4.1. Linear amplitude growth

The temporal evolution of the dimensional overall interface amplitude ( $a$ ) for different cases is presented in figure 3. Here, $a$ is defined as half of the horizontal distance between the tips of the bubble and spike, and the determination of its uncertainty, represented by error bars, is presented in § B.4. Also, $t^*$ denotes the duration of the start-up period, i.e. a phase during which the amplitude growth rate rises from zero and gradually reaches $\dot {a}_l$ under the drive of pressure perturbations (Lombardini & Pullin Reference Lombardini and Pullin2009), and $a^*$ is the corresponding $a$ at $t = t^*$ . At early times, $a$ exhibits an approximately linear growth in all cases, and $\dot {a}_l^e$ is obtained by performing a linear fit to these data.

Figure 3.

Temporal variations of perturbation amplitude in dimensional form for cases with different $A_t^0$ .

The classical impulsive model has been demonstrated to apply to weak-shock RMI at a small-amplitude single-mode light–heavy interface over a wide range of $A_t^0$ conditions, as reported by Chen et al. (Reference Chen, Xing, Wang, Zhai and Luo2023) (see table 2 herein). Notably, prior theoretical studies (Mikaelian Reference Mikaelian1994; Yang et al. Reference Yang, Zhang and Sharp1994; Velikovich et al. Reference Velikovich, Herrmann and Abarzhi2014) indicate that theoretically, the fundamental criterion for the model’s applicability is the low flow compressibility. This criterion not only remains valid for weak shocks, but can also hold at finite shock strengths, provided that $\gamma$ is sufficiently large. In the present study, the impulsive model is re-examined to evaluate the influence of strong flow compressibility arising from strong shocks and highly compressible fluids ( ${\textrm{CO}}_2$ , ${\textrm{SF}}_6$ and ${\textrm{C}}_4{\textrm{F}}_{10}$ ) on its applicability under diverse $A_t^0$ . It can be expressed as

(4.1) \begin{align} \dot {a}_l^i = ka_1 A_t^1 u^e_{si}. \end{align}

Here, $a_1=a_0(1 - u^e_{si}/v^e_{\textit{is}})$ , with $v^e_{\textit{is}}$ representing the experimentally measured velocity of the incident shock. Besides the semi-empirical impulsive model, it is particularly valuable to evaluate compressible linear theories (Richtmyer Reference Richtmyer1960; Wouchuk Reference Wouchuk2001a ; Napieralski et al. Reference Napieralski, Cobos, Velikovich and Huete2024), which are grounded in first principles and incorporate the full acoustic coupling history, as they provide a comprehensive description of compressible linear RMI. Notably, for conciseness, only the Richtmyer theory is considered here. Given the high similarity among the predictions of the compressible linear theories, the evaluation of the Richtmyer theory’s predictive capability can serve as a representative assessment of the others. The Richtmyer theory lacks an analytical solution and involves a set of nonlinear equations that need to be solved numerically. Details on this theory are available in the pioneering works of Richtmyer (Reference Richtmyer1960) and Yang et al. (Reference Yang, Zhang and Sharp1994).

Table 2.

Comparison of experimental linear amplitude growth rate ( $\dot {a}_l^e$ ) with the predictions of the impulsive model ( $\dot {a}_l^i$ ), Richtmyer theory ( $\dot {a}_l^r$ ), and compressible impulsive model ( $\dot {a}_l^{ci}$ ). Here, $E^{i/r/ce}=(\dot {a}_{l}^{i/r/ci}-\dot {a}_{l}^{e})/\dot {a}_{l}^{e}$ is the relative error between $\dot {a}_{l}^{i/r/ci}$ and $\dot {a}_{l}^{e}$ .

Values of $\dot {a}_l^e$ , $\dot {a}_l^i$ , and $\dot {a}_l$ predicted by the Richtmyer theory ( $\dot {a}_l^{r}$ ) for different cases are provided in table 2 for comparison. The relative error $E^{i/r}=(\dot {a}_{l}^{i/r}-\dot {a}_{l}^{e})/\dot {a}_{l}^{e}$ is introduced to quantify the deviation of $\dot {a}_{l}^{i/r}$ from $\dot {a}_{l}^{e}$ . The impulsive model provides an accurate prediction for case 0.20 ( $E^{i}=2.95\,\%\pm2.97\,\%$ ). However, for the other cases with higher $A_t^0$ , this model significantly overestimates $\dot {a}_l^e$ , and $E^{i}$ is larger at higher $A_t^0$ , reaching up to 71.95 % $\pm$ 3.46 % for case 0.71. Compared with the observations of Chen et al. (Reference Chen, Xing, Wang, Zhai and Luo2023), the present findings demonstrate that flow compressibility induced by strong shocks and intrinsically compressible fluids exerts a significant influence on the predictive capability of the impulsive model, which is consistent with the theoretical predictions reported by Mikaelian (Reference Mikaelian1994), Yang et al. (Reference Yang, Zhang and Sharp1994) and Velikovich et al. (Reference Velikovich, Herrmann and Abarzhi2014). In contrast, the Richtmyer theory yields favourable predictions for all cases, with $|E^{r}|$ remaining below 7.50 %.

The above results indicate that the accuracy of the impulsive model is highly sensitive to $A_t^0$ . However, it is noteworthy that variations in the composition of the downstream gas influence not only the density but also $\gamma$ , a parameter characterising the gas compressibility. The $\gamma$ value of the gas upstream of the interface ( $\gamma _u$ ) is 1.40, while the $\gamma$ value of the gas downstream of the interface ( $\gamma _d$ ) varies across cases: 1.29, 1.21, 1.15, 1.09 and 1.07 for cases 0.20, 0.35, 0.51, 0.67 and 0.71, respectively. Quantitative evaluation of the effects of $A_t$ and $\gamma _d$ on the impulsive model’s accuracy necessitates a reliable benchmark, either experimental or theoretical, that ideally allows one parameter to vary while the other remains fixed. However, gas combinations satisfying the requirements may not exist naturally, posing challenges for establishing an experimental benchmark. Consequently, the Richtmyer theory is adopted as a theoretical benchmark in this study, and its validity under various initial conditions is further verified through numerical simulations, with details provided in Appendix E. To compare the predictions from the impulsive model and Richtmyer theory, three series of cases with the same $\gamma _u$ (1.40) but different $\gamma _d$ (1.29, 1.15 and 1.07) are considered. Note that 1.29, 1.15 and 1.07 correspond to $\gamma _d$ in cases 0.20, 0.51 and 0.71, respectively, and for clarity, $\gamma _d$ of cases 0.35 and 0.67 are not considered. Each case series includes five values of $A_t^0$ (0.20, 0.35, 0.51, 0.67 and 0.71), corresponding to five experiments.

To facilitate comparison, we follow the approach of Mikaelian (Reference Mikaelian1994) to normalise $\dot {a}_l$ as $\alpha_l=\dot{a}_l/ka_0u_{si}$ , where $u_{si}$ is the velocity of the shocked interface. The values of $\alpha _l$ predicted by the impulsive model ( $\alpha _l^i$ ) and Richtmyer theory ( $\alpha _l^r$ ) are plotted in figure 4 as functions of the shock intensity, which is defined as $\varepsilon = 1 - p_0/p_1$ , with $p_0$ and $p_1$ representing the pressures of gases in front of and behind the incident shock, respectively. Here, $\varepsilon$ ranges from 0 to 1, indicating shock intensities from extremely weak to extremely strong. When $\gamma _u$ and $\gamma _d$ are similar, as illustrated in figure 4(a), $\alpha _l^i$ and $\alpha _l^r$ are in reasonable agreement across a wide range of $A_t^0$ and $\varepsilon$ values. In contrast, when $\gamma _u$ and $\gamma _d$ differ significantly, as shown in figures 4(b) and 4(c), $\alpha _l^i$ increasingly deviates from $\alpha _l^r$ with increasing $\varepsilon$ . This discrepancy occurs at lower $\varepsilon$ and is more pronounced when the difference between $\gamma _u$ and $\gamma _d$ is greater and $A_t^0$ is higher. In other words, the predictive capability of the impulsive model is related to three key parameters governing the compressibility of the flow: $\gamma _u / \gamma _d$ , $\varepsilon$ and $A_t^0$ . The impulsive model yields accurate predictions when $\gamma _d$ is close to $\gamma _u$ , but its accuracy deteriorates at high $\varepsilon$ and $A_t^0$ when the disparity between $\gamma _d$ and $\gamma _u$ is significant.

Figure 4.

Variations of dimensionless linear amplitude growth rate $\alpha _l$ with shock intensity $\varepsilon$ : theoretical predictions of the impulsive model and Richtmyer theory for the series of cases with $\gamma _d$ of (a) 1.29, (b) 1.15 and (c) 1.07.

The Richtmyer theory remains valid across both weak and strong compressibility regimes. However, its practical application is hindered by the need for tedious numerical computations. In contrast, the impulsive model offers simplicity and ease of use, but its accuracy deteriorates in highly compressible regimes. Therefore, developing a linear model that combines simplicity with robustness across a broad range of compressibility conditions is of considerable significance. Motivated by this need, we propose a modified impulsive model informed by experimental observations and theoretical predictions. The modified model, referred to as the compressible impulsive model, is formulated as

(4.2) \begin{align} \dot {a}_l^{ci} = \dot {a}_l^i \left \{\left [1-g\left (\frac {\gamma _u}{\gamma _d}\right )\right ] + g\left (\frac {\gamma _u}{\gamma _d}\right ) h(\varepsilon , A_t^0)\right \}\!, \end{align}

in which

(4.3) \begin{align} g\left (\frac {\gamma _u}{\gamma _d}\right ) = \frac {1}{1 + \textrm{e}^{-22\left (\frac {\gamma _u}{\gamma _d}-1.18\right )}} \quad \text{and} \quad h(\varepsilon , A_t^0) = \frac {1}{1+\left (\frac {6+8A_t^0}{1+15A_t^0}\right )\varepsilon ^{\frac {16+8A_t^0}{2+3A_t^0}}}. \end{align}

The ideas leading to such a model are elaborated as follows. A correction factor applied to $\dot {a}_l^i$ is introduced to account for the limitations of the impulsive model under varying compressibility conditions. This correction incorporates two primary functions, i.e. $g(\gamma _u / \gamma _d)$ and $h(\varepsilon , A_t^0)$ , that characterise the influence of three key parameters governing the compressibility of the flow: $\gamma _u / \gamma _d$ , $A_t^0$ and $\varepsilon$ . To preserve the high predictive accuracy of the impulsive model as $\gamma _u / \gamma _d$ approaches unity, $g(\gamma _u / \gamma _d)$ is constructed to vanish in this limit, thereby ensuring that $\dot {a}_l^{ci}$ reduces to $\dot {a}_l^i$ . Furthermore, to mitigate the increasing overprediction of $\dot {a}_l$ by the impulsive model as $\gamma _u / \gamma _d$ , $\varepsilon$ and $A_t^0$ increase, $h(\varepsilon , A_t^0)$ is defined to remain below unity and to decrease monotonically with increasing $\varepsilon$ and $A_t^0$ , while $g(\gamma _u / \gamma _d)$ is designed to increase and approach unity as $\gamma _u / \gamma _d$ increases. The comparison between $\dot {a}_{l}^{e}$ ( $\dot {a}_l^{r}$ ) and $\dot {a}_{l}^{ci}$ is presented in table 2 (table 3), in which $E^{ce}=(\dot {a}_{l}^{ci}-\dot {a}_{l}^{e})/\dot {a}_{l}^{e}$ ( $E^{cr}=(\dot {a}_{l}^{ci}-\dot {a}_{l}^{r})/\dot {a}_{l}^{r}$ ) denotes the relative error between $\dot {a}_{l}^{ci}$ and $\dot {a}_{l}^{e}$ ( $\dot {a}_{l}^{r}$ ). It is observed that $E^{ce}$ and $E^{cr}$ remain below 10 %, indicating that the compressible impulsive model maintains high predictive accuracy across a broad range of flow compressibility conditions.

Table 3.

Comparison of $\dot {a}_l$ predicted by the Richtmyer theory and the compressible impulsive model ( $\dot {a}_l^r$ and $\dot {a}_{l}^{ci}$ ). Here, $E^{cr}=(\dot {a}_{l}^{ci}-\dot {a}_l^r)/\dot {a}_l^r$ is the relative error between $\dot {a}_l^r$ and $\dot {a}_{l}^{ci}$ . Units of $\dot {a}_l^r$ and $\dot {a}_{l}^{ci}$ are m ${\textrm{s}}^{-1}$ .

Notably, although the compressible impulsive model is indeed simpler and computationally more efficient than the Richtmyer theory, given present computational capabilities, the runtime difference does not constitute a principal advantage. Instead, its principal merit over the Richtmyer theory, deriving predictions by solving a set of nonlinear equations numerically, lies in its explicit formulation, which establishes clear and explicit relationships between model predictions and key physical parameters, such as $ka_0$ , $\gamma _u / \gamma _d$ , $A_t^0$ and $\varepsilon$ . In other words, although the compressible impulsive model, a heuristic prescription, cannot be deemed sufficient for a comprehensive description of compressible linear RMI as it neglects the complete history of acoustic coupling, its simplicity facilitates a clearer understanding of the dependence of linear evolution on the initial conditions.

Figure 5.

Temporal variations of perturbation amplitude in dimensionless form for (a,b) case 0.20, (c,d) case 0.51, and (e,f) case 0.71: (a, c, e) overall interface; (b, d, f) bubble and spike.

4.2. Nonlinear amplitude evolution

As the interface enters the nonlinear evolution period, its profile becomes distinctly asymmetric, and the bubble and spike exhibit markedly different evolutionary behaviours. The determination of the amplitudes of the bubble and spike ( $a_b$ and $a_s$ ) and their associated uncertainties is described in detail in § B.4. The temporal variations of $a$ and $a_{b/s}$ in dimensionless form are illustrated in figure 5. Here, $t$ , $a$ and $a_{b/s}$ are non-dimensionalised as $\tau=k \dot {a}_{l}^{e}(t - t^*)$ , $\alpha=k(a - a^*)$ and $\alpha_{b/s}=k(a_{b/s} - a^*_{b/s})$ , respectively, in which $a^*_{b/s}$ denotes the value of $a_{b/s}$ at $t = t^*$ . Representative nonlinear models, including the ZS, SEA, MIK (Mikaelian Reference Mikaelian1994), DR (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010), ZG (Zhang & Guo Reference Zhang and Guo2016) and ZG-new (Zhang & Guo Reference Zhang and Guo2022) models, are considered for evaluation, with detailed expressions available in our previous work (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). A comparison between experimental and theoretical results reveals that the predictive performance of these models for cases 0.35 and 0.67 closely resembles that for cases 0.20 and 0.71, respectively. For clarity, three experimental cases (cases 0.20, 0.51 and 0.71), representing strong-shock RMI with low, moderate and high $A_t^0$ , respectively, are chosen for detailed discussion. Notably, the nonlinear models initialised using $\dot {a}_{l}$ can be regarded as effectively incompressible, since $\dot {a}_{l}$ describes the dominant early-time compressibility effects (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010), i.e. continuous pressure perturbations induced by disturbed transmitted and reflected waves. Thus these models, together with the systematic analyses of their predictive capabilities concerning weak-shock RMI over a broad range of $A_t^0$ reported by Chen et al. (Reference Chen, Xing, Wang, Zhai and Luo2023) (see figure 5 herein), constitute essential references for evaluating the influence of compressibility on nonlinear evolution dynamics.

For strong-shock RMI with low $A_t^0$ , all models yield reasonable predictions for the overall interface growth, as well as the individual evolution of the bubble and spike, which closely resembles the behaviour observed in the weak-shock RMI scenario (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). This finding highlights that the influence of compressibility effects, specifically shock proximity and secondary compression, is rather limited under such conditions, which can primarily be ascribed to the thermodynamic properties of the downstream gas, i.e. ${\textrm{CO}}_2$ . Its relatively large $\gamma$ and low molecular mass lead to a rapid separation of the transmitted shock from the interface. As a result, the shock-proximity effect and the associated secondary-compression effect remain weak, and accordingly, the predictive performance of the models is similar to that under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023).

For strong-shock RMI with moderate $A_t^0$ , the predictive performance of the models differs substantially from that under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). This indicates that the increase in $M$ introduces additional physical mechanisms that alter the nonlinear evolution dynamics. For the bubble, these models, which have been validated under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), underestimate the amplitude growth. This can be attributed to the fact that the shock-proximity and secondary-compression effects are significant in this scenario, yet they are not accounted for in these models. For the spike, the experimental results exhibit favourable agreement with the predictions of the ZS, DR and ZG-new models, but fall below the SEA model prediction, and above those of the MIK and ZG models. It is noteworthy that in the weak-shock RMI, the spike growth agrees well with the predictions of the MIK and ZG models that do not consider the spike-acceleration effect (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), suggesting that this effect is negligible or rather weak at moderate $A_t^0$ . The promotion of spike growth relative to weak-shock RMI can be attributed to the significant shock-proximity phenomenon, wherein the high-pressure region generated by the transmitted shock persists near the spike root for an extended duration, thereby accelerating the spike evolution. Consequently, the observed favourable prediction of the ZS, DR and ZG-new models for the experimental data is accidental: their overestimation of the spike-acceleration effect coincidentally describes the shock-proximity effect on spike growth. The MIK and ZG models, which have not considered the shock-proximity effect on spike growth, underestimate the spike evolution. In contrast, the SEA model still overestimates the spike evolution, owing to its excessive overestimation of the spike-acceleration effect (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010; Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023; Xing et al. Reference Xing, Chen, Li, Wang, Zhai and Luo2025).

For strong-shock RMI with high $A_t^0$ , a pronounced spike-acceleration effect is observed: the spike sustains a high growth rate throughout the experiment, causing the overall interface amplitude to seemingly exhibit weakly nonlinear growth, despite the interface having already entered a strongly nonlinear regime. For the bubble, as in the moderate $A_t^0$ scenario, amplitude growth is suppressed by shock-proximity and secondary-compression effects, leading models that are valid under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) to underestimate the growth. For the spike, the ZS, DR and ZG-new models that capture the spike-acceleration effect and perform well under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) also provide a reasonable prediction under strong-shock conditions, with only slight underestimation overall. This indicates that although the shock-proximity effect enhances spike growth, its influence is minor relative to the spike-acceleration effect, and thus does not fundamentally affect the predictive performance of the models. Moreover, the MIK and ZG models, which do not consider both spike-acceleration and shock-proximity effects, underestimate spike growth, whereas the SEA model overestimates it due to excessive prediction of the spike-acceleration effect.

Summarising the above analysis, under low- $A_t^0$ conditions, the shock-proximity phenomenon is not significant. As a result, the compressible effects on interface evolution are weak, and the evolution can be reasonably described by existing models. As $A_t^0$ increases to moderate and high magnitudes, the shock-proximity phenomenon becomes pronounced, and compressible effects significantly suppress the bubble growth. Additionally, the spike-acceleration and shock-proximity effects appear and promote the spike evolution. For the spike growth, the ZS, DR and ZG-new models yield reasonable predictions across all cases, whether physical or incidental. However, none of the existing models can accurately predict the bubble evolution under all $A_t^0$ conditions, as they do not consider compressibility effects that persist into the nonlinear phase.

A rigorous characterisation of the shock-proximity and secondary-compression effects under varying $A_t^0$ conditions is highly challenging. Thus we attempt to propose an empirical model for bubble evolution. The DR model not only captures the late-time $1/t$ behaviour of amplitude growth rate (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010), but also offers an accurate description of bubble evolution across a broad range of $A_t^0$ under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). Here, we extend the DR model by incorporating the compressibility effects introduced by strong shock waves. Based on the experimental results from the present work and a previous study (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), the modified DR model (J-DR model) is constructed as

(4.4) \begin{align} \begin{aligned} \dot {a}_{b}^{j-dr}= & N_{b}\dot {a}_l^{e} \frac {1+\left (1 - A_t^1\right ) N_{b}\tau }{1+C_{b} N_{b}\tau +\left (1 - A_t^1\right ) F_{b}(N_{b}\tau )^2}K_b, \\\text{in which} \ &\ C_{b}=[4.5 + A_t^1 + (2 - A_t^1) k a_1 ]/4, \quad F_{b}= 1 + A_t^1, \\& \ \ \ \ \ N_{b} = 1 - 0.6 \textrm{e}^{-8 \xi } - 0.05A_t^1(5-\xi ), \\K_b&=\frac {1}{1 + \textrm{e}^{3[\tau -f(\xi )]}} \quad \text{and} \quad f(\xi ) = 15 \xi ^2 + 1. \end{aligned} \end{align}

Notably, the expressions for $C_{b}$ and $F_{b}$ remain the same as those in the DR model. We next present the rationale behind the construction of the J-DR model.

To incorporate the shock-proximity effect, which primarily manifests in the early stage when the transmitted shock remains fairly close to the interface, $\dot {a}_l^{e}$ in the DR model is modified as $N_{b}\dot {a}_l^{e}$ , where $N_{b}$ is a coefficient with magnitude substantially less than unity. Experimental results indicate that the shock-proximity effect depends on two primary factors. The first factor is the degree of significance of the shock-proximity phenomenon, which can be quantified by $\xi=(v_{\textit{ts}}-u_{si})/u_{si}$ , where $v_{\textit{ts}}$ denotes the velocity of the transmitted shock. A smaller $\xi$ indicates a more pronounced shock-proximity phenomenon, and accordingly, $N_b$ is formulated as a monotonically increasing function of $\xi$ to reflect this trend. The second factor is the magnitude of $A_t^0$ : a higher $A_t^0$ leads to a slower early-time bubble growth. This phenomenon can be interpreted based on the following physical reasoning. The interface evolution is governed by the balance between fluid inertia and available kinetic energy. At higher $A_t^0$ , harmonics more rapidly transfer to the spike region, increasing the kinetic energy for spike growth (which is the intrinsic mechanism of the spike-acceleration effect), while reducing the kinetic energy available for bubble development. Meanwhile, the increased inertia contrast between the light and heavy fluids further reduces the ability of the light fluid to ‘push’ the heavy one. These result in slower bubble growth at higher $A_t^0$ conditions, and $N_b$ is accordingly formulated as a decreasing function of $A_t^1$ to capture this behaviour.

To capture the secondary-compression effect, which mainly manifests at a short period after the transverse shock interaction, a time-dependent function $K_b = 1/\{1+{\rm e}^{3[\tau -f(\xi )]}\}$ is incorporated into the model. Here, $f(\xi )$ represents the characteristic time at which transverse shock interaction occurs at the bubble tip; $K_b$ converges to 1 as $\tau - f(\xi ) \rightarrow -\infty$ , and to 0 as $\tau - f(\xi ) \rightarrow +\infty$ , with significant variation occurring only near $\tau = f(\xi )$ . In other words, at approximately $\tau = f(\xi )$ , $K_b$ rapidly transitions from approximately 1 to nearly 0, thus realising the description of the secondary-compression effect. The determination of $f(\xi )$ is informed by observations in prior RMI studies (Zhou Reference Zhou2017a , Reference Zhoub , Reference Zhou2024) that the secondary-compression effect is negligible in the weak-shock limit, and the transverse wave intersection occurs earlier when the shock-proximity phenomenon is more pronounced. In other words, $f(\xi )$ is expected to tend to infinity as $\xi \rightarrow +\infty$ , and decrease as $\xi$ decreases. Taking into account the constraints imposed by these conditions as well as the experimental results, $f(\xi )$ is empirically determined as $15 \xi ^2 + 1$ . As illustrated in figure 5, the J-DR model yields accurate predictions for bubble evolution across a broad range of $A_t^0$ conditions.

4.3. Modal evolution

Modal analysis is conducted to investigate the modal evolution of strong-shock RMI under diverse $A_t^0$ conditions. For clarity, only cases 0.20, 0.51 and 0.71, representing strong-shock RMI with low, moderate and high $A_t^0$ , respectively, are considered for detailed analysis. The experimental modal information is extracted by identifying interface contours at stages preceding the appearance of roll-up structures, i.e. before the interface becomes multi-valued, and subsequently applying a fast Fourier transform. The modal models proposed by Zhang & Sohn (Reference Zhang and Sohn1997) (ZSM model) and Vandenboomgaerde et al. (Reference Vandenboomgaerde, Gauthier and Mügler2002) (VM model) are considered as theoretical references. Notably, both models were originally developed for incompressible flow conditions, thus their predictions, as well as their predictive performance for weak-shock RMI over a broad range of $A_t^0$ , reported by Chen et al. (Reference Chen, Xing, Wang, Zhai and Luo2023) (see figures 6 and 7 therein), provide valuable benchmarks for assessing the influence of compressibility on modal evolution. For the first three harmonics ( $m_1$ , $m_2$ and $m_3$ ), the temporal variations of their amplitudes ( $a_{m1}$ , $a_{m2}$ and $a_{m3}$ ) in dimensionless form, obtained from experiments and predicted by the ZSM and VM models, are presented in figures 6(a), 6(c) and 6(e). Additionally, to facilitate the understanding of the relationship between modal evolution and interface morphology, the modal characteristics of the shocked interface contours at $\tau \approx 0.55$ in cases 0.20, 0.51 and 0.71 are presented in figures 6(b), 6(d) and 6(f), respectively. Here, $a_{mi}$ (with $i=1,2,3$ ) is normalised as $\alpha _{i}=k(a_{mi}-a_{mi}^*)$ , in which $a_{mi}^*$ denotes the corresponding $a_{mi}$ at $t=t^*$ .

Figure 6.

(a,c,e) Modal evolutions obtained from experiments and predicted by ZSM and VM models, and (b,d,f) modal information on the contour of shocked interface at $\tau \approx 0.55$ for (a,b) case 0.20, (c,d) case 0.51, and (e,f) case 0.71. Here, $\alpha _1$ , $\alpha _2$ and $\alpha _3$ are dimensionless amplitudes of the first three harmonics $m_1$ , $m_2$ and $m_3$ , and SI means shocked interface.

For case 0.20, as shown in figure 6(a), the ZSM and VM models yield similar predictions, and both capture the evolution of $m_1$ with reasonable accuracy. Moreover, $m_3$ grows at a significantly slower rate compared to $m_1$ and $m_2$ . These observations are consistent with previous findings under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). However, $m_2$ exhibits markedly faster growth than that in weak-shock RMI (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). In other words, although compressibility has a limited influence on the overall amplitude growth primarily governed by $m_1$ and $m_3$ , it nevertheless alters the interface morphology. This can be observed from the experimental shadowgraphs illustrated in figure 2: while the interface morphology overall resembles that under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), the bubble tip is noticeably flatter at late times. The flattening of the bubble, as shown in figure 6(b), results in the generation of $m_2$ that is out of phase with $m_1$ at positions near the bubble tip. As a result, the ZSM and VM models, which reasonably predict the evolution of $m_2$ under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), underestimate $\alpha _2$ .

Compared with case 0.20, the growth of $m_1$ in case 0.51 is slower, whereas $m_2$ and $m_3$ exhibit more rapid evolution, with $m_3$ reaching a considerable amplitude at late times, as shown in figure 6(c). The predictive performance of models differs markedly from that under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). As shown in figure 6(d), for the bubble, the shock-proximity and secondary-compression effects suppress its growth and lead to flattening, thereby attenuating the growth of $m_1$ while enhancing that of $m_2$ . The flattening and widening of the bubble simultaneously lead to the sharpening and narrowing of the spike. Meanwhile, the shock-proximity effect promotes the spike evolution. These result in the generation of $m_2$ and $m_3$ that are in phase with $m_1$ at positions near the spike tip. Accordingly, the ZSM and VM models, which start to overestimate the growth of $m_1$ from intermediate stages in weak-shock RMI (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), significantly overpredict the evolution of $m_1$ from the early stages under strong-shock conditions. For $m_2$ and $m_3$ , the ZSM model outperforms the VM model under weak-shock conditions (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) because their underlying assumptions dictate that the former is more accurate when higher-order harmonics are negligible, and less accurate when they are not, corresponding to weak-shock scenarios with low-to-moderate and high $A_t^0$ , respectively. However, under strong-shock conditions, the enhanced influence of compressibility on the growth of $m_2$ and $m_3$ results in the VM model outperforming the ZSM model for describing scenarios with moderate $A_t^0$ . For case 0.71, the compressibility effect on modal evolution is similar to that in case 0.51, only more pronounced. Accordingly, the ZSM and VM models overestimate the growth of $m_1$ from the early stages, and for $m_2$ and $m_3$ , the VM model, which performs well for describing weak-shock RMI with high $A_t^0$ (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), underestimates their growth.

Overall, the compressibility effects, i.e. the shock-proximity and secondary-compression effects, exert a significant influence on the modal evolution in strong-shock RMI, particularly under moderate and high $A_t^0$ conditions. Consequently, the ZSM and VM models proposed for RMI in incompressible flows are inapplicable, and the reasonable agreement between model predictions and experimental results in some cases should be regarded as coincidental.