3.2.1. Evolution law of once-shocked interface

Understanding the evolution law of the once-shocked interface is essential for investigating S-RMI since the pre-reshock interface serves as the ‘initial interface’ for reshock. For the linear amplitude growth rate of the once-shocked interface ($\dot {a}_{1}^{l}$), the impulsive model proposed by Richtmyer (Reference Richtmyer1960) is used to provide a theoretical reference ($\dot {a}_{1,{i}}^{l}$, with subscript ‘$i$’ denoting parameters predicted by the impulsive model) for the experimental result ($\dot {a}_{1,{e}}^{l}$). For the once-shocked interface, the impulsive model can be written as

(3.1)\begin{equation} \dot{a}_{1,{i}}^{l} = C_1 k a_0 A_1 u_{1,{o}}, \end{equation}

where $C_1= 1 - u_{1,{o}} / v_{1,{e}}^{i}$ is the first shock compression factor, with $v_{1}^{i}$ being the velocity of IS$_1$. Here $\dot {a}_{1,{e}}^{l}$ is determined by linearly fitting the early-time variation of the once-shocked interface amplitude ($a_1$) measured from experiments. As shown in table 2, $\dot {a}_{1,{e}}^{l}$ agrees well with $\dot {a}_{1,{i}}^{l}$.

Table 2.

Experimental and theoretical results for the evolution of the once-shocked interface: $a_{1,{zg}}^r$, pre-reshock amplitude predicted by the ZG model; $\dot {a}_{1,{e}}^{l}$ and $\dot {a}_{1,{i}}^{l}$, linear amplitude growth rates of the once-shocked interface obtained from experiments and predicted by impulsive model, respectively; $\dot {a}_{1,{e}}^{r}$ and $\dot {a}_{1,{zg}}^{r}$, pre-reshock amplitude growth rates obtained from experiments and predicted by ZG model, respectively; $k_{I} a_{I}$ and $k_{II} a_{II}$, scaled amplitudes of the fundamental and second-order modes, respectively.

Referring to previous studies (Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018; Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), the theoretical model constructed by Zhang & Guo (Reference Zhang and Guo2016) (ZG model) is employed to provide a theoretical reference for the nonlinear evolution law of the once-shocked interface, which can be written as

(3.2)\begin{equation} \dot{a}_{1,{zg}}^{b/s}(t) = \frac{\dot{a}_{1,{i}}^{l}}{1 + \theta_{1}^{b/s} k \dot{a}_{1,{i}}^{l}t}. \end{equation}

Here,

(3.3)\begin{equation} \theta_{1}^{b/s}=\frac{3}{4}\frac{(1 \pm A_1)(3 \pm A_1)}{3 \pm A_1 + \sqrt{2}(1 \pm A_1)^{1/2}} \frac{4(3 \pm A_1) + \sqrt{2}(9 \pm A_1)(1 \pm A_1)^{1/2}}{(3 \pm A_1)^2 + 2\sqrt{2} (3\mp A_1)(1\pm A_1)^{1/2}}, \end{equation}

with superscripts ‘$b$’ and ‘$s$’ denoting parameters of the bubble and spike, respectively, and subscript ‘$zg$’ representing parameters predicted by the ZG model. The ZG model is applied once the start-up process (Yang, Zhang & Sharp Reference Yang, Zhang and Sharp1994; Lombardini & Pullin Reference Lombardini and Pullin2009) concludes, and the duration of this process ($t_1^*$) as well as $a_1$ at $t=t_1^*$ ($a_1^*$) are calculated using methods proposed by Lombardini & Pullin (Reference Lombardini and Pullin2009). Figure 6(a) shows the temporal variations of $a_1$, obtained from experiments and predicted by the ZG model, in dimensionless form. Here, $t$ and $a_1$ are normalized as $\tau _1=k\dot {a}_{1,{i}}^{l} (t-t_1^*)$ and $\alpha _1=k (a_1-a_1^*)$, respectively. Moreover, the temporal variations of the bubble and spike amplitudes of the once-shocked interface in dimensionless form ($\alpha _{1}^b$ and $\alpha _{1}^s$) are shown in figure 6(b). The scaling collapses the results of all experiments, indicating that the once-shocked interface exhibits similar evolution laws across different cases. Besides, the ZG model provides excellent predictions for the evolutions of the overall, bubble and spike amplitudes of the once-shocked interface across all cases.

Figure 6.

Experimental and theoretical results of the amplitude evolutions of the once-shocked interface (a) and its bubble and spike (b).

Next, the characteristics of the pre-reshock interface are explored. Determining the pre-reshock interface profile is highly challenging in experiments using the explosion-driven scheme to generate successive shocks (Buttler et al. Reference Buttler2014a,Reference Buttlerb). In contrast, in the present work, the pre-reshock interface profile can be clearly captured. Fast Fourier transform is applied to obtain the modal information (amplitude, phase and frequency) of the pre-reshock interface. Since fast Fourier transform is inapplicable when the interface cannot be expressed by a single-valued function, only the results of the cases with reshock occurring at the linear, weakly nonlinear and moderately nonlinear stages are provided. In addition, since the magnitudes of the third-order and other higher-order modes are negligible relative to that of the fundamental mode before the roll-up structures emerge (Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018), only the results of the fundamental and second-order modes are presented. The wavenumbers (amplitudes) of the fundamental and second-order modes are represented as $k_{I}$ and $k_{II}$ ($a_{I}$ and $a_{II}$), respectively. Note that $k_{I}$ and $k_{II}$ are equivalent to $k$ and $2k$, respectively. Table 2 shows the ratios of the scaled amplitudes of the fundamental and second-order modes to that of the overall pre-reshock interface ($k_{I} a_{I}/ka_{1,{e}}^r$ and $k_{II} a_{II}/ka_{1,{e}}^r$). For cases with reshock occurring at the linear evolution stage of the once-shocked interface, $k_{II} a_{II}/ka_{1,{e}}^r$ is notably lower than $k_{I} a_{I}/ka_{1,{e}}^r$ and $ka_{1,{e}}^r$ still satisfies the small-amplitude criterion (i.e. $ka\ll 1$). In contrast, in cases where reshock occurs at the nonlinear evolution stage of the once-shocked interface, $k_{II} a_{II}/ka_{1,{e}}^r$ is generally higher than 0.2 and $ka_{1,{e}}^r$ deviates from the small-amplitude criterion. Overall, the evolution state of the pre-reshock interface reflects the extent to which it deviates from a small-amplitude single-mode one. In previous research on reflected-reshock RMI (Balasubramanian et al. Reference Balasubramanian, Orlicz, Prestridge and Balakumar2012; Reilly et al. Reference Reilly, McFarland, Mohaghar and Ranjan2015; Guo et al. Reference Guo, Cong, Si and Luo2022a,Reference Guo, Si, Zhai and Luob), it was found that the morphology and amplitude of the pre-reshock interface affect the post-reshock flow evolution. Therefore, it can be preliminarily inferred that in S-RMI, the shape effect of the pre-reshock interface would also influence the post-reshock perturbation development, warranting further detailed investigation.

Finally, the amplitude growth rate of the once-shocked interface at the onset of reshock, i.e. the pre-reshock amplitude growth rate ($\dot {a}_1^r$), which contributes to $\dot {a}_2^l$, is investigated. The experimental $\dot {a}_1^r$ ($\dot {a}_{1,{e}}^r$) is obtained by fitting the pre-reshock amplitude evolution of the once-shocked interface, and its theoretical reference ($\dot {a}_{1,{zg}}^{r}$) is calculated using the ZG model. Table 2 shows that $\dot {a}_{1,{zg}}^{r}$ closely matches $\dot {a}_{1,{e}}^r$, further indicating the effectiveness of using the ZG model to predict the evolution of the once-shocked interface. Additionally, $\dot {a}_{1,{zg}}^{r}$ is compared with $\dot {a}_{1,{i}}^{l}$. In cases where reshock occurs at the linear evolution stage of the once-shocked interface, the difference between $\dot {a}_{1,{zg}}^{r}$ and $\dot {a}_{1,{i}}^{l}$ is limited. However, in cases with reshock occurring at the nonlinear evolution stage of the once-shocked interface, the discrepancy between $\dot {a}_{1,{zg}}^{r}$ and $\dot {a}_{1,{i}}^{l}$ is significant, suggesting that the nonlinearity of the pre-reshock amplitude growth may also affect the post-reshock interface evolution. Overall, for S-RMI with reshock occurring at the nonlinear evolution stage of the once-shocked interface, further investigation is needed to ascertain whether and how the shape of the pre-reshock interface and its nonlinear amplitude growth affect the perturbation evolution of the twice-shocked interface.

3.2.2. Quasi-linear amplitude growth of twice-shocked interface

To predict $\dot {a}_2^l$, Mikaelian (Reference Mikaelian1985) first proposed the linear superposition model which assumes that both $\dot {a}_{1}^{l}$ and $\Delta \dot {a}$ can be predicted by the impulsive model and that $\dot {a}_2^l$ is equal to their linear superposition. The model can be written as

(3.4)\begin{equation} \dot{a}_{2,{ls}}^l=\dot{a}_{1,{i}}^{l}+\Delta \dot{a}_{i}=\dot{a}_{1,{i}}^{l}+C_2 k a_{1,zg}^r A_2 (u_{2,{o}} - u_{1,{o}}), \end{equation}

where $\Delta \dot {a}_{i}$ denotes $\Delta \dot {a}$ predicted by the impulsive model; $u_{2,{o}}$ is the velocity of the twice-shocked interface predicted by one-dimensional gas dynamics theory; $C_2=1-(u_{2,{o}}-u_{1,{o}})/(v_{2,{e}}^{i}-u_{1,{o}})$ is the reshock compression factor, with $v_{2,{e}}^{i}$ being the velocity of IS$_2$ obtained from experiment; and $a_{1,zg}^r$ represents $a_{1}^r$ predicted by the ZG model and can be written as

(3.5)\begin{equation} a_{1,zg}^r=a_1^*+\int_{t_1^*}^{\Delta t} \frac{{\dot a}_{1,{zg}}^{b}(t)+{\dot a}_{1,{zg}}^{s}(t)}{2} {\rm d} t. \end{equation}

As shown in tables 1 and 2, $u_{2,{o}}$ and $ka_{1,{zg}}^r$ agree well with their experimental counterparts. Notably, since the linear superposition model assumes that $\Delta \dot {a}$ can be predicted by the impulsive model, it is theoretically valid only when the pre-reshock interface still satisfies the small-amplitude criterion. Building upon the linear superposition model, Wang et al. (Reference Wang, Cao, Chen, Zhai and Luo2022) proposed a modified model that additionally considers the high-amplitude effect of the pre-reshock interface (H model). The H model can be expressed as

(3.6)\begin{equation} \dot{a}_{2,{h}}^l=\dot{a}_{1,{i}}^{l}+R_{dr} \Delta \dot{a}_{i}= \dot{a}_{1,{i}}^{l}+R_{dr} C_2 k a_{1,{zg}}^r A_2 (u_{2,{o}} - u_{1,{o}}), \end{equation}

where $R_{dr}=1/[1+(k a_{1,zg}^r/3)^{4/3}]$ is the theoretical reduction factor proposed by Dimonte & Ramaprabhu (Reference Dimonte and Ramaprabhu2010) for describing the high-amplitude effect in single-shock RMI at a single-mode interface. Note that other theoretical reduction factors proposed for single-shock RMI (Rikanati et al. Reference Rikanati, Oron, Sadot and Shvarts2003; Buttler et al. Reference Buttler2012; Probyn et al. Reference Probyn, Williams, Thornber, Drikakis and Youngs2021) are not considered in this study as their values closely approximate that of $R_{dr}$ (Wang et al. Reference Wang, Wang, Zhai and Luo2023b). The H model has been demonstrated to outperform the linear superposition model for predicting $\dot {a}_2^l$ in S-RMI with a weak second shock and $ka_{1,{e}}^r<2/3$ (Wang et al. Reference Wang, Cao, Chen, Zhai and Luo2022). Based on the H model, we propose a further modified model that accounts for the nonlinearity of the pre-reshock amplitude growth (NH model), which can be written as

(3.7)\begin{equation} \dot{a}_{2,{nh}}^l=\dot{a}_{1,zg}^{r}+R_{dr} \Delta \dot{a}_{i}= \dot{a}_{1,zg}^{r}+ R_{dr} C_2 k a_{1,zg}^r A_2 (u_{2,{o}} - u_{1,{o}}). \end{equation}

The $\dot {a}_{2}^{l}$ obtained from experiments ($\dot {a}_{2,{e}}^{l}$) and predicted by the linear superposition, H and NH models ($\dot {a}_{2,{ls}}^l$, $\dot {a}_{2,{h}}^l$ and $\dot {a}_{2,{nh}}^l$) are provided in table 3 for comparison. These models exhibit poorer performance when the nonlinearity of the pre-reshock interface evolution is more significant and/or the second shock is stronger. Moreover, none of these models are universally applicable under the considered reshock conditions. Nevertheless, the NH model generally offers the most accurate prediction for $\dot {a}_{2,{e}}^{l}$, highlighting the need to account for both the shape effect of the pre-reshock interface and the nonlinearity of its evolution. For cases VW, the secondary compression effect is weak, and the deviation of $\dot {a}_{2,{e}}^{l}$ from $\dot {a}_{2,{nh}}^l$ can directly reflect the applicability of $R_{dr}$ in describing the shape effect of the pre-reshock interface. In cases VW-0.35 and VW-0.70, the NH model accurately predicts $\dot {a}_{2,{e}}^{l}$, suggesting that $R_{dr}$ can well describe the pre-reshock interface shape effect when reshock occurs at the linear or weakly nonlinear evolution stage of the once-shocked interface. However, in cases VW-1.13 and VW-1.78, the NH model overestimates $\dot {a}_{2,{e}}^{l}$, indicating that $R_{dr}$ is inadequate to describe the shape effect of the pre-reshock interface when it deviates significantly from a small-amplitude single-mode one. This overestimation may be attributed to the non-single-mode profile of the pre-reshock interface (Liang et al. Reference Liang, Zhai, Ding and Luo2019; Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020). For cases W, the NH model predicts $\dot {a}_{2,{e}}^{l}$ well (poorly) when reshock occurs at the linear (nonlinear) evolution stage of the once-shocked interface. It is worth noting that $R_{dr}$ is close in magnitude to the reduction factor derived for RMI in incompressible flow (Velikovich & Dimonte Reference Velikovich and Dimonte1996). Therefore, the failure of $R_{dr}$ to describe the reduction of the experimental $\Delta {\dot {a}}$ ($\Delta {\dot {a}}_{e}$) relative to $\Delta {\dot {a}}_{i}$ in case W-0.71 should be attributed to the secondary compression effect, which is negligible in case W-0.32 (VW-0.70) due to the low $ka_{1,{e}}^r$ ($M_2$). For cases W-1.19 and W-1.88, the combined influence of the pre-reshock interface shape effect and the secondary compression effect on $\Delta {\dot {a}}_{e}$ results in $\dot {a}_{2,{e}}^{l}$ being significantly lower than $\dot {a}_{2,{nh}}^l$. In cases MS, the coupling of these two effects has a more significant influence on $\Delta {\dot {a}}_{e}$ compared with that in cases W, and the secondary compression effect is non-negligible even when reshock occurs at the linear evolution stage of the once-shocked interface. Overall, both the pre-reshock interface shape effect and the secondary compression effect suppress the post-reshock linear amplitude growth and they would couple with each other. Furthermore, $R_{dr}$ fails to accurately describe the reduction in $\Delta {\dot {a}}_{e}$ relative to $\Delta {\dot {a}}_{i}$ when either effect is significant.

Table 3.

Experimental and theoretical results related to the pre-reshock amplitude evolution and post-reshock linear amplitude growth: $\dot {a}_{2,{e}}^{l}$, $\dot {a}_{2,{ls}}^{l}$, $\dot {a}_{2,{h}}^{l}$ and $\dot {a}_{2,{nh}}^{l}$, $\dot {a}_{2}^{l}$ obtained from experiments and predicted by linear superposition, H and NH models, respectively; $a_{1,{zg}}^{r,b}$ and $a_{1,{zg}}^{r,s}$ ($\dot {a}_{1,{zg}}^{r,b}$ and $\dot {a}_{1,{zg}}^{r,s}$), amplitudes (amplitude growth rates) of the pre-reshock bubble and spike predicted by the ZG model, respectively; $\dot {a}_{2,{e}}^{l,b}$ and $\dot {a}_{2,{e}}^{l,s}$, post-reshock linear amplitude growth rates of the bubble and spike extracted from experiments, respectively; $\dot {a}_{2,{em}}^{l,b}$, $\dot {a}_{2,{em}}^{l,s}$ and $\dot {a}_{2,{em}}^{l}$, post-reshock linear amplitude growth rates of the bubble, spike and overall interface predicted by the EM model, respectively.

Developing a theoretical model capable of accurately predicting $\dot {a}_{2,{e}}^{l}$ under various reshock conditions is desirable. However, providing a rigorous description of the coupling of the pre-reshock interface shape effect and secondary compression effect is challenging. Therefore, we seek to propose an empirical model based on the present experimental results. Due to nonlinearity, the bubble and spike generally have different amplitudes and amplitude growth rates before reshock. Consequently, their post-reshock linear amplitude growth rates ($\dot {a}_{2}^{l,b}$ and $\dot {a}_{2}^{l,s}$) should also differ and require separate consideration. Building on the NH model, an empirical model considering the coupling of the pre-reshock interface shape effect and the secondary compression effect (EM model) is proposed, which can be written as

(3.8)\begin{align} \dot{a}_{2,{em}}^{l} &= \frac{\dot{a}_{2,{em}}^{l,b}+\dot{a}_{2,{em}}^{l,s}}{2} \nonumber\\ & = \frac{[\dot a_{1,{zg}}^{r,b}+R_{em}^b C_2 k a_{1,zg}^{r,b} A_2 (u_{2,o} - u_{1,o})]+[\dot a_{1,zg}^{r,s}+R_{em}^s C_2 k a_{1,zg}^{r,s} A_2 (u_{2,o} - u_{1,o})]} {2}, \end{align}

where

(3.9)$$\begin{gather} R_{em}^b = \frac{1}{1+\left\{ka_{1,zg}^{r,b}\dfrac{\sqrt {5 [1+(u_{2,o} - u_{1,o})/ (v_{2,o}^{t} - u_{2,o})]} }{2}\right\}^2}; \end{gather}$$

(3.10)$$\begin{gather}R_{em}^s = \frac{1}{1+\left\{ka_{1,zg}^{r,s}\dfrac{\sqrt {5 [1+(u_{2,o} - u_{1,o})/ (v_{2,o}^{t} - u_{2,o})]} }{4}\right\}^2}. \end{gather}$$

Here, $\dot {a}_{2,{em}}^{l,b}$ and $\dot {a}_{2,{em}}^{l,s}$ correspond to $\dot {a}_{2}^{l,b}$ and $\dot {a}_{2}^{l,s}$ predicted by the EM model, respectively; $a_{1,zg}^{r,b}$ and $a_{1,zg}^{r,s}$ ($\dot a_{1,zg}^{r,b}$ and $\dot a_{1,zg}^{r,s}$) represent the amplitudes (amplitude growth rates) of the pre-reshock bubble and spike predicted by the ZG model, respectively; $R_{em}^b$ and $R_{em}^s$ are the empirical factors utilized to describe the reduction in the reshock-induced linear amplitude growth rates of the bubble and spike, respectively; and $v_{2,o}^{t}$ denotes the velocity of TS$_2$ predicted by the one-dimensional gas dynamics theory. In $R_{em}^b$ ($R_{em}^s$), $ka_{1,zg}^{r,b}$ ($ka_{1,zg}^{r,s}$) is employed to depict the shape effect of the pre-reshock bubble (spike), while $(u_{2,o}- u_{1,o})/ (v_{2,o}^{t} - u_{2,o})$ is introduced to describe the secondary compression effect. Notably, the form of the latter is referenced to a parameter used to characterize the dependence of linear amplitude evolution on shock intensity in single-shock RMI studies (Holmes et al. Reference Holmes, Dimonte, Fryxell, Gittings, Grove, Schneider, Sharp, Velikovich, Weaver and Zhang1999; Glendinning et al. Reference Glendinning2003). Experimental $\dot {a}_{2}^{l,b}$ and $\dot {a}_{2}^{l,s}$ ($\dot {a}_{2,{e}}^{l,b}$ and $\dot {a}_{2,{e}}^{l,s}$) are determined via using the position of the unperturbed interface, calculated by the one-dimensional gas dynamics theory, to distinguish the bubble and spike. As shown in table 3, $\dot {a}_{2,{em}}^{l,b}$, $\dot {a}_{2,{em}}^{l,s}$ and $\dot {a}_{2,{em}}^{l}$ align well with $\dot {a}_{2,{e}}^{l,b}$, $\dot {a}_{2,{e}}^{l,s}$ and $\dot {a}_{2,{e}}^{l}$, respectively, demonstrating the validity of the EM model under various reshock conditions considered in this study.

The dependence of $\Delta {\dot {a}}$ on $ka_1^r$ is further analysed by introducing a correlation coefficient defined as

(3.11)\begin{equation} \psi=\frac{\Delta{\dot{a}}}{A_2 (u_{2,o} - u_{1,o})}. \end{equation}

The variation rate of $\psi$ with $ka_1^r$, i.e. $\partial \psi /\partial ka_1^r$, clearly illustrates the correlation between $\Delta {\dot {a}}$ and $ka_1^r$. If $|\partial \psi /\partial ka_1^r|$ is close to $0$ ($1$), it indicates a weak (strong) correlation between $\Delta {\dot {a}}$ and $ka_1^r$. Additionally, a positive (negative) $\partial \psi /\partial ka_1^r$ suggests a positive (negative) correlation between $\Delta {\dot {a}}$ and $ka_1^r$. The results for $\psi$ and $\partial \psi /\partial ka_1^r$ obtained from experiments ($\psi _{e}$ and $\partial \psi _{e}/\partial ka_1^r$), along with the corresponding predictions from the EM model ($\psi _{em}$ and $\partial \psi _{em}/\partial ka_1^r$), are shown in figure 7. It can be observed that the correlation between $\Delta {\dot {a}}$ and $ka_1^r$ changes from strongly positive to weakly positive and then to weakly negative as $ka_1^r$ increases, regardless of the intensity of the second shock. This indicates that $\Delta {\dot {a}}$ is non-monotonically related to $ka_1^r$, and that this relationship is primarily due to the shape effect of the pre-reshock interface. Additionally, the correlation between $\Delta {\dot {a}}$ and $ka_1^r$ is weaker when the second shock is stronger, due to the suppression of the secondary compression effect on post-reshock amplitude evolution.

Figure 7.

Variations of $\psi$ (a) and $\partial \psi /\partial ka_1^r$ (b) with $ka_1^r$.

3.2.3. Nonlinear amplitude growth of twice-shocked interface

Following a short period of linear growth, the amplitude of the twice-shocked interface enters the nonlinear evolution stage. Currently, a rigorous theory for describing the post-reshock nonlinear amplitude evolution is still lacking. In previous studies on S-RMI (Karkhanis et al. Reference Karkhanis, Ramaprabhu, Buttler, Hammerberg, Cherne and Andrews2017; Karkhanis & Ramaprabhu Reference Karkhanis and Ramaprabhu2019; Wang et al. Reference Wang, Cao, Chen, Zhai and Luo2022), some nonlinear models proposed for single-shock RMI have been utilized to predict the nonlinear perturbation growth of the twice-shocked interface. Wang et al. (Reference Wang, Cao, Chen, Zhai and Luo2022) found that the ZG model is applicable to predict the post-reshock amplitude evolution when the second shock is weak and reshock occurs at the linear evolution stage of the once-shocked interface. In this work, the ZG model is similarly adopted to offer a theoretical reference for the nonlinear perturbation growth of the twice-shocked interface, which can be rewritten as

(3.12)\begin{equation} \dot{a}_{2,{zg}}^{b/s}(t) = \frac{\dot{a}_{2,{em}}^{l,b/s}}{1 + \theta_2^{b/s} k \dot{a}_{2,{em}}^{l,b/s} t}, \end{equation}

where

(3.13)\begin{equation} \theta_2^{b/s}=\frac{3}{4}\frac{(1 \pm A_2)(3 \pm A_2)}{3 \pm A_2 + \sqrt{2}(1 \pm A_2)^{1/2}} \frac{4(3 \pm A_2) + \sqrt{2}(9 \pm A_2)(1 \pm A_2)^{1/2}}{(3 \pm A_2)^2 + 2\sqrt{2}(3\mp A_2)(1\pm A_2)^{1/2}}. \end{equation}

Figure 8 illustrates the temporal variations of the bubble and spike amplitudes of the twice-shocked interface ($a_2^b$ and $a_2^s$) in dimensionless form. The evolution time and amplitude of the bubble/spike are normalized as $\tau _2^{b/s}=k \dot {a}_{2,{em}}^{l,b/s}(t-t_{2}^{b*/s*})$ and $\alpha _2^{b/s}=k(a_2^{b/s}-a_2^{b*/s*})$, respectively, in which $t_{2}^{b*/s*}$ indicates the moment when the linear growth of $a_2^{b/s}$ starts in the experiment, and $a_2^{b*/s*}$ is the corresponding $a_2^{b/s}$ at $t=t_2^{b*/s*}$. As the ZG model does not account for the initial amplitude effect, its predictions for cases with similar second shock intensities but different $ka_{1,{e}}^r$ are highly consistent. Accordingly, only two theoretical lines, corresponding to the bubble and spike of the twice-shocked interface, respectively, are provided in each figure. The scaling effectively collapses the results of cases with similar second shock intensities. For cases VW and W, the ZG model accurately predicts the overall bubble and spike evolutions. However, for cases MS, the ZG model overestimates the bubble growth while underestimating the spike development from the intermediate to late stages. These results suggest that the ZG model can reasonably describe the post-reshock nonlinear amplitude evolution when the second shock is very weak/weak, and the secondary compression effect could alter the nonlinear evolution law of the twice-shocked interface when the second shock reaches a certain intensity. Specifically, the high pressure generated by interactions among transverse shock waves inhibits (promotes) the bubble (spike) growth, leading to the overestimation (underestimation) of the ZG model.

Figure 8.

Experimental and theoretical results on the amplitude evolutions of the bubble and spike of the twice-shocked interface: (a) cases VW, (b) cases W and (c) cases MS.

Further, we attempt to propose an empirical nonlinear model based on the ZG model and present experimental results, aiming to effectively describe the post-reshock nonlinear evolution law under all reshock conditions considered. For the evolutions of the bubble and spike of the twice-shocked interface, the empirical model (mZG model), obtained after many attempts, can be respectively written as

(3.14)\begin{equation} \dot{a}_{2,{mzg}}^{b}(t) = \frac{\dot{a}_{2,{em}}^{l,b}}{1 + \theta_2^b k \dot{a}_{2,{em}}^{l,b} F^b t} \end{equation}

and

(3.15)\begin{equation} \dot{a}_{2,{mzg}}^{s}(t) = \frac{\dot{a}_{2,{em}}^{l,s} F^s}{1 + \theta_2^s k \dot{a}_{2,{em}}^{l,s} F^s t}. \end{equation}

Here, $F^b=1+(u_{2,o}-u_{1,o})/3(v_{2,o}^t-u_{2,o})$ and $F^s=1+(u_{2,o}-u_{1,o})/5(v_{2,o}^t-u_{2,o})$ represent the empirical factors for describing the secondary compression effect on the post-reshock growths of the bubble and spike, respectively. As illustrated in figure 8, the mZG model accurately predicts the overall bubble and spike evolutions for all cases, demonstrating its ability to precisely describe the nonlinear evolution law of the twice-shocked interface under the reshock conditions considered.