3.2.1. Interface morphology

Schlieren images of the developments of the shock-impacted fluid layer for seven cases are shown in figure 3. The wave–interface interaction processes are similar to those in the undisturbed case, and the relevant descriptions are omitted here. Taking case 30-20-IP as an example, when CW impact SI$_2$, the phase inversion process of the perturbation on SI$_2$ has been completed, and TS$_2$ is away from SI$_2$ ($298\,\mathrm {\mu }\textrm {s}$). Just after CW impact SI$_2$, the perturbations on two interfaces can still be considered as single mode. Later, the multi-mode perturbations grow on SI$_1$ and SI$_2$ ($695\,\mathrm {\mu }\textrm {s}$). Finally, spikes and bubbles arise symmetrically ($1091\,\mathrm {\mu }\textrm {s}$). The sizes and shapes of the bubble and spike on SI$_1$ and SI$_2$ differ significantly. From the schlieren images, the interface profile is clear enough to extract its contour, and can still be described as a single-valued function so that the fast Fourier transform can be applied to obtain the amplitude of each mode (Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018a). The evolutions of the bubbles and spikes at the two interfaces are very sensitive to the value of $K$. When the first interface is disturbed at a higher mode than the second interface ($K>1$), the perturbation on SI$_2$ remains a quasi-single-mode profile, and the bubbles and spikes develop without inclination. However, the multi-mode perturbations on SI$_1$ appear more significant than those on SI$_2$, and the bubbles and spikes on SI$_1$ develop asymmetrically. Specifically, in case 20-60-IP, the middle bubble on SI$_1$ has a wider spread but a smaller amplitude than the side bubbles, and the spikes on SI$_1$ develop with an apparent inclination to the middle bubble. In case 30-60-IP, the middle spike has a greater scale than the side spikes, and the bubbles are inclined to the side spikes. Besides, vortices have already arisen on the side spikes, which indicate that nonlinearity occurs earlier on the side spike than the middle spike. In case 20-30-IP, the middle bubble has a narrower spread but a larger amplitude than the side bubbles, and the spikes are also inclined to the middle bubble. On the whole, if the bubble (spike) on SI$_1$ faces directly the spike (bubble) on SI$_2$, then it is deformed weakly. Otherwise, the bubble (spike) on SI$_1$ deforms greatly. When $K<1$, the first interface has a lower mode disturbance than the second one, and the multi-mode perturbations on SI$_2$ appear more significant compared with those on SI$_1$. In case 30-20-IP, the middle bubble on SI$_1$ has a larger amplitude than the side bubbles, and two spikes develop equally, with a flattened head. The middle spike on SI$_2$ has a larger amplitude and a narrower head compared with the side spikes, and two bubbles develop similarly. In cases 40-20-IP and 60-20-IP, a small middle spike on SI$_2$ occurs, and the shape of the side spikes on SI$_2$ is similar to the shape of the side bubbles on SI$_1$. For these two cases, the bubbles on SI$_1$ behave quite differently, which illustrates the importance of the wavenumber combination. In case 60-20-AP, in contrast, small side spikes occur on SI$_2$. In general, the shape of SI$_1$ is still dominated by its initial basic mode, while the shape of SI$_2$ obviously presents multi-mode.

Figure 3.

Schlieren images of the shock-induced fluid layer evolution for different cases. The symbols have the same meanings as in figure 2. Numbers denote time in $\mathrm {\mu }\textrm {s}$.

3.2.2. Linear growth of amplitude

Previous works (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995; Mikaelian Reference Mikaelian1996) have revealed that the amplitudes of a shocked heavy layer grow linearly after the compression stage and start-up process. Considering a planar fluid layer (with thickness $L$) that is composed of a fluid of density $\rho _1$ being surrounded by a fluid of density $\rho _2$, Jacobs et al. (Reference Jacobs, Jenkins, Klein and Benjamin1995) proposed the J-model to predict the perturbation growth rates of the shocked two interfaces. When the fluids are incompressible and irrotational, the equations that govern the growth of infinitesimal disturbances on the interfaces of the layer can be expressed as

(3.2)\begin{equation} \nabla^2\phi=0, \end{equation}

where $\phi$ is the velocity potential of the fluid, and the kinematic and pressure conditions on the interfaces are

(3.3)\begin{equation} \left. \begin{aligned} & \frac{{\partial} \eta}{{\partial} {t}}-\frac{{\partial} \phi}{{\partial}{y}}=0,\\ & \rho\,\frac{{\partial} \phi}{{\partial}{t}}-\rho g\eta+p=P_0. \end{aligned} \right\} \end{equation}

Here, $p$ is the pressure, $g$ is the acceleration, $\eta$ is the perturbation shape on the interfaces, and $P_0$ is a constant equalling the pressure at the interface in the unperturbed rest state.

As shown in figure 4, the periodic perturbations set in the layer are of the form

(3.4)\begin{equation} \left. \begin{aligned} \eta_1 & =a_1(t)\cos(kx),\\ \eta_2 & =a_2(t)\cos(kx), \end{aligned} \right\} \end{equation}

where $a_1(t)$ and $a_2(t)$ are the amplitudes of the first and second interfaces, respectively. Then the velocity potentials in the three fluids can be obtained and expressed as

(3.5)\begin{equation} \left. \begin{aligned} \phi_1 & =b_1(t)\,{\rm e}^{{-}k(\kern0.7pt y-L/2)}\cos(kx), \\ \phi_2 & ={-}\left[\frac{b_1+b_2}{2}\,\frac{\sinh(ky)}{\cosh(kL/2)} +\frac{b_1-b_2}{2}\,\frac{\cosh(ky)}{\sinh(kL/2)}\right]\cos(kx),\\ \phi_3 & ={-}b_2(t)\,{\rm e}^{k(\kern0.7pt y+L/2)}\cos(kx), \end{aligned} \right\} \end{equation}

where $b_1$ and $b_2$ are the functions of time in the velocity potentials. Substituting (3.4) and (3.5) into (3.3) and (3.2), the J-model (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995) can then be deduced as

(3.6)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}^{2}a_1}{{{\rm d}}t^{2}}+\frac{{{\rm d}}^{2}a_2}{{{\rm d}}t^{2}} & =kA_{{t}}g(a_1-a_2),\\ \frac{{{\rm d}}^{2}a_1}{{{\rm d}}t^{2}}-\frac{{{\rm d}}^{2}a_2}{{{\rm d}}t^{2}} & =kA_{{c}}g(a_1+a_2), \end{aligned} \right\} \end{equation}

where

(3.7)\begin{equation} \left. \begin{aligned} A_{{t}} & =\frac{\rho_1-\rho_2}{\rho_1\tanh (kL/2)+\rho_2},\\ A_{{c}} & =\frac{\rho_1-\rho_2}{\rho_1\coth (kL/2)+\rho_2}, \end{aligned} \right\} \end{equation}

are the two modified Atwood numbers.

Figure 4.

Schematic showing the configuration analysed using linear stability theory for the double-interface system, where $\rho _1$ and $\rho _2$ are the densities of the fluids inside and outside the layer, respectively, $\eta _1$ and $\eta _2$ represent the perturbation shapes of the two interfaces, $\phi _1$, $\phi _2$ and $\phi _3$ are the velocity potentials, $L$ is the initial layer thickness, and $g$ is the acceleration of the two interfaces.

In this work, $\rho _1=7.98\,\textrm {kg}\,\textrm {m}^{-3}$ ($\rho _2=1.78\,\textrm {kg}\,\textrm {m}^{-3}$) is the fluid density inside (outside) the layer after the reflected compression waves in the layer (CW) interact with the shocked second interface (SI$_2$). Here, $L$ is chosen as the distance between the two shocked interfaces (SI$_1$ and SI$_2$) after CW collide with SI$_2$, and can be expressed as

(3.8)\begin{equation} L=L_0\left(1-\frac{\Delta u_1}{v_2}\right)\left(1+\frac{\Delta u_2-\Delta u_1}{c_1}\right) \left(1-\frac{\Delta u_1+\Delta u_{{R}}-\Delta u_2}{c_2}\right), \end{equation}

where $\Delta u_{{R}}$ is the jump velocity of SI$_1$ induced by the reflected rarefaction waves in the layer (RW). In previous work (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995; Liang & Luo Reference Liang and Luo2021), the accelerations $g$ of two interfaces of the layer were considered the same for both RT and RM instabilities. However, when a shock encounters a gas layer with a finite thickness, the accelerations of the two interfaces are different. Assuming that $g_1$ and $g_2$ represent the accelerations of the first and second interfaces, respectively, and then substituting (3.4) and (3.5) into (3.3) and (3.2), yields

(3.9)\begin{equation} \left. \begin{aligned} \frac{{\rm d}{a}_1}{{\rm d}{t}} & ={-}kb_1,\\ \frac{{\rm d}{a}_2}{{\rm d}{t}} & ={-}kb_2, \end{aligned} \right\} \end{equation}

(3.10)\begin{equation} \left. \begin{aligned} \frac{{\rm d}{b}_1}{{\rm d}{t}}+\frac{{\rm d}{b}_2}{{\rm d}{t}} & ={-}A_{{t}}(g_1a_1-g_2a_2),\\ \frac{{\rm d}{b}_1}{{\rm d}{t}}-\frac{{\rm d}{b}_2}{{\rm d}{t}} & ={-}A_{{c}}(g_1a_1+g_2a_2). \end{aligned} \right\} \end{equation}

Combining (3.9) with (3.10), the J-model can be rewritten as

(3.11)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}^{2}a_1}{{{\rm d}}t^{2}}+\frac{{{\rm d}}^{2}a_2}{{{\rm d}}t^{2}} & =kA_{{t}}(g_1a_1-g_2a_2),\\ \frac{{{\rm d}}^{2}a_1}{{{\rm d}}t^{2}}-\frac{{{\rm d}}^{2}a_2}{{{\rm d}}t^{2}} & =kA_{{c}}(g_1a_1+g_2a_2). \end{aligned} \right\} \end{equation}

For the RM instability in this work, based on the experimental movements of the undisturbed interfaces shown in figure 2, the two interfaces in the layer will, respectively, experience two acceleration processes. For the first interface, the jump velocities induced by the incident shock (IS) and the rarefaction waves (RW) are $\Delta u_1$ and $\Delta u_{{R}}$, respectively. For the second interface, the jump velocities induced by the transmitted shock (TS$_1$) and the compression waves (CW) are $\Delta u_2$ and $\Delta u_{{C}}$, respectively. Due to the very short durations of the waves interacting with the interfaces, the Dirac $\delta$-function is always used to model the shock impacting the gas layer (Jacobs et al. Reference Jacobs, Jenkins, Klein and Benjamin1995; Liang & Luo Reference Liang and Luo2023a). Therefore, the accelerations $g_1$ and $g_2$ can be expressed as

(3.12)\begin{equation} \left. \begin{aligned} g_1 & =\Delta u_1\,\delta t+\Delta u_{{R}}\,\delta (t-t_{{R}}), \\ g_2 & =\Delta u_2\,\delta (t-t_1)+\Delta u_{{C}}\,\delta (t-t_{{C}}), \end{aligned} \right\} \end{equation}

where $t_1$, $t_{{R}}$ and $t_{{C}}$ have been defined in (3.1). Substituting (3.12) into (3.11) and integrating the equation, a new model for predicting the linear growth rate of the heavy layer after CW impact SI$_2$ is established:

(3.13)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}a_1}{{{\rm d}}t}+\frac{{{\rm d}}a_2}{{{\rm d}}t} & =kA_{{t}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}}-\Delta u_2\,a_2^0-\Delta u_{{C}}\,a_2^{{C}}),\\ \frac{{{\rm d}}a_1}{{{\rm d}}t}-\frac{{{\rm d}}a_2}{{{\rm d}}t} & =kA_{{c}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}}+\Delta u_2\,a_2^0+\Delta u_{{C}}\,a_2^{{C}}). \end{aligned} \right\} \end{equation}

According to the impulsive model (Richtmyer Reference Richtmyer1960) and MB model (Meyer & Blewett Reference Meyer and Blewett1972), in (3.13), $a_1^+$ is the amplitude of SI$_1$ just after being impacted by IS; $a_1^{{R}}$ is the amplitude of SI$_1$ just after being impacted by RW; $a_2^0$ is the average amplitude of the unshocked and shocked second interface; and $a_2^{{C}}$ is the average amplitude of SI$_2$ before and after being impacted by CW.

Note that (3.13) is established assuming that both interfaces have the same wavenumber. In the linear regime, no additional modes except the initial basic mode are generated even if interface coupling is present. In this work, due to the different initial perturbation wavenumbers on two interfaces ($k_1$ and $k_2$ for the first and second initial interfaces, respectively), the perturbation with $k_1$ ($k_2$) will be imprinted on SI$_2$ (SI$_1$) through such as interface coupling. In other words, both perturbation modes are present on SI$_2$ and SI$_1$ even in the linear regime. Therefore, it is necessary to predict the linear growth rate of $k_1$ on SI$_2$ ($k_2$ on SI$_1$). Because mode coupling in the linear regime is weak (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020), we attempt to establish a linear model by considering separately the perturbation growth of $k_1$ and $k_2$ on SI$_1$ (SI$_2$).

We first consider the perturbation growths of $k_1$ on SI$_1$ and SI$_2$. As sketched in figure 5(a), a single-mode perturbation with $k_1$ is imposed on II$_1$ and there is no perturbation on II$_2$. Because the jump velocities of the interfaces have been calculated as shown in table 2, here the amplitudes of the two interfaces at the specific times are provided to solve (3.13). After IS interacts with II$_1$, the amplitude of SI$_1$ ($a_1^+$) can be expressed as

(3.14)\begin{equation} a_1^+=a_1^-Z_1, \end{equation}

where $a_1^-$ and $Z_1=1-\Delta u_1/v_1$ are the initial amplitude of II$_1$ and the compression factor of IS, respectively, and $v_1$ is the velocity of IS. In previous work (Ishizaki & Nishihara Reference Ishizaki and Nishihara1997), after a shock interacts with a single-mode interface in the plane geometry, the transmitted shock will be seeded by the initial interface shape. If the initial amplitude of the interface is small, then the transmitted shock has a sinusoidal shape, but its amplitude decreases very quickly because of the pressure perturbation. If the initial amplitude of the interface is high, then the triple-shock configuration will be formed on the transmitted shock front, which is no longer a smooth one (Rikanati et al. Reference Rikanati, Oron, Sadot and Shvarts2003; Wang et al. Reference Wang, Wang, Zhai and Luo2023). The duration of such a transmitted shock recovering to planar will be longer. In the present work, due to the small amplitude of the initial first interface, the amplitude of the transmitted shock when it is generated in the layer (TS$_1$) is also small. It is believed that TS$_1$ will recover quickly to almost planar when it interacts with the second interface. As a result, SI$_2$ is still undisturbed ($a_2^0=0$), and planar RW aregenerated.

Figure 5.

Sketches of the interactions of TS$_1$ with II$_1$, RW with SI$_1$, and CW with SI$_2$: (a) II$_2$ is planar and II$_1$ is perturbed; (b) II$_1$ is planar and II$_2$ is perturbed.

The planar RW move upstream and interact with SI$_1$. Due to the stretching of RW, the perturbation amplitude of SI$_1$ ($a_1^{{R}}$) increases and is of the form

(3.15)\begin{equation} a_1^{{R}}=a_1^+R, \end{equation}

where $R=1+\Delta u_{{R}}/(c_1+c_2)$ is the stretching factor of RW. After RW collide with SI$_1$, CW are reflected and move to SI$_2$. Similar to TS$_1$, the amplitude of CW is also ignored in this work when CW encounter the planar SI$_2$ ($a_2^{{C}}=0$). In other words, considering a disturbed II$_1$ and a planar II$_2$ as shown in figure 5(a), because the amplitudes of both TS$_1$ and CW are ignored in this work, both TS$_1$ and CW will not seed the perturbation to the second interface, and SI$_2$ obtains the perturbation only through interface coupling. The specific amplitudes calculated are substituted into (3.13), and the linear growth rates of the perturbations with $k_1$ (the wavenumber of the initial first interface) on SI$_1$ and SI$_2$ can be obtained as

(3.16)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}a_1}{{{\rm d}}t} & =\frac{k_1[A_{{t}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}})+A_{{c}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}})]}{2},\\ \frac{{{\rm d}}a_2}{{{\rm d}}t} & =\frac{k_1[A_{{t}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}})-A_{{c}}(\Delta u_1\,a_1^++\Delta u_{{R}}\,a_1^{{R}})]}{2}. \end{aligned} \right\} \end{equation}

Figure 6 presents the amplitude developments of $k_1$ on SI$_1$ and SI$_2$ for $K>1$ (figures 6a,b) and $K<1$ (figures 6c,d). Note that the initial time ($t=0$) here is defined as the moment when the linear growth of perturbation starts (after CW compress SI$_2$), which is different from the definition earlier. On SI$_1$, the amplitude of $k_1$ grows fast, and nonlinearity decreases the amplitude growth rate soon. The model can predict the linear growth of the amplitude of $k_1$ on SI$_1$. The slight differences of the linear growth rates among cases are probably ascribed to the slight differences of the initial amplitudes and weak dependence upon $k_1$, as indicated in figure 7. On SI$_2$, due to the small initial amplitude of $k_1$, nonlinearity is weak and the linear model is sufficiently accurate to predict the amplitude evolution, except in cases 30-60-IP and 40-20-IP. In these two cases, the amplitudes of $k_1$ on SI$_2$ at late stages are promoted, as shown in figures 6(b,d), mainly because of mode coupling, i.e. the self-coupling of $k_2$ in case 30-60-IP generates $k_1$, and the mutual coupling of $k_2$ with $k_1$ in case 40-20-IP generates $k_1$. Besides, the linear growth rates of $k_1$ on SI$_2$ differ significantly between cases with different $k_1$, because they are strongly associated with $k_1$, as indicated in figure 7. Specifically, the linear growth rate of $k_1$ on SI$_2$ is greater for a smaller $k_1$. Relatively, the amplitude of $k_1$ on SI$_2$ is small (large) for $K>1$ ($K<1$).

Figure 6.

Comparisons of the amplitudes of $k_1$ and $k_2$ between experimental and theoretical results in cases (a,b) $K>1$, and (c,d) $K<1$. For the perturbations with the same wavelength, the predictions of the linear growth rate are the same, and only one theoretical line is provided. Here, the initial time is the moment when the linear growth of perturbation starts (after CW compress SI$_2$), and $a_0$ is the amplitude of the interface at the initial time, and similarly hereafter.

Figure 7.

Dependence of linear growth rates of mode amplitudes, predicted by (3.16) and (3.25), on initial wavelength.

If a single-mode perturbation with $k_2$ is imposed on II$_2$, while keeping II$_1$ planar ($a_1^+=0$), then the results will be different. As shown in figure 5(b), because both IS and II$_1$ are planar, TS$_1$ is also planar. Induced by TS$_1$, the average amplitude of II$_2$ and SI$_2$ just being shocked ($a_2^0$) can be calculated:

(3.17)\begin{equation} a_2^0=a_2^-(1+Z_2)/2, \end{equation}

where $a_2^-$ is the amplitude of II$_2$, $v_2$ is the velocity of TS$_1$, and $Z_2=1-\Delta u_2/v_2$ is the compression factor of TS$_1$. The amplitude of SI$_2$ reduces until phase inversion is finished.

The interaction of TS$_1$ with II$_2$ generates perturbed RW. Note that the amplitude of RW increases with time (Velikovich & Phillips Reference Velikovich and Phillips1996). The amplitude of the RW tail before it impacts SI$_1$ ($a_{{RW}}$) can be calculated by the model proposed by Velikovich & Phillips (Reference Velikovich and Phillips1996):

(3.18)\begin{equation} a_{{RW}}=a_2^-Z_2+k_2a_2^-c_1u_r(t_{{R}}-t_1), \end{equation}

where $u_{{r}}=\sqrt {(\gamma +1)(1-{c_2}/{c_1})/(\gamma -1)}$ is a factor of RW, and $\gamma$ is the adiabatic exponent of the gas before the RW front. The time and velocity parameters are all presented in table 2. Due to the RW stretching, the perturbation with $k_2$ arises on SI$_1$ just being impacted by RW, and its amplitude ($a_1^{{R}}$) can be expressed as

(3.19)\begin{equation} a_1^{{R}}=\frac{(a_2^-+a_{{RW}})(R-1)}{2}. \end{equation}

After perturbed RW collide with II$_1$, perturbed CW are formed, but are believed to recover to planar before they interact with SI$_2$ as mentioned before. The amplitude of SI$_2$ just before being impacted by CW ($a_2^{{C}-}$) differs significantly from $a_2^+$ (amplitude of SI$_2$ just after TS$_1$ impact) and can be expressed as

(3.20)\begin{equation} a_2^{{C}-}=a_2^-Z_2+V_2^0(t_{{C}}-t_1), \end{equation}

where $V_2^0$ is the perturbation growth rate of SI$_2$ before being impacted by CW, and can be calculated by the MB model (Meyer & Blewett Reference Meyer and Blewett1972) as

(3.21)\begin{equation} V_2^0={-}k_2a_2^0A^+\,\Delta u_2. \end{equation}

Then the average amplitude of SI$_2$ before and after being impacted by CW ($a_2^{{C}}$) is

(3.22)\begin{equation} a_2^{{C}}=a_2^{{C}-}(1+Z_{{C}})/2, \end{equation}

where $Z_{{C}}=(1-\Delta u_{{C}}/c_2)$ is the compression factor of CW. Substituting $a_1^+$, $a_2^0$, $a_1^{{R}}$ and $a_2^{{C}}$ into (3.13), the linear growth rates of perturbations with $k_2$ on SI$_1$ and SI$_2$ can be predicted. However, the predictions deviate from the experimental results, which is probably ascribed to the assumption of deducing (3.11) that the velocity potentials of the flow in the layer are the same. Although the different accelerations of the two interfaces are considered, the effects of the disturbed RW and CW also need to be considered. In other words, when considering a planar II$_1$ and disturbed II$_2$ as shown in figure 5(b), SI$_1$ obtains a perturbation through both the waves’ effect and interface coupling. Actually, the disturbed RW not only accelerates SI$_1$ but also obviously changes the perturbation growth. The impulsive model (Richtmyer Reference Richtmyer1960) is used to calculate the amplitude growth rate of SI$_1$ induced by the perturbed RW:

(3.23)\begin{equation} \frac{{{\rm d}}a_{1, {{R}}}}{{{\rm d}}t}=k_2a_1^{{R}}\,\Delta u_{{R}}\,A^+. \end{equation}

When CW impacts SI$_2$, the amplitude of $k_2$ develops significantly and the MB model (Meyer & Blewett Reference Meyer and Blewett1972) is used to predict the amplitude growth rate of SI$_2$ induced by CW:

(3.24)\begin{equation} \frac{{{\rm d}}a_{2, {\rm{C}}}}{{{\rm d}}t}={-}k_2a_2^{{C}}\,\Delta u_{{C}}\,A^+. \end{equation}

Combining (3.13), (3.23) with (3.24), the model that can predict the linear growth rate of the perturbation with $k_2$ (the wavenumber of the initial second interface) on SI$_1$ and SI$_2$ can be written as

(3.25)\begin{align} \left. \begin{aligned} \frac{{{\rm d}}a_1}{{{\rm d}}t} & =\frac{k_2[A_{{t}}(\Delta u_{{R}}\,a_1^{{R}}-\Delta u_2\,a_2^0-\Delta u_{{C}}\,a_2^{{C}})+A_{{c}}(\Delta u_{{R}}\,a_1^{{R}}+\Delta u_2\,a_2^0+\Delta u_{{C}}\,a_2^{{C}})]}{2}\\ & \quad +k_2a_1^{{R}}\Delta u_{{R}}\,A^+,\\ \frac{{{\rm d}}a_2}{{{\rm d}}t} & =\frac{k_2[A_{{t}}(\Delta u_{{R}}\,a_1^{{R}}-\Delta u_2\,a_2^0-\Delta u_{{C}}\,a_2^{{C}})-A_{{c}}(\Delta u_{{R}}\,a_1^{{R}}+\Delta u_2\,a_2^0+\Delta u_{{C}}\,a_2^{{C}})]}{2}\\ & \quad -k_2a_2^{{C}}\,\Delta u_{{C}}\,A^+. \end{aligned} \right\} \end{align}

The amplitude developments of $k_2$ on SI$_1$ and SI$_2$ are also presented in figure 6 for $K>1$ (figures 6a,b) and $K<1$ (figures 6c,d). On SI$_2$, the amplitude of $k_2$ grows first linearly and then nonlinearly. On SI$_1$, the linear model can well predict the amplitude growth of $k_2$. From figure 7, the linear growth rates of $k_2$ on both SI$_1$ and SI$_2$ are positively correlated with the wavelength of II$_2$ ($\lambda _2$). As a result, the linear growth rates of $k_2$ on both SI$_1$ and SI$_2$ are greater when $\lambda _2$ is larger. If the phases of two initial perturbations are opposite, then the linear models can also predict the growth rates of both $k_1$ and $k_2$.

3.2.3. Nonlinear growth of amplitude

As shown in figure 6, generally, only the amplitudes of $k_1$ (wavenumber of II$_1$) on SI$_1$, and $k_2$ (wavenumber of II$_2$) on SI$_2$, grow nonlinearly. The amplitude developments of perturbation with $k_1$ on SI$_1$ ($k_2$ on SI$_2$) in dimensionless form are shown in figure 8. On SI$_1$, the time and amplitude of $k_1$ are scaled as $\tau =k_1({{\textrm {d}}a_{1}^l}/{{\textrm {d}}t})t$ and $\alpha =k_1(a_1-a_1^0)$, where ${{\textrm {d}}a_{1}^l}/{{\textrm {d}}t}$ is the experimental linear growth rate of $k_1$, and $a_1^0$ is the amplitude of $k_1$ at $t=0$. On SI$_2$, the time and amplitude of $k_2$ are scaled as $\tau =k_2({{\textrm {d}}a_{2}^{l}}/{{\textrm {d}}t})t$ and $\alpha =k_2(a_2-a_2^0)$, respectively, where ${{\textrm {d}}a_{2}^{l}}/{{\textrm {d}}t}$ is the experimental linear growth rate of $k_2$, and $a_2^0$ is the amplitude of $k_2$ at $t=0$. The nonlinear amplitude growths of $k_1$ on SI$_1$ for cases collapse. However, for $k_2$ on SI$_2$, the amplitude growths in cases with $K>1$ are different from those in cases with $K<1$.

Figure 8.

Comparison of the dimensionless amplitudes of the perturbation with (a) $k_1$ on SI$_1$ and (b) $k_2$ on SI$_2$.

Zhang & Guo (Reference Zhang and Guo2016) previously proposed a nonlinear model (the ZG model) to predict the amplitude growth of a single interface. The ZG model can be described as

(3.26)\begin{equation} \frac{{{\rm d}}a_{{{b/s}}}^{{{ZG}}}}{{{\rm d}}t}(t)=\frac{{{\rm d}}a^{l}}{{{\rm d}}t}\,\frac{1}{1+\theta k\,\dfrac{{{\rm d}}a^{l}}{{{\rm d}}t}\,t}, \end{equation}

where $k$ is the wavenumber of the perturbation, ${{\textrm {d}}a^{l}}/{{\textrm {d}}t}$ is the linear growth rate of the perturbation after the waves impact, and

(3.27)\begin{equation} \theta=\frac{3}{4}\,\frac{(1+A^+)(3+A^+)}{3+A^++\sqrt{2(1+A^+)}}\times \frac{4(3+A^+)+\sqrt{2}(9+A^+)(1+A^+)^{1/2}}{(3+A^+)^2+2\sqrt{2}(3-A^+)(1+A^+)^{1/2}} \end{equation}

is a function for the Atwood number ($A^+$) of the shocked interface. The ZG model has been verified widely for a single-mode interface (Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018a; Guo et al. Reference Guo, Cong, Si and Luo2022), but it overestimates the nonlinear growth of $k_1$ on SI$_1$, as shown in figure 8(a). This indicates that in a heavy layer, interface coupling promotes nonlinearity of $k_1$ growth on SI$_1$, which is consistent with the observation in figure 3 that interface coupling greatly deforms the morphology of SI$_1$. For $k_2$ on SI$_2$, as shown in figure 8(b), the ZG model slightly overestimates the nonlinear amplitude growth for cases with $K<1$, whereas it generally predicts the nonlinear amplitude growth for cases with $K>1$. These indicate that interface coupling promotes nonlinearity of $k_2$ growth on SI$_2$ for cases with $K<1$, but it has a little effect on the amplitude growth of $k_2$ on SI$_2$ for cases with $K>1$. These conclusions are also consistent with the observations in figure 3.

The nonlinear model proposed by Sadot et al. (Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998) (the SEA model) has been modified empirically (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010; Mansoor et al. Reference Mansoor, Dalton, Martinez, Desjardins, Charonko and Prestridge2020) to predict the nonlinear growth of RM instability. The SEA model can be expressed as

(3.28)\begin{equation} \frac{{\rm d}a_{b/s}^{{{SEA}}}}{{\rm d}t}(t)=\frac{{\rm d}a^{l}}{{\rm d}t}\, \frac{1+k\,\dfrac{{\rm d}a^{l}}{{\rm d}t}\,t}{1+(1\pm A^+)k\,\dfrac{{\rm d}a^{l}}{{\rm d}t}\,t +C(1\pm A^+)/(1+A^+)\left(k\dfrac{{\rm d}a^{l}}{{\rm d}t}t\right)^2}, \end{equation}

where $C=1$ for $|A| \rightarrow 0$, and $C=1.5$ for $|A| > 0.5$. Based on the experimental results, the SEA model is modified empirically in the current work to predict the amplitude growths of two interfaces. The expression of the modified SEA model (mSEA model) can be described as

(3.29)\begin{equation} \frac{{{\rm d}}a_{{{b/s}}}^{{{mSEA}}}}{{{\rm d}}t}(t)=\frac{{{\rm d}}a^{l}}{{{\rm d}}t}\,\frac{1+\dfrac{(1\mp A^+)}{2}k\,\dfrac{{{\rm d}}a^{l}}{{{\rm d}}t}\,t}{1+(1\pm A^+)k\,\dfrac{{{\rm d}}a^{l}}{{{\rm d}}t}\,t+5(1+A^+)(1-A^+)\left(k\,\dfrac{{{\rm d}}a^{l}}{{{\rm d}}t}\,t\right)^2}. \end{equation}

The predictions of the mSEA model for $k_1$ on SI$_1$ and $k_2$ on SI$_2$ are shown in figure 8. For $k_2$ on SI$_2$ in cases $K<1$, and $k_1$ on SI$_1$, the mSEA model provides reasonable predictions.

3.2.4. Modal evolution

As described earlier, new modes are generated due to interface coupling on the interfaces except for the initial basic modes, then mode coupling arises. For the evolution of a multi-mode interface, a modal model with second-order accuracy was proposed by Haan (Reference Haan1991) – referred to as the Haan model hereafter – to calculate the amplitude growth of the coupling modes resulting from the addition or subtraction of the initial basic modes. The Haan model was then simplified by Remington et al. (Reference Remington, Weber, Marinak, Haar, Kilkenny, Wallace and Dimonte1994) with second-order accuracy to quantify mode coupling in the RT instability (denoted as the REM model) and modified by Luo et al. (Reference Luo, Liu, Liang, Ding and Wen2020) to predict the evolution of the multi-mode RM instability. However, all the previous models were solved and modified only for a single multi-mode interface but not for a gas layer. To quantify the mode-coupling effect in a shocked heavy layer, a new model is established by combining the REM model with (3.11)–(3.12). For the two different basic modes $k_1$ and $k_2$ on the initial interfaces considered in this work, the REM model (Remington et al. Reference Remington, Weber, Marinak, Haar, Kilkenny, Wallace and Dimonte1994) can be expressed as

(3.30)\begin{equation} a_{k_1 \pm k_2}={\mp} \frac{1}{2}(k_1\pm k_2)A^+(a_{k_1}^la_{k_2}^l), \end{equation}

where $a_{k_1}$ ($a_{k_2}$) is the amplitude of the perturbation with mode $k_1$ ($k_2$), $a_{k_1 \pm k_2}$ is the amplitude of the perturbation with coupling modes $k_1 \pm k_2$, and the superscript $l$ denotes the linear regime. The right-hand side of (3.30) represents the generation of coupling mode $k_1\pm k_2$ from initial modes $k_1$ and $k_2$. Taking the second derivative of (3.30) with time as in our previous work (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020), one can obtain

(3.31)\begin{equation} \frac{{{\rm d}}^{2}a_{k_1\pm k_2}}{{{\rm d}}t^{2}}={\mp} \frac{1}{2}\,A^+(k_1\pm k_2)\left(\frac{{{\rm d}}^{2}a_{k_1}^l}{{{\rm d}}t^{2}}\,a_{k_2}^l +\frac{{{\rm d}}^{2}a_{k_2}^l}{{{\rm d}}t^{2}}\,a_{k_1}^l +2\,\frac{{{\rm d}}a_{k_1}^l}{{{\rm d}}t}\,\frac{{{\rm d}}a_{k_2}^l}{{{\rm d}}t}\right). \end{equation}

For a fluid layer, to quantify the evolution of ${{\textrm {d}}^{2}a_{k_1}^l}/{{\textrm {d}}t^{2}}$ and ${{\textrm {d}}^{2}a_{k_2}^l}/{{\textrm {d}}t^{2}}$, (3.11) can be written as follows: for the first interface,

(3.32)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}^{2}a_{1,k_1}^l}{{{\rm d}}t^{2}} & =\frac{k_1}{2}\left[A_{{t},k_1}(g_1a_{1,k_1}^l-g_2a_{2,k_1}^l)+A_{{c},k_1}(g_1a_{1,k_1}^l+g_2a_{2,k_1}^l)\right],\\ \frac{{{\rm d}}^{2}a_{1,k_2}^l}{{{\rm d}}t^{2}} & =\frac{k_2}{2}\left[A_{{t},k_2}(g_1a_{1,k_2}^l-g_2a_{2,k_2}^l)+A_{{c},k_2}(g_1a_{1,k_2}^l+g_2a_{2,k_2}^l)\right]; \end{aligned} \right\} \end{equation}

for the second interface,

(3.33)\begin{equation} \left. \begin{aligned} \frac{{{\rm d}}^{2}a_{2,k_1}^l}{{{\rm d}}t^{2}} & =\frac{k_1}{2}\left[A_{{t},k_1}(g_1a_{1,k_1}^l-g_2a_{2,k_1}^l)-A_{{c},k_1}(g_1a_{1,k_1}^l+g_2a_{2,k_1}^l)\right],\\ \frac{{{\rm d}}^{2}a_{2,k_2}^l}{{{\rm d}}t^{2}} & =\frac{k_2}{2}\left[A_{{t},k_2}(g_1a_{1,k_2}^l-g_2a_{2,k_2}^l)-A_{{c},k_2}(g_1a_{1,k_2}^l+g_2a_{2,k_2}^l)\right]. \end{aligned} \right\} \end{equation}

Substituting (3.32)–(3.33) into (3.31), the following expressions can be obtained: for the first interface,

(3.34) \begin{align} \frac{{{\rm d}}^{2}a_{1, k_1\pm k_2}}{{{\rm d}}t^{2}}&={\mp} \frac{1}{2}\,A^+(|k_1\pm k_2|)\nonumber\\ &\qquad \left\{\vphantom{\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}}\frac{1}{2}\left[k_1A_{{{t}}, k_1}\left(g_1a_{1, k_1}^l-g_2a_{2, k_1}^l\right)+k_1A_{{{c}}, k_1}(g_1a_{1, k_1}^l+g_2a_{2, k_1}^l)\right]a_{1, k_2}^l\right.\nonumber\\ &\quad +\frac{1}{2}\left[k_2A_{{{t}}, k_2}(g_1a_{1, k_2}^l-g_2a_{2, k_2}^l)+k_2A_{{{c}}, k_2}(g_1a_{1,k_2}^l+g_2a_{2, k_2}^l)\right]a_{1, k_1}^l\nonumber\\ &\left.\quad {}+2\,\frac{{{\rm d}}a_{1, k_1}^l}{{{\rm d}}t}\,\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}\right\}; \end{align}

for the second interface,

(3.35) \begin{align} \frac{{{\rm d}}^{2}a_{2, k_1\pm k_2}}{{{\rm d}}t^{2}}&={\mp} \frac{1}{2}\,A^+(|k_1\pm k_2|)\nonumber\\ &\qquad \left\{\vphantom{\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}}\frac{1}{2}\left[k_1A_{{{t}}, k_1}(g_1a_{1, k_1}^l-g_2a_{2, k_1}^l)-k_1A_{{{c}}, k_1}(g_1a_{1, k_1}^l+g_2a_{2, k_1}^l)\right]a_{2, k_2}^l\right.\nonumber\\ &\quad +\frac{1}{2}\left[k_2A_{{{t}}, k_2}(g_1a_{1, k_2}^l-g_2a_{2, k_2}^l)-k_2A_{{{c}}, k_2}(g_1a_{1, k_2}^l+g_2a_{2, k_2}^l)\right]a_{2, k_1}^l\nonumber\\ &\left.\quad {}+2\,\frac{{{\rm d}}a_{2, k_1}^l}{{{\rm d}}t}\,\frac{{{\rm d}}a_{2, k_2}^l}{{{\rm d}}t}\right\}. \end{align}

Here, $a_{1, k_1}$ ($a_{2, k_1}$), $a_{1, k_2}$ ($a_{2, k_2}$) and $a_{1, k_1\pm k_2}$ ($a_{2, k_1\pm k_2}$) are the amplitudes of modes $k_1$, $k_2$ and $k_1\pm k_2$ on SI$_1$ (SI$_2$), respectively; $A_{{{t}}, k_1}$ ($A_{{{t}}, k_2}$) and $A_{{{c}}, k_1}$ ($A_{{{c}}, k_2}$) are the two modified Atwood numbers of perturbation with $k_1$ ($k_2$), which can be expressed by (3.7), and $g_1$ ($g_2$) is the acceleration of the first (second) interface calculated by (3.12). Substituting $g_1$ and $g_2$ into (3.34)–(3.35) and then integrating (3.34)–(3.35), a model that can quantify the mode-coupling effect in a shocked layer is obtained and expressed as follows: for the first interface,

(3.36) \begin{align} \frac{{{\rm d}}a_{1, k_1\pm k_2}}{{{\rm d}}t}&={\mp} \frac{1}{2}\,A^+(|k_1\pm k_2|)\nonumber\\ &\qquad \left\{\vphantom{\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}}\frac{1}{2}\left[k_1A_{{{t}}, k_1}\left(\Delta u_1\,a_{1, k_1}^+a_{1, k_2}^++\Delta u_{{R}}\,a_{1, k_1}^{{R}}a_{1, k_2}^{{R}}-\Delta u_2\,a_{2, k_1}^0a_{1, k_2}^+{-}\Delta u_{{C}}\,a_{2, k_1}^{{C}}a_{1, k_2}^{{C}}\right)\right.\right.\nonumber\\ &\left.\quad {}+k_1A_{{{c}}, k_1}\left(\Delta u_1\,a_{1, k_1}^+a_{1, k_2}^++\Delta u_{{R}}\,a_{1, k_1}^{{R}}a_{1, k_2}^{{R}}+\Delta u_2\,a_{2, k_1}^0a_{1, k_2}^++\Delta u_{{C}}\,a_{2, k_1}^{{C}}a_{1, k_2}^{{C}}\right)\right]\nonumber\\ &\quad +\frac{1}{2}\left[k_2A_{{{t}}, k_2}\left(\Delta u_1\,a_{1, k_2}^+a_{1, k_1}^++\Delta u_{{R}}\,a_{1, k_2}^{{R}}a_{1, k_1}^{{R}}-\Delta u_2\,a_{2, k_2}^0a_{1, k_1}^+{-}\Delta u_{{C}}\,a_{2, k_2}^{{C}}a_{1, k_1}^{{C}}\right)\right.\nonumber\\ &\left.\quad{}+k_2A_{{{c}}, k_2}\left(\Delta u_1\,a_{1, k_2}^+a_{1, k_1}^++\Delta u_{{R}}\,a_{1, k_2}^{{R}}a_{1, k_1}^{{R}}+\Delta u_2\,a_{2, k_2}^0a_{1, k_1}^++\Delta u_{{C}}\,a_{2, k_2}^{{C}}a_{1, k_1}^{{C}}\right)\right]\nonumber\\ &\left.\quad{}+2\int \frac{{{\rm d}}a_{1, k_1}^{{{mSEA}}}}{{{\rm d}}t}\,\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}\,{\rm d}t\right\}; \end{align}

for the second interface,

(3.37) \begin{align} \frac{{{\rm d}}a_{2, k_1\pm k_2}}{{{\rm d}}t}&={\mp} \frac{1}{2}\,A^+(|k_1\pm k_2|)\nonumber\\ &\qquad \left\{\vphantom{\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}}\frac{1}{2}\left[k_1A_{{{t}}, k_1}\left(\Delta u_1\,a_{1, k_1}^+a_{2,k_2}^0+\Delta u_{{R}}\,a_{1,k_1}^{{R}}a_{2,k_2}^{{R}}-\Delta u_2\,a_{2, k_1}^0a_{2,k_2}^0-\Delta u_{{C}}\,a_{2,k_1}^{{C}}a_{2,k_2}^{{C}}\right)\right.\right.\nonumber\\ &\left.\quad -k_1A_{{{c}}, k_1}\left(\Delta u_1\,a_{1, k_1}^+a_{2,k_2}^0+\Delta u_{{R}}\,a_{1,k_1}^{{R}}a_{2,k_2}^{{R}}+\Delta u_2\,a_{2, k_1}^0a_{2,k_2}^0+\Delta u_{{C}}\,a_{2,k_1}^{{C}}a_{2,k_2}^{{C}}\right)\right]\nonumber\\ &\quad +\frac{1}{2}\left[k_2A_{{{t}}, k_2}\left(\Delta u_1\,a_{1, k_2}^+a_{2,k_1}^0+\Delta u_{{R}}\,a_{1,k_2}^{{R}}a_{2,k_1}^{{R}}-\Delta u_2\,a_{2, k_2}^0a_{2,k_1}^0-\Delta u_{{C}}\,a_{2,k_2}^{{C}}a_{2,k_1}^{{C}}\right)\right.\nonumber\\ &\left.\quad -k_2A_{{{c}}, k_2}\left(\Delta u_1\,a_{1, k_2}^+a_{2,k_1}^0+\Delta u_{{R}}\,a_{1,k_2}^{{R}}a_{2,k_1}^{{R}}+\Delta u_2\,a_{2, k_2}^0a_{2,k_1}^0+\Delta u_{{C}}\,a_{2,k_2}^{{C}}a_{2,k_1}^{{C}}\right)\right]\nonumber\\ &\left.\quad {}+2\int\frac{{{\rm d}}a_{2, k_1}^l}{{{\rm d}}t}\,\frac{{{\rm d}}a_{2, k_2}^{{{ZG}}}}{{{\rm d}}t}\,{\rm d}t\right\}. \end{align}

Here, $a_1^+$ ($a_1^{{R}}$) represents the amplitude of the first interface just being impacted by IS (RW), and $a_2^0$ ($a_2^{{C}}$) is the average amplitude of the second interface before and after being impacted by TS$_1$ (CW), as defined before. In this work, $a_{1, k_2}^+$, $a_{2, k_1}^0$, $a_{2, k_1}^{{R}}$ and $a_{2, k_1}^{{C}}$ are all equal to zero, and other specific amplitudes can be calculated through the linear theory mentioned in § 3.2.2. In the integral terms on the right-hand sides of (3.34) and (3.35), ${{\textrm {d}}a_{1, k_2}^l}/{{\textrm {d}}t}$ and ${{\textrm {d}}a_{2, k_1}^l}/{{\textrm {d}}t}$ can be calculated by (3.16) and (3.25), but ${{\textrm {d}}a_{1, k_1}^l}/{{\textrm {d}}t}$ and ${{\textrm {d}}a_{2, k_2}^l}/{{\textrm {d}}t}$ are substituted respectively with ${{\textrm {d}}a_{1, k_1}^{{{mSEA}}}}/{{\textrm {d}}t}$ expressed in (3.29) and ${{\textrm {d}}a_{2, k_2}^{{{ZG}}}}/{{\textrm {d}}t}$ expressed in (3.26), because the growths of $a_{1, k_1}$ and $a_{2, k_2}$ quickly enter the nonlinear stage.

For cases 20-30-IP and 30-20-IP, modes $k_1\pm k_2$ and harmonics have no influence on the growths of $a_{2, k_1}$ and $a_{1, k_2}$. As a result, the linear models can provide accurate predictions for these two amplitude developments. From figure 3, a nearly perfect single-mode profile of SI$_2$ is observed in case 20-30-IP, which means that mode coupling on SI$_2$ is very weak, i.e. $a_{2, k_1\pm k_2}$ should be very small and the predictions are ignored. Besides, $a_{2, k_1\pm k_2}$ in case 30-20-IP are also very limited. Therefore, only the developments of $a_{1, k_1\pm k_2}$ in these two cases are predicted by (3.36) and discussed here. As presented in figure 9, the theories provide good predictions for the amplitude developments of these coupling modes. Note that the deviation between experiment and theoretical model for the developments of $a_{1,k_1-k_2}$ for the 30-20 case seems significant, which is caused mainly by the relatively small range of the vertical axis in figure 9. Considering the error bars, the model can reasonably predict the experimental results for $a_{1,k_1-k_2}$.

Figure 9.

Comparison of the developments of $a_{1, k_1\pm k_2}$ between experimental measurements and theoretical predictions in cases 20-30-IP and 30-20-IP.

For cases 60-20-IP (20-60-IP) and 60-20-AP, the coupling mode $k_2-k_1$ ($k_1-k_2$) can be described as the second-order harmonic of $k_1$ ($k_2$). For case 40-20-IP (30-60-IP), $k_2=2k_1$ ($k_1=2k_2$). To fully quantify the modal evolutions, the amplitude growths of the harmonics are also considered after constructing (3.36) and (3.37) for $k_1\pm k_2$. Based on the perturbation expansion method, the ZS model was proposed (Zhang & Sohn Reference Zhang and Sohn1997) to calculate the amplitude growth of the harmonics for a single-mode interface. Here, the ZS model is considered for the amplitude growth of the second-order harmonics: for the first interface,

(3.38)\begin{align} \left. \begin{aligned} a_{1, 2k_1} & =\frac{1}{2}\,A^+k_1\,\frac{{{\rm d}}a_{1, k_1}^l}{{{\rm d}}t}\,t^2-\frac{1}{12}\,k_1^3\left(\frac{{{\rm d}}a_{1, k_1}^l}{{{\rm d}}t}\right)^2\left[4(A^+)^3\left(\frac{{{\rm d}}a_{1, k_1}^l}{{{\rm d}}t}\right)^2t^4+3A^+a_{1,k_1}^{{R}}t^2\right],\\ a_{1, 2k_2} & =\frac{1}{2}\,A^+k_2\,\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}\,t^2-\frac{1}{12}\,k_2^3\left(\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}\right)^2\left[4(A^+)^3\left(\frac{{{\rm d}}a_{1, k_2}^l}{{{\rm d}}t}\right)^2t^4+3A^+a_{1,k_2}^{{R}}t^2\right]; \end{aligned} \right\} \end{align}

for the second interface,

(3.39)\begin{align} \left. \begin{aligned} a_{2, 2k_1} & =\frac{1}{2}\,A^+k_1\,\frac{{{\rm d}}a_{2, k_1}^l}{{{\rm d}}t}\,t^2-\frac{1}{12}\,k_1^3\left(\frac{{{\rm d}}a_{2, k_1}^l}{{{\rm d}}t}\right)^2\left[4(A^+)^3\left(\frac{{{\rm d}}a_{2, k_1}^l}{{{\rm d}}t}\right)^2t^4+3A^+a_{2,k_1}^{{C}}t^2\right],\\ a_{2, 2k_2} & =\frac{1}{2}\,A^+k_2\,\frac{{{\rm d}}a_{2, k_2}^l}{{{\rm d}}t}\,t^2-\frac{1}{12}\,k_2^3\left(\frac{{{\rm d}}a_{2, k_2}^l}{{{\rm d}}t}\right)^2\left[4(A^+)^3\left(\frac{{{\rm d}}a_{2, k_2}^l}{{{\rm d}}t}\right)^2t^4+3A^+a_{2,k_2}^{{C}}t^2\right]. \end{aligned} \right\} \end{align}

For the single-mode RM instability, previous work (Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018a) has shown that the amplitudes of harmonics higher than third order grow slowly and can be ignored. Therefore, only the second-order harmonic is considered in this work. The linear and nonlinear developments of amplitudes of the basic modes ($a_{1, k_1}$ and $a_{2, k_2}$) can be well predicted through (3.16), (3.25), (3.26) and (3.29). Considering the amplitudes of other modes, (3.36)–(3.39) are used to provide predictions for modal evolutions.

For cases 20-60-IP, 60-20-IP and 60-20-AP, the linear models can also well predict the growths of $a_{1, k_2}$ and $a_{2, k_1}$. The experimental amplitudes of mode $k_1+k_2$ agree with the predictions by (3.36)–(3.37), as given in figures 10(a,b). However, for predicting the amplitude of mode $|k_1-k_2|$, the effect of harmonics needs to be considered because $|k_1-k_2|=2k_1$ or $2k_2$. Through adding (3.38)–(3.39) to (3.36)–(3.37), the amplitude growths of $|k_1-k_2|$ can be well predicted. For cases 30-60-IP and 40-20-IP, because $k_1=2k_2$ and $k_2=2k_1$, the second-order harmonics and mode-coupling mode ($|k_1-k_2|$) directly affect the growth of $a_{2, k_1}$ and $a_{1, k_2}$, therefore the linear models fail to predict the growths of $a_{1, k_2}$ and $a_{2,k_1}$ at late stages. Combining (3.16), (3.25) with (3.36)–(3.39), the predictions for $a_{1, k_2}$ and $a_{2, k_1}$ in these two cases are provided. In case 40-20-IP, the amplitude growth of $k_2$ on SI$_1$ even experiences a phase inversion process because the harmonic induced by $k_1$ develops faster than the linear growth of $k_2$ at late stages. To predict the developments of $a_{1, k_1\pm k_2}$ and $a_{2, k_1\pm k_2}$, (3.36)–(3.37) are adopted and ${{\textrm {d}}a_{1, k_2}^l}/{{\textrm {d}}t}$ (${{\textrm {d}}a_{2, k_1}^l}/{{\textrm {d}}t}$) is replaced with ${{\textrm {d}}a_{1, k_2}}/{{\textrm {d}}t}$ (${{\textrm {d}}a_{2, k_1}}/{{\textrm {d}}t}$), which is calculated by combining the linear model with the mode-coupling model and harmonic model.

Figure 10.

Comparisons of the amplitudes of $k_1\pm k_2$ measured from experiments with models in (a,b) cases 20-60, 60-20 and 60-20-AP, and (c,d) cases 30-60 and 40-20.

In summary, through combining the linear models (3.16) and (3.25), and nonlinear models (3.26) and (3.29), with modal models (3.36)–(3.39), a complete theory that can reasonably predict the amplitude growths of the basic modes ($a_{1, k_1}$ and $a_{2,k_2}$), the modes generated by interface coupling ($a_{2, k_1}$ and $a_{1, k_2}$), and the modes generated by mode coupling ($a_{k_1\pm k_2}$) is established. By comparing the results in figures 6 and 10, we see that mode coupling is significant (ignorable) for the growth of $a_{2, k_1}$ and $a_{1, k_2}$ ($a_{1, k_1}$ and $a_{2,k_2}$).

3.2.5. Freeze-out of the second interface growth

In previous work, freeze-out of amplitude growth (i.e. the amplitude growth stagnates) was realized by compressibility (Mikaelian Reference Mikaelian1994), the second successive shock impact (Charakhch'yan Reference Charakhch'yan2001) or the weak reflected shock/rarefaction wave impact (Chen et al. Reference Chen, Wang, Zhai and Luo2023a,Reference Chen, Xing, Wang, Zhai and Luob). In this work, if the first interface is planar and the second interface is single-mode, then amplitude freeze-out of the second interface may be realized through interface coupling according to (3.25). In other words, amplitude freeze-out of a single-mode heavy–light perturbation could be realized through interface coupling by adding a planar light–heavy interface in front of the perturbed interface. Evidently, the condition for the amplitude freeze-out of the second interface is ${{\textrm {d}}a_2}/{{\textrm {d}}t}=0$, i.e.

(3.40) \begin{align} 0&=\frac{1}{2}\left\{A_{{t}}\left(\Delta u_{{R}}\,\frac{1+Z_2+k_2c_1u_{{r}}(t_{{R}}-t_1)}{2}\,(R-1)-\Delta u_2\,\frac{1+Z_2}{2}\right.\right.\nonumber\\ &\left.\quad {}-\Delta u_{{C}} \left(Z_2-k_2\,\frac{1+Z_2}{2}\,A^+\,\Delta u_2\,(t_{{C}}-t_1)\right)(1+Z_{{C}})/2\right)\nonumber\\ &\quad -A_{{c}}\left(\Delta u_{{R}}\,\frac{1+Z_2+k_2c_1u_{{r}}(t_{{R}}-t_1)}{2}\,(R-1)+\Delta u_2\,\frac{1+Z_2}{2}\right. \nonumber\\ &\left.\left.\quad {}+\Delta u_{{C}} \left(Z_2-k_2\,\frac{1+Z_2}{2}\,A^+\,\Delta u_2\,(t_{{C}}-t_1)\right) (1+Z_{{C}})/2\right)\right\}\nonumber\\ &\quad-\Delta u_{{C}}\,A^+\left(Z_2-k_2\,\frac{1+Z_2}{2}\,A^+\,\Delta u_2\,(t_{{C}}-t_1)\right)(1+Z_{{C}})/2. \end{align}

Here, all the parameters except $k_2$ are dependent on the shock intensity, gas properties and $L_0$, but are independent of the initial amplitude. Based on (3.40), when the shock intensities and gas properties are fixed, $k_2$ and $L_0$ satisfy the specific relationship to achieve the amplitude freeze-out of the second interface. According to this relationship, an experiment with $k_2=20$ mm, $k_2a_2^-=0.083$ and $L_0$ = 27 mm is designed and conducted. The shock intensity and gas properties are the same as the other cases in this work. The typical schlieren images of the perturbed SF$_6$–air interface are provided in figure 11(a), in which SI$_1$ is not shown after $453\,\mathrm {\mu }\textrm {s}$. After CW impacts SI$_2$ ($453\unicode{x2013} 969\,\mathrm {\mu }\textrm {s}$), the amplitude of SI$_2$ changes very slowly, and amplitude freeze-out is almost achieved. Note that if the first interface is perturbed and the second interface is planar, then it is impossible to achieve the amplitude freeze-out of the first interface through changing $L_0$ based on (3.16).

Figure 11.

(a) Schlieren images showing the evolution of the SF$_6$–air interface. (b) The temporal variations of the amplitude of SI$_2$ after the phase inversion is finished at $t=t_0$.

The amplitude development of the second interface after the phase inversion is shown in figure 11(b), in which $t_0$ is the time when the phase inversion is completed. The amplitude of SI$_2$ grows linearly and can be well predicted by the MB model before CW arrive. This also proves that when II$_1$ is planar, interface coupling does not affect the perturbation evolution on SI$_2$ before CW impact. After CW impact SI$_2$, the amplitude is compressed first and then rises for a short while because interface coupling needs a start-up process. When the start-up process is finished, the amplitude almost settles in a steady-state value, and freeze-out is almost realized (the tiny increase of the amplitude may be attributed to the negligible error of the theory). As a result, amplitude freeze-out of a single-mode SF$_6$–air interface is almost realized experimentally through generating a planar air–SF$_6$ interface upstream with a suitable distance.

Through interface coupling between two successive heavy–light interfaces, the amplitude freeze-out of a heavy–light interface was realized numerically by Liang & Luo (Reference Liang and Luo2023a). However, the initial amplitudes of two successive interfaces need to satisfy a specific relationship that is difficult to realize in ICF. In the present work, the first light–heavy interface is planar, and amplitude freeze-out of the second perturbed heavy–light interface is achieved just by simply manipulating the distance of two interfaces. In previous work for the target designs in ICF (Qiao & Lan Reference Qiao and Lan2021), the glow discharge polymer plastic (CH, with density 1.05 g cm$^{-3}$) was used as the outermost ablator layer, while the high-density carbon (HDC, with density 3.48 g cm$^{-3}$) was kept as the main ablator. Through constructing the outer CH-HDC interface, the hydrodynamic instabilities on the interface between HDC and the deuterium–tritium ice (DT ice, with density 0.26 g cm$^{-3}$) can be reduced significantly. The CH-HDC interface (light–heavy) and the HDC-DT interface (heavy–light) construct a heavy layer. The decreasing instabilities on the inner HDC-DT interface coincide with the finding in this work that the amplitude freeze-out of the second interface can be realized in a shocked heavy layer. These findings suggest that in ICF, through generating a light–heavy interface with a suitable distance upstream from the interface separating the ablator material and DT ice, the RM instability may be effectively suppressed.