2$) in Stage I, $v_{1}^*$ denotes the speed of the SI$_1$ in Stage II, $v_{2}^+$ denotes the speed of the SI$_2$ in Stage III, $v_{fn}$ denotes the asymptotic velocity of the two interfaces, $t_{2}^0$ denotes the specific time when the TS$_2$ impacts II$_2$, $t_{1}^*$ denotes the specific time when the RF$_2$ impacts SI$_1$ and $t_{2}^*$ denotes the specific time when the RWH impacts SI$_2$. RWT denotes the rarefaction wave tail. The units for speed and time are m s$^{-1}$ and $\mathrm {\mu }$s, respectively.

4.2. Characteristics of reverberating waves

The waves reverberating inside an LMH fluid layer are sketched in figure 5(a–d). To have a clear description of the interaction of a reverberating wave and interface, we first define several specific times including $t_2^0$ when the TS$_1$ impacts the II$_2$, $t_1^*$ when the RF$_2$ impacts the SI$_1$ and $t_2^*$ when the rarefaction wave head (RWH) impacts the SI$_2$. Then $t_2^0$, $t_1^*$ and $t_2^*$ can be evaluated as

(4.1)\begin{equation} \left. \begin{aligned} & t_2^0=\frac{2h_0}{v_{{TS_1}}},\\ & t_1^*=t_2^0+\frac{2h_0-t_2^0\Delta v_1}{\Delta v_1-v_{{RF}_2}},\\ & t_2^*=t_1^*+\frac{2h_0-t_2^0\Delta vv_1+(t_1^*-t_2^*)(\Delta v_2-\Delta v_1)}{v_{{RWH}}-\Delta v_2}, \end{aligned} \right\} \end{equation}

in which $v_{{RF}_2}$ represents the speed of the RF$_2$, $v_{{RWH}}$ represents the speed of the RWH. The values of $v_{{RF}_2}$, $v_{{RWH}}$ and $\Delta v_2$ are calculated according to the 1-D gas dynamics theory (Drake Reference Drake2018), and then $t_2^0$, $t_1^*$ and $t_2^*$ are obtained, as listed in table 2.

Figure 5.

Sketches of (a) the interaction of the TS$_1$ and II$_2$ at $t_2^0$, (b) the interaction of the RF$_2$ and SI$_1$ at $t_1^*$, (c) the interaction of the RW and SI$_2$ at $t_2^*$ and (d) the motions of the SI$_1$ and SI$_2$ when $t>t_2^+$.

After the RF$_2$ impacts the SI$_1$, the speed of the SI$_1$ decreases from $\Delta v_1$ to $v_1^*$. The values of $v_1^*$ can be calculated according to the 1-D gas dynamics theory (Drake Reference Drake2018), as listed in table 2. After the RW impact the SI$_2$, the speed of the SI$_2$ reduces from $v_2^\alpha$ to $v_2^+$. The interaction of the RW and a fast/slow interface (i.e. SI$_2$) is different from the interaction of the RW and a slow/fast interface previously investigated (Liang & Luo Reference Liang and Luo2021a). Based on the 1-D gas dynamics theory (Velikovich & Phillips Reference Velikovich and Phillips1996; Drake Reference Drake2018), $v_2^+$ can be calculated as

(4.2)\begin{equation} v_2^+{=}\Delta v_2+\frac{2c_5}{\gamma_5-1}[p_{31}^{(\gamma_5-1)/2\gamma_5}-1], \end{equation}

where $\gamma _5$ is the specific heat ratio of gas C, $c_5$ is the sound speed of gas C after the TS$_1$ impacts the II$_2$. The only unknown parameter $p_{31}$, i.e. the ratio of the pressure of the flow after the RWT leaves the SI$_2$ ($p_3$) to the pressure of the flow between the RF$_2$ and SI$_2$ ($p_1$), as sketched in figure 5(b,d), satisfies

(4.3) \begin{equation} \frac{c_5}{c_1}=\frac{(\gamma_5-1)[p_{31}^{(\gamma_1-1)/2\gamma_1}-2p_{21}^{(\gamma_1-1)/2\gamma_1}+1]}{(\gamma_1-1)[1-p_{31}^{(\gamma_5-1)/2\gamma_5}]}, \end{equation}

where $\gamma _1$ is the specific heat ratio of gas B, $c_1$ is the sound speed of gas B after the TS$_1$ impacts the II$_2$ and $p_{21}$ is the ratio of the pressure of the flow between the RWT and SI$_1$ ($p_2$) to $p_1$. Then the $v_2^+$ can be calculated by solving (4.2)–(4.3), as listed in table 2.

The number of reverberating waves is infinite. However, the consecutive changes of pressure and velocity imparted by these reverberations decrease continuously, eventually converging to a well-defined asymptotic 1-D post-shock state. The influences of the reverberating waves on the 1-D and 2-D dynamics of the two interfaces should be considered (Aglitskiy et al. Reference Aglitskiy2006) until the two interfaces enter the asymptotic post-shock state. A simple and advantageous method is adopted to judge whether the higher-order reverberations should be accounted for or not. The interaction of an IS with a semi-infinite interface separating two fluids with $\rho _1$ and $\rho _3$ is considered, and the intermediate layer is ignored. Then, the final post-shock velocity, $v_{{fn}}$, can be calculated according to the 1-D gas dynamics theory (Drake Reference Drake2018). As listed in table 2, the asymptotic velocity $v_{{fn}}$ is the same as $v_1^*$ and is 0.1 % smaller than $v_2^+$. Therefore, it is reasonable to regard that, after the RWT leaves the SI$_2$, the motions of the two interfaces enter the asymptotic state. Unlike the sudden acceleration imposed by a shock wave on an interface, the interaction of the RW and an interface is generally regarded as a continuous process (Li & Book Reference Li and Book1991; Li, Kailasanath & Book Reference Li, Kailasanath and Book1991; Morgan, Likhachev & Jacobs Reference Morgan, Likhachev and Jacobs2016; Morgan et al. Reference Morgan, Cabot, Greenough and Jacobs2018; Liang et al. Reference Liang, Zhai, Luo and Wen2020b; Liang & Luo Reference Liang and Luo2021a). The length of the RW ($L_{RW}$), the interaction time of the RW with SI$_2$ ($T_{{RW}}$) and the average deceleration imposed on the SI$_2$ by the RW ($\bar {g}_{{RW}}$) are separately deduced as

(4.4)\begin{equation} \left. \begin{aligned} & L_{RW}=(v_{{RWH}}-v_{{RWT}})(t_2^*-t_1^*),\\ & T_{{RW}}=\frac{2L_{RW}}{2u_{{RWT}}-\Delta v_2-v_2^+},\\ & \bar{g}_{{RW}}=\frac{v_2^+{-}\Delta v_2}{T_{{RW}}}, \end{aligned} \right\} \end{equation}

where $u_{{RWT}}$ denotes the speed of the RWT. The values of $L_{RW}$, $T_{{RW}}$ and $\bar {g}_{{RW}}$ in all cases are calculated according to (4.4), as listed in table 3. As the layer thickness increases, both the $L_{RW}$ and $T_{{RW}}$ increase, whereas the $|\bar {g}_{{RW}}|$ decreases.

Table 3.

Physical parameters of the RW inside an LMH fluid layer. Here, $L_{{RW}}$ denotes the length of the RW at $t_2^*$; $T_{{RW}}$ denotes the interaction time of the RW and SI$_2$, and $\bar {g}_{{RW}}$ denotes the average deceleration imposed on the SI$_2$ by the RW. The units for length, time and deceleration are mm, $\mathrm {\mu }$s and $10^6\times$ m s$^{-2}$, respectively.

It is evident that the motions of the two interfaces of a shocked LMH fluid layer can be divided into several stages, as illustrated in figure 4(b). Specifically, the first interface's motion can be separated into two stages: Stage I. Uniform motion with the $\Delta v_1$ during $t_1^*>t>0$; Stage II. Uniform motion with the $v_1^*$ during $t>t_1^*$. The second interface's motion can be divided into three stages: Stage I. Uniform motion with the $\Delta v_2$ during $t_2^*>t>t_2^0$; Stage II. Deceleration with the $\bar {g}_{{RW}}$ during $t_2^+>t>t_2^*$ with $t_2^+=t_2^*+T_{{RW}}$; Stage III. Uniform motion with the $v_2^+$ during $t>t_2^+$. In fact, the motions of the two interfaces can be described by the derived specific times (4.1) and interface velocities ((4.2)–(4.4)), which actually form a general 1-D theory for the interface motion. The predictions of the 1-D theory for the first interface motions in Stages I and II are marked with black and orange solid lines in figure 4(a,b), and they agree well with the experimental results. The predictions of the 1-D theory for the second interface motions in Stages I and III are marked as black and purple dashed lines, and they agree well with the experimental results. Due to the limited duration of Stage II for the second interface (e.g. $T_{{RW}}$ is only around 1 $\mathrm {\mu }$s in this study), the second interface motions in Stage II are not represented in figure 4(a,b).

5.1. Experimental results

Schlieren images of the shock-induced quasi-2-D LMH fluid-layer evolution are shown in figure 6(a–f). Taking the L50-IP case as an example, the deformations of the two interfaces are discussed in detail. After the IS impacts the perturbed II$_1$, the rippled RF$_1$ is reflected outside the fluid layer, and the rippled TS$_1$ moves towards the II$_2$ (53 $\mathrm {\mu }$s). Meanwhile, the perturbation on the SI$_1$ increases due to the vorticity deposited by the IS on the SI$_1$. After the TS$_1$ impacts the perturbed II$_2$, the rippled TS$_2$ is refracted outside the fluid layer, and the rippled RF$_2$ is reflected inside the fluid layer. Meanwhile, the perturbation on the SI$_2$ increases due to the vorticity deposited by the TS$_1$ on the SI$_2$ (237 $\mathrm {\mu }$s). Finally, all waves are refracted outside the fluid layer, and the two interfaces evolve with the same phase as their initial perturbations. In addition, different from the large $h_0$ cases, portions of the two interfaces collide when $h_0$ is limited. For example, bubbles of the two interfaces collide at 1284 $\mathrm {\mu }$s in the L10-IP case, and the bubble of the SI$_1$ collides with the spike of the SI$_2$ at 509 $\mathrm {\mu }$s in the L10-AP case.

Figure 6.

Schlieren images of the shock-induced quasi-2-D LMH fluid-layer evolution in cases (a) L10-IP, (b) L10-AP, (c) L30-IP, (d) L30-AP, (e) L50-IP and ( f) L50-AP. Here, ‘bub’ and ‘spi’ denote bubble and spike, respectively. Numbers denote time in $\mathrm {\mu }$s.

The experimental amplitude growth rate of the first interface ($\dot {a}_1^{{exp}}$) and that of the second interface ($\dot {a}_2^{{exp}}$) in Stage I are acquired by linearly fitting the experimental data, as listed in table 4. In the three in-phase cases, as $h_0$ decreases, both $v_1^{{exp}}$ and $\dot {a}_2^{{exp}}$ increase. However, in the three anti-phase cases, as $h_0$ decreases, both $\dot {a}_1^{{exp}}$ and $|\dot {a}_2^{{exp}}|$ decrease. The results indicate that the RMI of an LMH fluid layer is influenced by the initial layer thickness and phases of the two interfaces, i.e. the interface-coupling effect. Due to the interface-coupling effect on the RMI of the second interface in the L10-AP case, an opposite sign of the amplitude growth rate of a fast/slow interface to its initial perturbation amplitude is observed. For a semi-infinite fast/slow Richtmyer–Meshkov (RM) unstable interface, the sign of its amplitude growth rate should be the same as the sign of its initial amplitude (Richtmyer Reference Richtmyer1960). In general, the amplitude growth rate of a semi-infinite slow/fast RM unstable interface has the opposite sign to its initial amplitude, which is ascribed to the phase reversal (Brouillette Reference Brouillette2002). The phase reversal of a shocked fast/slow interface (the second interface in the L10-AP case) is called ‘abnormal phase reversal’ in this study.

Table 4.

The linear amplitude growth rates of the two interfaces of an LMH fluid layer, where $\dot {a}_1^{{exp}}$ ($\dot {a}_2^{{exp}}$) denotes the experimental amplitude growth rate of the first (second) interface in Stage I, $\dot {a}_1^l$ ($\dot {a}_2^l$) denotes the amplitude growth rate of the first (second) interface in Stage I calculated with (2.14) (2.15); $\psi _1$ ($\psi _2$) quantifies the interface-coupling effect on the RMI of the first (second) interface calculated with (5.1); $\dot {a}_1^{{*}}$ denotes the amplitude growth rate of the first interface at $t_1^*$ calculated with (5.2) and $\dot {a}_2^{{+}}$ denotes the amplitude growth rate of the second interface at $t_2^+$ calculated with (5.4)–(5.6).

The values of $\dot {a}_1^l$ and $\dot {a}_2^l$ are calculated with the LL model and listed in table 4, and a good agreement with the experimental results is found. It is noted that $\dot {a}_2^l$ also has the opposite sign to $a_2(0)$ in the L10-AP case.

5.2. The interface-coupling effect

To quantify the interface-coupling effect on the RMI of an arbitrary fluid layer, we first propose a normalised growth rate ($\psi _n$) which is defined as the ratio of the amplitude growth rate of the $n$th interface calculated with the LL model to that calculated with the impulsive theory (Richtmyer Reference Richtmyer1960),

(5.1)\begin{equation} \left. \begin{aligned} & \psi_1=\frac{(R_1+1)(2R_3\xi+\xi^2+1)-(R_1+1)(R_3-1)(1-\xi^2)/[\epsilon(R_1-1)]}{2\xi(R_1R_3+1)+(R_1+R_3)(\xi^2+1)},\\ & \psi_2=\frac{\epsilon(R_1-1)(R_3+1)(\xi^2-1)/(R_3-1)+(R_3+1)(2R_1\xi+\xi^2+1)}{2\xi(R_1R_3+1)+(R_1+R_3)(\xi^2+1)}. \end{aligned} \right\} \end{equation}

The values of $\psi _1$ and $\psi _2$ in all the LMH layer cases are listed in table 4. It is found that $\psi _1>1$ and $\psi _2>1$ in the three in-phase cases, and $\psi _1<1$ and $|\psi _2|<1$ in the three anti-phase cases, indicating that the interface-coupling effect leads to the two in-phase interfaces being more unstable than the semi-infinite RMI, and the two anti-phase interfaces being more stable than the semi-infinite RMI, under the LMH fluid-layer condition. The conclusion is opposite to the A/B/A-type fluid-layer counterpart, i.e. the interface-coupling effect results in the two in-phase interfaces of an A/B/A-type fluid layer being more stable, and the two anti-phase interfaces being more unstable (Liang & Luo Reference Liang and Luo2021a, Reference Liang and Luo2022).

It is proved that the interface-coupling effect is determined by the density ratios of the fluids outside the layer to the fluid inside the layer ($R_1$ and $R_3$), the post-shock amplitude ratio of the two interfaces ($\epsilon$) and the dimensionless post-shock layer thickness ($Z_Lkh_0$). The values of $\psi _{n}$ of an LMH fluid layer vs $Z_Lkh_0$ under different $R_1$, $R_3$ and $\epsilon$ conditions are calculated based on (5.1), as shown in figure 7(a–d), respectively.

Figure 7.

Value of $\psi _{n}$ of the $n$th interface vs $Z_Lkh_0$ under (a) different $R_1$ conditions, (b) different $R_3$ conditions, (c) different $\epsilon$ ($>0$) conditions and (d) different $\epsilon$ ($<0$) conditions. Hollow circles represent the instability freeze out of the first or second interface.

In figure 7(a), we plot $\psi _1$ and $\psi _2$ as functions of $Z_Lkh_0$ for the case $R_1$ varying from 0.1 to 1.0, $R_3=2.0$ and $\epsilon =1.0$. When $Z_Lkh_0<2$, $\psi _1>1$ and $\psi _2>1$ indicate that the interface-coupling effect results in the two interfaces of an LMH fluid layer being more unstable if the two interfaces are initially in phase. Specifically, when $R_1\leq (1/R_3)$ ($R_1>(1/R_3)$), $\psi _1\leq \psi _2$ ($\psi _1>\psi _2$), indicating the interface-coupling effect destabilises the second interface more (less) than the first interface. When $Z_Lkh_0>2$, $\psi _1$ and $\psi _2$ converge at 1, revealing that the interface-coupling effect has a negligible influence on the RMI of a fluid layer.

In figure 7(b), we plot $\psi _1$ and $\psi _2$ as functions of $Z_Lkh_0$ for the case $R_3$ varying from 1.0 to 10.0, $R_1=0.5$ and $\epsilon =1.0$. It is found that, when $R_3\leq (1/R_1)$ ($R_3>(1/R_1)$), $\psi _1\leq \psi _2$ ($\psi _1>\psi _2$), indicating the interface-coupling effect destabilises the second interface more (less) than the first interface, which agrees well with the findings in figure 7(a).

In figure 7(c), we plot $\psi _1$ and $\psi _2$ as functions of $Z_Lkh_0$ for the case $\epsilon$ varying from 0.1 to 10.0, $R_1=0.5$ and $R_3=2.0$. It is found that when $3.0>\epsilon >0$, $\psi _1>1$ and $\psi _2>1$, indicating that the interface-coupling effect results in both interfaces being more unstable. Specifically, the interface-coupling effect promotes the RMI of the first interface more than the second interface counterpart (i.e. $\psi _1>\psi _2$) when $1.0>\epsilon >0$, and the interface-coupling effect promotes the RMI of the second interface more than the first interface counterpart (i.e. $\psi _1<\psi _2$) when $3.0>\epsilon >1.0$. In addition, when $\epsilon >3.0$, $\psi _1<1$ and $\psi _2>1$, demonstrating that the interface-coupling effect leads to the first interface being more stable, and the second interface being more unstable. Overall, the two in-phase interfaces of an LMH fluid layer are generally destabilised by the interface-coupling effect, but the first interface might be stabilised if the $\epsilon$ is large.

In figure 7(d), we plot $\psi _1$ and $\psi _2$ as functions of $Z_Lkh_0$ for the case $\epsilon$ varying from $-$0.1 to $-$10.0, $R_1=0.5$ and $R_3=2.0$. It is found that, when $0>\epsilon >-2.0$ and $Z_Lkh_0<2$, $\psi _1<0$ and $\psi _2>0$, indicating that the interface-coupling effect results in the abnormal phase reversal of the first interface. The interface-coupling effect leads to the first interface being more unstable when $0>\epsilon >-0.75$ and the second interface being more stable when $0>\epsilon >-0.5$. When $\epsilon <-2.0$, $1>\psi _1>0$ and $\psi _2<0$, demonstrating that the interface-coupling effect causes the first interface to be more stable, and abnormal phase reversal of the second interface. Moreover, the instability freeze out of the first interface or the second interface occurs when $\psi _1=0$ or $\psi _2=0$, as marked with hollow circles in figure 7(d).

In summary, (i) when $Z_Lkh_0<2$, the interface-coupling effect has a non-negligible influence on the RMI of a fluid layer; (ii) the interface-coupling effect destabilises (stabilises) the $n$th interface when $|\psi _n|>1$ ($|\psi _n|<1$); (iii) the two in-phase interfaces of an LMH fluid layer are generally destabilised by the interface-coupling effect, but the first interface might be stabilised if $\epsilon$ is large; (iv) the instability freeze out and the abnormal phase reversal of the $n$th interface occurs when $\psi _n=0$ and $\psi _n<0$, respectively.

5.3. Effect of reverberating waves

As stated in the 1-D cases, there exist reverberating waves inside the layer which in turn will interact with the two interfaces. In the 2-D cases, reverberating waves will impose additional vorticity and/or accelerations on the interface in its Stage II.

In Stage II for the first interface, the RF$_2$ impulsively impacts the SI$_1$ and deposits the additional vorticity on the first interface, causing the post-shock amplitude growth rate of the first interface, i.e. $\dot {a}_1^*$, to be different from $\dot {a}_1^l$. Because of the baroclinic mechanism created by the misalignment of the density gradient ($\boldsymbol {\nabla }\rho$) and the pressure gradient ($\boldsymbol {\nabla } p$), the RF$_2$ deposits vorticity with an opposite direction to the vorticity deposited by the IS on the first interface, as sketched in figure 8(a). Therefore the RF$_2$ imposes an additional RMI on the first interface and leads to the first interface being more stable. Since the first interface evolves in the linear regime before the RF$_2$ impacts it, it is reasonable to adopt the re-shock impulsive theory proposed by Mikaelian (Reference Mikaelian1985) to modify the LL model and calculate $a_1^*$ as

(5.2)\begin{equation} \dot{a}_1^{*}=\dot{a}_1^{l}+ka_1^{*}A_1(v_1^*-\Delta v_1), \end{equation}

with $a_1^{*}$ denoting the first interface amplitude at $t_1^*$

(5.3)\begin{equation} a_1^{*}=Z_1a_1(0)+\dot{a}_1^{l}t_1^*. \end{equation}

The values of $\dot {a}_1^{*}$ in all cases are listed in table 4.

Figure 8.

Sketches of the interaction of the RF$_2$ and SI$_1$ (a) and the interaction of the RW and SI$_2$ (b). The blue (red) arc with an arrow in (a) illustrates the vorticity deposition induced by the IS (RF$_2$) on the first interface. The blue (orange) arc with an arrow in (b) represents the vorticity deposition induced by the TS$_1$ (RW) on the second interface. The purple (green) arrows represent the pressure (density) gradient $\boldsymbol {\nabla } p$ ($\boldsymbol {\nabla }\rho$).

In Stage II for the second interface, the RW continuously deposit additional vorticity on the second interface, causing the post-rarefaction amplitude growth rate of the second interface, i.e. $\dot {a}_2^+$, to be different from $\dot {a}_2^l$. The RW induce vorticity deposition with the opposite direction to the vorticity deposited by the TS$_1$ on the second interface, as sketched in figure 8(b). Therefore, the RW impose RTS on the second interface, leading to the second interface being more stable. Here, we adopt the potential flow model proposed by Zhang & Guo (Reference Zhang and Guo2016) considering the universal curves of all spikes and bubbles at any density ratio, to quantify the RTS imposed by the RW on the second interface and modify the LL model. The expressions for the second interface bubble/spike amplitude growth rate ($\dot {a}_{2b/2s}^{*}$) in Stage II are

(5.4)\begin{equation} \ddot{a}_{2b/2s}^{*}=\alpha_{b/s}k[(\dot{a}_{2b/2s}^{*})^{2}-(\dot{a}_{2b/2s}^{qs})^2], \end{equation}

with

(5.5)\begin{equation} \dot{a}_{2b/2s}^{qs}=\sqrt{\frac{A_2|\bar{g}_{{RW}}|}{3k}\frac{8}{(1\pm A_2)(3\pm A_2)}\frac{\left[3\pm A_2+\sqrt{2(1\pm A_2)}\right]^{2}}{\left[4(3\pm A_2)+\sqrt{2(1\pm A_2)}(9\pm A_2)\right]}}, \end{equation}

and

(5.6)\begin{equation} \alpha_{b/s}=\frac{3}{4}\frac{(1\pm A_2)(3\pm A_2)}{\left[3\pm A_2+\sqrt{2(1\pm A_2)}\right]}\frac{\left[4(3\pm A_2)+\sqrt{2(1\pm A_2)}(9\pm A_2)\right]}{\left[\sqrt{3\pm A_2}+2\sqrt{2(1\pm A_2)}(3\mp A_2)\right]}. \end{equation}

The upper (lower) sign of $\pm$ and $\mp$ applies to the bubble (spike). Note that the interface-coupling effect is considered since the initial $\dot {a}_{2b/2s}^{*}$ equals $\dot {a}_2^l$ at $t_2^*$. The average values of $\dot {a}_{2b}^{*}$ and $\dot {a}_{2s}^{*}$ at $t_2^+$, i.e. $\dot {a}_2^+$, in all cases are calculated based on (5.4)–(5.6), as listed in table 4. The value of $\dot {a}_2^+$ is only $0.1\,\mathrm {m}\,\mathrm {s}^{-1}$ in the L10-AP case, which indicates that the second interface's instability is close to being frozen due to both the interface-coupling effect and the RTS induced by the RW. In summary, the reverberating waves inside an LMH fluid layer, including the RF$_2$ and RW, stabilise the two interfaces.

5.4. Nonlinear model for a shocked LMH fluid layer

The time-varying amplitudes of the two interfaces are measured and are shown in figures 9(a) and 9(b). These clearly show a nonlinear behaviour after a certain time. For the first interface, time is scaled as $\tau _1=k\dot {a}_1^lt$, and amplitude is scaled as $\zeta _1=k(a_1-Z_1a_1(0))$. For the second interface, time is scaled as $\tau _2=k|\dot {a}_2^l|(t-t_2^0)$ and amplitude is scaled as $\zeta _2=k(|a_2|-|Z_2a_2(0)|)$. The dimensionless $a_1$ converges in all cases, indicating that the initial layer-thickness and amplitude combination has limited influence on the nonlinear behaviour of the first interface perturbation growth. The dimensionless $a_2$ converges in all cases except for the L10-AP case (due to the abnormal phase reversal of the second interface in the L10-AP case). The LL model's predictions for $a_1$ and $a_2$ by considering the interface-coupling effect and the additional instabilities induced by the reverberating waves are shown with black lines in figure 9(a,b), and they agree well with all the experimental results before dimensionless time 0.4. Due to the nonlinearity effect on the RMI (Velikovich & Dimonte Reference Velikovich and Dimonte1996; Zhang & Sohn Reference Zhang and Sohn1997; Nishihara et al. Reference Nishihara, Wouchuk, Matsuoka, Ishizaki and Zhakhovsky2010), $a_1$ and $a_2$ deviate from the LL model's predictions after dimensionless time 0.4. Therefore, the nonlinearity effect on the RMI of an LMH fluid layer should be considered in Stage II for the first interface and in Stage III for the second interface.

Figure 9.

Comparisons of the dimensionless amplitudes of the first interface (a), and the second interface (b). Black and orange lines represent the theoretical predictions for the two interface amplitudes based on the LL model and the LL-DR model, respectively.

The nonlinearity of single-mode RMI generally can be interpreted by the perturbation expansion model (Zhang & Sohn Reference Zhang and Sohn1997; Vandenboomgaerde, Gauthier & Mügler Reference Vandenboomgaerde, Gauthier and Mügler2002), the vortex model (Jacobs & Sheeley Reference Jacobs and Sheeley1996; Peng et al. Reference Peng, Yang, Wu and Xiao2021), the potential flow model (Hecht, Alon & Shvarts Reference Hecht, Alon and Shvarts1994; Mikaelian Reference Mikaelian1998, Reference Mikaelian2003; Sadot et al. Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998; Zhang Reference Zhang1998; Goncharov Reference Goncharov2002; Sohn Reference Sohn2003; Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010; Zhang & Guo Reference Zhang and Guo2016; Zhang, Deng & Guo Reference Zhang, Deng and Guo2018), etc. First, the perturbation expansion model (Zhang & Sohn Reference Zhang and Sohn1997; Vandenboomgaerde et al. Reference Vandenboomgaerde, Gauthier and Mügler2002) is applicable for quantifying the development of high-order harmonics and describing the interface deformation in the weakly nonlinear regime. Since the effective time scope of the perturbation expansion model is short, we do not utilise such a model to quantify the nonlinearity of RMI. Second, the vortex model (Jacobs & Sheeley Reference Jacobs and Sheeley1996; Peng et al. Reference Peng, Yang, Wu and Xiao2021) was derived with the assumption that the ‘mushroom’ structure on the spike is the manifestation of the vortex core induced by the initial thin vortex sheet on the interface. In other words, there should be enough time for the vortex on the interface to merge into point vortices. The vortex model is applicable for the RMI in the late nonlinear regime and, therefore, we do not adopt such a model to quantify the nonlinearity of the RMI. Third, most potential flow models (Hecht et al. Reference Hecht, Alon and Shvarts1994; Mikaelian Reference Mikaelian1998, Reference Mikaelian2003; Zhang Reference Zhang1998; Goncharov Reference Goncharov2002; Sohn Reference Sohn2003) established two decades ago are suitable for quantifying the asymptotic growth trend of the nonlinear RMI. The potential flow model deduced by Sadot et al. (Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998) is the pioneering one covering both the linear growth rate and the nonlinearity of the RMI. However, tuning parameters $C$ and $E$ are adopted in such a model to match experimental results, and their exact values at intermediate Atwood numbers remain uncertain (Mansoor et al. Reference Mansoor, Dalton, Martinez, Desjardins, Charonko and Prestridge2020). The potential flow models recently proposed by Zhang & Guo (Reference Zhang and Guo2016) and Zhang et al. (Reference Zhang, Deng and Guo2018) have attracted much attention and have been validated by several independent experiments and simulations. Nevertheless, such potential flow models are available when the initial amplitude–wavelength ratio is small ($<0.04$).

The potential flow model proposed by Dimonte & Ramaprabhu (Reference Dimonte and Ramaprabhu2010), i.e. the DR model, considering various amplitude–wavelength ratios and density ratios, has been adopted to successfully quantify the nonlinearity of RMI of SF$_6$ gas layers (Liang & Luo Reference Liang and Luo2021a) and helium gas layers (Liang & Luo Reference Liang and Luo2022). Here, we extend the LL model to a nonlinear one by combining it with the DR model to describe the nonlinearity of RMI. The expressions of the combined model (i.e. LL–DR model) for the first interface bubble/spike amplitude growth rate ($\dot {a}_{1b/1s}^{nl}$) and the second interface bubble/spike amplitude growth rate ($\dot {a}_{2b/2s}^{nl}$) are

(5.7)\begin{equation} \left. \begin{aligned} & \dot{a}_{1b/1s}^{nl}=\frac{\dot{a}_{1}^{*}\left[1+(1\mp A_1)k\dot{a}_{1}^{*}t\right]}{1+C_{1b/1s}k\dot{a}_{1}^{*}t+(1\mp A_1)F_{1b/1s}(k\dot{a}_{1}^{*}t)^2},\\ & \dot{a}_{2b/2s}^{nl}=\frac{\dot{a}_{2}^{+}\left[1+(1\mp A_2)k|\dot{a}_{2}^{+}|t\right]}{1+C_{2b/2s}k|\dot{a}_{2}^{+}|t+(1\mp A_2)F_{2b/2s}(k|\dot{a}_{2}^{+}|t)^2}, \end{aligned} \right\} \end{equation}

with

(5.8)\begin{equation} \left. \begin{aligned} & C_{1b/1s}=\frac{4.5\pm A_1+(2\mp A_1)kZ_1a_1(0)}{4},\\ & C_{2b/2s}=\frac{4.5\pm A_2+(2\mp A_2)k|Z_2a_2(0)|}{4}, \end{aligned} \right\} \end{equation}

and

(5.9a,b)\begin{equation} F_{1b/1s}=1\pm A_1,\quad F_{2b/2s}=1\pm A_2. \end{equation}

The upper (lower) sign of $\pm$ and $\mp$ in (5.7)–(5.9a,b) applies to the bubble (spike). The LL–DR model predictions for $a_1$ and $a_2$ are shown with orange lines in figures 9(a) and 9(b), respectively, and they agree well with the experimental results.