3.1. Quasi-one-dimensional experimental results and analysis

Schlieren images of the shock-induced quasi-1-D helium 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 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 rarefaction waves and the TS$_1$ are generated (due to the low density of the test gas and the restricted resolution of images, it is challenging to distinguish the TS$_1$), and the shocked first interface (SI$_1$) begins to move forwards ($33\ \mathrm {\mu }\textrm {s}$). Then the TS$_1$ impacts the II$_2$, and the TS$_2$ moves outside the fluid layer followed by the shocked second interface (SI$_2$) ($116\ \mathrm {\mu }\textrm {s}$). Meanwhile, shocks are reflected and go forth and back inside the light-fluid layer since both the SI$_1$ and SI$_2$ are fast/slow interfaces relative to the motions of the reverberated shocks. Finally, all waves refract outside the fluid layer, and the interfaces at two sides move at the same speed ($783\ \mathrm {\mu }\textrm {s}$).

Figure 2. Schlieren images of the shock-induced quasi-1-D helium gas 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; TS$_2$ denotes the transmitted shock; and RW denotes rarefaction waves. Numbers indicate time in microseconds.

We subtracted the background of raw experimental images, then utilised image processing software to obtain the greyscale values of the processed experimental images. Later, we acquired the coordinates of the interface contours by judging the pixels with greyscale values smaller than a specific value (generally, we chose 50). The average coordinates of an interface were obtained by calculating the average $x$ coordinates at every $y$ coordinate. 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. The pixel size of schlieren images introduces the experimental measurement uncertainty. The size of error bars equals the size of symbols in figures. 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 $u_{{SI}_n}/u_{1}^\alpha$. The dimensionless displacements and velocities of the first (second) interface converge in all $L_0$ cases, indicating that we can utilise a general 1-D theory applicable to arbitrary layer thickness to describe the motions of the two interfaces of a light-fluid layer.

Figure 3. The dimensionless displacements (a) and velocities (b) of the first interface (filled symbols) and the second interface (open symbols). Black, purple and orange solid (dashed) lines represent the 1-D theory predictions for the first (second) interface motions in stages $\alpha$, $\beta$ and $\sigma$, respectively.

The reverberated shocks and the timings when the shocks impact the two interfaces are defined in figure 4. Based on the 1-D gas dynamics theory (Drake Reference Drake2018), the Mach numbers of the RF$_2^\alpha$, RF$_1^\alpha$, RF$_2^\beta$ and RF$_1^\beta$ are derived as 1.036, 1.012, 1.004 and 1.002, respectively. Although the reflected shocks are rather weak, the induced instantaneous changes in velocities of the two interfaces cannot be ignored.

Figure 4. Sketches of (a) the TS$_1$ impacting II$_2$ at $t_2^\alpha$, (b) the reflected shock (RF$_2^\alpha$) impacting SI$_1$ at $t_1^\beta$, (c) the reflected shock (RF$_1^\alpha$) impacting SI$_2$ at $t_2^\beta$, (d) the reflected shock (RF$_2^\beta$) impacting SI$_1$ at $t_1^\sigma$, (e) the reflected shock (RF$_1^\beta$) impacting SI$_2$ at $t_2^\sigma$ and (f) the motions of the SI$_1$ and SI$_2$ when $t>t_2^\sigma$, where $u_1^\alpha$ ($u_2^\alpha$), $u_1^\beta$ ($u_2^\beta$) and $u_1^\sigma$ ($u_2^\sigma$) are the first (second) interface velocities in stages $\alpha$, $\beta$ and $\sigma$, respectively.

The motions of the first (second) interface can be separated into three stages. Stage $\alpha$: uniform motion with $u_1^\alpha$ ($u_2^\alpha$) during $t_1^\beta >t>0$ ($t_2^\beta >t>t_2^\alpha$); stage $\beta$: uniform motion with $u_1^\beta$ ($u_2^\beta$) during $t_1^\sigma >t>t_1^\beta$ ($t_2^\sigma >t>t_2^\beta$); and stage $\sigma$: uniform motion with $u_1^\sigma$ ($u_2^\sigma$) when $t>t_1^\sigma$ ($t>t_2^\sigma$). In this work, the interface velocities in all cases are derived based on the 1-D gas dynamics theory (Drake Reference Drake2018) as $u_1^\beta =111\pm 1\ \mathrm {m}\ \mathrm {s}^{-1}$, $u_1^\sigma =107\pm 1\ \mathrm {m}\ \mathrm {s}^{-1}$, $u_2^\beta =105\pm 1\ \mathrm {m}\ \mathrm {s}^{-1}$ and $u_2^\sigma =106\pm 1\ \mathrm {m}\ \mathrm {s}^{-1}$. Because $u_1^\sigma \approx u_2^\sigma$, it is reasonable to regard that when $t>t_2^\sigma$ all waves refract outside the light-fluid layer, and the two interfaces move at the same speed. The specific times $t_2^\alpha$, $t_1^\beta$, $t_2^\beta$, $t_1^\sigma$ and $t_2^\sigma$ are separately deduced as

(3.1)\begin{equation} \left.\begin{gathered} t_2^\alpha=\frac{L_0}{u_{t1}},\quad t_1^\beta=t_2^\alpha+\frac{L_0-t_2^\alpha u_1^\alpha}{u_1^\alpha-u_{{RF}_2}^ \alpha},\quad t_2^\beta=t_1^\beta+\frac{L_0-t_2^\alpha u_1^\alpha+(t_1^\beta-t_2^\alpha)(u_2^\alpha-u_1^\alpha)}{ u_{{RF}_1}^\alpha-u_2^\alpha},\\ t_1^\sigma=t_2^\beta+\frac{L_0-t_2^\alpha u_1^\alpha+(t_1^\beta-t_2^\alpha)(u_2^\alpha-u_1^\alpha)+(t_2^\beta-t_1^\beta)(u_2^\alpha-u_1^\beta)}{u_1^\beta-u_{{RF}_2}^\beta}, \\ t_2^\sigma=t_1^\sigma+\frac{L_0\!-\!t_2^\alpha u_1^\alpha+(t_1^\beta\!-\!t_2^\alpha)(u_2^\alpha-u_1^\alpha)\!+\!(t_2^\beta-t_1^\beta)(u_2^\alpha\!-\!u_1^\beta)+(t_1^\sigma\!-\!t_2^\beta)(u_2^\beta\!-\!u_1^\beta)}{u_{{RF}_1}^\beta-u_2^\beta}, \end{gathered}\right\} \end{equation}

and their values are listed in table 2 for all cases. Although the specific times $t_2^\alpha$, $t_1^\beta$, $t_2^\beta$, $t_1^\sigma$ and $t_2^\sigma$ increase as $L_0$ increases, the dimensionless times $t_2^\alpha u_1^\alpha /L_0$, $t_1^\beta u_1^\alpha /L_0$, $t_2^\beta u_1^\alpha /L_0$, $t_1^\sigma u_1^\alpha /L_0$ and $t_2^\sigma u_1^\alpha /L_0$ are the same in different $L_0$ cases, indicating that the layer thickness does not influence the reflected shocks’ motions. As a result, a general 1-D theory is adopted to describe the movements of the two interfaces based on the derived interface velocities and specific times in all stages. The predictions of the 1-D theory for the interface motions in stages $\alpha$, $\beta$ and $\sigma$ are marked with black, purple and orange lines, respectively, as shown with solid lines for the first interface and dashed lines for the second interface in figure 3, and agree well with the experimental results.

Table 2. The specific times in the interaction of a shock wave and a helium gas layer. The unit for time is $\mathrm {\mu }\textrm {s}$.

The time-varying layer thickness $L$ ($=x_{{SI}_2}-x_{{SI}_1}$) is calculated and shown in figure 5. The layer thickness is scaled as $L/L_0$. Before all reflected shocks refract outside the fluid layer ($t< t_2^\sigma$), $L$ decreases gradually and finally reaches a saturated value of $0.77L_0$ in all cases. The prediction of the 1-D theory adopted in this work well agrees with the experimental data in all stages, as shown with the solid black line in figure 5.

Figure 5. The time-varying dimensionless quasi-1-D helium gas layer thickness. The solid black line represents the 1-D theory prediction.

3.2. Quasi-two-dimensional experimental results and analysis

Schlieren images shown in figure 6 are for shock-induced quasi-2-D helium gas layer evolution. Taking the L50-AP case as an example, the deformations of the two interfaces are discussed in detail. After the IS impacts the perturbed II$_1$, the rippled rarefaction waves are reflected outside the fluid layer and the rippled TS$_1$ moves towards the II$_2$ ($34\ \mathrm {\mu }\textrm {s}$). Meanwhile, the perturbation on the SI$_1$ decreases due to the phase-reversal process (Brouillette Reference Brouillette2002). After the TS$_1$ impacts the perturbed II$_2$, the rippled TS$_2$ moves outside the fluid layer, and the perturbation on the SI$_2$ gradually increases ($117\ \mathrm {\mu }\textrm {s}$). Meanwhile, the perturbation on the SI$_1$ reduces to zero, indicating the end of the phase-reversal process. Later, the perturbation on the SI$_1$ gradually increases ($451\ \mathrm {\mu }\textrm {s}$), and finally the two interfaces evolve similarly ($784\ \mathrm {\mu }\textrm {s}$). Under the light-fluid-layer condition, the final phase of the SI$_2$ in the three in-phase cases is the opposite to that of the SI$_1$, whereas the final phase of the SI$_2$ in the three anti-phase cases is the same as that of the SI$_1$. This observation is the same as for a heavy-fluid-layer counterpart (Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020; Liang & Luo Reference Liang and Luo2021a). Especially, contrary to the general knowledge of a classical single-mode interface that the spike is sharp and the bubble is flat, the spike of the SI$_1$ is flat and the bubble of the SI$_1$ is sharp in the L10-IP case. In addition, the spike heads of two interfaces almost collide with each other in the L10-IP case at a later time ($788\ \mathrm {\mu }\textrm {s}$).

Figure 6. Schlieren images of the shock-induced quasi-2-D helium gas 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 head and the bubble head of the first and second interfaces, respectively. The time-varying amplitudes of the two interfaces are measured from experiments and shown in figure 7. For the first interface, the amplitude is scaled as $\eta _1=k(|a_1|-|Z_1a_1^0|)$ with a compression factor $Z_1$ $(=1-u_1^\alpha /u_{s})$ of 0.66 in all cases; and time is scaled as $\tau _1=k|v_1^{MB}|t$, in which $v_1^{MB}$ is the linear amplitude growth rate calculated with the modified impulsive theory (Meyer & Blewett Reference Meyer and Blewett1972):

(3.2)\begin{equation} v_1^{MB}=\frac{(Z_1+1)a_1^0Au_1^\alpha}{2}. \end{equation}

In this work, $v_1^{MB}$ equals $-15.4\ \textrm {m}\ \textrm {s}^{-1}$ in all cases. For the second interface, the amplitude is scaled as $\eta _2=k(|a_2|-|Z_2a_2^0|)$ with a compression factor $Z_2$ $(=1-u_2^\alpha /u_{t1})$ of 0.89 in all cases; and time is scaled as $\tau _2=k|v_2^{R}|(t-t_2^\alpha )$, in which $v_2^{R}$ is the linear amplitude growth rate calculated with the impulsive theory (Richtmyer Reference Richtmyer1960):

(3.3)\begin{equation} v_2^{R}=Z_2a_2^0Au_2^\alpha. \end{equation}

In this work, $v_2^{R}$ equals $11.2\ \textrm {m}\ \textrm {s}^{-1}$ in the three in-phase cases and $-11.2\ \textrm {m}\ \textrm {s}^{-1}$ in the three anti-phase cases. In the three in-phase cases, as $L_0$ decreases, both the dimensionless $a_1$ and $a_2$ decrease. On the contrary, in the three anti-phase cases, as $L_0$ decreases, both the dimensionless $a_1$ and $a_2$ increase. Moreover, except for the large $L_0$ cases (i.e. cases L50-IP and L50-AP), the dimensionless $a_1$ and $a_2$ in the anti-phase cases are larger than those in the corresponding in-phase cases with the same $L_0$, which is ascribed to the interface-coupling effect on the RMI of a fluid layer (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995; Mikaelian Reference Mikaelian1996; Liang et al. Reference Liang, Liu, Zhai, Si and Wen2020; Liang & Luo Reference Liang and Luo2021a). It is concluded that a light-fluid layer consisting of two initially anti-phase interfaces is more unstable than a light-fluid layer consisting of two initially in-phase interfaces and a semi-infinite single-mode interface, especially when the initial fluid layer is thin.

Figure 7. Comparisons of the dimensionless amplitudes of the first interface (a) and the second interface (b) in all cases.

According to the above analysis of the shock-induced quasi-1-D helium gas layer evolution, the growths of $a_1$ and $a_2$ are separated into three stages. First of all, in stage $\alpha$, the vorticity deposition induced by the IS (TS$_1$) on the first (second) interface dominates the RMI of the first (second) interface. Jacobs et al. (Reference Jacobs, Jenkins, Klein and Benjamin1995) introduced linear solutions (J model) for describing the amplitude growth rates of the first interface ($v_1^{\alpha }$) and the second interface ($v_2^{\alpha }$) of a thin SF$_6$ gas curtain with a jump velocity of $\Delta u$ imposed by a shock wave as

(3.4)\begin{equation} \left.\begin{gathered} v_1^{\alpha}=\frac{k\Delta u\left[A_t(a_1^0-a_2^0)+A_c(a_1^0+a_2^0)\right]}{2},\\ v_2^{\alpha}=\frac{k\Delta u\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:

(3.5a,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 impulsive theory (Richtmyer Reference Richtmyer1960). Unlike the impulsive theory, the J model adopts the pre-shock parameters and ignores the shock compression effect. Also, for the first interface, both pre- and post-shock parameters should be considered (Meyer & Blewett Reference Meyer and Blewett1972) since the first interface is a slow/fast one relative to the motion of the IS. Different from a thin SF$_6$ gas curtain investigated by Jacobs et al. (Reference Jacobs, Jenkins, Klein and Benjamin1995), the jump velocities of the two interfaces of a light-fluid layer (i.e. $u_1^\alpha$ and $u_2^\alpha$) are somewhat different. Here, we modify the J model for the first interface according to the modified impulsive theory (Meyer & Blewett Reference Meyer and Blewett1972) and the second interface based on the impulsive theory (Richtmyer Reference Richtmyer1960). Then the modified J model (mJ model) considering different jump velocities for the two interfaces is expressed as

(3.6)\begin{equation} \left.\begin{gathered} v_1^{\alpha}=\frac{ku_1^\alpha\left\{A_t\left[(Z_1+1)a_1^0/2-Z_2a_2^0\right]+A_c\left[(Z_1+1)a_1^0/2+Z_2a_2^0\right]\right\}}{2},\\ v_2^{\alpha}=\frac{ku_2^\alpha\left\{A_t\left[(Z_1+1)a_1^0/2-Z_2a_2^0\right]-A_c\left[(Z_1+1)a_1^0/2+Z_2a_2^0\right]\right\}}{2}, \end{gathered}\right\} \end{equation}

with two new modified Atwood numbers:

(3.7a,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 $Z_L=1-u_1^\alpha /v_{t1}$ is introduced considering the compression of the shock wave on the layer thickness, and it equals 0.84 in all cases. The values of $A_t$, $A_c$, $v_1^{\alpha }$ and $v_2^{\alpha }$ are listed in table 3. In cases L50-IP and L50-AP, $A_t=A_c=A$, indicating the interface-coupling effect has a limited influence on the RMI of a light-fluid layer when $kL_0\geqslant 5.24$. As $L_0$ decreases, the discrepancy between $A_t$ and $A_c$ increases, and, therefore, the interface-coupling effect on RMI is more and more prominent. The predictions of the mJ model are shown with solid black lines for the first interface and dashed black lines for the second interface in figure 8 for all cases, and one can find that the predictions agree well with the experimental results in stage $\alpha$.

Figure 8. Comparisons of the amplitudes of the first interface (red symbols) and the second interface (blue symbols) measured from experiments with theories in cases (a) L10-IP, (b) L10-AP, (c) L30-IP, (d) L30-AP, (e) L50-IP and (f) L50-AP. The solid (dashed) black lines represent the predictions of the mJ model in stage $\alpha$ for the first (second) interface amplitudes. The solid (dashed) light purple lines represent the predictions of the M model in stage $\beta$ for the first (second) interface amplitudes. The solid (dashed) orange, green and dark purple lines represent the predictions of the M model, DR model and mDR model in stage $\sigma$ for the first (second) interface amplitudes, respectively. Here $n=1$ for the first interface and $n=2$ for the second interface, and similarly hereinafter.

Table 3. Physical parameters of a light-fluid layer, where $A_t$ and $A_c$ denote the new modified Atwood numbers calculated with (3.7a,b); $v_1^{\alpha }$ ($v_2^{\alpha }$) denotes the first (second) interface amplitude growth rate in stage $\alpha$ calculated with (3.6); $t_1^{rev}$ denotes the end time of the first interface's phase reversal calculated with (3.8); $v_1^{\beta }$ ($v_2^{\beta }$) denotes the first (second) interface amplitude growth rate in stage $\beta$ calculated with (3.9a,b); and $v_1^{\sigma }$ ($v_2^{\sigma }$) denotes the first (second) interface amplitude growth rate in stage $\sigma$ calculated with (3.11a,b). The units for time and velocity are $\mathrm {\mu }\textrm {s}$ and $\textrm {m}\ \textrm {s}^{-1}$, respectively.

In stage $\beta$, the RF$_2^{\alpha }$ ($\textrm {RF}_1^{\alpha }$) deposits additional vorticity on the first (second) interface, leading to the primary post-reshock amplitude growth rate of the first (second) interface, i.e. $v_1^\beta$ ($v_2^\beta$), different from $v_1^\alpha$ ($v_2^\alpha$). For the first interface, since it experiences a phase-reversal process early, its phase when the $\textrm {RF}_2^{\alpha }$ impacts it decides the effect of the $\textrm {RF}_2^{\alpha }$ on its instability. The end time of the first interface's phase reversal ($t_1^{rev}$) can be evaluated as

(3.8)\begin{equation} t_1^{rev}=Z_1a_1^0/|v_1^{\alpha}|, \end{equation}

and the values of $t_1^{rev}$ in all cases are listed in table 3. On comparing $t_1^{rev}$ with $t_1^{\beta }$ in table 2, it can be found that $t_1^{rev}>t_1^{\beta }$ under $L_0=10$ and 30 mm conditions and $t_1^{rev}< t_1^{\beta }$ under $L_0=50$ mm conditions. Because of the baroclinic mechanism created by the misalignment of the density gradient ($\boldsymbol {\nabla }\rho$) and the pressure gradient ($\boldsymbol {\nabla } p$), the $\textrm {RF}_2^{\alpha }$ deposits vorticity with opposite direction to the one deposited by the IS on the first interface, as sketched in figure 9(a). On the contrary, the $\textrm {RF}_2^{\alpha }$ impacts the first interface after the first interface's phase reversal under $L_0=50$ mm conditions, depositing vorticity with the same direction as that deposited by the IS on the first interface, as sketched in figure 9(b). As a result, the $\textrm {RF}_2^{\alpha }$ leads to the first interface being more stable if $t_1^{rev}>t_1^{\beta }$, and being more unstable if $t_1^{rev}< t_1^{\beta }$. For the second interface, the $\textrm {RF}_1^{\alpha }$ induces vorticity deposition with the same direction as that deposited by the TS$_1$ on the second interface; therefore the $\textrm {RF}_1^{\alpha }$ certainly destabilises the second interface. Because the two interfaces evolve in the linear stage before the reflected waves ($\textrm {RF}_2^{\alpha }$ and $\textrm {RF}_1^{\alpha }$) separately impact them, it is reasonable to adopt the re-shock impulsive theory proposed by Mikaelian (Reference Mikaelian1985) (M model) to deduce the primary post-reshock amplitude growth rates as

(3.9a,b)\begin{equation} v_1^{\beta}=v_1^{\alpha}+ka_1^{\beta}A(u_1^\beta-u_1^\alpha),\quad v_2^{\beta}=v_2^{\alpha}-ka_2^{\beta}A(u_2^\beta-u_2^\alpha), \end{equation}

where $a_1^{\beta }$ denotes the first interface amplitude at $t_1^\beta$ and $a_2^{\beta }$ denotes the second interface amplitude at $t_2^\beta$, and they are deduced as

(3.10a,b)\begin{equation} a_1^{\beta}=Z_1a_1^0+v_1^{\alpha}t_1^\beta,\quad a_2^{\beta}=Z_2a_2^0+v_2^{\alpha}(t_2^\beta-t_2^\alpha). \end{equation}

The values of $v_1^{\beta }$ and $v_2^{\beta }$ are listed in table 3. The predictions of the M model in stage $\beta$ are shown with solid light purple lines for the first interface and dashed light purple lines for the second interface in figure 8 for all cases, and agree well with the experimental results.

Figure 9. Sketches of (a) the interaction of the $\textrm {RF}_2^\alpha$ and SI$_1$ under $L_0=10$ and 30 mm conditions, (b) the interaction of the $\textrm {RF}_2^\alpha$ and SI$_1$ under $L_0=50$ mm condition and (c) the interaction of $\textrm {RF}_1^\alpha$ and SI$_2$. The blue (red) arc with an arrow in (a,b) illustrates the vorticity deposition induced by the IS ($\textrm {RF}_2^\alpha$) on the first interface. The blue (red) arc with an arrow in (c) illustrates the vorticity deposition induced by the $\textrm {TS}_1$ ($\textrm {RF}_1^\alpha$) on the second interface. The purple (green) arrows represent the pressure (density) gradient $\boldsymbol {\nabla } p$ ($\boldsymbol {\nabla }\rho$).

In stage $\sigma$, the $\textrm {RF}_2^{\beta }$ ($\textrm {RF}_1^{\beta }$) deposits additional vorticity on the first (second) interface, resulting in a secondary post-reshock amplitude growth rate, i.e. $v_1^\sigma$ ($v_2^\sigma$), different from $v_1^\beta$ ($v_2^\beta$). Because the condition that $t_1^{rev}>t_1^{\beta }$ is only satisfied in the two $L_0=10$ mm cases, the $\textrm {RF}_2^{\beta }$ leads to the first interface being more stable under $L_0=10$ mm conditions, and being more unstable under $L_0=30$ and 50 mm conditions. Similar to stage $\beta$, the $\textrm {RF}_1^{\beta }$ certainly destabilises the second interface. Here, we replace the primary post-reshock parameters in the M model with the secondary post-reshock parameters in stage $\sigma$ to derive $v_1^\sigma$ and $v_2^\sigma$ as

(3.11a,b)\begin{equation} v_1^{\sigma}=v_1^{\beta}+ka_1^{\sigma}A(u_1^\sigma-u_1^\beta),\quad v_2^{\sigma}=v_2^{\beta}-ka_2^{\sigma}A(u_2^\sigma-u_2^\beta), \end{equation}

where $a_1^{\sigma }$ denotes the first interface amplitude at $t_1^\sigma$ and $a_2^{\sigma }$ denotes the second interface amplitude at $t_2^\sigma$:

(3.12a,b)\begin{equation} a_1^{\sigma}=a_1^{\beta}+v_1^{\beta}(t_1^\sigma-t_1^\beta),\quad a_2^{\sigma}=a_2^{\beta}+v_2^{\beta}(t_2^\sigma-t_2^\beta). \end{equation}

The values of $v_1^{\sigma }$ and $v_2^{\sigma }$ are listed in table 3. The predictions of the M model in stage $\sigma$ are shown with solid orange lines for the first interface and dashed orange lines for the second interface in figure 8 for all cases, and agree with the experimental results in early regimes. 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), the amplitude growths of the two interfaces deviate from the M model predictions at a later time. Therefore, the nonlinearity effect on the RMI of a light-fluid layer should be considered in stage $\sigma$.

Here, we adopt the model of Dimonte & Ramaprabhu (Reference Dimonte and Ramaprabhu2010) (DR model) by considering various amplitude-to-wavelength ratios and density ratios to quantify the nonlinearity effect. The expressions of the DR model for the first interface spike/bubble amplitude growth rate ($v_{1s/1b}$) and the second interface spike/bubble amplitude growth rate ($v_{2s/2b}$) are

(3.13)\begin{equation} \left.\begin{gathered} v_{ns/nb}=\frac{v_{n}^{\sigma}\left[1+(1\mp A)k|v_{n}^{\sigma}|t\right]}{1+C_{ns/nb}k|v_{n}^{\sigma}|t+(1\mp A)F_{s/b}(k|v_{n}^{\sigma}|t)^2},\\ C_{ns/nb}=\frac{4.5\pm A+(2\mp A)k|a_{n}^\sigma|}{4},\quad F_{s/b}=1\pm A, \end{gathered}\right\} \end{equation}

where $n=1$ for the first interface and $n=2$ for the second interface, and the upper (lower) sign of $\pm$ and $\mp$ in (3.13) applies to the spike (bubble). The DR model predictions for $a_1$ and $a_2$ are shown with solid green lines and dashed green lines, respectively, in figure 8 and agree well with all experimental results except for the L10-IP and L30-IP cases. The experimental growths of $a_1$ and $a_2$ in the two cases (the L10-IP and L30-IP cases) are lower than the DR model predictions, which can be ascribed to the collision of the two interfaces’ spikes when the initial fluid layers are thin.

We regard that when the sum of the spike amplitudes of the two interfaces predicted by the DR model reaches the saturated layer thickness (0.77$L_0$) in cases L10-IP and L30-IP, $v_{1s}=0$ and $v_{2s}=0$. Then, the bubble amplitude growths of the two interfaces determine the instability developments. The predictions of the modified DR model (mDR model) for $a_1$ and $a_2$ in cases L10-IP and L30-IP are shown with solid dark purple lines and dashed dark purple lines in figures 8(a) and 8(c), respectively, and agree well with experimental results at a later time.

3.3. Comparison between light-fluid layer and heavy-fluid layer

Finally, the two interface amplitude growths under the light-fluid-layer and heavy-fluid- layer conditions (Liang & Luo Reference Liang and Luo2021a) are compared, as shown in figure 10. The incident shock strength, initial layer thicknesses and amplitude combinations are the same, but the Atwood number is the opposite between the light-fluid-layer and heavy-fluid-layer cases.

Figure 10. Comparisons of the two interface amplitude growths under the light-fluid-layer condition and heavy-fluid-layer condition in cases (a) L10-IP, (b) L10-AP, (c) L30-IP, (d) L30-AP, (e) L50-IP and (f) L50-AP.

(i) On comparing with light-fluid-layer cases, the heavy-fluid layer's first interface amplitude deviates from its second interface amplitude more evidently. The rarefaction waves inside a heavy-fluid layer induce the additional Rayleigh–Taylor instability (RTI) (Rayleigh Reference Rayleigh1883; Taylor Reference Taylor1950) and the decompression effect on the first interface, and the compression waves inside a heavy-fluid layer induce the additional Rayleigh–Taylor stabilisation (RTS) and the compression effect on the second interface (Liang & Luo Reference Liang and Luo2021a). As a result, the two interface amplitude growths obviously deviate from each other. Since the reflected shocks reverberating inside a light-fluid layer are rather weak, the light-fluid layer's two interface amplitude growths are similar. Further, the first (second) interface amplitude growth rate in stage $\alpha$ under the light-fluid-layer condition, $v_1^{{light}}$ ($v_2^{{light}}$), and that under the heavy-fluid-layer condition, $v_1^{{heavy}}$ ($v_2^{{heavy}}$), are calculated according to (3.6) in this study and (3.9) in our previous work (Liang & Luo Reference Liang and Luo2021a), respectively.

(ii) The ratio of $v_1^{{light}}$ to $v_1^{{heavy}}$ versus $kL_0$ is shown in figure 11(a). Because the jump velocity imposed by the IS on the light-fluid layer's first interface is larger than the heavy-fluid-layer counterpart, $v_1^{{light}}$ is obviously larger than $v_1^{{heavy}}$ especially if the two interfaces are initially in phase. However, due to the RTI and the decompression effect induced by the rarefaction waves on the heavy-fluid layer's first interface, the interface's instability is prominently enlarged, and, therefore, the late-time $a_1$ of a heavy-fluid layer is even greater than that of a light-fluid layer.

Figure 11. (a) The ratio of the first interface amplitude growth rate in stage $\alpha$ under the light-fluid-layer condition ($v_1^{{light}}$) to that under the heavy-fluid-layer condition ($v_1^{{heavy}}$) versus $kL_0$ and (b) the ratio of the second interface amplitude growth rate in stage $\alpha$ under the light-fluid-layer condition ($v_2^{{light}}$) to that under the heavy-fluid-layer condition ($v_2^{{heavy}}$) versus $kL_0$.

(iii) The ratio of $v_2^{{light}}$ to $v_2^{{heavy}}$ versus $kL_0$ is shown in figure 11(b). The jump velocities imposed by the TS$_1$ on the second interface of a light-fluid layer and a heavy-fluid layer are similar. Therefore, the difference between $v_2^{{light}}$ and $v_2^{{heavy}}$ is mainly ascribed to the interface-coupling effect. Despite $v_2^{{light}}$ being larger than $v_2^{{heavy}}$ if the two interfaces are initially in phase, $v_2^{{light}}$ and $v_2^{{heavy}}$ approach the same value when $kL_0$ is more extensive. If the two interfaces are initially anti-phase, $v_2^{{light}}< v_2^{{heavy}}$ when $kL_0<3.5$ and $v_2^{{light}}>v_2^{{heavy}}$ when $kL_0>3.5$. Except for case L10-AP, due to the RTS and the compression effect, $a_2$ under the heavy-fluid-layer condition is generally lower than that under the light-fluid-layer condition. In case L10-AP, the two interfaces of a heavy-fluid layer coincide. Therefore, the heavy-fluid layer's second interface is destabilised by the first interface counterpart (Liang & Luo Reference Liang and Luo2021a), and $a_2$ is larger under the heavy-fluid-layer condition than under the light-fluid-layer condition.