3.1. Quasi-1-D experimental results and analysis

Schlieren images of the shock-induced quasi-1-D $\textrm {SF}_6$ gas layer evolution are shown in figures 2(a)–2(c) for $L_0=10$, 30 and 50 mm, respectively. The moment when the IS impacts the average position of $\textrm {II}_1$ 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 $\textrm {II}_1$, the $\textrm {TS}_1$ is generated, and the shocked first interface ($\textrm {SI}_1$) begins to move forwards ($58\ \mathrm {\mu }\textrm {s}$). Then the $\textrm {TS}_1$ impacts the $\textrm {II}_2$ ($274 \ \mathrm {\mu }\textrm {s}$), and the $\textrm {TS}_2$ moves outside the fluid layer followed by the shocked second interface ($\textrm {SI}_2$). Meanwhile, the RW is reflected inside the fluid layer and moves towards the $\textrm {SI}_1$ ($458\ \mathrm {\mu }\textrm {s}$), because the second interface is a slow/fast interface relative to the RW motion. Subsequently, the RW impacts and accelerates the $\textrm {SI}_{1}$. Meanwhile, the CW is reflected inside the fluid layer ($658\ \mathrm {\mu }\textrm {s}$) and moves towards the $\textrm {SI}_2$ because the first interface is a slow/fast interface relative to the CW motion. Later, the CW impacts and accelerates the $\textrm {SI}_2$. After the CW passes the $\textrm {SI}_2$, the waves reflected inside the fluid layer are feeble and therefore ignored in this work. Finally, all waves refract outside the fluid layer and the interfaces at two sides of the fluid layer move with the same speed ($958\ \mathrm {\mu }\textrm {s}$). Notably, owing to the limited $L_0$ in the L10-1D case, the two interfaces coalesce to one at a late time ($958\ \mathrm {\mu }\textrm {s}$).

Figure 2.

Schlieren images of the shock-induced quasi-1-D $\textrm {SF}_6$ gas-layer evolution in cases (a) L10-1D, (b) L30-1D and (c) L50-1D. Here $\textrm {SI}_1$ ($\textrm {SI}_2$) denotes the shocked first (second) interface, $\textrm {TS}_1$ denotes the transmitted shock after the IS impacts the $\textrm {II}_1$ and $\textrm {TS}_2$ denotes the transmitted shock after the $\textrm {TS}_1$ impacts the $\textrm {II}_2$. Numbers denote time in microseconds.

The interface displacements ($x_{SI_n}$) and velocities ($u_{SI_n}$) of the two interfaces are measured from experiments ($n=1$ for the first interface and $n=2$ for the second interface), as shown in figures 3(a) and 3(b), respectively. We measure the positions of the top, middle and bottom points of both interfaces from each schlieren image and calculate the average value of the three-point positions as the interface position. The pixel size of schlieren images introduces the experimental measurement uncertainty, which is 0.4 mm in this work. The size of error bars equals the size of symbols representing the experimental results. Time is scaled as $tu_{t1}/L_0$, interface displacement is scaled as $(x_{SI_n}-x_{01})/L_0$ with $x_{01}$ the initial position of the first interface and interface velocity is scaled as $u_{SI_n}/\Delta u_1$, respectively. The dimensionless displacements and velocities of the first (second) interface converge in all $L_0$ cases, especially the interface velocity increment of the first (second) interface during the interaction of the RW (CW) with the first (second) interface, which indicates that we can establish a general 1-D theory applicable to an arbitrary fluid-layer thickness to describe the behaviour of the two interfaces.

Figure 3.

The dimensionless displacements (a) and velocities (b) of the first interface (solid symbols) and the second interfaces (hollow symbols). Black, yellow and purple solid (dash-dotted) lines represent the 1-D theory predictions for the first (second) interface in stages I, II and III, respectively. The light blue zones in the insets indicate the interaction of the RW (CW) with the $\textrm {SI}_1$ ($\textrm {SI}_2$). The black dashed lines in the insets represent the scale of measurement error.

First, we define the specific times when the RW and CW are initially generated as $t^0_{r}$ and $t^0_{c}$, the time when the RW impacts the $\textrm {SI}_1$ as $t^*_{r}$ and the time when the CW impacts the $\textrm {SI}_2$ as $t^*_{c}$, as sketched in figure 4, respectively. Then $t^0_{r}$, $t^0_{c}$, $t^*_{r}$ and $t^*_{c}$ can be calculated as

(3.1)\begin{equation} \left.\begin{gathered} t^0_r=\frac{L_0}{u_{t1}},\\ t^0_c=t^*_r=t^0_r+\frac{L_0(1-\Delta u_1/u_{t1})}{c_1},\\ t^*_c=t^*_r+\frac{L_0}{c_1^*}\left[1-\frac{\Delta u_1}{u_{t1}}-\frac{(1-\Delta v_1/v_{t1})(\Delta u_2-\Delta u_1)}{c_1}\right], \end{gathered}\right\} \end{equation}

where $c_1$ and $c^*_1$ are the sound speeds of the test gas between the $\textrm {SI}_1$ with $\textrm {TS}_1$, and between the $\textrm {SI}_2$ with the RW tail, respectively. Based on 1-D gas dynamics theory (Han & Yin Reference Han and Yin1993), $c_1$ and $c^*_1$ are $148\pm 1\ \textrm {m}\ \textrm {s}^{-1}$ and $146\pm 1\ \textrm {m}\ \textrm {s}^{-1}$, respectively. Then $t^0_{r}$, $t^0_{c}$, $t^*_{r}$ and $t^*_{c}$ are derived in all cases, as listed in table 2. As $L_0$ increases, $t^0_{r}$, $t^0_{c}$, $t^*_{r}$ and $t^*_{c}$ increase. However, the dimensionless times, i.e. $t^0_{r}v_{t1}/L_0$, $t^0_{c}v_{t1}/L_0$, $t^*_{r}v_{t1}/L_0$ and $t^*_{c}v_{t1}/L_0$, equal 1.00, 1.78, 1.78 and 2.68 in all cases, which indicates that the motions of waves inside the fluid layer are not influenced by the fluid-layer thickness.

Figure 4.

Sketches of (a) the interaction of $\textrm {TS}_1$ and $\textrm {II}_2$, (b) the interaction of the RW and $\textrm {SI}_1$ and (c) the interaction of the CW and $\textrm {SI}_2$. Here,$c_1$ represents the sound speed of the test gas between the $\textrm {SI}_1$ with $\textrm {TS}_1$, $c_1^*$ denotes the sound speed of the test gas between the $\textrm {SI}_2$ with RW tail, $c_5$ denotes the sound speed of air after the IS impacts $\textrm {II}_1$, $p_1$ denotes the pressure of the flow behind the $\textrm {TS}_1$, $p_2$ denotes the pressure of the flow behind the RW and $p_3$ denotes the pressure of the flow behind the CW tail.

Table 2.

Physical parameters of an $\textrm {SF}_6$ fluid layer calculated by the 1-D theory established in this work (3.4) and (3.5). Here $t^0_r$ ($t^0_c$) denotes the time when RW (CW) is initially generated, $t^*_r$ ($t^*_c$) denotes the time when the RW (CW) impacts the first (second) interface, $L_r$ ($L_c$) denotes the streamwise length of the RW (CW) at $t^*_r$ ($t^*_c$), $\delta t_r$ ($\delta t_c$) denotes the interaction time of the RW (CW) with the first (second) interface, $\overline {g_r}$ ($\overline {g_c}$) denotes the average acceleration imposed on the first (second) interface by the RW (CW) and $t_r^+$ ($t_c^+$) denotes the time when the RW (CW) tail leaves the first (second) interface. The units for time, length and acceleration are microseconds, millimetres and $10^6\ \mathrm {m}\ \mathrm {s}^{-2}$, respectively.

Second, as sketched in figure 4, we define the streamwise length of the RW when its head impacts $\textrm {SI}_1$ as $L_r$, the streamwise length of the CW when its head impacts $\textrm {SI}_2$ as $L_c$, the interaction time of the RW with $\textrm {SI}_1$ as $\delta t_r$, the interaction time of the CW with $\textrm {SI}_2$ as $\delta t_c$, the average acceleration that the RW imposes on the $\textrm {SI}_1$ as $\overline {g_r}$ and the average acceleration that the CW imposes on the $\textrm {SI}_2$ as $\overline {g_c}$, respectively. Notably, although the acceleration imposed by the RW on the $\textrm {SI}_1$ during $\delta t_r$, and the acceleration imposed by the CW on the $\textrm {SI}_2$ during $\delta t_c$ are not constant (Morgan et al. Reference Morgan, Cabot, Greenough and Jacobs2018; Liang et al. Reference Liang, Zhai, Luo and Wen2020b), to simplify the analysis, the accelerations are taken as constant (Morgan, Likhachev & Jacobs Reference Morgan, Likhachev and Jacobs2016; Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020a), which can be regarded as a first-order approximation. To calculate the $L_r$, $L_c$, $\delta t_r$, $\delta t_c$, $\overline {g_r}$ and $\overline {g_c}$, the final velocity ($u_{fn}$) of the fluid layer, i.e. the speed of the $\textrm {SI}_1$ after the RW accelerates it, should be derived at first. Based on 1-D gas dynamics theory (Velikovich & Phillips Reference Velikovich and Phillips1996; Liang et al. Reference Liang, Zhai, Luo and Wen2020b),

(3.2)\begin{equation} u_{fn}=\Delta u_1-\frac{2c_1}{\gamma_1-1}\left[p_{21}^{(\gamma_1-1)/2\gamma_1}-1\right]+ \frac{\sqrt{2}c_1(p_{31}/p_{21}-1)}{\sqrt{\gamma_1(\gamma_1+1)p_{31}/p_{21}}}, \end{equation}

where $\gamma _1$ is the specific heat ratio of the test gas inside the fluid layer and equals 1.1, $p_{21}$ is the ratio of the pressure of the flow behind the $\textrm {TS}_2$ ($p_2$) to the pressure of the flow behind the $\textrm {TS}_1$ ($p_1$), as sketched in figure 4 and which equals 0.85. The only unknown parameter $p_{31}$, i.e. the ratio of the pressure of the flow behind the CW tail ($p_3$) to $p_1$, satisfies

(3.3)\begin{equation} \frac{c_5}{c_1}=\frac{\gamma_5-1}{p_{31}^{(\gamma_5-1)/2\gamma_5}-1}\left\{\frac{2}{\gamma_1-1} \left[p_{21}^{(\gamma_1-1)/2\gamma_1}-1\right]-\frac{\sqrt{2}(p_{31}/p_{21}-1) p_{21}^{(\gamma_1-1)/2\gamma_1}}{\sqrt{(\gamma_1+1)\gamma_1p_{31}/p_{21}+(\gamma_1-1)\gamma_1}}\right\}, \end{equation}

where $c_5$ is the sound speed of air after the IS impacts $\textrm {II}_1$ and equals $369\pm 2\ \textrm {m}\ \textrm {s}^{-1}$ and $\gamma _5$ is the specific heat ratio of air and equals 1.4. Then the $p_{31}$ and $u_{fn}$ can be calculated as 0.90 and $101\pm 1\ \textrm {m}\ \textrm {s}^{-1}$ by solving (3.2) and (3.3). Subsequently, we can deduce the expressions for $L_r$, $\delta t_r$ and $\overline {g_r}$ as

(3.4)\begin{equation} \left.\begin{gathered} L_r=\frac{L_0(1-\Delta u_1/u_{t1})(\gamma_1+1)(\Delta u_2-\Delta u_1)}{2c_1},\\ \delta t_r=\frac{2L_r}{\gamma_1(\Delta u_1-\Delta u_2)+u_{fn}-\Delta u_1+2c_1},\\ \overline{g_r}=\frac{u_{fn}-\Delta u_1}{\delta t_r}. \end{gathered}\right\} \end{equation}

In this work, because the waves reflected inside the fluid layer after the CW impacts the $\textrm {SI}_2$ are ignored, the sound speed of the test gas between the $\textrm {SI}_1$ with the CW tail equals $c_1^*$, as sketched in figure 4(c). Then the expressions for $L_c$, $\delta t_c$ and $\overline {g_c}$ are

(3.5)\begin{equation} \left.\begin{gathered} L_c=\delta t_r\left[u_{fn}+c_1^*+\frac{1}{2}(\Delta u_1-\Delta u_2)\right]-(t_c^*-t_r^*)(u_{fn}-\Delta u_2)-\frac{1}{2}\overline{g_r}(\delta t_r)^2,\\ \delta t_c=\frac{2L_c}{(u_{fn}-\Delta u_2)+2c_1^*},\\ \overline{g_c}=\frac{u_{fn}-\Delta u_2}{\delta t_c}. \end{gathered}\right\} \end{equation}

The values of $L_r$, $L_c$, $\delta t_r$, $\delta t_c$, $\overline {g_r}$ and $\overline {g_c}$ are listed in table 2. As $L_0$ increases, $L_r$, $L_c$, $\delta t_r$ and $\delta t_c$ increase, but $\overline {g_r}$ and $\overline {g_c}$ decrease, which indicates the strengths of the RW and CW become weaker.

In summary, we establish a general 1-D theory to describe the motions of the first interface and second interface in three stages. For the first interface: (I) uniform motion with $\Delta u_1$ during $t^*_r>t>0$; (II) acceleration with $\overline {g_r}$ during $t_r^+>t>t^*_r$ with $t_r^+=t^*_r+\delta t_r$ and (III) uniform motion with $u_{fn}$ during $t>t_r^+$. For the second interface: (I) uniform motion with the $\Delta u_2$ during $t^*_c>t>t^0_r$, (II) acceleration with $\overline {g_c}$ during $t_c^+>t>t^*_c$ with $t_c^+=t^*_c+\delta t_c$, and (III) uniform motion with $u_{fn}$ during $t>t^*_c+\delta t_c$. The values of $t_r^+$ and $t_c^+$ are listed in table 2. The predictions of the 1-D theory for the interface motions in stages I, II and III are marked with black, yellow and purple lines, respectively, as represented with solid lines for the first interface and dash-dotted lines for the second interface in figures 3(a) and 3(b), which agree well with the experimental results.

3.2. Quasi-2-D experimental results and analysis

Schlieren images of six kinds of the shock-induced quasi-2-D $\textrm {SF}_6$ fluid-layer evolution are shown in figures 5(a)–5(f). Except for the actual timescale, we also provide a dimensionless timescale, i.e. $\tau _R$ (${=}kv_1^Rt$, in which $v_1^R$ is the Richtmyer growth rate calculated with (3.6)). Taking the L30-IP case as an example, the deformations of the two interfaces are discussed in detail. After the IS impacts the perturbed $\textrm {II}_1$, the perturbation on the $\textrm {SI}_1$ grows, and a rippled $\textrm {TS}_1$ moves towards $\textrm {II}_2$ ($58\ \mathrm {\mu }\textrm {s},\ \tau _R=0.05$). After the $\textrm {TS}_1$ impacts the perturbed $\textrm {II}_2$, the rippled RW is reflected inside the fluid layer and moves towards $\textrm {SI}_1$ ($258\ \mathrm {\mu }\textrm {s},\ \tau _R=0.21$). Meanwhile, the perturbation on $\textrm {SI}_2$ decreases gradually owing to the phase reversal process (Brouillette Reference Brouillette2002). Subsequently, after the RW impacts $\textrm {SI}_1$, the rippled CW is reflected inside the fluid layer and moves towards $\textrm {SI}_2$, and the perturbation on $\textrm {SI}_2$ nearly disappears at $389\ \mathrm {\mu }\textrm {s}$, $\tau _R=0.32$). Later, the perturbation on $\textrm {SI}_2$ increases with the phase opposite to $\textrm {SI}_1$ ($673\ \mathrm {\mu }\textrm {s},\ \tau _R=0.56$). Finally, vortices appear on the spike of $\textrm {SI}_1$ ($956\ \mathrm {\mu }\textrm {s},\ \tau _R=0.79$), which are ascribed to the KHI. The final phase of the $\textrm {SI}_2$ in the three in-phase cases is opposite to that of the $\textrm {SI}_1$, whereas the final phase of the $\textrm {SI}_2$ in the three anti-phase cases is the same as that of the $\textrm {SI}_1$. The two interfacial deformations in the L50-IP and L50-AP cases are similar to the corresponding L30-IP and L30-AP cases. Nevertheless, the late-time interface morphologies of the two interfaces in the L10-IP and L10-AP cases are different from the other four large $L_0$ cases. In the L10-IP case at $959\ \mathrm {\mu }\textrm {s}$ ($\tau _R=0.79$), bubbles of the two interfaces coalesce while spikes grow oppositely with their sizes much smaller than the other four large $L_0$ cases. In the L10-AP case at $965\ \mathrm {\mu }\textrm {s}$ ($\tau _R=0.80$), all structures of the two interfaces coalesce. The whole fluid layer evolves into a ‘sinuous’ shape, which qualitatively agrees with the shock-induced gas-curtain morphology reported previously (Jacobs et al. Reference Jacobs, Klein, Jenkins and Benjamin1993; Budzinski et al. Reference Budzinski, Benjamin and Jacobs1994; Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995).

Figure 5.

Schlieren images of the shock-induced quasi-2-D $\textrm {SF}_6$ fluid-layer evolution in cases (a) L10-IP, (b) L10-AP, (c) L30-IP, (d) L30-AP, (e) L50-IP and (f) L50-AP. Numbers denote time in microseconds.

The amplitudes of the first and second interfaces, $a_1$ and $a_2$, are defined as half of the streamwise distance between the spike tip with the bubble tip of the first interface and second interface, respectively. We measure the positions of two bubble tips and one spike tip of the first interface from each schlieren image, and calculate the $a_1$ by the average of two bubble tip positions minus the spike tip position. We measure the positions of two bubble (spike) tips and one spike (bubble) tip of the second interface in the three in-phase (anti-phase) cases from each schlieren image, and calculate the $a_2$ by the average of two bubble (spike) tip positions minus the spike (bubble) tip position. The amplitude growths on the two interfaces are divided into three stages based on the 1-D theory established in this work: I. Linear stage ($t^*_r>t>0$ for the first interface, $t^*_c>t>t^0_r$ for the second interface), II. Early nonlinear stage ($t^+_r>t>t^*_r$ for the first interface, $t^+_c>t>t^*_c$ for the second interface) and III. Late nonlinear stage ($t>t^+_r$ for the first interface, $t>t^+_c$ for the secondinterface).

In the linear stage, only RMI decides the amplitude growths of the two interfaces. The linear growth rates of the amplitude of the two interfaces, $v_1^{exp}$ and $v_2^{exp}$, are acquired by linear fitting the two interface amplitudes measured from experiments, as listed in table 3. For cases with the same $L_0$, the $v_1^{exp}$ and $v_2^{exp}$ are larger in the three anti-phase cases than those in the three in-phase cases. For cases with the same amplitude combinations, as the $L_0$ increases, the $v_1^{exp}$ and $v_2^{exp}$ increase in the three in-phase cases, but decrease in the three anti-phase cases. In order to determine the difference between the classical RMI of a semi-infinite single-mode interface with the RMI of a heavy fluid layer, the Richtmyer impulsive theory (Richtmyer Reference Richtmyer1960) is adopted to calculate the first interface amplitude growth rate ($v_1^R$), and the Meyer–Blewett impulsive theory (Meyer & Blewett Reference Meyer and Blewett1972) is utilised to obtain the second interface amplitude growth rate ($v_2^{MB}$) as

(3.6)\begin{equation} \left.\begin{gathered} v_1^R=Z_1a_1^0k\Delta u_1A,\\ v_2^{MB}={-}\frac{(Z_2+1)a_2^0k\Delta u_2A}{2}, \end{gathered}\right\} \end{equation}

with two compression factors $Z_1=1-\Delta u_1/v_s$ and $Z_2=1-\Delta u_2/v_{t1}$. The values of $v_1^R$ and $v_2^{MB}$ in all cases are listed in table 3. Here $v_1^{exp}$ is smaller than $v_1^R$ in the three in-phase cases, and $v_2^{exp}$ is larger than $v_2^{MB}$ in the three anti-phase cases. The difference between the experimental result and the corresponding theoretical result is pronounced when $L_0$ is limited, which agrees with previous findings (Ott Reference Ott1972). As a result, the coupling between the two interfaces of a fluid layer, i.e. the interface-coupling effect, plays a vital role in influencing RMI development (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995; Mikaelian Reference Mikaelian1996; Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020a), especially when the initial fluid-layer thickness is small.

Table 3.

The amplitude growth rates of the two interfaces in different cases. Here, $v_1^{exp}$ ($v_2^{exp}$) denotes the linear amplitude growth rate of the first (second) interface measured from experiments, $v_1^{R}$ ($v_2^{MB}$) denotes the linear amplitude growth rate of the first (second) interface calculated with the Richtmyer (Meyer–Blewett) impulsive theory (3.6), $v_1^{l}$ ($v_2^{l}$) denotes the linear amplitude growth rate of the first (second) interface calculated with the mJ model (3.9) and (3.10a,b) and $v_{1b/1s}^{+}$ ($v_{2b/2s}^{+}$) denotes the early nonlinear bubble/spike amplitude growth rate of the first (second) interface at $t_r^+$ ($t_c^+$) calculated with the mZG model (3.11) and (3.12). The unit for velocity is $\textrm {m}\ \mathrm {s}^{-1}$.

Jacobs et al. (Reference Jacobs, Jenkins, Klein and Benjamin1995) introduced a linear solution (J model) for describing the amplitude growth rates of the first interface ($v_1^l$) and second interface ($v_2^l$) by considering the interface-coupling effect on the RMI,

(3.7)\begin{equation} \left.\begin{gathered} v_1^l=\frac{k\Delta u_1\left[A_t(a_1^0-a_2^0)+A_c(a_1^0+a_2^0)\right]}{2},\\ v_2^l=\frac{k\Delta u_2\left[A_t(a_1^0-a_2^0)-A_c(a_1^0+a_2^0)\right]}{2}, \end{gathered}\right\} \end{equation}

with two modified Atwood numbers,

(3.8a,b)\begin{equation} A_t=\frac{\rho_2-\rho_1}{\rho_2\tanh(kL_0/2)+\rho_1}\quad \mathrm{and}\quad A_c=\frac{\rho_2-\rho_1}{\rho_2\coth(kL_0/2)+\rho_1}. \end{equation}

When $L_0\rightarrow \infty$, $A_t=A_c=A$ and the J model reduces to the Richtmyer impulsive theory. Notably, the amplitude growth rate of one interface is influenced by the initial amplitude of the other interface, which indicates the interface-coupling effect. The J model adopts the pre-shock parameters and ignores the shock compression effect. Here, we modify the J model for the first interface by considering the IS compression and for the second interface by considering the $\textrm {TS}_1$ compression. Then the modified J model (mJ model) can be expressed as

(3.9)\begin{equation} \left.\begin{gathered} v_1^l=\frac{k\Delta u_1\left\{A_t\left[Z_1a_1^0-(Z_2+1)a_2^0/2\right]+A_c\left[Z_1a_1^0+(Z_2+1)a_2^0/2\right]\right\}}{2},\\ v_2^l=\frac{k\Delta u_2\left\{A_t\left[Z_1a_1^0-(Z_2+1)a_2^0/2\right]-A_c\left[Z_1a_1^0+(Z_2+1)a_2^0/2\right]\right\}}{2}, \end{gathered}\right\} \end{equation}

with two new modified Atwood numbers,

(3.10a,b)\begin{equation} A_t=\frac{\rho_2-\rho_1}{\rho_2\tanh(Z_LkL_0/2)+\rho_1}\quad \mathrm{and} \quad A_c=\frac{\rho_2-\rho_1}{\rho_2\coth(Z_LkL_0/2)+\rho_1}, \end{equation}

where a new compression factor $Z_L=1-\Delta u_1/v_{t1}$ is introduced. The values of $v_1^l$ and $v_2^l$ are listed in table 3, which agree well with the experimental results. As a result, the mJ model is applicable for describing the interface-coupling effect on the RMI in the linear stage.

The time-varying nonlinear amplitudes of the two interfaces are shown in figures 6(a) and 6(b), respectively. For the first interface, the time is scaled as $\tau _1=kv_1^l(t-t_r^*)$, and the amplitude is scaled as $\eta _1=k(a_1-a_1^*)$ with $a_1^*$ the amplitude of the first interface at $t_r^*$. For the second interface, the time is scaled as $\tau _2=k|v_2^l|(t-t_c^*)$ and the amplitude is scaled as $\eta _2=k(|a_2|-|a_2^*|)$ with $a_2^*$ the amplitude of the second interface at $t_c^*$. The predictions of the mJ model, as shown with black lines in figures 6(a) and 6(b), underestimate the dimensionless $a_1$ but overestimate the dimensionless $a_2$ when $\tau _1>0$ and $\tau _2>0$, respectively. Therefore, the interaction of the RW with the first interface leads to the first interface being more unstable, and the interaction of the CW with the second interface results in the second interface being more stable. In addition, for the first interface, the dimensionless $a_1$ converges in all cases, which indicates that the fluid-layer thickness and amplitude combinations have limited influence on the nonlinear behaviour of the first interface perturbation growth. For the second interface, although the dimensionless $a_2$ converges in the four large $L_0$ cases, the dimensionless $a_2$ in the two $L_0=10$ mm cases are different and larger than the other four large $L_0$ cases at a late time, which indicates that the nonlinear behaviour of the second interface perturbation growth is influenced by the amplitude combination and the fluid-layer thickness when the $L_0$ is limited.

Figure 6.

Comparisons of the dimensionless amplitudes of (a) the first interface and (b) the second interface. Black, yellow and purple lines represent the theoretical predictions for two interface amplitudes in the linear stage (3.9) and (3.10a,b), early nonlinear stage (3.11)–(3.15) and late nonlinear stage (3.16)–(3.18), respectively. The black dashed lines in the insets represent the scale of measurement error.

In the early nonlinear stage, during the interaction of the RW and the first interface, as sketched in figure 7(a), the light fluids outside the fluid layer accelerate the heavy fluids inside the fluid layer, which leads to the additional RTI of the first interface. In contrast, during the interaction of the CW and the second interface, the heavy fluids inside the fluid layer accelerate the light fluids outside the fluid layer, leading to the additional RTS of the second interface. The additional RTI increases the amplitude growth rate of the first interface (de Frahan et al. Reference de Frahan, Movahed and Johnsen2015; Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020a) but the additional RTS decreases the amplitude growth rate of the second interface. Here, we modify the ZG model Zhang & Guo (Reference Zhang and Guo2016) by considering the universal curves of all spikes and bubbles at any density ratio, to quantify the additional RTI of the first interface and the additional RTS of the second interface. To simplify the analysis, we consider that the RTI of the first interface and the RTS of the second interface are induced by the 1-D RW and CW, respectively. Then the expressions of the modified ZG model, i.e. the mZG model, for the first interface bubble/spike amplitude growth rate ($v_{1b/1s}^{en}$) and the second interface bubble/spike amplitude growth rate ($v_{2b/2s}^{en}$) are

(3.11)\begin{equation} \left.\begin{gathered} \frac{\mathrm{d}v_{1b/1s}^{en}}{\mathrm{d}t}={-}\alpha_{b/s}k\left[(v_{1b/1s}^{en})^{2}-(v_{1b/1s}^{qs})^2\right],\\ \frac{\mathrm{d}v_{2b/2s}^{en}}{\mathrm{d}t}=\frac{a_2^0}{|a_2^0|}\alpha_{b/s}k\left[(v_{2b/2s}^{en})^{2}-(v_{2b/2s}^{qs})^2\right], \end{gathered}\right\} \end{equation}

with

(3.12)\begin{equation} \left.\begin{gathered} v_{1b/1s}^{qs}=\sqrt{\frac{A\overline{g_{r}}}{3k}\frac{8}{(1\pm A)(3\pm A)}\frac{\left[3\pm A+\sqrt{2(1\pm A)}\right]^{2}}{\left[4(3\pm A)+\sqrt{2(1\pm A)}(9\pm A)\right]}},\\ v_{2b/2s}^{qs}=\sqrt{\frac{A\overline{g_{c}}}{3k}\frac{8}{(1\pm A)(3\pm A)}\frac{\left[3\pm A+\sqrt{2(1\pm A)}\right]^{2}}{\left[4(3\pm A)+\sqrt{2(1\pm A)}(9\pm A)\right]}}, \end{gathered}\right\} \end{equation}

and

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

The upper (lower) sign of $\pm$ and $\mp$ in all equations of this work applies to the bubble (spike). Note that the interface-coupling effect is considered because the initial $v_{1b/1s}^{en}$ equals $v_1^l$ at $t^*_r$, and the initial $v_{2b/2s}^{en}$ equals $v_2^l$ at $t^*_c$. The final bubble/spike amplitude growth rates of the two interfaces at the end of the early nonlinear stage, i.e. the bubble/spike amplitude growth rate of the first interface ($v_{1b/1s}^{+}$) at $t_r^+$, and the bubble/spike amplitude growth rate of the second interface ($v_{2b/2s}^{+}$) at $t_c^+$, are calculated with the mZG model (3.11) and (3.12), as listed in table 3. The differences between $v_{1b/1s}^{+}$ with $v_1^l$ are prominent in all cases, which indicates the effect of the RTI induced by the RW on the first interface perturbation growth is essential, regardless of a large or small fluid-layer thickness. The differences between $v_{2b/2s}^{+}$ with $v_2^l$ are prominent when the $L_0$ is limited, which indicates as the fluid-layer thickness decreases, the RTS induced by the CW plays an increasingly important role on influencing the second interface perturbation growth.

Figure 7.

Sketches of (a) the interaction of the RW with $\textrm {SI}_1$ and (b) the interaction of the CW with $\textrm {SI}_2$.

In addition, the RW firstly accelerates the bubble, then the spike of the first interface, as shown in figure 7(a). The direction of the acceleration that the RW imposes on the bubble is the same as the bubble amplitude growth. Hence, the RW decompresses the bubble before it impacts the spike of the first interface. The streamwise distance between the bubble tip with the spike tip of the first interface is $2a_1^*$, the relative velocity between the first interface and the RW head is $c_1$, and the first interface speed increment induced by the RW is ($u_{fn}-\Delta u_1$). Therefore, the time-varying RW decompression effect on the first interface bubble amplitude ($a_r$) is

(3.14)\begin{equation} a_r=\frac{(u_{fn}-\Delta u_1)(t-t_r^*+a_1^*/c_1)}{2}, \end{equation}

with $(t_r^*+a_1^*/c_1)>t>(t_r^*-a_1^*/c_1)$. The CW also first accelerates the bubble, then the spike of the second interface, as shown in figure 7(b). However, the direction of the acceleration that the CW imposes on the bubble is the opposite of the bubble amplitude growth. Hence the CW compresses the bubble before it impacts the spike of the first interface. The streamwise distance between the bubble tip with the spike tip of the second interface is $2a_2^*$, the relative velocity between the second interface and the CW head is $c_1^*$, and the second interface speed increment induced by the CW is ($u_{fn}-\Delta u_2$). Therefore, the time-varying CW compression effect on the second interface bubble amplitude ($a_c$) is

(3.15)\begin{equation} a_c={-}\frac{(u_{fn}-\Delta u_2)(t-t_c^*+a_2^*/c_1^*)}{2}, \end{equation}

with $(t_c^*+a_2^*/c_1^*)>t>(t_c^*-a_2^*/c_1^*)$.

Based on the previous analysis, it is concluded that in the early nonlinear stage, the first interface evolution is dominated by the RMI, RTI and the decompression induced by the RW; and the second interface evolution is dominated by the RMI, RTS and the compression caused by the CW. The prediction for the $a_1$ by considering the additional RTI using the ZG model and the decompression induced by the RW, as shown with yellow lines in figure 6(a), agrees well with all experimental results. The prediction for the $a_2$ by considering the additional RTS using the ZG model and the compression induced by the CW, as shown with yellow lines in figure 6(b), agrees well with all experimental results except for the two cases with $L_0=10$ mm. In the L10-IP and L10-AP cases, the CW impacts the second interface before the end of the phase reversal of the second interface owing to the limited $L_0$. Therefore, the decrement of the dimensionless $a_2$ in the early nonlinear stage is mainly ascribed to the phase reversal process imposed by the RMI.

In the late nonlinear stage, after the RW tail leaves the first interface, the first interface moves with a constant speed, and therefore only the RMI dominates the first interface evolution. Similarly, after the CW tail leaves the second interface, only the RMI dominates the second interface evolution. Because the amplitudes of the two interfaces, $a_1^+$ and $a_2^+$, at the end of the early nonlinear stage violate the small-amplitude prerequisite (i.e. $ka_1^+\ll 1$ and $|ka_2^+|\ll 1$), the high-amplitude effect on the nonlinear behaviour of the RMI development is non-negligible. Here, we modify the DR model (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010) by considering various amplitude-wavelength ratios and density ratios to describe the late nonlinear RMI in this study. The expressions of the modified DR model, i.e. mDR model, for the first interface bubble/spike amplitude growth rate ($v_{1b/1s}^{ln}$) and the second interface bubble/spike amplitude growth rate ($v_{2b/2s}^{ln}$) are

(3.16)\begin{equation} \left.\begin{gathered} v_{1b/1s}^{ln}=\frac{v_{1b/1s}^{+}\left[1+(1\mp A)kv_{1b/1s}^{+}t\right]}{1+C_{1b/1s}kv_{1b/1s}^{+}t+(1\mp A)F_{b/s}(kv_{1b/1s}^{+}t)^2},\\ v_{2b/2s}^{ln}=\frac{v_{2b/2s}^{+}\left[1+(1\mp A)k|v_{2b/2s}^{+}|t\right]}{1+C_{2b/2s}k|v_{2b/2s}^{+}|t+(1\mp A)F_{b/s}(k|v_{2b/2s}^{+}|t)^2}, \end{gathered}\right\} \end{equation}

with

(3.17)\begin{equation} \left.\begin{gathered} C_{1b/1s}=\frac{4.5\pm A+(2\mp A)ka_{1b/1s}^+}{4},\\ C_{2b/2s}=\frac{4.5\pm A+(2\mp A)k|a_{2b/2s}^+|}{4}, \end{gathered}\right\} \end{equation}

and

(3.18)\begin{equation} F_{b/s}=1\pm A. \end{equation}

The mDR model predictions for $a_1$ and $a_2$ are shown in figures 6(a) and 6(b) with purple lines, respectively, and agree well with all experimental results for the first interface, and the four large $L_0$ cases for the second interface. As the RTI and decompression effect induced by the RW promote the first interface perturbation growth, and the RTS and compression effect caused by the CW suppress the second interface perturbation growth, the $a_1$ is larger than the $a_2$ in all cases. In addition, bubbles of the two interfaces coalesce in the L10-IP case, and all structures of two interfaces combine in the L10-AP case at a late time, as shown in figures 5(e) and 5(f). As a result, the second interface perturbation growths are promoted by the first interface counterpart in the L10-IP and L10-AP cases, and which exceed the mDR model predictions. Overall, the $a_2$ in the L10-IP and L10-AP cases is smaller than the $a_1$ but larger than the mDR model prediction.