3.2.1. Shock-compression stage

Following the impact of IS $_1$ on the initial interface at $t=0$ , $|a_1|$ decreases continuously until the completion of the shock–interface interaction (Richtmyer Reference Richtmyer1960). The expressions for the duration of the interaction ( $t_1$ ) and the amplitude variation during this stage were proposed by Richtmyer (Reference Richtmyer1960), and can be written as

(3.1) \begin{equation} t_1=2\,|a_0|/v_1^{\textit{is}}, \end{equation}

(3.2) \begin{equation} a_1(t)=a_0+\Delta u_{1}\,t/2. \end{equation}

These relations, collectively referred to as the R model hereafter, have been extensively validated (Zhou Reference Zhou2024), and were incorporated into the FS model.

3.2.2. Start-up stage

During this stage, $\dot {a}_{1}(t)$ rises from zero and gradually reaches an asymptotic value $\dot {a}_{1}^{+}$ under the drive of pressure perturbation (Yang, Zhang & Sharp Reference Yang, Zhang and Sharp1994). Lombardini & Pullin (Reference Lombardini and Pullin2009) proposed an analytical model to forecast the characteristic time of the start-up process for a once-shocked light–heavy interface. For a once-shocked heavy–light interface, Yang et al. (Reference Yang, Zhang and Sharp1994) deduced a compressible linear theory to predict the perturbation evolution in the start-up period. However, this theory needs to be solved numerically. Recently, following the approach of Lombardini & Pullin (Reference Lombardini and Pullin2009), Li et al. (Reference Li, Chen, Zhai and Luo2024) deduced an analytical solution for predicting the duration of the start-up process for a once-shocked heavy–light interface, which can be expressed as

(3.3) \begin{equation} t_{2}-t_{1}=\dfrac {1}{k}\left (\dfrac {1-A_{1}}{v_{1}^{rw}+\Delta u_{1}}+\dfrac {1+A_{1}}{v_{1}^{\textit{ts}}-\Delta u_{1}}\right ). \end{equation}

Here, $t_{2}$ denotes the end moment of the start-up period, while $v_{1}^{rw}$ represents the velocity of RW $_1$ . In addition, Li et al. (Reference Li, Chen, Zhai and Luo2024) proposed that the amplitude growth in the start-up stage can be described as

(3.4) \begin{equation} \dot {a}_{1}(t)=\dfrac {2\dot {a}_{1}^{+}}{(1+A_{1})\coth \big(kL^{\textit{ts}}_1\big)+(1-A_{1})\coth \big(kL^{rw}_1\big)} \quad (t_1\lt t\lt t_2), \end{equation}

where

(3.5) \begin{equation} \begin{cases} L^{\textit{ts}}_1= \big(v_{1}^{\textit{ts}}-\Delta u_{1}\big)(t-t_{1}),\\ L^{rw}_1=\big(v_{1}^{rw}+\Delta u_{1}\big)(t-t_{1}), \end{cases} \end{equation}

are the displacements of TS $_1$ and RW $_1$ , respectively.

3.2.3. Linear stage

When $t\gt t_2$ , the amplitude evolution is commonly regarded as entering the linear stage, during which $\dot {a}_{1}(t)$ remains nearly constant at $\dot {a}_{1}^{+}$ . However, the linear stage is an approximate description of a period where nonlinearity exists but remains limited, and defining the demarcation point between the linear and nonlinear stages is challenging. Therefore, the FS model considers the end of the start-up period as the beginning of both the linear and nonlinear stages. In other words, the linear stage is considered to be of negligible duration in the FS model. The irrotational WN model (WNi model) (Wouchuk & Nishihara Reference Wouchuk and Nishihara1997; Wouchuk Reference Wouchuk2001), validated for scenarios where the shock intensity (defined as $1-p_{in}/p_{be}$ , where $p_{in}$ and $p_{be}$ represent the pressures of fluids in front of and behind the shock, respectively) is lower than 0.5 (Li et al. Reference Li, Chen, Zhai and Luo2024), was adopted to predict $\dot {a}_{1}^{+}$ . The WNi model can be written as

(3.6) \begin{equation} \dot {a}_1^{+}=k a_{0} \left [\dfrac {1-A_{1}}{2}\left (1+\dfrac {v^{rw}_{1}}{v_{1}^{\textit{is}}}\right )\left (u_{1}-\Delta u_{1}\right ) +\Delta u_{1} \dfrac {1+A_{1}}{2}\left (1-\dfrac {v^{\textit{ts}}_{1}}{v_{1}^{\textit{is}}}\right )\right ], \end{equation}

where $u_{1}$ represents the velocity of the fluid behind IS $_{1}$ . The system (3.3)–(3.6), collectively referred to as the Li model, was incorporated into the FS model.

3.2.4. Weakly nonlinear stage

As high-order harmonics emerge and become significant, $\dot {a}_{1}(t)$ decreases progressively, signifying that the interface evolution has entered the weakly nonlinear stage (Zhou Reference Zhou2024). Zhai et al. (Reference Zhai, Chen, Xing, Li, Cao, Wang and Luo2025) experimentally demonstrated that the DR model (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010) accurately describes the weakly nonlinear evolution of a once-shocked SF $_6$ –air interface, and therefore adopted it to construct the FS model. The DR model can be expressed as

(3.7) \begin{equation} \begin{cases} \dot {a}_{1}(t)=\big[\dot {a}^{b}_{1}(t)+\dot {a}^{s}_{1}(t)\big]/2, \\ \dot {a}^{b/s}_{1}(t)=\dot {a}_1^{+}\dfrac {1+k\dot {a}_1^{+} (t-t_2)(1\mp |A_1|)}{1+E^{b/s}k\dot {a}_1^{+} (t-t_2)+(1\mp |A_1|)F^{b/s}k^2 \dot {a}_1^{+2} (t-t_2)^2}, \end{cases} \end{equation}

where superscripts $b$ and $s$ represent bubble and spike, respectively, $E^{b/s}=[4.5\pm |A_1|+(2\mp |A_1|)(1-\Delta u_1/v_1^{\textit{is}})k|a_0|]/4$ and $F^{b/s}=1\pm |A_1|$ .

Figure 6.

Overall amplitude evolutions of once-shocked heavy–light interfaces with various $A_1$ . Symbols denote experimental results; dashed and solid lines represent theoretical predictions by the FS and mFS models, respectively. Error bars represent potential errors in extracting the interface amplitude, with a length equal to half the width of the interface in the experimental schlieren image, and the same hereafter.

The comparison between the present experimental results and corresponding theoretical predictions by the FS model, composed of (3.1)–(3.7), is illustrated in figure 6. It can be observed that the FS model provides reasonable predictions for the entire amplitude evolution process under low and moderate $|A_1|$ conditions, as well as for the evolution of the SF $_6$ –He interface from the shock-compression to linear stages. However, this model fails to accurately predict the weakly nonlinear evolution of the SF $_6$ –He interface. Notably, the R and Li models have been demonstrated to be valid over a wide range of $A_1$ conditions (Xing et al. Reference Xing, Chen, Li, Wang, Zhai and Luo2025). Therefore, the failure of the FS model should be attributed to the limitation of the DR model. In our previous work (Xing et al. Reference Xing, Chen, Li, Wang, Zhai and Luo2025), an empirical nonlinear model (mSEA model) was proposed based on existing models and extensive experimental results (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023; Xing et al. Reference Xing, Chen, Li, Wang, Zhai and Luo2025), which can be described as

(3.8) \begin{equation} \begin{cases} \dot {a}_{1}(t)=\big[\dot {a}^{b}_{1}(t)+\dot {a}^{s}_{1}(t)\big]/2, \\ \dot {a}^{b/s}_{1}(t)=\dot {a}_1^{+}\dfrac {1+k\dot {a}_1^{+} (t-t_2)}{1+(2\mp |A_1|)(1.07\pm |A_1|)k\dot {a}_1^{+} (t-t_2)+\dfrac {1\pm |A_{1}|}{1+|A_{1}|}\dfrac {1}{2\pi C_1}k^2 \dot {a}_1^{+2} (t-t_2)^2}, \end{cases} \end{equation}

where $C_1$ is given by $1/3\pi$ for cases with $|A_{1}|\geqslant 0.5$ , and $1/2\pi$ for cases where $|A_{1}|\rightarrow 0$ . The mSEA model has been shown to effectively describe the weakly nonlinear evolution across the ranges of $A_1$ from $-0.95$ to $-0.35$ , and from 0.30 to 0.86. Therefore, we can obtain a modified FS model (mFS model) by substituting the DR model with the mSEA model. The overall amplitude evolution predicted using the mFS model is presented in figure 6. The close agreement between the mFS model predictions and experimental results suggests that the mFS model is capable of reliably predicting the complete amplitude evolution of once-shocked heavy–light interfaces with various $A_1$ .

3.3.1. Linear amplitude growth

To describe the linear evolution following reshock, Mikaelian (Reference Mikaelian1985) proposed a linear superposition principle, which assumes that the post-reshock linear amplitude growth rate ( $\dot {a}_{2}^{+}$ ) is equal to the linear superposition of the pre-reshock amplitude growth rate ( $\dot {a}_{2}^{-}$ ) and the additional growth rate induced by reshock ( $\Delta \dot {a}$ ). This principle has since served as a theoretical basis for numerous investigations. Specifically, the Mik-L model proposed by Mikaelian (Reference Mikaelian1985) assumes that $\dot {a}_{2}^{+}$ equals the sum of $\dot {a}_{2}^{-}$ and $\Delta \dot {a}$ calculated using the impulsive model. As a result, the Mik-L model is applicable only for scenarios with a light–heavy interface, and is expected to fail when reshock occurs at the nonlinear evolution stage of the once-shocked interface. Building upon the linear superposition principle, Charakhch’yan (Reference Charakhch’yan2000, Reference Charakhch’yan2001) proposed the Cha-L model for predicting $\dot {a}_{2}^{+}$ . However, the Cha-L model does not include a relation/model to predict the amplitude evolution of the once-shocked interface, and instead relies on $\dot {a}_{2}^{-}$ extracted from numerical simulations or experiments. Thus its application is significantly constrained. Additionally, the Cha-L model is expected to be applicable only when reshock occurs at strongly nonlinear stages, as it assumes that $\Delta \dot {a}$ is independent of the pre-reshock perturbation characteristics. To address the limitations of the previous models, Zhai et al. (Reference Zhai, Chen, Xing, Li, Cao, Wang and Luo2025) proposed a new model (LS model) to predict $\dot {a}_{2}^{+}$ of an SF $_6$ –air interface subjected to reshock occurring at any stage prior to the onset of strong nonlinearity. The LS model can be expressed as

(3.9) \begin{equation} \dot {a}_{2,ls}^{+}=\dot {a}_{2,fs}^{-}+\Delta \dot {a}_{wni}, \end{equation}

where

(3.10) \begin{align} \Delta \dot {a}_{wni}&=k a_{2,fs}^{-} \left [\dfrac {1-A_{2}}{2}\left (1+\dfrac {v^{rw}_{2}+\Delta u_{1}}{v_{2}^{\textit{is}}-\Delta u_{1}}\right )(u_{2}-\Delta u_{1}-\Delta u_{2}) \right. \nonumber \\ & \quad + \left. \Delta u_{2} \dfrac {1+A_{2}}{2}\left (1-\dfrac {v^{\textit{ts}}_{2}-\Delta u_{1}}{v_{2}^{\textit{is}}-\Delta u_1}\right )\right ] \end{align}

represents $\Delta \dot {a}$ predicted by the WNi model; $\dot {a}_{2,fs}^{-}$ and $a_{2,fs}^{-}$ correspond to $\dot {a}_{2}^{-}$ and the pre-reshock amplitude ( $a_{2}^{-}$ ) predicted by the FS model, respectively; $v_2^{rw}$ and $u_2$ denote the velocities of RW $_2$ and the fluid behind IS $_{2}$ , respectively; $A_2$ represents the post-reshock Atwood number.

Table 3.

Experimental and theoretical results related to the post-reshock linear amplitude evolution: $\dot {a}_{2,e}^{+}$ , $\dot {a}^{+}_{2,ls}$ and $\dot {a}^{+}_{2,mls}$ are post-reshock linear amplitude growth rates obtained from experiments and predicted by the LS and mLS models, respectively; $ka^{-}_{2,mfs}$ and $\dot {a}^{-}_{2,mfs}$ are pre-reshock dimensionless amplitude and amplitude growth rate predicted by the mFS model, respectively.

Values of $\dot {a}_{2}^{+}$ extracted from experiments ( $\dot {a}_{2,e}^{+}$ ) and predicted by the LS model are presented in table 3. It can be observed that the LS model reasonably forecasts $\dot {a}_{2,e}^{+}$ for cases 0.36-553, 0.36-1043 and 0.67-235, whereas it performs poorly in predicting $\dot {a}_{2,e}^{+}$ for cases 0.67-566, 0.83-202 and 0.83-529. The failure of the LS model for case 0.67-566 should be attributed to the fact that $ka_2^{-}$ no longer satisfies the small-amplitude criterion, which is a fundamental assumption for the validity of the WNi model. Extensive research (Rikanati et al. Reference Rikanati, Oron, Sadot and Shvarts2003; Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010; Probyn et al. Reference Probyn, Williams, Thornber, Drikakis and Youngs2021; Wang et al. Reference Wang, Wang, Zhai and Luo2023a ,Reference Wang, Wang, Zhai and Luo b ) has demonstrated that the high initial amplitude leads to the practical linear amplitude growth rate being lower than the predictions from models deduced based on the small-amplitude assumption – a phenomenon generally referred to as the high-amplitude effect. Several reduction factors have been proposed to describe this effect (Rikanati et al. Reference Rikanati, Oron, Sadot and Shvarts2003; Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010; Probyn et al. Reference Probyn, Williams, Thornber, Drikakis and Youngs2021; Wang et al. Reference Wang, Wang, Zhai and Luo2023b ). Among them, the one proposed by Dimonte & Ramaprabhu (Reference Dimonte and Ramaprabhu2010) ( $R_{\textit{dr}}$ ) has been demonstrated to effectively describe the high-amplitude effect on the evolution of a once-shocked heavy–light interface (Wang et al. Reference Wang, Wang, Zhai and Luo2023a ). In contrast, for case 0.83-202, where $ka_2^{-}$ still satisfies the small-amplitude criterion, the failure of the LS model should be attributed to the fact that the FS model loses accuracy under high $|A_1|$ conditions, as demonstrated in § 3.2. For case 0.83-529, the failure of the LS model results from the combined influence of the high-amplitude effect and the limitation of the FS model.

Here, we propose a modified model (mLS model) to predict $\dot {a}_{2}^{+}$ by combining the linear superposition principle, WNi model, mFS model and $R_{\textit{dr}}$ , expressed as

(3.11) \begin{equation} \dot {a}_{2,mls}^{+}=\dot {a}_{2,mfs}^{-}+\Delta \dot {a}_{wni}\times R_{\textit{dr}}=\dot {a}_{2,mfs}^{-}+\dfrac {\Delta \dot {a}_{wni}} {1+\big (|ka_{2,mfs}^{-}|/3\big )^{4/3}}, \end{equation}

where

(3.12) \begin{align} \Delta \dot {a}_{wni}&=k a_{2,mfs}^{-} \left [\dfrac {1-A_{2}}{2}\left (1+\dfrac {v^{rw}_{2}+\Delta u_{1}}{v_{2}^{\textit{is}}-\Delta u_{1}}\right )(u_{2}-\Delta u_{1}-\Delta u_{2}) \right. \nonumber \\ & \quad + \left.\Delta u_{2} \dfrac {1+A_{2}}{2}\left (1-\dfrac {v^{\textit{ts}}_{2}-\Delta u_{1}}{v_{2}^{\textit{is}}-\Delta u_1}\right )\right ], \end{align}

and $\dot {a}_{2,mfs}^{-}$ and $a_{2,mfs}^{-}$ correspond to $\dot {a}_{2}^{-}$ and $a_{2}^{-}$ predicted by the mFS model, respectively. As presented in table 3, the mLS model is capable of accurately predicting the linear amplitude evolution of a twice-shocked heavy–light interface under diverse $A_1$ and $\Delta t$ conditions.

3.3.2. Complete amplitude evolution

Understanding the post-reshock linear amplitude evolution is essential. However, achieving a full-process prediction of the twice-shocked interface evolution remains a more desirable objective. Existing models, including the Mik-L, Cha-L, LS and mLS models, are limited to predicting only $\dot {a}_{2}^{+}$ , without capturing the full temporal evolution. We therefore aim to develop a theoretical model that accurately describes the complete post-reshock evolution. Analogous to the development of a once-shocked interface, the evolution of a twice-shocked interface can be broadly divided into the shock-compression stage, start-up stage, linear stage, weakly nonlinear stage, strongly nonlinear stage and turbulent mixing stage. In this study, we focus exclusively on the evolution preceding the strongly nonlinear stage. The time-varying amplitude growth rate and amplitude of the twice-shocked interface are represented by $\dot {a}_{2}(t)$ and $a_{2}(t)$ , respectively. The details of the model construction are provided below.

Shock-compression stage: The velocity of IS $_2$ relative to the once-shocked interface is typically much greater than $\dot {a}_{2}^{-}$ . Consequently, as IS $_2$ propagates across the interface, the interface can be approximated as moving with velocity $\Delta u_1$ without undergoing any self-evolution. Accordingly, the duration of the shock-compression stage ( $t_3-\Delta t$ ) and $a_{2}(t)$ in this stage can also be predicted using the R model:

(3.13) \begin{equation} t_3-\Delta t=2a_{2,mfs}^{-}/\big(v_2^{\textit{is}}-\Delta u_1\big), \end{equation}

(3.14) \begin{equation} a_2(t)=a_{2,mfs}^{-}-\Delta u_2 \big(t-\Delta t\big)/2. \end{equation}

Start-up stage: The twice-shocked interface enters the start-up evolution stage ( $t_3\lt t\lt t_4$ ) once IS $_2$ has completely traversed the interface. Following the idea of linear superposition, the start-up evolution of the twice-shocked interface is considered as a process wherein $\dot {a}_{2}(t)$ varies from $\dot {a}_{2}^{-}$ to $\dot {a}_{2}^{+}$ under the drive of pressure perturbations introduced by the perturbed TS $_2$ and RW $_2$ . Accordingly, the duration of the start-up process of the twice-shocked interface ( $t_4 - t_3$ ) and $\dot {a}_{2}(t)$ in this stage can be described by the reformulated Li model:

(3.15) \begin{equation} t_{4}-t_{3}=\dfrac {1}{k}\left (\dfrac {1-A_{2}}{v_{2}^{rw}+\Delta u_{1}+\Delta u_{2}}+\dfrac {1+A_{2}}{v_{2}^{\textit{ts}}-\Delta u_{1}-\Delta u_{2}}\right ), \end{equation}

(3.16) \begin{equation} \dot {a}_{2}(t)=\dot {a}_{2}^{-}+\dfrac {2\,\Delta \dot {a}}{(1+A_{2})\coth \big(kL^{\textit{ts}}_2\big)+(1-A_{2})\coth \big(kL^{rw}_2\big)}, \end{equation}

where $L^{\textit{ts}}_2$ and $L^{rw}_2$ denote the displacements of TS $_2$ and RW $_2$ , respectively, which can be expressed as

(3.17) \begin{equation} \begin{cases} L^{\textit{ts}}_2= \big(v_{2}^{\textit{ts}}-\Delta u_{1}-\Delta u_{2}\big)(t-t_{3}),\\ L^{rw}_2=\big(v_{2}^{rw}+\Delta u_{1}+\Delta u_{2}\big)(t-t_{3}). \end{cases} \end{equation}

Linear stage: Similarly, the linear evolution stage of the twice-shocked interface is considered to have a negligible duration, during which $\dot {a}_{2}$ remains constant at $\dot {a}_{2}^{+}$ . The mLS model, which has been validated earlier, is employed to predict $\dot {a}_{2}^{+}$ . Specifically, $\dot {a}_{2}^{-}$ and $\Delta \dot {a}$ in (3.16) are given by $\dot {a}_{2,mfs}^{-}$ and $\Delta \dot {a}_{wni} \times R_{\textit{dr}}$ , respectively.

Weakly nonlinear stage: Currently, there is still a lack of nonlinear models specifically proposed for the twice-shocked interface. Our previous study (Wang et al. Reference Wang, Cao, Chen, Zhai and Luo2022) demonstrated that for a light–heavy interface configuration, the evolution of a twice-shocked interface can be reasonably predicted by models developed for once-shocked interfaces, provided that the reshock occurs prior to the strongly nonlinear stage. This suggests that it is potentially feasible to treat a twice-shocked heavy–light interface as a once-shocked one with equivalent post-shock amplitude and linear growth rate. Based on this idea, the weakly nonlinear evolution of twice-shocked heavy–light interfaces with varying $A$ is also described using the mSEA model reformulated as

(3.18) \begin{equation} \begin{cases} \dot {a}_{2}(t)=\big[\dot {a}^{b}_{2}(t)+\dot {a}^{s}_{2}(t)\big]/2, \\ \dot {a}^{b/s}_{2}(t)=\dot {a}_{2,mls}^{+} \dfrac {1+k\dot {a}_{2,mls}^{+} (t-t_4)}{1+(2\mp |A_2|)(1.07\pm |A_2|)k\dot {a}_{2,mls}^{+} (t-t_4)+\dfrac {1\pm |A_{2}|}{1+|A_{2}|}\dfrac {1}{2\pi C_2}k^2 \dot {a}_{2,mls}^{+2} (t-t_4)^2}, \end{cases} \end{equation}

where $C_2$ is given by $1/3\pi$ for cases with $|A_{2}|\geqslant 0.5$ , and $1/2\pi$ for cases with $|A_{2}|\rightarrow 0$ . For clarity, the combination of (3.11)–(3.18) is hereafter referred to as the DS model. As illustrated in figure 8, the DS model offers accurate predictions for the amplitude evolution from the instant IS $_2$ hits the once-shocked interface to the weakly nonlinear evolution stage of the twice-shocked interface under diverse $A$ and $\Delta t$ conditions.

In conclusion, two models, namely the mFS and DS models, have been developed based on existing theoretical models and physically justified assumptions. Experimental validation confirms that, when combined, they can reliably predict the complete perturbation evolution of S-RMI across a wide range of $A$ . In the following, we will employ these models to achieve the controllable suppression of RMI via a same-side reshock under low, moderate and high $|A|$ conditions.

Figure 8.

Amplitude evolution of twice-shocked interface under diverse $A$ and $\Delta t$ conditions. Symbols and solid lines denote experimental results and theoretical predictions by the DS model, respectively.

Figure 9.

Schematics of seven possibilities for S-RMI at an initial heavy–light interface without Atwood-number reversal ( $A_0\lt 0$ and $A_1\lt 0$ ).

Figure 10.

Parameter conditions corresponding to each perturbation evolution possibility for scenarios with various $A$ . Open circles denote the experimental cases that have been discussed in § 3, whereas solid circles represent the supplementary experimental runs.