3. Results and discussion

A quasi-1-D experiment corresponding to a uniform cylindrical interface interacting with a cylindrical divergent shock is first performed to examine the background flow, for which there is no theoretical solution. Various shock Mach numbers are considered. Detailed parameters corresponding to the initial conditions for cases U1-U4 are listed in table 1, where the Atwood number ($A$) is defined as $A = (\rho _2 - \rho _1)/(\rho _2 + \rho _1)$ with $\rho _1$ and $\rho _2$ being the gas densities on the left and right sides of the interface, respectively. Sequences of schlieren images illustrating the movements of the interface and the shock for cases U1 and U4 are displayed in figure 2. Case U1 is taken as an example to detail the motion process. Time origin in this work is defined as the moment at which the incident shock arrives at the mean position of the interface. At the beginning ($-26\,\mathrm {\mu }$s), an incident cylindrical shock (ICS) together with a cylindrical unperturbed interface (UI) is clearly observed. Later, the ICS collides with the air/SF$_6$ interface, bifurcating into an inward-moving reflected shock (RS) and an outward-moving TS ($40\,\mathrm {\mu }$s). After that, the shocked interface (SI) moves downstream following the TS. Due to the shock impact, the soap film is atomized into small droplets with a diameter of approximately $30\,\mathrm {\mu }$m. Previous studies (Ding et al. Reference Ding, Si, Yang, Lu, Zhai and Luo2017; Liu et al. Reference Liu, Liang, Ding, Liu and Luo2018; Zhang et al. Reference Zhang, Ding, Si and Luo2023) have demonstrated a negligible influence of the soap droplets on the interface evolution. The SI maintains a cylindrical shape during the experimental time, which indicates a negligible influence of boundary layer on the interface motion. The mass fraction of SF$_6$ inside the layer is determined based on 1-D gas dynamics theory, and the detailed estimation process has been reported by Zhang et al. (Reference Zhang, Ding, Si and Luo2023). It is found that as the shock is intensified, the relative velocity between the TS and the interface decreases, indicating a closer proximity between them.

Figure 2.

Schlieren images showing the motions of waves and the interface for cases U1 and U4: Ma, the Mach number of the incident shock; ICS, the incident cylindrical shock; UI, the initial unperturbed interface; SI, the shocked interface; TS, the transmitted shock; RS, the reflected shock. The unit of numbers is $\mathrm {\mu }$s.

Dimensionless variations of the interface displacement with time are plotted in figure 3(a), where two cases with lower Mach numbers reported by Li et al. (Reference Li, Ding, Zhai, Si, Liu, Huang and Luo2020) are also given. Here, time is scaled as $\Delta Vnt/R_0$ with $\Delta V$ being the initial velocity of the SI and the interface displacement as $n(R-R_0)/R_0$ with R being the radius of the interface. For all cases, the interface moves uniformly during the early period following the shock passage, and later presents an evident deceleration. The stronger the incident shock, the larger the interface deceleration. The normalized trajectories of the interface collapse at the early stage for all cases, but deviate at the late stage due to different deceleration. It is desirable to derive an analytical solution for the interface trajectory, which is an important step towards the understanding and modelling of the divergent RM instability. Unfortunately, the unsteady, non-uniform flow behind the divergent shock greatly impedes the theoretical derivation.

Figure 3.

(a) Trajectories of the interface for all cases. (b) The time duration ($\Delta t$) between the moments at which the shock and the interface pass through position $R$.

In our recent work, an analytical solution for the trajectory of a fluid element in divergent geometry was derived under the incompressible flow assumption, which is expressed as

(3.1)\begin{equation} r=[r_0^2+2C(t-t_0)]^{1/2}. \end{equation}

Here $t_0$ is the initial time, $r_0=r(t_0)$ and $C=u_{0}r_0$ with $u_{0}=u(t_0)$, where u is the flow velocity in the radial direction. Note the non-uniform postshock pressure field has not been taken into account in (3.1). Here, provided the initial shock strength $M = M_0$ at $r = r_0$, the variation of the divergent shock Mach number with radius $r$ can be calculated according to the Chester–Chisnell–Whitham relation (Whitham Reference Whitham1958). To quantify the effect of a non-uniform pressure field on the interface motion, an assumption that the postshock pressure gradient remains invariant at each radius is adopted. Incorporating the pressure gradient (calculated by Rankine–Hugoniot conditions) into the 1-D momentum equation in a cylindrical coordinate system, the interface trajectory predicted by (3.1) can be modified. It is found that this modified model gives a reasonable prediction of the interface trajectory for all cases. Particularly, higher-Mach-number cases ($Ma=1.6$ and 1.7) present better agreement with the prediction. An interpretation is given below. Figure 3(b) gives the time duration ($\Delta t$) between the instances at which the divergent shock and the SI pass through the position $R$. A notable reduction in $\Delta t$ is observed when increasing the Mach number. It means that for a stronger shock, the postshock flow at radius $R$ undergoes a shorter period of variation before the arrival of the interface. This makes the assumption of invariance of postshock pressure gradient more reasonable.

The divergent RM instability at a single-mode interface under four shock strengths (cases S1–S4) is then examined. Detailed parameters corresponding to the initial conditions for each case are listed in table 1. The initial amplitude of the interface is $a_0 = 1.5$ mm and the azimuthal mode number is $n = 36$, corresponding to an amplitude-to-wavelength ratio of 0.054. The other initial conditions remain the same as the unperturbed case. Developments of the wave patterns and the interface for these cases are illustrated in figure 4. Case S1 ($Ma=1.3$) is taken as an example to detail the evolution process. At the beginning ($-9\,\mathrm {\mu }$s), an ICS together with a sinusoidal perturbed interface (PI) are clearly observed. Then, the ICS collides with the PI, bifurcating immediately into sine-like transmitted and RSs ($41\,\mathrm {\mu }$s). During this process, the interface suffers a quick drop in amplitude due to shock compression. Subsequently, the SI moves downstream with its amplitude increasing persistently. As time proceeds, the TS propagates forwards with a gradually decaying amplitude and finally recovers to a uniform cylindrical shock ($491\,\mathrm {\mu }$s). It is worth noting that no evident spike and bubble structures are observed even at the late stage when intensifying the incident shock, which differs from the planar RM instability (Sadot et al. Reference Sadot, Rikanati, Oron, Ben-Dor and Shvarts2003; Motl et al. Reference Motl, Oakley, Ranjan, Weber, Anderson and Bonazza2009). This phenomenon exists uniquely in divergent RM instability.

Figure 4.

The developments of wave patterns and interfacial morphologies for all cases. The symbols are the same as those in figure 2. The unit of the numbers is $\mathrm {\mu }$s.

Normalized variations of the interface amplitude with time for all cases are plotted in figure 5(a). The amplitude is normalized as $\alpha = n(a - a^+_0 )/R_0$ and the time as $\tau =n \dot {a}_0 (t-t_0^+ )/R_0$, where $a^+_0$ refers to the postshock amplitude, $t^+_0$ to the time just after the shock passage and $\dot {a}_0$ to the initial growth rate. The solid line in figure 5(a) refers to the linear prediction of Bell (Reference Bell1951) and the dashed line to the prediction of Epstein (Reference Epstein2004) considering compressibility effect. The present experiments afford a longer observation time for the interface development as compared with that of Li et al. (Reference Li, Ding, Zhai, Si, Liu, Huang and Luo2020) such that the instability growth at a later stage can be examined. The amplitude variation curves for these cases collapse quite well at the early stage, but deviate from each other at $\tau > 1.0$. For all cases, the growth rate decays gradually with time. The stronger the incident shock, the quicker the growth rate decay. Particularly, for the $Ma=1.6$ and $Ma=1.7$ cases, the interface amplitude decreases at the late stage (i.e. a negative growth rate), which indicates the presence of some factors stabilizing the instability.

Figure 5.

(a) Temporal variations of the interface amplitude and (b) comparison between the nonlinear predictions (Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015) considering uniform motion (filled symbol) and deceleration (hollow symbol) for harmonic growths. The solid line in (a) refers to the linear prediction of Bell (Reference Bell1951) and the dashed line to the prediction of Epstein (Reference Epstein2004) considering compressibility effect.

The initial growth rates, obtained with a linear fit of experimental data, are 9.6, 10.2, 13.5 and 16.4 m s$^{-1}$ for cases S1, S2, S3 and S4. The growth rates predicted by the incompressible linear model (Bell Reference Bell1951) are 16.1, 21.2, 28.6 and 47.0 m s$^{-1}$ for cases S1–S4, respectively. The lower growth rate in experiment than prediction is primarily attributed to compressibility effect (i.e. pressure disturbances behind the reflected and TSs play a role). This is similar to the finding in convergent RM instability (Ding et al. Reference Ding, Si, Yang, Lu, Zhai and Luo2017). Differing from the weak shock case, in which compressibility effect is weak at late stages, for a moderate shock ($Ma=1.6$ and $1.7$), the TS is closer to the interface (observed in the unperturbed case) and compressibility may be pronounced at late stages. Thus, the effect of compressibility on the instability growth is first assessed. Assuming the fluids on both sides of the interface are inviscid and irrotational and also undergo uniform compression (i.e. $\dot {\rho }_2/\rho _2 = \dot {\rho }_1/\rho _1$ with $\dot{\rho_1} (\dot{\rho_2}$) being the first derivative of $\rho_1 (\rho_2)$ with time), Epstein (Reference Epstein2004) proposed a linear model for the instability growth at a perturbed cylindrical interface:

(3.2)\begin{equation} \left(-\frac{\dot{\rho}}{\rho}+\frac{{\rm d}}{{\rm d} t}\right) \frac{{\rm d}}{{\rm d} t}(a \rho R)=n A a \rho \frac{{\rm d}^2 R}{{\rm d} t^2}, \end{equation}

where $\dot {\rho }/\rho = \dot {\rho }_2/\rho _2 \approx \dot {\rho }_1/\rho _1$ with $\rho$ being the average density of fluids on both sides of the interface, and $\alpha$ is the amplitude of the perturbed interface. Letting $c = -\dot {\rho }/\rho$, the above formula is simplified to

(3.3)\begin{equation} \frac{{\rm d}^2 a}{{\rm d} t^2}+\left(\frac{2 \dot{R}}{R}-c\right) \frac{{\rm d} a}{{\rm d} t}-\left[(n A-1) \frac{\ddot{R}}{R}+ \frac{\dot{R}}{R} c+\frac{{\rm d} c}{{\rm d} t}\right] a=0. \end{equation}

Here, $R(t)$ is given by the function fitted from the measured radius of the corresponding UI, $\dot {R}$ and $\ddot {R}$ are the first and second derivatives of interface radius with time, respectively. For $c = -\dot {\rho }/\rho = 0$, the equation reduces to the Bell (Reference Bell1951) model for incompressible fluids. Here, the volume enclosed by the shock and the interface can be measured from schlieren images and thus the density variation rate $c$ is available. Then, substituting the postshock parameters into (3.3), the instability growth with compressibility effect is obtained. As shown in figure 5(a), both compressible and incompressible models give a reasonable prediction of the instability growth within the measurement error. The predictions of the two models nearly collapse, which indicates a weak influence of compressibility on the instability growth for the parameter space considered in this work. Thus, the incompressible model is suitable for the present experiments. Note disturbances at a divergent shock decay more quickly than the convergent and planar shocks, which is a reason for the weak compressibility effect here.

Previous studies (Li et al. Reference Li, Ding, Zhai, Si, Liu, Huang and Luo2020; Zhang et al. Reference Zhang, Ding, Si and Luo2023) showed that, in addition to the common regimes in planar RM instability, the growth of divergent RM instability at the linear stage is also affected by geometric divergence and RT effect. It is realized that these two effects could also affect the nonlinear growth of divergent RM instability, which has never been reported. To quantify the degree of nonlinearity, the growths of harmonics are obtained by performing the Fourier analysis of the interface morphologies extracted from the schlieren images. The Fourier analysis, which is applicable only to interfaces represented as single-valued functions, becomes invalid at the late stage when the spikes roll up. To obtain as much data as possible, the small curled structures at the spike head are removed via postprocessing. In this way, the growths of harmonics at a relatively later time can be obtained. As shown in the inserts of figure 6(a), a virtual interface composed of the first three harmonics obtained nearly collapses with the extracted morphology, which demonstrates the reliability of the Fourier analysis. Temporal variations of the amplitudes of the first three harmonics are plotted in figure 6. Higher-order harmonics that present a negligibly small amplitude during the experimental time are ignored. For cases S1 and S2 ($Ma=1.3, 1.4$), the amplitude of each harmonic increases persistently with a gradually decaying growth rate at the early stage and later freezes out. For cases S3 and S4 ($Ma=1.6, 1.7$), the amplitude of each harmonic drops after saturation. The considerable amplitude of the second and third harmonics suggests that the instability has entered into a nonlinear stage. The freeze-out or decay of the second and third harmonics at the late stage indicates that nonlinearity is suppressed by a certain factor, particularly evident in stronger shock cases. The interface amplitude measured from experiment (filled circle) and that of the virtual interface composed of the harmonics (hollow circle) are also given in figure 6, and they collapse well.

Figure 6.

Comparisons between the experiment and the nonlinear prediction for the growths of the first three harmonics ($a_1$, $a_2$ and $a_3$) for cases S1 (a), S2 (b), S3 (c) and S4 (d). Filled symbols denote the amplitude of interface in experiment and hollow symbols denote the amplitude of a virtual interface composed of the first three harmonics.

Under the incompressible, inviscid, irrotational fluid assumption, a third-order weakly nonlinear model has been derived by Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015). Nevertheless, it is difficult to derive an analytical solution for the nonlinear growth of divergent RM instability at an interface with arbitrary radial motion. In this work, to predict the nonlinear growths of harmonics, the model of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015) is solved numerically by substituting the parameters in the corresponding unperturbed case (i.e. displacement $R$, velocity $\dot {R}$ and acceleration $\ddot {R}$) into the model. Comparisons of the growths of harmonics between the experiment and the nonlinear prediction considering interface deceleration are given in figure 6. The model predicts well the growths of the second and third harmonics for all cases. Also, it gives a good prediction of the first harmonic at the early stage, but overestimates the late-stage growth. A possible reason is that the prediction of the first harmonic takes only the third-order feedback into account, and higher-order harmonics are ignored. Both the freeze-out (cases S1 and S2) and amplitude reduction (cases S3 and S4) of the harmonics are well reproduced by the nonlinear model. To elucidate this phenomenon, comparison between the nonlinear predictions considering uniform (i.e. $\ddot {R}=0$) and non-uniform motions for case S1 is made. As illustrated in figure 5(b), the interface deceleration greatly inhibits the growth of each harmonic, which manifests the significant role of RT effect in divergent RM instability, leading to very weak nonlinearity during the whole experimental time. Particularly, higher-Mach-number cases present weaker nonlinearity due to stronger RT effect there. This is consistent with the observation in schlieren images, where a distinct vortex pair is noticeable at the spike tip for cases S1–S2, but diminishes for cases S3–S4. This abnormal linear growth regime has never been discovered. Various decelerations in cases S1–S4 cause different degrees of suppression of the amplitude growth, which explains the deviation in amplitude growth among these cases (figure 5a). Note without interface deceleration (i.e. RT stability), the cylindrical RM instability could present evident nonlinearity (Matsuoka & Nishihara Reference Matsuoka and Nishihara2006). Under the incompressible flow assumption, geometric divergence causes a gradual reduction in interface amplitude and a continuous increment in wavelength, resulting in a low amplitude–wavelength ratio. This is also an important factor responsible for the longer linear stage observed in divergent RM instability than the planar and convergent counterparts. Despite the linear regime, geometric divergence effect that is inherent in divergent RM instability causes a continuous decay in growth rate, and thus the amplitude growth is no longer linear in time as in the planar RM instability. The present analysis gives an in-depth understanding of the regimes of divergent RM instability. The finding indicates that the coexistence of compressibility and nonlinearity at late stages of RM instability under strong shocks, that poses a great challenge to theoretical treatment, can be avoided for the divergent situation. This is crucial for modelling the divergent RM instability under strong shocks.