1. Introduction
The impulsive acceleration of a perturbed interface separating fluids of different densities, typically due to a shock wave traversing the interface, results in the growth of interfacial perturbations. This instability was first studied theoretically by Richtmyer (Reference Richtmyer1960) and later experimentally by Meshkov (Reference Meshkov1969). It is therefore known as the Richtmyer–Meshkov (RM) instability. The RM instability development is generally driven by baroclinic vorticity produced on the interface due to misalignment of the pressure gradient (provided by the shock wave) and density gradient between the two materials. The perturbation amplitude grows linearly at the early stage under certain conditions. When these conditions are not satisfied, the amplitude exhibits nonlinear growth and, given sufficient initial energy, eventually the flow transitions to a turbulent mixing state. The RM instability is an important phenomenon occurring in many applications. For example, in inertial confinement fusion (ICF) reactions, the mixing induced by the RM instability (created by surface non-uniformities at the fuel–shell interface or by non-uniform radiation drive) results in the contamination of the fuel and limits the fusion energy gain (Lindl et al. Reference Lindl, Landen, Edwards, Moses and Team2014; Chu et al. Reference Chu, Wang, Qi, Xu and Li2022). An understanding of the various stages of these instabilities is an essential requirement of ICF efforts. In supersonic combustion, the RM instability increases the mixing between the fuel and oxidizer, enhancing combustion efficiency (Yang, Kubota & Zukoski Reference Yang, Kubota and Zukoski1993). A better understanding of the RM instability could allow for improved design of combustors that optimizes mixing for combustion.
In previous studies, a single-mode interface has been widely used to investigate the RM instability because of its simple configuration. Experimentally, the single-mode interface was created by using the nitrocellulose membrane technique (Meshkov Reference Meshkov1969; Jourdan & Houas Reference Jourdan and Houas2005), membrane-less technique (Jacobs Reference Jacobs1992; Jacobs et al. Reference Jacobs, Klein, Jenkins and Benjamin1993, Reference Jacobs, Jenkins, Klein and Benjamin1995; Balakumar et al. Reference Balakumar, Orlicz, Ristorcelli, Balasubramanian, Prestridge and Tomkins2012; Morgan et al. Reference Morgan, Aure, Stockero, Greenough, Cabot, Likhachev and Jacobs2012; Vandenboomgaerde et al. Reference Vandenboomgaerde, Souffland, Mariani, Biamino, Jourdan and Houas2014) or soap-film technique (Luo, Wang & Si Reference Luo, Wang and Si2013; Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018), and the shocked interface behaviours have been thoroughly studied. Theoretically, various single-mode models for different evolution stages based on several assumptions have been proposed, e.g. models for the linear stage (Richtmyer Reference Richtmyer1960; Meyer & Blewett Reference Meyer and Blewett1972; Wouchuk Reference Wouchuk2001; Luo et al. Reference Luo, Wang and Si2013), early nonlinear stage (Zhang & Sohn Reference Zhang and Sohn1996, Reference Zhang and Sohn1997; Mariani et al. Reference Mariani, Vandenboomgaerde, Jourdan, Souffland and Houas2008) and late nonlinear stage (Goncharov Reference Goncharov2002; Sohn Reference Sohn2003; Jacobs & Krivets Reference Jacobs and Krivets2005).
In practical applications, however, random multi-mode perturbations commonly exist on initial interfaces. After a shock wave acceleration, complicated mechanisms, such as high-order harmonic generation and bubble merger process (Sadot et al. Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998; Niederhaus & Jacobs Reference Niederhaus and Jacobs2003; Leinov et al. Reference Leinov, Malamud, Elbaz, Levin, Ben-Dor, Shvarts and Sadot2009; Orlicz, Balasubramanian & Prestridge Reference Orlicz, Balasubramanian and Prestridge2013; Pandian, Stellingwerf & Abarzhi Reference Pandian, Stellingwerf and Abarzhi2017; Elbaz & Shvarts Reference Elbaz and Shvarts2018; Mohaghar et al. Reference Mohaghar, Carter, Pathikonda and Ranjan2019), are generally involved. Previous studies have shown that in the multi-mode RM instability, each mode develops independently at the linear stage. When the mode amplitude develops comparable to its wavelength, the evolution enters the nonlinear stage, and mode coupling becomes very significant (Haan Reference Haan1989, Reference Haan1991; Ofer et al. Reference Ofer, Alon, Shvarts, McCrory and Verdon1996). Finally, under the assumption that there are no longer any larger}-wavelength perturbations yet to saturate, the interface development enters the bubble merging stage, where smaller bubbles are gradually swallowed up by larger bubbles (Alon et al. Reference Alon, Hecht, Ofer and Shvarts1995; Sadot et al. Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998; Guo et al. Reference Guo, Zhai, Si and Luo2019).
Experiments of a dual-mode RM instability under high-Mach-number conditions have been performed (Di Stefano et al. Reference Di Stefano, Malamud, Kuranz, Klein and Drake2015a,Reference Di Stefano, Malamud, Kuranz, Klein, Stoeckl and Drakeb), and the results indicated that new modes are generated from mode coupling, and their amplitudes grow and saturate over time. The dual-/multi-mode RM instability under weak shock conditions in planar geometry was also investigated to explore the dependence of interface development on its initial shape (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020; Liang et al. Reference Liang, Liu, Zhai, Ding, Si and Luo2021a). The growth rate of the multi-mode RM instability was found to strongly rely on the wavenumber, phase and amplitude of initial constituent modes. The interaction of a small-wavelength multi-mode perturbation with a large-wavelength inclined interface perturbation was investigated using three-dimensional (3-D) simulations (McFarland et al. Reference McFarland, Reilly, Black, Greenough and Ranjan2015). It was shown that the flow has a distinct memory of the initial conditions that are present in both large-scale-driven entrainment measures and small-scale-driven mixing measures. A similar conclusion was also drawn by Mohaghar et al. (Reference Mohaghar, Carter, Musci, Reilly, McFarland and Ranjan2017) in their experimental work using density and velocity statistics. Further, the effect of initial perturbations on the evolution of the inclined RM turbulent mixing layer was investigated by Mohaghar, McFarland & Ranjan (Reference Mohaghar, McFarland and Ranjan2022) through 3-D simulations. The results indicated that the multi-mode perturbations produce a completely different mixing mechanism compared with the single-mode case.
Compared with the planar RM instability, the convergent RM instability is more closely related to ICF. In the convergent RM instability, both radial and angular directions are involved, and the perturbation development is associated with more mechanisms. Bell (Reference Bell1951) and Plesset (Reference Plesset1954) first analysed the early-time growth of the Rayleigh–Taylor (RT) instability (Rayleigh Reference Rayleigh1883; Taylor Reference Taylor1950) in cylindrical and spherical geometries, and found that the perturbation growth rate varies with interface radius, referred to as the Bell–Plesset (BP) effect. Several nonlinear models (Mikaelian Reference Mikaelian2005b; Liu, He & Yu Reference Liu, He and Yu2012; Liu et al. Reference Liu, Yu, Ye, Wang and He2014; Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015) revealed that the BP effect suppresses nonlinearity and extends the linear stage longer than that in the planar configuration, as demonstrated in laser-driven experiments (Fincke et al. Reference Fincke, Lanier, Batha, Hueckstaedt, Magelssen, Rothman, Parker and Horsfield2005). Besides, for continuous radial flow behind a convergent shock, the interface as a whole is in a non-uniform pressure field, which inevitably introduces the RT effect (Lanier et al. Reference Lanier, Barnes, Batha, Day, Magelssen, Scott, Dunne, Parker and Rothman2003; Lombardini, Pullin & Meiron Reference Lombardini, Pullin and Meiron2014; Ding et al. Reference Ding, Si, Yang, Lu, Zhai and Luo2017; Samulski et al. Reference Samulski, Srinivasan, Manuel, Masti, Sauppe and Kline2022). Therefore, coupling of the BP effect, RT effect and multiple impacts (shocks reflected back and forth between the interface and convergence centre) greatly increases the complexity of the convergent RM instability.
Experimental measurements of single-mode perturbation amplitude in the convergent RM instability were performed by Ding et al. (Reference Ding, Si, Yang, Lu, Zhai and Luo2017), and the RT stabilization caused by interface deceleration was found to reduce the growth rate. By reasonably evaluating the RT stabilization, a modified model based on the Bell equation was proposed to predict the perturbation growth before reshock. Luo et al. (Reference Luo, Zhang, Ding, Si, Yang, Zhai and Wen2018) found that the long-term effect of the RT stabilization even leads to a phase inversion on the single-mode light–heavy interface before reshock provided that the initial interface has sufficiently small perturbations. Further, Zhai et al. (Reference Zhai, Zhang, Zhou, Ding and Wen2019) numerically investigated the RT effect on the phase inversion observed in experiments (Luo et al. Reference Luo, Zhang, Ding, Si, Yang, Zhai and Wen2018), considering effects of the initial amplitude, azimuthal mode number and reshock timing. A novel shock tube was designed to study the nonlinear features of the convergent single-mode RM instability (Luo et al. Reference Luo, Li, Ding, Zhai and Si2019). The shock tube employs a concave–oblique–convex wall profile which first transforms a planar shock into a cylindrical arc, then gradually strengthens the cylindrical shock along the oblique wall and finally converts it back into a planar one. Therefore, the new facility can realize the analysis of nonlinearity of the convergent RM instability by eliminating the RT effect and reshock. The perturbation amplitude experiences a linear growth much longer than that in planar geometry, which is mainly ascribed to the counteraction between the promotion by the BP effect and the suppression by nonlinearity.
Previous studies associated with the convergent RM instability mainly focused on the development of a single-mode perturbation. However, the interaction of a convergent shock with multi-mode interfaces is more relevant to ICF. Relative to the convergent single-mode RM instability, mode-coupling effects are introduced in the convergent multi-mode RM instability. Kumar, Hornung & Sturtevant (Reference Kumar, Hornung and Sturtevant2003) studied the growth of a multi-mode initial interface, formed by sandwiching a polymeric membrane between wire-mesh frames, in a conical geometry, and found that the turbulent mixing zone has a relatively larger growth rate than that in the planar counterpart. Mode coupling in the convergent RM instability of a dual-mode interface has been numerically investigated by Zhou et al. (Reference Zhou, Ding, Zhai, Cheng and Luo2020). The first-mode amplitude growth was found to depend heavily on the second mode, and the amplitude signs of harmonics generated for all possible dual-mode configurations were obtained. Besides, the second-order weakly nonlinear solutions for a dual-mode interface subjected to a uniformly radical motion, i.e. a pure BP environment, were obtained by Guo, Cheng & Li (Reference Guo, Cheng and Li2020). Until now, experiments related to the convergent multi-mode RM instability have been scarce mainly because a uniformly convergent shock and a well-defined initial interface are difficult to generate in experiments and complicated mechanisms are involved. In our previous work, a smoothly convergent shock was generated based on shock dynamics theory (Zhai et al. Reference Zhai, Liu, Qin, Yang and Luo2010; Zhan et al. Reference Zhan, Li, Yang, Zhu and Yang2018; Luo et al. Reference Luo, Li, Ding, Zhai and Si2019), and a well-defined initial multi-mode interface was formed by the soap-film technique (Liang et al. Reference Liang, Liu, Zhai, Ding, Si and Luo2021a; Guo et al. Reference Guo, Cong, Si and Luo2022). These provide us an opportunity to explore the developments of multi-mode interfaces accelerated by a convergent shock.
In the convergent RM instability, the BP effect always exists. During the coasting stage of an ICF implosion (Craxton et al. Reference Craxton, Anderson, Boehly, Goncharov, Harding, Knauer, McCrory, McKenty, Meyerhofer and Myatt2015), the shell moves at a constant velocity, which means that perturbations on the shell evolve in an approximately pure BP environment. Therefore, it is necessary to explore the pure BP growth of multi-mode perturbations. However, the RT effect, BP effect, mode coupling and reshock are generally coupled together, and it is difficult to decouple them. It is unclear as to how the BP effect and mode coupling behave in the convergent multi-mode RM instability. Starting from a simple configuration, i.e. a dual-mode interface, interaction of a convergent shock with a dual-mode air–SF
$_6$ interface is experimentally investigated in this work. The experiments are conducted in the same convergent shock tube as that used by Luo et al. (Reference Luo, Li, Ding, Zhai and Si2019). In this convergent shock tube, the RT effect and reshock are avoided, and compressibility is also negligible. Therefore, the BP and mode-coupling effects can be highlighted. Six kinds of dual-mode air–SF
$_6$ interfaces with different initial perturbation spectra are formed to explore the dependency of shocked flow on initial conditions. The objectives of present work are twofold. The first is to evaluate the mode-coupling effects on the growth rate of each mode in convergent geometry. The second is to demonstrate the different roles of the BP effect and mode coupling in dual-mode perturbation developments.
In the following, the experimental methods are first provided. Then, the flow features are qualitatively described and the mode amplitude growth for each dual-mode case is presented and compared with the single-mode counterpart to highlight the mode-coupling effects. Finally, the predictions of the mode amplitude growth are presented to evaluate the BP effect.
2. Experimental methods
The dual-mode interface, as sketched in figure 1(a), is superimposed by two cosine perturbations: 
$r(\theta )=R_{0}+a_0\cos (m_1\theta +\phi _1)+a_0\cos (m_2\theta +\phi _2)$, where 
$R_{0} = 270$ mm refers to the mean radius of initial interface, 
$a_{0} = 1.0$ mm is the initial amplitude of each fundamental mode, 
$m_{i}$ (
$i=1, 2$) is the azimuthal mode number and 
$\phi _{i}$ (
$i=1, 2$) is the initial phase. The dual-mode interfaces can be divided into two groups based on the phase difference between fundamental modes: in-phase and anti-phase. Each group includes three mode combinations: 
$(m_1, m_2) = (48, 24)$, (72, 24) and (72, 48). Therefore, the effects of relative phase and mode combination on the shocked dual-mode interface evolution can be studied. To facilitate comparison and to obtain the background flow field, three single-mode cases corresponding to the three fundamental modes involved and one undisturbed case are also considered. The detailed initial conditions for each dual-mode case (denoted by relative phase 
$m_1$–
$m_2$) and each single-mode case (denoted by SM 
$m$) are listed in table 1.
Sketches of a convergent shock interacting with a dual-mode interface (a) and of the interface formation process (b). IS, incident convergent shock; II, initial interface.
Initial conditions of dual- and single-mode interfaces studied in the present work. Here, 
$m_{i}$, 
$\phi _{i}$, 
$a_{0}/\lambda _{m_{i}}$ (
$i=1, 2$) refer to the initial mode number, phase and amplitude–wavelength ratio of the corresponding constituent mode, respectively; 
$A^{+}$ is the post-shock Atwood number (
$A^+=(\rho _2^+-\rho _1^+)/(\rho _2^++\rho _1^+)$ with 
$\rho _2^+$ and 
$\rho _1^+$ being the post-shock densities of air and SF
$_6$); 
$\psi$ is the SF
$_6$ mass fraction on the downstream side of the interface; 
$v_{i}$ is the the velocity of the incident convergent shock just before it meets the interface; 
$\Delta v$ is the the interface velocity jump by shock impact; 
$v_{f}$ is the reflected shock velocity; and 
$v_{t}$ is the transmitted shock velocity. The units for length and velocity are mm and m s
$^{-1}$, respectively.
The soap-film technique (Luo et al. Reference Luo, Dong, Si and Zhai2016; Liang et al. Reference Liang, Liu, Zhai, Ding, Si and Luo2021a; Guo et al. Reference Guo, Cong, Si and Luo2022) is used to form the well-defined dual-mode interfaces. Before interface formation, the transparent devices (devices A and B) shown in figure 1(b) are manufactured by combining two transparent acrylic plates (3.0 mm in thickness) with pedestals (7.0 mm in height). Two identical constraint strips (0.50 mm in width and 0.25 mm in thickness) are attached to the pre-carved grooves on device A to constrain the soap film. The ratio of the total height of the constraint strips protruding into the flow field to the height of the whole flow field is smaller than 10 %, and thus the constraint strips have negligible effects on the shocked flow (Wang et al. Reference Wang, Wang, Zhai and Luo2022). To generate the soap-film interface, the constraint strips are wetted by the soap solution (78 % pure water, 2 % sodium oleate and 20 % glycerin by mass) in advance, then a rectangular brush with the soap solution attached is pulled carefully along the constraint strips and finally the soap-film interface is generated.
To create an air–SF
$_6$ interface, air in device A needs to be replaced by SF
$_6$. As illustrated in figure 1(b), a thin membrane is used to seal device A, and two pipes are inserted into device A through the membrane. The SF
$_6$ is injected into device A through the inlet pipe and air is exhausted through the outlet pipe. An oxygen concentration detector is placed at the outlet pipe to monitor the oxygen concentration in device A. Once the volume fraction of oxygen is reduced to less than 0.5 %, the pipes and thin membrane are removed to avoid shock reflection, and experiments are conducted immediately. The time interval is short enough and will not introduce additional uncertainty of gas concentration (Luo et al. Reference Luo, Li, Ding, Zhai and Si2019). According to one-dimensional gas dynamics theory, the gas components on both sides of the interface, as listed in table 1, are determined by velocities of the incident and transmitted shock waves measured from experimental schlieren images.
Experiments are conducted in a convergent shock tube with a smooth concave–oblique–convex wall designed to modify the shock shape (Luo et al. Reference Luo, Li, Ding, Zhai and Si2019; Li et al. Reference Li, Ding, Zhai, Si, Liu, Huang and Luo2020). The present design enables a strong convergent shock near the convergence centre to exit the convergent section without shock focusing and reflecting. Thus, interface deceleration and reshock are eliminated, and contributions from the BP effect and mode coupling to the perturbation growth are isolated. The postshock flow is recorded by high-speed schlieren photography. The frame rate of the camera (FASTCAM SA-Z, Photron Limited) is 50 400 frames per second, and the spatial resolution of each image is 0.285 mm pixel
$^{-1}$. The ambient pressure and temperature are 
$101.3 \pm 0.1$ kPa and 
$294.7 \pm 2$ K, respectively. The Mach number of the convergent shock just before it meets the interface is 
$M_s = 1.26 \pm 0.01$.
Before performing the experiments, the boundary-layer effect is first evaluated. It is found from the schlieren images that no obvious turbulence characteristics appear on the evolving interface. In addition, the Reynolds numbers (
$Re$) at four moments are discussed here. Note that the calculation methods of 
$Re$ almost involve the vorticity field information, which is unable to be obtained in the present experiments. In this work, 
$Re$ for the RM instability turbulence is calculated, trying to give a rough judgement on whether there is a transition. The Reynolds number is calculated by 
$Re=h\dot {h}/\nu _{mix}$, where 
$h$ is half of the mixing width, 
$\dot {h}$ is the growth rate of 
$h$ and 
$\nu _{mix}$ is the mixing kinematic viscosity of the two gases. The mixing kinematic viscosity can be obtained by using (
$\mu _1+\mu _2)/(\rho _1+\rho _2$), where 
$\mu _1$ and 
$\mu _2$ (
$\rho _1$ and 
$\rho _2$) are the viscosities (densities) of the mixed gases at the left and right sides of the interface, respectively (Mohaghar et al. Reference Mohaghar, Carter, Musci, Reilly, McFarland and Ranjan2017; Guo et al. Reference Guo, Cong, Si and Luo2022). According to one-dimensional gas dynamics theory, 
$\nu _{mix}$ is calculated as 
$2.79\times 10^{-6}$. The values of 
$Re$ at four moments are given in table 2. As stated by Dimotakis (Reference Dimotakis2000), achieving 
$Re\thicksim 10^4$ is conjectured to be necessary to observe a mixing transition and is a more restrictive condition beyond conditions for turbulent transition. However, the Reynolds numbers in our experiments have not yet developed to meet the necessary conditions for a mixing transition. As a result, a laminar model is suitable for calculating the boundary-layer thickness in our experiments. In this work, three laminar models are used to calculate the boundary-layer thickness, and their expressions are given in table 3. In this table, 
$r_{m}=100$ mm refers to the maximum distance that the interface moves in the radial direction, 
$\rho = 1.204$ kg m
$^{-3}$ (
$\rho = 6.143$ kg m
$^{-3}$) and 
$\mu = 1.83 \times 10^{-5}$ Pa s (
$\mu = 1.6 \times 10^{-5}$ Pa s) are the density and viscosity coefficient of pure air (SF
$_6$), respectively, and 
$\Delta v = 93.4$ m s
$^{-1}$ represents the postshock flow velocity. The results calculated show that the boundary-layer thickness during the time studied is much smaller than the inner height of the test section (7.0 mm), and it has a limited effect on the interface movements.
The values of Reynolds number at different moments.
The boundary-layer thickness calculated from different laminar models.
Then any 3-D effects of the soap-film interface are evaluated. Taking case I72-24 as an example, the interface amplitudes at the boundary slice (
$z = \pm 3.5$ mm) and symmetry slice (
$z = 0$ mm) are 2.0 and 1.907 mm according to previous work (Luo et al. Reference Luo, Wang and Si2013; Liang et al. Reference Liang, Liu, Zhai, Si and Luo2021b). The absolute difference between these two amplitudes is 0.093 mm, which is much smaller than the size of a single pixel in the schlieren images. Therefore, 3-D effects may be considered negligible.
3. Results and discussion
3.1. Interface morphologies and flow features
Developments of the unperturbed and single-mode interfaces are similar to those in the work of Luo et al. (Reference Luo, Li, Ding, Zhai and Si2019) and, therefore, the related descriptions are omitted. Distinct schlieren images showing developments of the in-phase dual-mode interfaces are presented in figure 2. The time origin (
$t = 0\ \mathrm {\mu }$s) is defined as the moment when the IS arrives at the average position of the II, i.e. 
$r=R_0$. Taking case I48-24 as an example, before IS impact (
$t = -44 \mathrm {\mu }$s), the interface profile looks quite thick because the interface in the schlieren image is covered by two dual-mode strips. When IS passes across II, it bifurcates into a downstream-moving TS and an upstream-propagating RS (
$t = 35\ \mathrm {\mu }$s). The SI leaves its original location, and its profile is thin and clean. This indicates that the diffusion layer (Jacobs & Krivets Reference Jacobs and Krivets2005) and three-dimensionality (Luo et al. Reference Luo, Zhang, Ding, Si, Yang, Zhai and Wen2018) of the interface are greatly reduced. At early times, the interface is still single-valued and the shape of the original perturbation is visible. As time proceeds, the perturbation width, defined as the difference between the maximum and minimum radii of the points on the interface in the radial direction, increases continuously. Eventually, finger-like bubbles (lighter fluid penetrating into heavier fluid) and spikes (heavier fluid penetrating into lighter fluid) arise (
$t = 591\unicode{x2013}1007\ \mathrm {\mu }$s) due to increasing nonlinearity. In this case, no prominent vortices arise during the time studied, and four nearly equal bubbles occupy the flow.
Experimental schlieren images illustrating the in-phase dual-mode interface evolution. RS, reflected shock; TS, transmitted shock; SI, shocked interface. Other symbols have the same meaning as those in figure 1. Numbers denote the time in 
$\mathrm {\mu }$s.
For the other two cases, some spikes start to roll up with a pair of vortices formed on their necks (
$t =1073\ \mathrm {\mu }$s in case I72-24 for example), causing the interface to become multi-valued. In case I72-24, there are two large bubbles and four small bubbles, and an arrangement of a large bubble with two small bubbles on both sides is observed. In case I72-48, there are four large bubbles and two small bubbles, and an arrangement of a small bubble with two large bubbles on both sides is observed. Spikes between the large and small bubbles skew towards the large bubbles, which is indicative of the bubble merger process (Sadot et al. Reference Sadot, Erez, Alon, Oron, Levin, Erez, Ben-Dor and Shvarts1998; Alon et al. Reference Alon, Hecht, Mukamel and Shvarts1994, Reference Alon, Hecht, Ofer and Shvarts1995). The flow diversity demonstrates that the initial mode combination significantly affects the perturbation morphology. As spikes grow, they seem to be dragged. As discussed earlier, the boundary-layer effect can be neglected. Actually, the spikes move forward followed by the soap-film droplets. After the shock wave impact, the soap film breaks into small droplets immediately. These droplets cannot catch up with the high-speed flow, which leads to the black lines along the line of sight in the schlieren images. The relationship between the size of the soap droplets and the shock strength was investigated by Ranjan et al. (Reference Ranjan, Niederhaus, Oakley, Anderson, Bonazza and Greenough2008) and Cohen (Reference Cohen1991). According to their work, the mean radius of the soap droplets is estimated to be 30 
$\mathrm {\mu }$m for the shock with 
$M_s \thicksim 1.3$. Moreover, Liang et al. (Reference Liang, Zhai, Ding and Luo2019) investigated the interaction of a planar air–SF
$_6$ interface with a planar shock, and found a negligible effect of the soap droplets on the interface motion.
Figure 3 shows developments of the anti-phase dual-mode interfaces. In case A48-24, large bubble and small bubble are arranged alternately, and again no prominent vortices arise (
$t = 1004 \mathrm {\mu }$s). In case A72-24 , the flow is occupied by four large bubbles and a small bubble, and four large bubbles are distributed symmetrically on both sides of the small bubble (
$t = 1050\ \mathrm {\mu }$s). Spikes between the two large bubbles are less prominent than those between the large and small bubbles. In case A72-48, there are also four large bubbles and a small bubble, and similarly, four large bubbles are distributed symmetrically on both sides of the small bubble (
$t = 1041 \mathrm {\mu }$s). However, spikes between the two large bubbles are more prominent than those between the large and small bubbles, which is contrary to case A72-24. Actually, the flow features in case A72-48 are quite similar to those in case I72-48. For these anti-phase cases, spikes between the large and small bubbles also skew towards the large bubble, whereas spikes between the two large bubbles develop without an inclination. Comparison with the in-phase cases indicates that the flow features also depend heavily on the initial phase difference.
Schlieren images illustrating the anti-phase dual-mode interface evolution.
3.2. Mixing width and mode amplitude growths
Variation of the dimensionless perturbation width (
$w$) of each dual-mode case is given in figure 4. The time is scaled as 
$\tau _{w}=({m}/{R_{0}}) v^{lin}_{w} (t-t^{*})$, where 
$v^{lin}_{w}$ is the experimental linear growth rate of the width and 
$t^{*}$ the time when compression phase ends. The width is normalized as 
$\bar {w}=({m}/{R_{0}})(w-w^{*})$, where 
$w^{*}$ is the perturbation width at 
$t=t^{*}$. The perturbation widths of single-mode cases are also provided for comparison to evaluate the mode-coupling effects. Generally, the perturbation width of each dual-mode case develops similarly to its single-mode counterpart at early times, but grows slower from the middle to late stages. This indicates that mode coupling occurs from the middle stage and suppresses the perturbation width development. A similar conclusion was also drawn by Miles et al. (Reference Miles, Edwards, Blue, Hansen, Robey, Drake, Kuranz and Leibrandt2004). Liang et al. (Reference Liang, Liu, Zhai, Ding, Si and Luo2021a) reported in their planar multi-mode RM instability study that although the mode-coupling effects reduce global mixing, they enhance local mixing. The perturbation width seems to be affected less significantly by initial phase difference. Miles et al. (Reference Miles, Edwards, Blue, Hansen, Robey, Drake, Kuranz and Leibrandt2004) also concluded that the phase-difference effect is limited during the linear and early nonlinear stages, but becomes prominent during the deep nonlinear stage.
Comparison of dimensionless mixing width for single- and dual-mode cases.
Distinct interfacial morphologies facilitate the recognition of interface contours, and a spectrum analysis can therefore be performed. By using an image processing program (Guo et al. Reference Guo, Cong, Si and Luo2022; Liang & Luo Reference Liang and Luo2022), the interface profile is first extracted. Taking case I72-48 as an example, as shown in figure 5, the white dotted lines overlayed on two schlieren images at different moments are the interface profiles extracted for Fourier analysis. Although the very small vortex structures on the spikes are not identified, the main features of the interface are well captured. Then, by applying the fast Fourier transform, amplitude developments of the fundamental modes are obtained, as shown in figure 6. The dimensionless time is 
$\tau = ({24}/{R_{0}}) v^{lin} (t-t^{*})$, with 
$v^{lin}$ being the experimental linear growth rate of the mode amplitude. The amplitude is scaled as 
$\alpha = ({24}/{R_{0}}) (a-a^{*})$, where 
$a^{*}$ is the mode amplitude at 
$t=t^{*}$. Although Fourier analysis becomes strictly invalid once the vortex rolls up, the interfaces are still analysed through fast Fourier transform before the vortex is further developed to obtain more information of the fundamental mode amplitude. This treatment may lead to probable misidentification of extremely high-order mode amplitudes, but has a limited effect on the fundamental low-order mode amplitude. To explore the mode-coupling effects in dual-mode interface evolution, the first-order mode amplitudes in the corresponding single-mode cases are also provided.
Comparison of the interface profile extracted with the actual interface at two different moments for case I72-48.
Comparison of dimensionless amplitudes of the fundamental modes for all dual-mode cases. The first-order mode amplitudes in corresponding single-mode cases are also provided.
As shown in figure 6(a), the amplitude of the mode with mode number 24 (
$m$24 mode for short) in case I48-24 is much smaller than that in the single-mode case, i.e. the 
$m$24 mode amplitude growth is suppressed by mode coupling. On the contrary, the 
$m$24 mode amplitude in case A48-24 is slightly larger than that in the single-mode case, i.e. the 
$m$24 mode amplitude growth is promoted by mode coupling. This implies that the role of mode coupling is affected by the phase difference of the two fundamental modes. Comparing case I48-24 with case I72-24, the 
$m$24 mode amplitude growth is suppressed in the former but is promoted in the latter, which indicates that the role of mode coupling is also affected by the mode number combination. In ICF, the lower-
$m$ mode amplitude growth is generally severe and takes a long time to saturate, which will result in severe deformation of the hotspot interface and sharp decrease of the hotspot volume (Dittrich et al. Reference Dittrich2014). Through designing the initial perturbation spectrum, one may expect to suppress the amplitude growth of the lower-
$m$ mode. In figure 6(b), the 
$m$48 mode amplitude growth in case A72-48 is almost consistent with that in the single-mode case during the time studied, which indicates that the mode-coupling effects on this mode evolution are negligible. Similar phenomena are found in the amplitude growths of the 
$m$48 mode in case I72-48 (although it shows a slight suppression after 
$\tau \sim 0.25$) and the 
$m$72 mode in case A72-24, as given in figure 6(c). In short, the mode-coupling effects in convergent geometry are still strongly dependent upon the initial perturbation spectrum.
To predict the amplitude signs of the modes generated in convergent geometry, based on the second-order solution derived by Haan (Reference Haan1991) in planar geometry, Zhou et al. (Reference Zhou, Ding, Zhai, Cheng and Luo2020) proposed a modified model, named the modified Haan model, which is given as
\begin{equation} a_{k_{j}\pm k_{i}}\approx{\pm} \tfrac{1}{2}(k_{j}\pm k_{i})a_{k_{i}}^{L}a_{k_{j}}^{L}, \end{equation}
where 
$k$ is the wavenumber and 
$a_{k}^{L}$ is the mode-
$k$ amplitude. From the amplitude signs of the modes generated, as provided in table 4, the mode-coupling effects on the amplitude growths of the fundamental modes can be demonstrated. For example, in case I48-24, the harmonic with mode number of 24 generated by coupling between fundamental modes has a negative amplitude, suppressing the 
$m$24 amplitude growth. In planar geometry (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020), the authors found that for the initial dual-mode interface with combination of 
$k$ and 
$k/3$ modes, mode coupling has a negligible influence on the growth of each basic wave. In convergent geometry, however, the 
$m$72 amplitude growth in case I72-24 is greatly suppressed, and the 
$m$24 amplitude growth is slightly promoted, as shown in figure 6. In case I72-24, coupling between positive 
$m$72 and positive 
$m$24 will generate the harmonics of positive 
$m$96 and negative 
$m$48. Further, coupling between negative 
$m$48 and positive 
$m$24 will generate negative 
$m$72 and positive 
$m$24, and coupling between positive 
$m$96 and positive 
$m$24 will also generate negative 
$m$72. As a result, the 
$m$72 amplitude growth is greatly suppressed, whereas the 
$m$24 amplitude growth is slightly promoted. For clarity, the amplitude signs of fundamental modes resulting from the feedback of second-order harmonics are listed in table 5. In other words, the feedback of second-order harmonics to fundamental modes becomes prominent in convergent geometry. Because the experimental time in this work is similar to that in previous work (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020), we can conclude that the BP effect promotes the occurrence of mode coupling, and the feedback of high-order modes to fundamental mode also arises earlier in convergent geometry than that in its planar counterpart.
Amplitude signs of the modes generated by mode coupling.
Amplitude signs of fundamental modes resulting from the feedback of second-order harmonics. The 
$[i,j]$ indicates mode coupling between the 
$i$ mode and 
$j$ mode before there is a saturated mode, because a saturated mode 
$k$ has no contribution to the generation of lower-
$k$ mode, and can be affected only by coupling of two lower-
$k$ modes (Ofer et al. Reference Ofer, Alon, Shvarts, McCrory and Verdon1996). The 
$(i,j)$ indicates mode coupling between the 
$i$ mode and 
$j$ mode during the whole time studied. The arrow 
$\downarrow$ means that the amplitude growth is inhibited.
Note that the 
$m$72 mode amplitude growth in the I/A72-48 cases almost stagnates at the late stage, but such a phenomenon has not been observed in the I/A72-24 cases. From the perspective of energy transfer, due to the presence of the 
$m$48 mode which acts as a transitional role between the 
$m$24 and 
$m$72 modes, energy is more easily transferred from higher-
$m$ modes to lower-
$m$ modes. Therefore, the 
$m$72 mode amplitude growth in the I/A72-48 cases saturates much earlier than in the I/A72-24 cases. For multi-mode cases, a collective band saturation amplitude 
$S(k)$ was introduced by Haan (Reference Haan1989). The mode whose amplitude reaches 
$S(k)$ is considered to be saturated. Provided that the initial spectrum is a smooth one, i.e. all neighbouring mode amplitudes are roughly equal initially, they showed that local structures with much larger amplitude than that of the individual mode will be formed by neighbouring modes. The local structure encounters large kinematic drag, leading to a much lower saturation amplitude 
$S(k)$ than 
$0.1\lambda$ (
$\lambda$ is the mode wavelength) for the individual mode (Shvarts et al. Reference Shvarts, Alon, Ofer, McCrory and Verdon1995; Ofer et al. Reference Ofer, Alon, Shvarts, McCrory and Verdon1996). This means that the constituent modes in multi-mode cases with such an initial spectrum are more likely to enter the nonlinear stage due to mode coupling. However, Ofer et al. (Reference Ofer, Alon, Shvarts, McCrory and Verdon1996) pointed out that for the initial dual-mode spectrum, 
$S(k)$ of the mode should still be taken as 
$0.1\lambda$ until there are saturated modes at the late stage. After that, the mode-coupling effects gradually become prominent, and the low-order and high-order harmonics are continuously generated and accumulated, which enables the concept of collective band saturation amplitude to take effect again.
The influence of mode coupling on saturation time (the time when a given mode amplitude grows to 
$0.1\lambda$) of main modes is shown in figure 7. In all dual-mode cases, the higher-
$m$ fundamental mode amplitude always reaches saturation first. The two fundamental modes’ amplitudes in the I/A72-48 cases are both saturated within the experimental time, whereas only the higher-
$m$ fundamental mode amplitude saturates in the other cases. The dimensionless saturation times when the modes just reach their saturation amplitudes are also marked in figure 7. To highlight the mode-coupling effects, the same method is used to obtain the dimensionless saturation time of the fundamental mode in the single-mode cases. Comparison of dimensionless saturation times for different cases is provided in table 6. Note that the fundamental mode 
$m$72 amplitude saturates at 
$\tau \sim 0.18$, but before this time the mode-coupling effects on its growth are less significant, as shown in figure 6. As a result, the evolution before saturation is almost the same as that in the single-mode case, resulting in the same saturation time. Similarly, the saturation times of mode 
$m$48 in the I/A72-48 cases are consistent with the single-mode results. However, for mode 
$m$48 in the I/A48-24 cases, amplitude saturation is not achieved by the time that mode coupling occurs (
$\tau \sim 0.15$). Because mode coupling suppresses the 
$m$48 mode amplitude growth, its saturation time is therefore delayed relative to the single-mode case. In short, the fundamental mode combination affects the saturation time, but the initial phase difference has a limited effect on the saturation time.
The absolute values of main modes’ amplitudes comparing with their 
$0.1\lambda$ in all cases. The ‘saturation zone’ means that the mode reaches its saturated amplitude (
$0.1\lambda$), and the borders are determined by variation of the saturated mode amplitude (
$0.1\lambda$) with the mode number at initial and final moments during the time studied. Different symbols represent different modes. For each mode, two amplitudes at initial and final moments are provided. The dimensionless times marked correspond to the times when the modes have just reached their saturation amplitudes.
The dimensionless saturation times of modes in single- and dual-mode cases. The dashes mean that the corresponding fundamental mode does not saturate or exist in this case.
To determine the dominant modes in dual-mode cases, the evolution of the average mode number is studied. The definition of the average mode number is given as
\begin{equation} \langle m \rangle =\left.\sum_{m}(a^{2}_{m}\times m)\right/ \sum_{m}a^{2}_{m}, \end{equation}
where 
$m$ refers to all main modes on the spectrum and 
$a_{m}$ represents the amplitude of mode 
$m$. Generally, the average mode number development can be divided into three stages marked with different colour blocks in figure 8. At stage I, the average mode number decreases, because the fundamental mode amplitude reduces due to shock compression. At stage II, the average mode number increases because of incipient development of the fundamental modes after shock impact. At stage III, the average mode number gradually reduces as time proceeds in all cases except I48-24. The reduction of the average mode number indicates that energy in the spectrum is generally transferred from higher-
$m$ modes to lower-
$m$ modes, and finally the lower-
$m$ modes dominate the spectrum. This is well known as the inverse cascade process, corresponding to the bubble merger process. Shvarts et al. (Reference Shvarts, Alon, Ofer, McCrory and Verdon1995) numerically discussed the variation of the average wavenumber in 3-D planar multi-mode RT instability, and found that the spectrum becomes increasingly peaked towards lower k and the mean wavenumbers decrease in time as in the two-dimensional (2-D) case. That is, the 3-D case is shown to share many qualitative features with the 2-D case, such as an inverse cascade of large structure generation. The inverse cascade process is believed to be a common feature of multi-mode flows in both 3-D and 2-D flows. This work gives the quantitative description of the inverse cascade process from the perspective of the average mode number in the 2-D dual-mode RM instability. In case I48-24, the bubble merger process is absent as shown in figure 2. Thus its average mode number does not decrease, and the higher-
$m$ fundamental mode is dominant. This conclusion is consistent with the observation in figure 7. In short, the average mode number reduction at stage III is ascribed to amplitude saturation of higher-
$m$ modes and generation of lower-
$m$ modes due to mode coupling.
Development of average mode number of spectrum in all cases. The colour blocks represent three different stages.
3.3. Theoretical prediction of the mode amplitude
3.3.1. Theoretical models for planar geometry
The RT effect and reshock are eliminated in the current convergent geometry, which provides a possibility to evaluate the BP effect on the fundamental mode development. In planar geometry, Ofer et al. (Reference Ofer, Alon, Shvarts, McCrory and Verdon1996) solved the modal model proposed by Haan (Reference Haan1991), and obtained a second-order solution for multi-mode RT instability, which is written as
\begin{equation} a_{k}(t)=a_{k}^{lin}(t)-\tfrac{1}{2}kA \left(\sum_{k'}a_{k'}^{lin}(t)a_{k+k'}^{lin}(t)- \tfrac{1}{2}\sum_{k'< k}a_{k'}^{lin}(t)a_{k-k'}^{lin}(t)\right), \end{equation}
where 
$k, k' \in \mathbb {R^{+}}$ and 
$a_{k}^{lin}(t)$ is the 
$k$ mode amplitude at the linear stage. The second term on the right-hand side of (3.3) represents the contributions of mode 
$(k+k')$ and mode 
$(k-k')$ to 
$k$ mode generation. To apply (3.3) to the RM instability, through replacing the constant acceleration 
$g$ with an impulsive acceleration 
$\delta t\Delta v$, Luo et al. (Reference Luo, Liu, Liang, Ding and Wen2020) deduced a dual-mode modal model, named the RM-Ofer model. By applying the RM-Ofer model to a dual-mode RM instability with fundamental 
$k$ and 
$k/2$ modes, the growth rates of the fundamental modes and the harmonic generated can be given as
\begin{equation} \left.
\begin{array}{c@{}}
v_{k/2}(t)=v^{lin}_{k/2}-\tfrac{1}{2}kA^{+}
\left[\left(\sqrt{2}+\tfrac{1}{2}\right)v^{lin}_{k}a^{+}_{k/2}+v^{lin}_{k}v^{lin}_{k/2}t +\tfrac{1}{2}v^{lin}_{k/2}a^{+}_{k}\right],\\
v_{k}(t)=v^{lin}_{k}+\tfrac{1}{2}kA^{+}(2v^{lin}_{k/2}a^{+}_{k/2}
+v^{lin}_{k/2}v^{lin}_{k/2}t),\\
v_{3k/2}(t)=\tfrac{3}{8}kA^{+}[2v^{lin}_{k/2}v^{lin}_{k}t
+v^{lin}_{k/2}a^{+}_{k}+(1+\sqrt{2})v^{lin}_{k}a^{+}_{k/2}],
\end{array}\right\} \end{equation}
where 
$a^{+}_{k}$ is the postshock amplitude of mode 
$k$. The RM-Ofer model considers mode coupling, but ignores nonlinearity. Therefore, it fails to predict the mode amplitude growth when nonlinearity is prominent (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020). To evaluate nonlinearity, a planar dual-mode model combining the nonlinear single-mode model proposed by Zhang & Guo (Reference Zhang and Guo2016) with the RM-Ofer model, named the mZG-Ofer model, has been proposed by Luo et al. (Reference Luo, Liu, Liang, Ding and Wen2020). The mZG-Ofer model is expressed as
\begin{align}
\left. \begin{array}{c@{}}
v_{k/2}(t)=\dfrac{v^{lin}_{k/2}-\frac{1}{2}kA^{+}\left[\left(\sqrt{2}+\tfrac{1}{2}\right)v^{lin}_{k}a^{+}_{k/2}+v^{lin}_{k}v^{lin}_{k/2}t+\tfrac{1}{2}v^{lin}_{k/2}a^{+}_{k}\right]}
{1+\hat{a}k\left\{v^{lin}_{k/2}-\tfrac{1}{2}kA^{+}\left[\left(\sqrt{2}+\frac{1}{2}\right)v^{lin}_{k}a^{+}_{k/2}+v^{lin}_{k}v^{lin}_{k/2}t+\frac{1}{2}v^{lin}_{k/2}a^{+}_{k}\right]\right\}t},
\\
v_{k}(t)=\dfrac{v^{lin}_{k}+\tfrac{1}{2}kA^{+}(2v^{lin}_{k/2}a^{+}_{k/2}+v^{lin}_{k/2}v^{lin}_{k/2}t)}{1+\hat{a}k[v^{lin}_{k}+\tfrac{1}{2}kA^{+}(2v^{lin}_{k/2}a^{+}_{k/2}+v^{lin}_{k/2}v^{lin}_{k/2}t)]t},
\\
v_{3k/2}(t)=\dfrac{\tfrac{3}{8}kA^{+}[2v^{lin}_{k/2}v^{lin}_{k}t+v^{lin}_{k/2}a^{+}_{k}+(1+\sqrt{2})v^{lin}_{k}a^{+}_{k/2}]}{1+\tfrac{3}{8}\hat{a}k^{2}A^{+}[2v^{lin}_{k/2}v^{lin}_{k}t+v^{lin}_{k/2}a^{+}_{k}+(1+\sqrt{2})v^{lin}_{k}a^{+}_{k/2}]\}t},\\
\hat{a}=\tfrac{3}{4}\dfrac{(1-A^{+})(3-A^{+})}{3-A^{+}+\sqrt{2}(1-A^{+})^{{1}/{2}}}\dfrac{4(3-A^{+})+\sqrt{2}(9-A^{+})(1-A^{+})^{{1}/{2}}}{(3-A^{+})^{2}+2\sqrt{2}(3+A^{+})(1-A^{+})^{{1}/{2}}}.
\end{array}\right\} \end{align}
The mZG-Ofer model considers both mode coupling and nonlinearity, and its applicability in predicting the mode amplitude nonlinear growth of dual-mode interfaces subjected to planar shock waves has been verified (Luo et al. Reference Luo, Liu, Liang, Ding and Wen2020).
Comparison of the experimental results with theoretical predictions from the mZG-Ofer model is given in figure 9. The model gives a good prediction for the higher-
$m$ fundamental mode amplitude growth, but it greatly underestimates the lower-
$m$ fundamental mode amplitude growth. This indicates that the BP effect cannot be ignored, and it has a stronger effect on the lower-
$m$ mode growth than the higher-
$m$ mode. A similar conclusion was also drawn in previous theoretical work on single-mode perturbations (Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015), and our work provides additional experimental evidence. In addition, the mZG-Ofer model generally provides good predictions for amplitude growth rates of the modes generated because they generally have high mode numbers. In experiments, the amplitude signs of 
$m_{1}+m_{2}$ mode and 
$m_{1}-m_{2}$ mode generated in in-phase (anti-phase) cases are positive (negative) and negative (positive), respectively, which are consistent with the theoretical predictions from the modified Haan model, as shown in table 4. As a result, the amplitude signs of the modes generated by mode coupling in convergent geometry can also be predicted by Haan's solution proposed for planar geometry.
Variations of amplitudes of fundamental modes and modes generated with time. Theoretical results from the mZG-Ofer model, Guo's model and PM model are shown to compare with experimental results. Here 
$m_{+}$ and 
$m_{-}$ refer to 
$m_{1}+m_{2}$ mode and 
$m_{1}-m_{2}$ mode.
3.3.2. Theoretical models for convergent geometry
In convergent geometry, according to previous work (Bell Reference Bell1951; Mikaelian Reference Mikaelian2005a), the linear solution of a single-mode growth rate in a pure BP environment can be expressed as
\begin{equation} \dot{a}(t)=v_{0}C_{r}^{2}, \end{equation}
where 
$v_{0}=a^{+}_{0}\Delta v (mA-1)/R_{0}$ is the initial growth rate and 
$C_{r}=R_{0}/R$ is the convergence ratio with 
$R=R_{0}+Vt$ (
$V$ is the constant interface velocity). To quantify nonlinearity in the single-mode interface evolution, the expression for third-order feedback to the fundamental mode was obtained by Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015), which is given as
\begin{align}
a_{f}&=a^{3}_{0}\frac{1}{8}\left(3\frac{m^{2}}{R^{2}}-\frac{A}{R}\frac{m}{R}\right)
\left(1-\frac{1}{C_{r}}\right)^{2}+a^{2}_{0}v_{0}
\frac{1}{24}t \left[(7-8A^{2})\frac{m^{2}}{R^{2}}
-\frac{19Am-18}{R^{2}}\right]C_{r}\nonumber\\ & \quad
+a^{2}_{0}v_{0}\frac{1}{24}t\left\{2\left[(8A^{2}-10)
\frac{m^{2}}{R^{2}}+7\frac{A}{R}\frac{m}{R}-\frac{3}{R^{2}}\right]-
\left[(8A^{2}-13)\frac{m^{2}}{R^{2}}-5\frac{A}{R}
\frac{m}{R}+\frac{12}{R^{2}}\right]\frac{1}{C_{r}}\right\}\nonumber\\
&\quad
+a_{0}v^{2}_{0}\frac{1}{24}t^{2}\left\{\left[(A^{2}+2)
\frac{m^{2}}{R^{2}}-22\frac{A}{R}\frac{m}{R}
+\frac{27}{R^{2}}\right]C^{2}_{r}+2\left[(5A^{2}-5)
\frac{m^{2}}{R^{2}}+13\frac{A}{R}\frac{m}{R}-
\frac{3}{R^{2}}\right]C_{r}\right\}\nonumber\\ &\quad
-a_{0}v^{2}_{0}\frac{1}{24}t^{2}\left[(11A^{2}-5)
\frac{m^{2}}{R^{2}}-2\frac{A}{R}\frac{m}{R}\right]
+v^{3}_{0}t^{3}\left(\frac{1}{120}A^{2}\frac{m^{2}}{R^{2}}
-\frac{11}{60}A\frac{1}{R}\frac{m}{R}+\frac{3}{8}
\frac{1}{R^{2}}\right)C^{3}_{r}\nonumber\\ &\quad
-v^{3}_{0}t^{3}\left\{\left[\left(\frac{1}{60}A^{2}+\frac{1}{12}\right)
\frac{m^{2}}{R^{2}}-\frac{11}{30}\frac{A}{R}\frac{m}{R}\right]C^{2}_{r}+
\left[\left(\frac{19}{120}A^{2}-\frac{1}{24}\right)
\frac{m^{2}}{R^{2}}+\frac{1}{60}\frac{A}{R}
\frac{m}{R}\right]C_{r}\right\}.
\end{align}The nonlinear effects on the single-mode perturbation growth in a pure BP environment were found to be reasonably evaluated by (3.7) (Luo et al. Reference Luo, Li, Ding, Zhai and Si2019). Recently, by solving the governing equations for multi-mode perturbation growth on a cylindrically convergent interface, Guo et al. (Reference Guo, Cheng and Li2020) obtained second-order weakly nonlinear solutions for dual-mode perturbations in a pure BP environment, which are expressed as
\begin{equation} \left.\begin{array}{c}
a_{1,m_{1}}=a_{m_{1}}^{0}+\dot{a}^{0}_{m_{1}}C_{r}t
,\\
a_{1,m_{2}}=a_{m_{2}}^{0}+\dot{a}^{0}_{m_{2}}C_{r}t
,\\ \begin{aligned} a_{+} &
=(a_{m_{2}}^{0}\dot{a}^{0}_{m_{1}}+a_{m_{1}}^{0}
\dot{a}^{0}_{m_{2}})\dfrac{1}{2R}[A(m_{1}+m_{2})-1](C_{r}-1)t\\
& \quad +\dot{a}^{0}_{m_{1}}\dot{a}^{0}_{m_{2}}
\dfrac{1}{6R}[A(m_{1}+m_{2})(C^{2}_{r}-4C_{r})-3C^{2}_{r}]t^{2},
\end{aligned}\\ \begin{aligned} a_{-} &
=\dfrac{1}{2R}[A(m_{1}-m_{2})(a_{m_{1}}^{0}
\dot{a}^{0}_{m_{2}}-a_{m_{2}}^{0}\dot{a}^{0}_{m_{1}})\\ &
\quad -(a_{m_{1}}^{0}\dot{a}^{0}_{m_{2}}+a_{m_{2}}^{0}
\dot{a}^{0}_{m_{1}})](C_{r}-1)t\\ & \quad
+\dot{a}^{0}_{m_{1}}\dot{a}^{0}_{m_{2}}
\dfrac{1}{6R}[A(m_{1}-m_{2})(C^{2}_{r}+2C_{r})-3C^{2}_{r})]t^{2},
\end{aligned} \end{array}\right\} \end{equation}
where 
$a_{1,m}$ refers to the linear solutions for the dual-mode BP growth, 
$a_{m}^{0}$ is the initial perturbation amplitude of the 
$m$ mode and 
$\dot {a}^{0}_{m}$ is the initial growth rate. The subscripts 
$+$ and 
$-$ represent the modes 
$m_{1}+m_{2}$ and 
$m_{1}-m_{2}$, respectively. This model is named Guo's model hereafter. Since the model solves the second-order governing equations, it only gives a linear solution to the growth of the fundamental mode, and does not consider mode coupling. However, for the modes 
$m_{1}+m_{2}$ and 
$m_{1}-m_{2}$, the solution does consider both the BP effect and mode coupling.
The predictions from Guo's model are illustrated in figure 9. For the mode generated, the model only provides good predictions to the amplitude growth at the early stage, because the amplitude growth has entered the strongly nonlinear stage at late times. For the fundamental mode, the model behaves differently. In the I/A48-24 and I/A72-24 cases, the model generally predicts well the lower-
$m$ mode amplitude growth, but it greatly overestimates the higher-
$m$ mode amplitude growth at the early stage. This implies that the higher-
$m$ (lower-
$m$) fundamental mode amplitude growth is more significantly affected by mode coupling (BP effect). In the I/A72-48 cases, however, both higher- and lower-
$m$ modes are overestimated by the model, probably because nonlinearity that gradually becomes significant after mode saturation has not been considered in the model. From the discussion on the mode saturation time, the higher-
$m$ mode amplitude reaches saturation earlier, and the amplitudes of both fundamental modes saturate in the I/A72-48 cases. In short, for the fundamental mode which reaches (does not reach) its saturation amplitude, its amplitude growth cannot (can) be predicted by Guo's model.
3.3.3. The present model
To reasonably evaluate the amplitude growths of the fundamental modes and modes generated, the BP effect and mode coupling as well as nonlinearity should be considered in the model. In this work, the amplitude of mode 
$m$ is considered to be composed of two parts, i.e.
\begin{equation} a_{m}=a^{lin}_{m}+a^{nonlin}_{m}, \end{equation}
where 
$a^{lin}_{m}$ is the linear amplitude and 
$a^{nonlin}_{m}$ is the nonlinear feedback from the third-order harmonic to mode 
$m$. Amplitude 
$a^{lin}_{m}$ is calculated by
\begin{equation} a^{lin}_{m}=\int_{t_{0}^{+}}^{t} v_{mc}C^{2}_{r} \,{\rm d} t, \end{equation}
where 
$v_{mc}$ is the growth rate given by (3.4), 
$t_{0}^{+}$ is the time just after the shock passage and 
$a^{nonlin}_{m}$ is obtained through substituting 
$v_0$ and 
$a_0$ by 
$v_{mc}$ and 
$a_{mc}$ respectively in (3.7). Note that the wavenumber 
$k$ is calculated as 
$m/R$ in convergent RM instability. For lower-
$m$ modes, nonlinearity is negligible, and, thus, only the first term in (3.9) is retained. Note that (3.9) can degenerate to 
$a_{mc}$ calculated by (3.4) when the BP effect and nonlinearity are absent, i.e. 
$C_r=1$, in which case the limit of large 
$R$ as 
$m/R \rightarrow k$ is adopted and the second term in (3.9) is equal to zero. It can also reduce to (3.6) when both mode coupling and nonlinearity are not considered, i.e. 
$v^{lin}_{m}a^{+}_{m}$ and 
$v^{lin}_{m}v^{lin}_{m}$ in (3.4) and the second term in (3.9) is equal to zero. In addition, when 
$m\rightarrow 0$, the items related to mode coupling in (3.9) tend to zero, and (3.9) will degenerate to (3.6). This verifies the conclusion that lower-
$m$ mode amplitude growth is more heavily affected by the BP effect. After the mode amplitude reaches saturation, nonlinearity is significant, but (3.7), from which 
$a^{nonlin}_{m}$ is obtained, is only valid for weakly nonlinear stages (Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015). For multi-mode initial spectra, Ofer et al. (Reference Ofer, Alon, Shvarts, McCrory and Verdon1996) considered that the mode amplitude growth obeys the following relation after saturation:
\begin{equation} a^{sat}_{m}(t)=a^{t_{sat}}_{m}\left[1+\log \left(\frac{a^{lin}_{m}(t)}{a^{lin}_{m}(t_{sat})}\right)\right], \end{equation}
where 
$a^{sat}_{m}(t)$ is the amplitude of mode 
$m$ after saturation, 
$a^{t_{sat}}_{m}$ is the saturation amplitude (here given by 
$0.1\lambda$) and 
$t_{sat}$ is the time at which saturation occurs. This post-saturation relation has been verified in convergent geometry (Milovich et al. Reference Milovich, Amendt, Marinak and Robey2004; El Rafei et al. Reference El Rafei, Flaig, Youngs and Thornber2019). In this work, the post-saturation relation is adopted in the model to evaluate the saturated mode's nonlinear behaviour. To sum up, (3.9)–(3.11) constitute the present model (named the PM model).
The PM model indicates that the mode amplitude develops as a 
$1/t$ decay after saturation, which is consistent with the late bubble growth rate of multi-mode RM instability given by both potential flow models (Alon et al. Reference Alon, Hecht, Mukamel and Shvarts1994, Reference Alon, Hecht, Ofer and Shvarts1995; Oron et al. Reference Oron, Arazi, Kartoon, Rikanati, Alon and Shvarts2001) and vortex models (Rikanati, Alon & Shvarts Reference Rikanati, Alon and Shvarts1998). The results in figure 9 show that the PM model generally gives good predictions for the amplitude growths of the fundamental modes with both lower and higher mode numbers. Relatively, the PM model provides a better prediction for the amplitude growth of the fundamental mode with a higher (lower) mode number than Guo's (mZG-Ofer) model. For the fundamental mode with a higher mode number, the PM model slightly overestimates the experimental result. First, the fundamental mode with a higher mode number is less affected by the BP effect which always promotes the perturbation growth. Second, the RM-Ofer model is only accurate to second-order precision and does not consider the feedback from the high-order harmonics. However, the feedback from the second-order harmonics always inhibits the development of the fundamental mode with a higher mode number, as indicated in table 5. For the fundamental mode with a lower mode number, the feedback behaves differently, and it is difficult to determine its role. In addition, the PM model greatly improves the predictions for the amplitude growths of the saturated modes relative to Guo's model. For the modes generated, the PM model provides good predictions at the early stage, but slightly overestimates at the late stage, probably because the BP effect is involved in the model.
In summary, for the development of the fundamental mode which does not reach its saturation amplitude, its amplitude growth is more affected by the BP effect than mode coupling. For the development of the fundamental mode which reaches its saturation amplitude, its amplitude growths are more significantly affected by mode coupling than the BP effect. After the mode reaches its saturation amplitude, the amplitude growth can be described by the post-saturation relation given by Ofer et al. (Reference Ofer, Alon, Shvarts, McCrory and Verdon1996).
4. Conclusion
Shock-tube experiments on the development of a dual-mode air–SF
$_6$ interface subjected to a convergent shock wave are carried out. The convergent shock tube is specially designed with a tail opening to eliminate the RT effect and reshock, and to highlight the BP effect on mode amplitude growth. Six kinds of dual-mode interface with different initial perturbation spectra that are produced by changing the mode combination and phase difference between fundamental modes are considered. The results show that the flow features are strongly dependent upon the initial spectrum. At the late stage, the flow is generally dominated by bubbles with different sizes. The number and arrangement of large bubbles and small bubbles as well as the inclination of spikes are closely associated with the mode combination and phase difference.
The developments of perturbation mixing width show that for dual-mode case, mode coupling arises from the middle stage and suppresses the width growth compared with the single-mode counterpart. Depending upon the initial spectrum, the fundamental mode amplitude growth is either promoted, inhibited or unaffected by mode coupling. In general, mode coupling inhibits the amplitude growth of the fundamental mode with a higher mode number, but causes different behaviours for the amplitude growth of the fundamental mode with a lower mode number. The fundamental mode amplitude variation can be explained by the feedback of the high-order modes. The BP effect promotes the occurrence of mode coupling, and the feedback of high-order modes to fundamental mode also arises earlier in convergent geometry than that in its planar counterpart. Moreover, the amplitude growth is found to saturate earlier for the fundamental mode with a higher mode number, and the saturation time is affected by mode coupling. By introducing the concept of the average mode number, the flow is found to be dominated by the fundamental mode with a lower mode number.
By incorporating mode coupling and nonlinearity into the planar dual-mode model, a model is established to predict the amplitude growth of the fundamental mode. The model provides good (poor) predictions for the amplitude growth of the fundamental mode with a higher (lower) mode number, which indicates that the amplitude growth of the fundamental mode with a lower mode number is more significantly affected by the BP effect which is not included in the model. Further, an existing convergent dual-mode model (Guo et al. Reference Guo, Cheng and Li2020) considering the BP effect and weak nonlinearity is used to predict the amplitude growths of the fundamental modes and modes generated. For the modes generated, the model can predict the amplitude growths at the early stage. For the fundamental mode with a lower mode number that does not reach its saturation amplitude, the model provides a good prediction, whereas for the fundamental mode with a higher mode number that reaches its saturation amplitude, the prediction is less satisfactory. This indicates that the fundamental mode with a higher (lower) mode number is more heavily affected by mode coupling (BP effect). Finally, the amplitude developments of the fundamental modes and modes generated can be reasonably well predicted by combining the planar mode-coupling model, the linear BP model and the third-order feedback model. In particular, for the fundamental mode that reaches its saturation amplitude, the post-saturation relation given by Ofer et al. (Reference Ofer, Alon, Shvarts, McCrory and Verdon1996) is introduced into the model to achieve a better prediction.
This work can provide a support for designing the initial mode combination when one wants to manipulate the mode amplitude growth. In future work, development of multi-mode perturbations subjected to a convergent shock wave will be investigated and a more complicated mode-coupling mechanism will be explored.
Funding
This work was supported by the National Natural Science Foundation of China (nos. 12022201, 91952205 and 12102425) and Youth Innovation Promotion Association CAS.
Declaration of interests
The authors report no conflict of interest.
