3.1. Flow features and interface morphology

Figure 3.

Experimental schlieren photographs of the evolutions of shocked heavy–light single-mode interfaces with different $A_{1}$ . Numbers represent time in microseconds.

Experimental schlieren images of the evolutions of shocked heavy–light single-mode interfaces with different $A_1$ are shown in figure 3. Case −0.66 is taken as an example to illustrate the detailed evolution process. The temporal origin ( $t = 0$ μs) is defined as the moment the incident shock hits the initial interface. It is noteworthy that the initial interface appears thick due to the sinusoidal strips embedded in the interface formation device for constraining the soap film. For clarity, the shadows from these strips are removed from the images through image processing once the shocked interface has moved away. Moreover, the width of the shocked interface in the experimental schlieren images generally increases over the evolution process. This phenomenon is primarily attributed to the gradual detachment of broken soap-film droplets from the interface, resulting in a relatively significant density gradient in the flow field adjacent to the interface. When the incident shock impacts the initial interface, a transmitted shock wave and reflected rarefaction waves, moving downstream and upstream, respectively, are generated. The phase inversion of the shocked interface has been completed at $t=139$ μs. After the phase inversion, the amplitude and asymmetry of the shocked interface increase gradually (139–759 μs), followed by the formation of distinct spike and bubble structures (759 μs), indicating the generation of high-order harmonics. A comparison of experiments with different $A_1$ reveals that the nonlinear features of the interface evolution, such as spike and bubble structures, develop more rapidly when $|A_{1}|$ is higher.

Since the major interface configuration can be clearly extracted from the schlieren photographs, it is a rare opportunity to evaluate the predictive capability of perturbation solutions (Zhang & Sohn Reference Zhang and Sohn1997b ; Velikovich et al. Reference Velikovich, Herrmann and Abarzhi2014) for describing the interface shape evolution. Given the complexity of the perturbation solutions, we consider the solutions for the first four modes, with their detailed expressions available in previous works (Zhang & Sohn Reference Zhang and Sohn1997b ; Velikovich et al. Reference Velikovich, Herrmann and Abarzhi2014). Additionally, the solutions are in the form of polynomials in time and tend to diverge rapidly, hence we focus only on the early-time interface evolution. For clarity, we present the results of cases −0.35, −0.66, −0.86 and −0.95 here. Interface contours extracted from experiments (black lines) and predicted by the perturbation solutions (blue lines) are depicted in figure 4, in which $x=0$ and $y=0$ correspond to the horizontal coordinate of the mean position of the initial interface and the vertical coordinate of the tip of the bubble in the middle, respectively. For case −0.35, the perturbation solutions reasonably predict the interface shape and also the amplitude. For cases −0.66, −0.86 and −0.95, the perturbation solutions predict poorly both the interface shape and amplitude, with the discrepancies becoming more pronounced as $|A_1|$ increases. The failure of the perturbation solutions for cases with medium to very high $|A_1|$ should be attributed to that $\dot {a}$ is positively related to $|A_1|$ and these solutions diverge rapidly with the increase of $ka$ (Jacobs & Krivets Reference Jacobs and Krivets2005).

Figure 4.

Interface contours extracted from experiments (black lines) and predicted by perturbation solutions (blue lines): ( $a$ ) case −0.35; ( $b$ ) case −0.68; ( $c$ ) case −0.86; ( $d$ ) case −0.95.

3.2. Amplitude evolution in start-up period

The perturbation evolution during the start-up stage establishes the initial conditions for subsequent amplitude growth. Therefore, exploring the evolution law of RMI during the start-up stage is essential. In H-L RMI, the start-up process (i.e. $\dot {a}$ increases gradually and reaches $\dot {a}_l$ when the process ends) is accompanied by the phase-inversion process (i.e. the absolute value of perturbation amplitude $|a|$ decreases gradually until reaching zero), both beginning at the onset of the shock impact.

Figure 5.

Amplitude variation in the start-up period for different cases. Symbols and lines: experimental and theoretical results, respectively. Error bars represent potential errors in extracting the interface amplitude for a single experiment, with a length equal to half the width of the interface in the experimental schlieren image, and the same hereafter.

As the YZS theory (Yang et al. Reference Yang, Zhang and Sharp1994) requires numerical solutions, the analytical Li model (Li et al. Reference Li, Chen, Zhai and Luo2024) is employed to provide a theoretical reference for the amplitude evolution in the start-up period, with its expressions detailed in Appendix A for clarity. Experimental and theoretical results of the amplitude variation in the start-up period for various cases are provided in figure 5. The coordinates of the starting point of the theoretical line are ( $t_1$ , $a_1$ ), where $t_1=2|a_{0}|/v_{is}^{e}$ represents the duration of the shock–interface interaction process and $a_1$ is the corresponding $a$ at $t=t_1$ , with $v_{is}^{e}$ denoting the experimental $v_{is}$ . Good agreement is reached between the experimental and theoretical results, indicating that the Li model is valid for describing the start-up evolution of H-L RMI across a wide range of $A_1$ conditions. The phase-inversion process is completed when $|a|$ decreases to zero, while the end moment of the start-up process is the $t$ coordinate of the last point of the Li model prediction since its duration is exactly the effective prediction time of the model. According to the Li model prediction, the temporal order of the terminations of the start-up and phase-inversion processes appears to be correlated with $A_1$ : the start-up process terminates before (after) the phase-inversion process in cases with relatively low (high) $|A_1|$ , i.e. cases −0.35 and −0.55 (cases −0.66, −0.75, −0.86 and −0.95). Parameter $a$ varies from $a_1$ to 0 during the phase-inversion process. Accordingly, the relative magnitude of $|a_1|$ and the absolute value of the amplitude variation in the start-up process ( $|a_{st}|$ ) determine which process terminates first. Both $|a_{st}|$ and $|a_1|$ are related to $A_1$ , $S$ and specific ratios of gases u and d ( $\gamma _{\textrm {u}}$ and $\gamma _{\textrm {d}}$ ). However, the relationships are highly complex and nonlinear because they are correlated with parameters that need to be obtained by solving the Riemann problem, such as the velocities of waves and interface. As a result, it is challenging to specify the relationship between $A_1$ and the temporal order of the terminations of the start-up and phase-inversion processes and the underlying physical mechanism.

3.3. Linear amplitude evolution

According to previous studies (Yang et al. Reference Yang, Zhang and Sharp1994; Wang et al. Reference Wang, Wang, Zhai and Luo2023; Li et al. Reference Li, Chen, Zhai and Luo2024), the MB model is theoretically valid for predicting $\dot {a}_l$ only under conditions where $ka_0$ is small and compressibility is weak. In the present experiments, $ka_0$ satisfies the small-amplitude criterion and the shock intensity is low. Therefore, these experiments are appropriate for evaluating the MB model over a wide range of $A_{1}$ values. The MB model can be expressed as

(3.1) \begin{equation} \dot {a}_l^{m}=k A_{1} \Delta u^{e} a_{0}\dfrac {1+C_{c}}{2}=k A_{1} \Delta u^{e} \dfrac {a_{0}+a_{1}}{2}, \end{equation}

where $C_{c}=1-\Delta u^{e}/v_{is}^{e}$ is the shock compression factor. Besides the empirical MB model, it is also desirable to evaluate the analytical linear models (Wouchuk & Nishihara Reference Wouchuk and Nishihara1996, Reference Wouchuk and Nishihara1997; Wouchuk Reference Wouchuk2001). As noted by Li et al. (Reference Li, Chen, Zhai and Luo2024), the difference between the predictions of the WN and WNi models is negligible when $S \lt 0.6$ . Because the value of $S$ in all current experiments is below 0.55, only the WNi model is considered here. The detailed expression of the WNi model is provided in Appendix A.

Experimental $\dot {a}_{l}$ ( $\dot {a}_{l}^{e}$ ), obtained by linearly fitting the amplitude evolution shortly after the start-up stage, and the theoretical counterparts predicted by the MB and WNi models ( $\dot {a}_l^{m}$ and $\dot {a}_l^{w}$ ) are presented in table 2 for comparison. The WNi model predicts the experimental results well under all considered $A_1$ conditions. In contrast, the MB model performs poorly in predicting $\dot {a}_{l}^{e}$ for case −0.35. Notably, the MB model was empirically developed based on numerical simulations. However, Meyer & Blewett (Reference Meyer and Blewett1972) did not consider simulations with $|A_1|\lt 0.5$ , which explains the MB model’s failure to predict $\dot {a}_{l}^{e}$ in case −0.35.

Table 2.

Comparison of experimental and theoretical linear amplitude growth rates. Here, $\dot {a}_{l}^{e}$ , $\dot {a}_{l}^{m}$ and $\dot {a}_{l}^{w}$ : linear amplitude growth rates obtained from experiments and predicted by MB and WNi models, respectively; $E^{m/w}=(\dot {a}_{l}^{m/w}-\dot {a}_{l}^{e})/\dot {a}_{l}^{e}$ : relative error between $\dot {a}_{l}^{e}$ and $\dot {a}_{l}^{m/w}$ . The units of linear amplitude growth rates are m s $^{-1}$ .

3.4. Weakly nonlinear amplitude evolution

Figure 6.

Temporal variations of perturbation amplitude in dimensionless form for cases with different $A_{1}$ : ( $a$ ) overall interface; ( $b$ ) bubble and spike. Two dashed reference lines have a slope of 1 and represent the scenario where nonlinearity is absent and the bubble and spike maintain an amplitude growth rate of $\dot {a}_{l}^{e}$ . The increase in the length of the error bars is attributed to the gradual widening of the interface observed in the experimental schlieren images.

Temporal variations of $a$ and $a_{b/s}$ (subscripts ‘ $b$ ’ and ‘ $s$ ’ represent bubble and spike, respectively) in dimensionless form for different cases are shown in figures 6( $a$ ) and 6( $b$ ), respectively. Here, $t$ , $a$ and $a_{b/s}$ are normalized 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, with $t^*$ representing the end moment of the start-up stage and $a^*$ ( $a^*_{b/s}$ ) being the corresponding $a$ ( $a_{b/s}$ ) at $t =t^*$ . The results of cases with $|A_1|\lt 0.8$ collapse onto each other but deviate from those of cases −0.86 and −0.95. This indicates that H-L RMI with $|A_1|\lt 0.8$ follows similar evolution laws, whereas H-L RMI with $|A_{1}|\gt 0.8$ exhibits significantly different behaviour. Specifically, the amplitude growth rate of the spike, which continuously decreases in cases with $|A_1|\lt 0.8$ , increases at the beginning of the weakly nonlinear period in cases −0.86 and −0.95. These results confirm that the spike acceleration is significant when $|A_{1}|$ is high, as reported in a previous numerical study (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010). To our knowledge, this is the first direct experimental observation of the spike acceleration in H-L RMI. Compared to the spike, the evolution law of the bubble is less sensitive to $|A_1|$ , which should be attributed to the insensitivity of the asymptotic curvature of the bubble to $|A_1|$ (Zhang & Guo Reference Zhang and Guo2022).

For a small-initial-amplitude shocked interface, previous research (Zhou Reference Zhou2017a ,Reference Zhou b , Reference Zhou2024) suggests that its evolution can be approximately divided into five successive stages: start-up stage ( $\dot {a}$ increases gradually), linear stage ( $\dot {a}$ remains almost unvaried), weakly nonlinear stage ( $\dot {a}$ decreases, with bubble and spike structures becoming significant), strongly nonlinear stage (roll-up structures emerge) and turbulent mixing stage (multiscale vortices appear and materials are mixed intensely). For cases with moderate to very high $|A_1|$ (cases −0.55, −0.66, −0.75, −0.86 and −0.95), as shown in figure 3, the asymmetry of the shocked interface increases gradually after the phase inversion, followed by the formation of distinct spike and bubble structures. This indicates that the interface evolution in these cases has entered the weakly nonlinear regime within the effective experimental time. For case −0.35, however, the shocked interface remains a quasi-single-mode profile throughout the experiment. As shown in figure 6( $b$ ), two dashed reference lines with a slope of 1 are included for comparison. These reference lines represent the scenario where nonlinearity is absent and the bubble and spike maintain an amplitude growth rate of $\dot {a}_{l}^{e}$ . Both the amplitude growths of the bubble and spike deviate significantly from the reference lines at small $\tau$ , indicating that the interface evolution in case −0.35 has also entered the weakly nonlinear stage within the effective experimental time. Subsequently, a comparative analysis of the experimental results and predictions from typical nonlinear models is performed. Nonlinear models including the ZS, SEA, MIK, DR, ZG, ZG-New and mDR models are considered, with detailed expressions available in previous work (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). A comparison of the experimental and theoretical results reveals that the predictive capabilities of the considered nonlinear models for cases −0.55, −0.66 and −0.75 are highly similar. For clarity, four typical experimental cases (cases −0.35, −0.66, −0.86 and −0.95), referred to as H-L RMI with low, moderate, high and very high $|A_{1}|$ , respectively, are chosen for further discussion.

Figure 7.

Comparisons between experimental and theoretical results for dimensionless $a$ ( $a,c,e,g$ ) and dimensionless $a_{b/s}$ ( $b,d,f,h$ ). ( $a,b$ ) case −0.35; ( $c,d$ ) case −0.66; ( $e,f$ ) case −0.86; ( $g,h$ ) case −0.95.

Table 3.

Summary of the predictive capabilities of the considered existing nonlinear models and newly proposed model (mSEA model) for the amplitude evolutions of the overall interface, bubble and spike under different $A_1$ conditions. L-H and H-L: L-H RMI and H-L RMI, respectively; ✔ represents that the model is capable of providing reasonable predictions for the experimental results, both theoretically and practically or coincidentally in practice; ✗ indicates that the model is practically incapable of providing a reasonable prediction; $-$ indicates that there are no available experimental results for comparison.

Comparisons between the temporal amplitude variations obtained from experiments and predicted by models for cases −0.35, −0.66, −0.86 and −0.95 are presented in figure 7. For case −0.35, the ZG and mDR models accurately predict the experimental results, while the other models overestimate the growths of $a$ and $a_{b/s}$ . For case −0.66, similarly, the ZG and mDR models provide accurate predictions for the experimental results. The MIK model predicts the evolutions of $a$ and $a_b$ poorly but accurately forecasts the spike evolution. In contrast, the ZS, SEA, DR, and ZG-New models provide reasonable predictions for the bubble evolution but overestimate the growths of $a$ and $a_s$ . For case −0.86, the SEA model (ZG model) overestimates (underestimates) the growths of $a$ and $a_s$ while predicting the evolution of $a_b$ well. The MIK model provides a reasonable prediction for the overall amplitude evolution, which is a coincidence caused by its underestimation of spike evolution and overestimation of bubble growth. In contrast, the ZS, DR, ZG-New and mDR models predict the growths of $a$ and $a_{b/s}$ accurately. For case −0.95, the SEA and mDR models (ZG model) overestimate (underestimates) the growths of $a$ and $a_s$ while accurately predicting the evolution of $a_b$ . The MIK model overestimates the bubble evolution while underestimating the growths of $a$ and $a_s$ . The ZS model predicts the overall amplitude evolution well, which is a coincidence caused by its overestimation of spike evolution and underestimation of bubble growth. In contrast, the DR and ZG-New models accurately predict the evolutions of $a$ and $a_{b/s}$ .

The predictive capabilities of the considered nonlinear models for L-H RMI (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) and H-L RMI under various $|A_{1}|$ conditions are summarized in table 3. It is found that only the MIK model exhibits different predictive capabilities for H-L RMI and L-H RMI under low-, moderate- and high- $|A_1|$ conditions. The RMI evolution can be viewed as a process that is initially compressible and subsequently becomes incompressible (Zhang & Sohn Reference Zhang and Sohn1997a ). The effect of flow compressibility on interface evolution becomes negligible when the start-up process ends (Yang et al. Reference Yang, Zhang and Sharp1994; Li et al. Reference Li, Chen, Zhai and Luo2024). Thus, the termination of the start-up stage can be regarded as the demarcation point between the compressible and incompressible periods. The origin in figure 7 represents the termination point of the start-up stage, which means the compressible period has been ‘subtracted’, and the experimental results presented include only the data in the incompressible period. In the incompressible period, both L-H RMI and H-L RMI are driven solely by baroclinic vorticity, with the major discrepancy being the difference in $a^*$ caused by the phase-inversion process existing in H-L RMI. For all models except the MIK model, the predictions are insensitive to $a^*$ , which explains why they have similar predictive capabilities for L-H RMI and H-L RMI. Notably, the analyses on their predictive capabilities for L-H RMI with low, moderate and high $|A_1|$ (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) are also applicable to H-L RMI with similar conditions, and the relevant discussions are omitted in this work. For the MIK model, one of its key assumptions is that the amplitude growth enters the nonlinear regime when $ka=1/3$ , i.e. $\dot {a}$ remains $\dot {a}_{l}$ when $ka\lt 1/3$ . Accordingly, the value of $ka^*$ has a significant influence on the predictions of the MIK model. In a previous study of L-H RMI (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), the value of $ka^*$ is approximately 1/3. In contrast, in the present work on H-L RMI, the value of $ka^*$ is much smaller than 1/3, or even negative, due to the presence of an additional phase-inversion process. This explains why the MIK model exhibits different predictive capabilities for H-L RMI and L-H RMI under low-, moderate- and high- $|A_1|$ conditions.

For H-L RMI with very high $|A_1|$ , the ZS and mDR models, which accurately describe H-L and L-H RMI with high $|A_1|$ , exhibit poor predictive capabilities. Under the condition of $|A_{1}|\gt \sqrt {(ka_{1})^{2}+1/2}\approx 0.75$ , the ZS model is derived using the $P^{0}_{1}$ Padé approximation. This approximation retains only a partial contribution of the third-order term of $a$ rather than the complete contribution. The coefficient of the neglected third-order term of $a$ is $A_{1}^{2}-1/2$ . Therefore, for RMI with $|A_1|\gt 0.75$ , the prediction of the ZS model is less accurate when $|A_{1}|$ is higher. This may explain why the ZS model predicts the results of case −0.86 well but performs poorly in predicting the amplitude evolution in case −0.95. For the mDR model constructed empirically based on experiments of L-H RMI with low to high $|A_1|$ , as previously mentioned (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), it is invalid when $|A_{1}|$ approaches 1 due to the limitation of the coefficient $E=(3-2|A_1|)/(9-9|A_1|)$ . Therefore, the failure of the mDR model under very-high- $|A_1|$ conditions is reasonable and foreseeable. Aside from the ZS and mDR models, the predictive capabilities of the other models for H-L RMI with very high $|A_1|$ are similar to those for H-L RMI with high $|A_1|$ , and the analyses of model predictive capabilities for the latter are also applicable to the former.

As illustrated in table 3, none of the considered models is universally applicable to RMI over the range of $A_{1}$ from $-0.95$ to $0.86$ . Given the complex nature of the RMI evolution, developing a rigorous theoretical model applicable over a wide range of $A_{1}$ conditions is challenging. Therefore, we seek to propose an empirical model based on the experimental results from the present work and the previous study on L-H RMI (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). To construct the empirical model, we first analyse the expressions of the existing nonlinear models. Other than the ZS and ZG-New models, the expressions of the considered nonlinear models follow a unified form:

(3.2) \begin{equation} \dot {a}_{b/s}=\dot {a}_l^e\dfrac {1+D_{b/s}k\dot {a}_l^e (t-t^*)}{1+E_{b/s}k\dot {a}_l^e (t-t^*)+F_{b/s}k^2 {\dot {a}_l^e}{}^2 (t-t^*)^2}, \end{equation}

where $D_{b/s}$ , $E_{b/s}$ and $F_{b/s}$ are constants or parameters related to initial conditions such as $|A_1|$ and $|a_1|$ . In the considered nonlinear models, $E_{b/s}$ is always non-zero, and these models can be classified into two groups according to whether $D_{b/s}$ and $F_{b/s}$ are zero. In the early stage, (3.2) could be approximated as

(3.3) \begin{equation} \dot {a}_{b/s}= \begin{cases} \dot {a}_{l}^{e}[1+(D_{b/s}-E_{b/s})k\dot {a}_{l}^{e}(t-t^*)]&\text {when } D_{b/s}\neq 0\text {, }F_{b/s}\neq 0\\ \dot {a}_{l}^{e}[1-E_{b/s}k\dot {a}_{l}^{e}(t-t^*)]&\text {when } D_{b/s}=0\text {, }F_{b/s}=0.\\ \end{cases} \end{equation}

In the late stage, (3.2) can be approximated as

(3.4) \begin{equation} \dot {a}_{b/s}= \begin{cases} \dfrac {D_{b/s}}{F_{b/s}kt}&\text {when } D_{b/s}\neq 0\text {, }F_{b/s}\neq 0\\[6pt] \dfrac {1}{E_{b/s}kt}&\text {when } D_{b/s}=0\text {, }F_{b/s}=0.\\ \end{cases} \end{equation}

When $D_{b/s}$ and $F_{b/s}$ are non-zero, the model prediction for early-time evolution is mainly influenced by $D_{b/s}$ and $E_{b/s}$ , while the model prediction of late-time interface development is primarily affected by $D_{b/s}$ and $F_{b/s}$ . In the special case where $D_{b/s}$ is non-zero constant, $E_{b/s}$ and $F_{b/s}$ independently govern the model predictions of the early- and late-time amplitude evolutions, respectively. Additionally, when $D_{b/s}$ and $F_{b/s}$ are zero, the evolution behaviour at both early and late stages is governed by $E_{b/s}$ . Notably, obtaining an empirical model which reasonably describes the Atwood-number dependence of both the early- and late-time evolution laws by modifying only one coefficient is challenging. Besides, modification based on a model with $D_{b/s}$ being non-zero non-constant is rather complex. Therefore, it is desirable to construct a new model on the basis of a model with $D_{b/s}$ being non-zero constant. The only model that satisfies these conditions is the SEA model, which is chosen as the base model for modification. The coefficients of the SEA model can be expressed as

(3.5) \begin{equation} \begin{cases} D_{b/s}^{sea}=1,&\\[5pt] E_{b/s}^{sea}=1\pm |A_{1}|,&\\[5pt] F_{b/s}^{sea}=\dfrac {1\pm |A_{1}|}{1+|A_{1}|}\dfrac {1}{2\pi C},&\\ \end{cases} \end{equation}

where $C=1/3\pi$ for $|A_{1}|\geqslant 0.5$ and $C=1/2\pi$ for $|A_{1}|\rightarrow 0$ . The upper and lower signs correspond to the bubble and spike, respectively, and similarly hereafter.

The idea of model construction is elaborated as follows. The maximum dimensionless time of experiments in the present work and previous study (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) does not exceed 2.5. In other words, the interface evolution in all considered experiments is within the weakly nonlinear period. Therefore, the new empirical model (mSEA model) is constructed by modifying $E_{b/s}^{sea}$ of the SEA model that governs the early-time evolution behaviour, while keeping the other coefficients unchanged. The SEA model overestimates the amplitude evolutions of the overall interface and spike for all present experiments with negative $A_1$ , consistent with the observation in previous work on RMI with positive $A_1$ (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). This should be attributed to its overestimation of the spike acceleration (Dimonte & Ramaprabhu Reference Dimonte and Ramaprabhu2010). Thus, the value of the modified $E_{s}^{sea}$ ( $E_{s}^{msea}$ ) needs to be larger than that of $E_{s}^{sea}$ under all $A_1$ conditions, in order to reasonably describe the evolution of spike. For the bubble, as mentioned earlier, its evolution law is less sensitive to $A_1$ compared with spike. Accordingly, the value of the modified coefficient $E_{b}^{sea}$ ( $E_{b}^{msea}$ ) should be less sensitive to $A_1$ compared with that of $E_{s}^{msea}$ . After numerous attempts, $E_{b,s}^{msea}$ satisfying these two requirements and also aligning with the data of all 13 experiments (with $A_1$ values of −0.95, −0.86, −0.75, −0.66, −0.55, −0.35, 0.30, 0.44, 0.52, 0.61, 0.68, 0.72 and 0.86) reported in the present work and prior study (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) is obtained as $(2\mp |A_{1}|)(1.07\pm |A_{1}|)$ .

As shown in figure 7, the mSEA model reasonably predicts the amplitude evolutions of H-L RMI with low, moderate, high and very high $|A_1|$ , and its predictive capabilities are presented in table 3 for comparison. Relatively, the mSEA model predicts bubble evolution generally better than spike development. This is because bubble evolution, which is insensitive to $A_1$ , is easier to model over a wide range of $A_1$ than spike development. To more comprehensively evaluate the predictive capability of the mSEA model, nine other experiments reported in the present work and previous study (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023), with respective $A_1$ values of −0.75, −0.55, 0.30, 0.44, 0.52, 0.61, 0.68, 0.72 and 0.86, are also considered. Comparisons between the results of experiments with $A_1$ values of −0.75 and −0.55 (0.30, 0.44, 0.52, 0.61, 0.68, 0.72 and 0.86) and corresponding mSEA model predictions are shown in figure 8 (figure 9). Overall, the mSEA model reasonably predicts the amplitude evolutions of the overall interface, bubble and spike for all 13 experiments, indicating that it applies to describe the weakly nonlinear evolution of RMI with $A_1$ ranging from −0.95 to −0.35 and from 0.30 to 0.86.

Figure 8.

Comparisons between experimental results of cases −0.55 and −0.75 (symbols) and corresponding theoretical predictions by the mSEA model (lines): dimensionless $a$ ( $a$ ) and dimensionless $a_{b/s}$ ( $b$ ).

Figure 9.

Comparisons between experimental results of L-H RMI with $A_1$ of 0.30, 0.44, 0.52, 0.61, 0.68, 0.72 and 0.86 (symbols) and corresponding theoretical predictions by the mSEA model (lines): dimensionless $a$ ( $a$ ) and dimensionless $a_{b/s}$ ( $b$ ).

The experimental results in the present study and previous work (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023) indicate that the spike acceleration is dependent on $A_1$ and is the main reason for the different amplitude evolution patterns under different $A_1$ conditions. However, the critical $A_1$ for its occurrence remains unclear. Since the mSEA model reasonably predicts the spike evolution over a wide range of $A_1$ conditions, we can infer the approximate value for the critical $A_1$ based on this model. The sign that the spike acceleration is occurring is that $\dot {a}_{s}$ increases at the beginning of the nonlinear phase ( $t=t^*$ ), rather than decreases, i.e. the time derivative of $\dot {a}_{s}$ ( $\ddot {a}_{s}$ ) is positive at $t=t^*$ . $\ddot {a}_{s}$ predicted by the mSEA model at $t=t^*$ can be expressed as

(3.6) \begin{equation} \ddot {a}_{s}^{msea}(t=t^*)=k\dot {a}_{l}^{e2}\big(D_{s}^{msea}- E_{s}^{msea}\big)= k\dot {a}_{l}^{e2}(|A_1|^2+0.93|A_1|-1.14). \end{equation}

The value of $\ddot {a}_{s}^{msea}(t=t^*)$ is positive when $|A_1| \gt 0.70$ . Thus, the critical $A_1$ for L-H and H-L configurations are approximately 0.70 and −0.70, respectively, i.e. the spike acceleration is expected to occur if $A_1 \gt 0.70$ or $A_1 \lt -0.70$ . The spike acceleration is indeed observed in cases −0.95 and −0.86 and the L-H RMI experiment with $A_1 = 0.86$ (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). However, no evident spike acceleration is observed in case −0.75 and the L-H RMI experiment with $A_1 = 0.72$ (Chen et al. Reference Chen, Xing, Wang, Zhai and Luo2023). This may be because the spike acceleration is weak when $|A_1|$ is slightly higher than 0.70, and thus its influence is rapidly overshadowed by the nonlinearity.