3. Results and discussion

To check the divergent shock tube and also understand the non-uniform flow feature in the divergent RMI, an undisturbed cylindrical air–SF$_6$ interface impacted by the cylindrically divergent shock is first considered. As shown in figure 2(a), when the incident shock, IS, strikes the undisturbed initial interface, II, it bifurcates into an outward-moving transmitted shock, TS, and an inward-moving reflected shock (not visible in schlieren images due to its very weak intensity). Here, the incident and transmitted shock fronts match well with the circular arcs (dashed lines) of the same centre, which verifies the feasibility of the present method. As time proceeds, the shocked interface, SI, moves outwards following the transmitted shock. It is seen that the wall boundary layer has a limited influence on the movement of the near-wall interface, and the whole interface maintains a nearly cylindrical shape during the experimental time. Variations of displacements of the waves and interface versus time are plotted in figure 2(b). After the shock impact, the interface moves outwards at a nearly constant speed of approximately 91 m s$^{-1}$ ($t< 500\ \mathrm {\mu }$s) and later undergoes a noticeable deceleration, which will cause RT stability or instability for a perturbed interface (Ding et al. Reference Ding, Si, Yang, Lu, Zhai and Luo2017). The transmitted shock also presents a distinct deceleration, which indicates a gradual attenuation of the divergent shock. Therefore, the present interface deceleration is induced by a gradually decaying divergent shock, which is quite different from the convergent case. In the convergent shock tube, the convergent shock becomes stronger with time such that the pressure behind the convergent shock becomes higher, which decelerates the interface movement. We also note that both cases are in an unsteady flow and, consequently, there are no analytic solutions. Assuming incompressible and steady subsonic flow, it can be easily found that the flow deceleration due to expansion is $\textrm {d} v/\textrm {d} t=-v^2/r$ at radius $r$ and time $t$ from the volume conservation. As compared in figure 2(b), the steady theory can roughly or qualitatively estimate the interface deceleration in a divergent geometry, but it overestimates the interface deceleration in experiment ($\sim$10 % at 900 $\mathrm {\mu }$s).

Figure 2.

(a) The interaction of a divergent shock with an unperturbed air–SF$_6$ interface. The numbers have units $\mathrm {\mu }$s. Dashed lines stand for circular arcs with the same centre as the cylindrical shock front. (b) Trajectories of the waves and interface. In panel (a): IS, incident shock; II, initial interface; SI, shocked interface; TS, transmitted shock. Dash–dotted lines represent the uniform movements of transmitted shock and shocked interface, and the solid line is the trajectory of the interface predicted by a steady theory.

For the present experiment, the length Reynolds number of the boundary layer of the air (SF$_6$) flow in the divergent test section is calculated to be $4.8\times 10^5$ ($2.8 \times 10^6$). Although this value approaches the upper limit of the critical Reynolds number of transition ($3.5\times 10^4 \sim 5.0\times 10^6$) (Reshotko Reference Reshotko1976; Saric, Reed & Kerschen Reference Saric, Reed and Kerschen2002), the boundary layer in experiment is laminar or at least not fully turbulent during the time of interest. This can be demonstrated by the schlieren images shown in figure 3 where the material interface close to the wall presents clear and ordered structures even at late stages. Hence, the post-shock boundary layer can be assumed to be laminar and incompressible, and its thickness ($\delta ^*$) can be estimated by

(3.1)\begin{equation} \delta^*=1.72\sqrt{\frac{\mu x}{\rho\varDelta v}}. \end{equation}

The maximum distance travelled by the interface during the experimental time is measured to be $x\approx$ 80 mm. The viscosity coefficient and the density of pure air (SF$_6$) under the experimental conditions are $\mu = 1.83 \times 10^{-5}\ \textrm {Pa}\,\textrm {s} (= 1.60 \times 10^{-5}\ \textrm {Pa}\,\textrm {s})$ and $\rho = 1.204\ \textrm {kg m}^{-3} (= 6.143\ \textrm {kg m}^{-3})$, respectively. The velocity of the post-shock flow is $\varDelta v\approx 91$ m s$^{-1}$. According to (3.1), the maximum thickness of the boundary layer is calculated to be approximately 0.19 mm in the air flow (0.08 mm in the SF$_6$ flow), which is much smaller than the inner height of the test section (7.0 mm). This indicates a negligible influence of the boundary layer on the interface development.

Figure 3.

Evolution of a single-mode air–SF$_6$ interface subjected to a cylindrically exploding shock for all cases. The symbols are the same as those in figure 2. The numbers have units $\mathrm {\mu }$s.

Then, five single-mode interfaces with different amplitudes and wavelengths are considered. Detailed parameters corresponding to the initial conditions for each case (denoted by $a_0-n$) are listed in table 1, where the Atwood number 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-hand sides of the interface, respectively. Specially, case 0-0 denotes the unperturbed case. Note in experiments, SF$_6$ gas could cross the soap film and contaminate the pure air on the other side of the interface. With the measured speeds of the incident shock, transmitted shock and shocked interface, mass fractions of SF$_6$ on the left- and right-hand sides of the interface can be estimated by one-dimensional gas dynamics theory. As indicated in table 1, gas contamination in the present experiments is limited and the discrepancy in Atwood number among all cases is very small.

Table 1.

Parameters corresponding to the initial conditions for each case (denoted by $a_0-n$) with $a_0$ and $n$ being the initial amplitude and azimuthal mode number of the interface, respectively. Specially, case 0-0 denotes the unperturbed case. Here, $A$ refers to the Atwood number, $\lambda$ to the wavelength and $v_0$ to the amplitude growth rate at the early stage. Here, MF(SF$_6$)$_L$ and MF(SF$_6$)$_R$ denote the mass fractions of SF$_6$ on the left- and right-hand sides of the interface, respectively. The velocities of the incident shock and shocked interface are $v_i$ and $\varDelta v$, respectively.

Representative interface morphologies and wave patterns illustrating the instability development processes for five cases are displayed in figure 3. The time origin is defined as the moment when the incident shock arrives at the mean position of initial interface. Prior to shock-interface interaction, the initial interface presents a perfect single-mode shape for all cases (first column), which demonstrates the feasibility and reliability of the current experimental method. As the incident divergent shock encounters the initial interface, it splits into a corrugated transmitted shock propagating downstream and an upstream-moving reflected shock. Here, distortions on the transmitted shock decay quickly with time, which is different from the shock propagation in a convergent geometry. Later, the outgoing interface deforms continuously driven by the baroclinic vorticity deposited along it. It is observed that the present interface deformation is much slower than its counterpart in a planar or convergent geometry. This is mainly ascribed to the circumferential stretching caused by geometric expansion. Particularly, the asymmetric bubble-spike structure is not fully developed even at very late times, which indicates a very weak nonlinearity there. For cases with a larger initial amplitude-to-wavelength ratio, although the roll-up of spike appears earlier, the whole structure is also much less distorted than the planar or convergent counterpart.

Normalized variations of the perturbation amplitude with time are plotted in figure 4, where the predictions of linear and nonlinear theories are also given for comparison. The amplitude is normalized as $\alpha = n(a - a^+_0)/R_0$, and the time is scaled as ${\tau = nv_0(t - t^+_0)/R_0}$. Here, $t^+_0$ is the time just after the shock passage, $a^+_0$ is the post-shock amplitude and $v_0$ is the experimental initial growth rate. The dimensionless experimental data for all cases collapse quite well, which illustrates a universal growth behaviour of the divergent RMI. After the shock impact, the perturbation amplitude first drops suddenly to a smaller value due to shock compression, and then increases continuously in a nearly linear manner ($\tau < 0.4$). Later, the growth rate of perturbation amplitude reduces evidently. We stress that such a reduction in growth rate is mainly ascribed to the geometric divergence rather than nonlinearity, which will be addressed hereinafter. At late stages, the perturbation amplitude almost stops growing and maintains a nearly constant value. Such an instability freeze-out exists for all cases considered in this work. Therefore, we claim it is a universal phenomenon for divergent RMI.

Figure 4.

Normalized variations of the overall perturbation amplitude. The dashed (dash–dot) line denotes the linear theory of Bell (Reference Bell1951) without (with) the RT effect, and the solid line denotes the nonlinear theory of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015). The dotted line refers to the growth of the third harmonic and the dash–dot–dot line to the third-order feedback to the basic mode.

Under the assumption of incompressible inviscid flow, Bell (Reference Bell1951) obtained a linear theory for the cylindrical RMI, which can be written as

(3.2)\begin{equation} \ddot{a}+2\frac{\dot{R}}{R}\dot{a}-(nA-1)\frac{\ddot{R}}{R}a=0,\end{equation}

where $R(t)$ is the time-dependent radius of the moving interface, $\dot {R}$ ($\ddot {R}$) is the first (second) derivative of radius with time and $\dot {a}$ ($\ddot {a}$) is the first (second) derivative of perturbation amplitude with time. The development of the cylindrical RMI can be divided into two phases: the shock-driven phase (during the shock impact) and the undriven growth phase (after the shock impact). The first phase yields a post-shock growth rate of perturbation amplitude, which can be obtained by integrating equation (3.2) once from $t_0$ (time instant when the shock meets the interface) to $t^+_0$ (shortly after the shock impact). Note that during this phase, the interface is motionless, i.e. $R(t)=R_0$, and the shock impact can be approximated as an impulse function, i.e. $\ddot {R}=\delta (t) \varDelta V$. Thus, the post-shock growth rate can be readily obtained as follows:

(3.3)\begin{equation} \dot{a}_0=\frac{(nA-1)\varDelta Va^+_0}{R_0}.\end{equation}

Here, the post-shock amplitude $a^+_0$ is adopted for approximating the compressibility effect, which is similar to the treatment of Richtmyer (Reference Richtmyer1960). Equation (3.3) illustrates that the cylindrical RMI possesses an initial growth rate $\dot {a}_0$ immediately after the shock impact ($t^+_0$). The growth rate of perturbation amplitude at an arbitrary evolution time $t_1$ after the shock impact can be obtained by integrating equation (3.2) from $t^+_0$ (with an initial growth rate $\dot {a}_0$) to $t_1$, which is expressed as

(3.4)\begin{equation} \dot a(t_1)=\frac{{R_0}^2}{{R^2(t_1)}}\dot{a_0}+\frac{1}{R^2(t_1)} \int_{t^+_0}^{t_1}{(nA-1)a(t)R(t)\ddot{R}(t)\, \textrm{d} t}.\end{equation}

Then, an analytical solution for the perturbation amplitude at an arbitrary time $t_2$ is available by integrating equation (3.4) from $t^+_0$ to $t_2$ as follows:

(3.5)\begin{align} a(t_2)=a^+_{0}+\dot{a_0}{R_0}^2\int_{t^+_0}^{t_2}{\frac{1}{R^2(t_1)}\, \textrm{d} t_1}+ \int_{t^+_0}^{t_2}\left[\frac{1}{R^2(t_1)}\int_{t^+_0}^{t_1}(nA-1)a(t)R(t)\ddot{R}(t)\, \textrm{d} t\right]\, \textrm{d} t_1.\end{align}

The second term on the right-hand side of (3.5) represents the pure RMI combined with the geometric convergence/divergence effect, and the third term denotes the RT effect caused by the flow acceleration or deceleration.

The present experimental results offer a rare opportunity to examine the validity of Bell theory for the divergent RMI. Considering that the discrepancies in the incident shock Mach number and the Atwood number among all cases in the experiments are very small, the interface trajectory for the unperturbed case (figure 2) can approximately represent the $R(t)$ for the perturbed cases. By substituting the $R(t)$ from the unperturbed case into (3.5), we can obtain the theoretical prediction of Bell theory for a perturbed case. As shown in figure 4, the Bell prediction (obtained from 3.5) assuming a steady flow (i.e. a uniform velocity of the interface) agrees reasonably with the experimental results at the early stage, but deviates later ($\tau > 0.8$). When taking the flow deceleration (indicated in the unperturbed case) into account, a good agreement between the theory and experiment is again achieved for the whole evolution process. Moreover, the instability freeze-out at late stages is also well captured by the theory. This demonstrates that the late-stage instability stagnation is mainly ascribed to the counteraction between the positive growth caused by pure RMI and the negative growth induced by geometric divergence and RT stabilization. To the authors’ knowledge, this is the first direct examination of Bell theory for the divergent RMI.

To assess the nonlinearity effect, the third-order weakly nonlinear model of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015), which assumes a small perturbation at a cylindrical interface subjected to a uniform radial motion, is employed to estimate the present perturbation growth. A brief description of this theory is given below. For the problem of two incompressible, inviscid, irrotational and immiscible fluids in an arbitrary radial motion, a velocity potential can be introduced for each fluid which satisfies the Laplace equation. Considering the kinematic and kinetic boundary conditions (i.e. the normal component of the velocity of each fluid at the interface should equal the normal velocity of the interface and also the pressure is continuous across the interface), another three governing equations for the velocity potentials can be obtained. Expanding the perturbation displacement and the velocity potentials into a power series, the first-, second- and third-order governing equations are available, and each is expressed as a second-order ordinary differential equation. Note that it is very difficult to get a general solution for the perturbation growth at a cylindrical interface of an arbitrary implosion (explosion) history. By assuming a uniformly moving interface, the solution process is greatly simplified and the first-, second- and third-order analytical solutions can be derived. Here, we only present the first- and second-order solutions (the third-order expressions are very long and not given here),

(3.6)\begin{gather} a_{1,1}=a_{0}+\dot{a_{0}}tC_{r}, \end{gather}

(3.7)\begin{gather}a_{2,2}=a_{0}\dot{a_{0}}t\left(A\frac{n}{R}-\frac{1}{2}\frac{1}{R}\right)(C_{r}-1) +\dot{a_{0}}^{2}t^{2}\left[\left(\frac{1}{6}A\frac{n}{R}-\frac{1}{4} \frac{1}{R}\right)C_{r}^{2}-\frac{2}{3}A\frac{n}{R}C_{r}\right], \end{gather}

(3.8)\begin{gather}a_{2,0}=-\left(a_{0}\dot{a_{0}}t\frac{1}{2}\frac{1}{R}C_{r}+\dot{a_{0}}^2t^{2} \frac{1}{4}\frac{1}{R}C_{r}^2\right), \end{gather}

where $C_r = R_0/R(t)$ is the convergence ratio, $a_{1,1}$ is the amplitude of the fundamental mode, $a_{2,2}$ is the amplitude of the second harmonic and $a_{2,0}$ is the second-order feedback to the zero-order mode (i.e. the radius of the unperturbed interface). For more details, the reader is referred to the original work of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015).

As shown in figure 4, there exists only a minor difference between the linear and nonlinear predictions even at very late stages, which indicates a very weak nonlinearity for the divergent RMI. This finding is completely different from that of convergent RMI where nonlinearity is found to be strong and behaves evidently (Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015; Luo et al. Reference Luo, Li, Ding, Zhai and Si2019). The growth of the third harmonic together with its feedback to the primary mode, predicted by the nonlinear model of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015), is also shown in figure 4 (the growth of the second harmonic produces no influence on the whole perturbation growth and is not given). It is found that at early stages the growth of the third harmonic in divergent geometry is much smaller than that in a planar or convergent geometry (Luo et al. Reference Luo, Li, Ding, Zhai and Si2019), which provides a further demonstration of weak nonlinearity. Although nonlinearity increases at late stages, the positive growth of the third harmonic approximately counteracts its negative feedback to the basic mode, which suggests a negligible influence of nonlinearity on the growth of overall perturbation amplitude. This explains well the reasonable prediction of linear Bell theory for the whole evolution process of the divergent RMI.

The high-fidelity experimental images obtained enable a reliable extraction of the interfacial contours. Taking case 3-24 as an example, interfacial profiles at seven typical time instants are extracted from schlieren images by an image processing program as shown in figure 5(a). By a spectrum analysis of the interfacial profiles, growths of the first, second and third harmonics are available for all cases. As shown in figure 5(b), the primary mode presents the same increasing tendency as the overall perturbation amplitude, i.e. first grows linearly, then slows down and eventually freezes out. Growths of the second and third harmonics are far slower, which indicates a dominant role of the primary mode at weakly nonlinear stage. The third harmonic maintains a very small amplitude during the experimental time, which implies that the energy of basic mode is mainly transferred to the second harmonic. Considerable agreement between the nonlinear prediction and the experiment for the growths of the first three harmonics is achieved for $\tau < 1.0$. Therefore, a direct validation of the nonlinear model of Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015) is performed for the divergent RMI.

Figure 5.

(a) Interfacial profiles extracted from schlieren images at seven moments (units are $\mathrm {\mu }$s) for case 3-24. (b) Normalized variations of the amplitudes of the first three harmonics with time. The nonlinear model is Wang et al. (Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015).

In this work, the flow regimes of the divergent RMI are analysed by comparing the linear and nonlinear models (Bell Reference Bell1951; Wang et al. Reference Wang, Wu, Guo, Ye, Liu, Zhang and He2015) with the experimental results under a limited range of initial parameters (i.e. different initial perturbation amplitudes and wavelengths under a single Mach number and Atwood). For the real ICF application, the initial conditions such as the shock Mach number, the perturbation shape and the Atwood number are different from the present ones, and the instability growth is much more complicated. Therefore, the validity of the present model should be studied using more cases with different initial parameters including shock Mach number and Atwood number. Design and manufacture of a new divergent shock tube for generating an incident strong divergent shock is ongoing. For a very large density ratio in ICF, usually corresponding to a gas–solid interface, new interface formation methods and flow diagnostic techniques should be developed. We hope that the RMI under a strong divergent shock at a large density ratio interface can be reported in the near future.