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The interaction of a planar shock wave with a polygonal $\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}{\\mathrm{N}}_2$ volume surrounded by ${\\mathrm{SF}}_6$ is investigated experimentally and numerically. Three polygonal interfaces (square, triangle and diamond) are formed by the soap film technique developed in our previous work, in which thin pins are introduced as angular vertexes to connect adjacent sides of polygonal soap films. The evolutions of the shock-accelerated polygonal interfaces are then visualized by a high-speed schlieren system. Wave systems and interface structures can be clearly identified in experimental schlieren images, and agree well with the numerical ones. Quantitatively, the movement of the distorted interface, and the length and height of the interface structures are further compared and good agreements are achieved between experimental and numerical results. It is found that the evolution of these polygonal interfaces is closely related to their initial shapes. In the square interface, two vortices are generated shortly after the shock impact around the left corner and dominate the flow field at late stages. In the triangular and diamond cases, the most remarkable feature is the small ‘ ${\\mathrm{SF}}_6$ jet’ which grows constantly with time and penetrates the downstream boundary of the interface, forming two independent vortices. These distinct morphologies of the three polygonal interfaces also lead to the different behaviours of the interface features including the length and height. It is also found that the velocities of the vortex pair predicted from the theory of Rudinger and Somers (J. Fluid Mech., vol. 7, 1960, pp. 161–176) agree with the experimental ones, especially for the square case. Typical free precursor irregular refraction phenomena and the transitions among them are observed and analysed, which gives direct experimental evidence for wave patterns and their transitions at a slow/fast interface. The velocities of triple points and shocks are experimentally measured. It is found that the transmitted shock near the interface boundary has weakened into an evanescent wave.

The development of a non-periodic $\\text{air}\\text{/}\\text{SF}_{6}$ gaseous interface subjected to a planar shock wave is investigated experimentally and theoretically to evaluate the effects of the non-periodic portions of the interface on the Richtmyer–Meshkov instability. Experimentally, five kinds of discontinuous chevron-shaped interfaces with or without non-periodic portions are created by the extended soap film technique. The post-shock flows and the interface morphologies are captured by schlieren photography combined with a high-speed video camera. A periodic chevron-shaped interface, which is multi-modal (81 % fundamental mode and 19 % high-order modes), is first considered to evaluate the impulsive linear model and several typical nonlinear models. Then, the non-periodic chevron-shaped interfaces are investigated and the results show that the existence of non-periodic portions significantly changes the balanced position of the initial interface, and subsequently disables the nonlinear model which is applicable to the periodic chevron-shaped interface. A modified nonlinear model is proposed to consider the effects of the non-periodic portions. It turns out that the new model can predict the growth of the shocked non-periodic interface well. Finally, a method is established using spectrum analysis on the initial shape of the interface to separate its bubble structure and spike structure such that the new model can apply to any random perturbed interface. These findings can facilitate the understanding of the evolution of non-periodic interfaces which are more common in reality.

We report the first experiments on divergent shock-driven Richtmyer–Meshkov instability (RMI) at well-controlled single-mode interfaces. These experiments are performed in a novel divergent shock tube designed by shock dynamics theory. Generally, the perturbation growth can be divided into three successive stages: linear growth, quick reduction in growth rate and instability freeze-out. It is observed that the growth rate at each stage is far lower than its counterpart in planar or convergent geometry due to geometric divergence. We also found that nonlinearity is much weaker than that in planar or convergent RMI, and has a negligible influence on the overall amplitude growth even at late stages when it has become strong. This weak nonlinear effect is because the growth of the third harmonic counteracts its feedback to the fundamental mode. As a consequence, the linear theory of Bell (report no. LA-1321) accounting for geometric divergence and Rayleigh–Taylor (RT) stabilization caused by flow deceleration can reasonably predict the present results from early to late stages. The instability freeze-out at late times is ascribed to the negative growth induced by geometric divergence and RT stabilization, and is also well reproduced by the linear theory.

Interaction of a planar shock wave with a discontinuous SF6 elliptic gas cylinder surrounded by air is investigated. Special attention is given to the effects of aspect ratio on wave pattern, interface evolution, and material mixing. An ideal discontinuous two-dimensional gas cylinder is created by the soap film technique in experiments, and the shocked flow is captured by schlieren photography combined with a high-speed video camera. The surface of the gas cylinder is clear enough to observe the shock motions, and the distinct interface boundaries allow us to extract more details. As aspect ratio varies, the shock focusing process is quite different. For the prolate gas cylinder, an inward jet is produced although an internal shock focusing firstly occurs. The inward jet has never been observed in membraneless prolate ellipse experiments probably because the inward jet is so faint due to less vorticity generation on membraneless interface that it is difficult to be observed. For the oblate gas cylinder, a secondary vortex pair, which has not been described clearly in previous work, is derived from the downstream interface. The material lines at early stages are extracted from experiments, which grow faster as aspect ratio increases. The interfacial area, the mean volume fraction and the mixing rate are presented from computations, and the results show that the increase of aspect ratio promotes the mixing between gases.

AbstractDevelopments of two-dimensional single-mode light/heavy interfaces driven by convergent shock waves are numerically investigated, focusing on the effect of the Atwood number on the Rayleigh–Taylor stabilization, the compressibility and the nonlinearity. Five different test gases, including CO2 , Kr, R22, R12 and SF6 , are considered with air as the ambient gas. It is clarified for the first time that the unperturbed interface begins to decelerate when the shock focuses at the convergence center, and the acceleration during the deceleration phase is proportional to the Atwood number. During the first reshock, the interface moves outwards with a deceleration until it starts moving inwards. When the initial interface is weakly disturbed, a more obvious amplitude reduction is observed for the case with a larger Atwood number before the reshock, which means that the Rayleigh–Taylor stabilization is stronger. To assess the effect of the Atwood number on the compressibility and the nonlinearity, three models, including a linear incompressible model, a nonlinear incompressible model and a linear compressible model, are adopted to predict the amplitude growth before the reshock. The results show that the nonlinearity is weak, and is almost not influenced by the Atwood number before the reshock. The compressibility, however, greatly changes the amplitude growth. As the Atwood number increases, the compressibility plays a less significant role in the amplitude growth because a heavier gas is harder to be compressed. Although a gas with a larger specific heat ratio is also difficult to be compressed, the specific heat ratio plays a minor role to the compressibility relative to the Atwood number. During the reshock, the amplitude grows linearly until the nonlinearity in the cases with large Atwood numbers is strong enough to reduce the amplitude growth rate.Graphic abstract

The converging Richtmyer–Meshkov (RM) instability on single- and dual-mode N2 / SF6 interfaces is studied by an upwind conservation element and solution element solver. An unperturbed case is first considered, and it is found that the shocked interface undergoes a long-term deceleration after a period of uniform motion. The evolution of single-mode interface at the early stage exhibits an evident nonlinearity, which can be reasonably predicted by the nonlinear model of Wang et al. (Phys Plasmas 22: 082702, 2015). During the deceleration stage, the perturbation amplitude drops quickly and even becomes a negative (phase inversion) before the reshock due to the Rayleigh–Taylor (RT) stabilization. After the reshock, the interface experiences a phase inversion again or does not, depending on the reshock time. The growth of the second-order harmonic in the deceleration stage clearly reveals the competition between the RT effect and the nonlinearity. For dual-mode interfaces, the growth of the first mode (wavenumber k1 ) relies heavily on the second mode (wavenumber k2 ) due to the mode coupling effect. Specifically, for cases where k2 is an even or odd multiple of k1 , the growth of the first mode is inhibited or promoted depending on its initial amplitude sign and the phase difference between two basic waves, while for cases where k2 is a non-integer multiple of k1 , the second mode has negligible influence on the first mode. Through a systematic study, signs of perturbation amplitudes of the generated k2-k1 and k2+k1 waves are obtained for all possible dual-mode configurations, which are reasonably predicted by a modified Haan model (Phys Fluids B 3: 2349–2355, 1991).

The regular reflection to Mach reflection ($\\text{RR}\\rightarrow \\text{MR}$) transition in inviscid perfect air for shock reflection over convex and straight wedges is investigated. Provided that the variation of shock intensity only has a second-order effect on the wave transition, the possible cases for the occurrence of the$\\text{RR}\\rightarrow \\text{MR}$transition for curved shock reflection over a wedge are discussed. For a planar shock reflecting over a convex wedge, four different flow regions are classified and the mechanism of the disturbance propagation is interpreted. It is found that the flow-induced rarefaction waves exist between disturbances generated from neighbouring positions and isolate them. For a curved shock reflecting over a convex wedge, although the distributions of the flow regions are different from those in planar shock reflection, the analysis for the planar shock case can be extended to curved shock cases as long as the wedge is convex. When a diverging shock reflects off a straight wedge, the flow-induced rarefaction waves are absent. However, the disturbances generated earlier cannot overtake the reflection point before the pseudo-steady criterion is satisfied. In the cases considered, the flow in the vicinity of the reflection point will not be influenced by the unsteady flow caused by the shock intensity and the wedge angle variations. This is clearly a local property of the shock–wall interaction, no matter what the history of the shock trajectory is. For validation, extensive inviscid numerical simulations are performed, and the numerical results show the reliability of the pseudo-steady criterion for predicting the$\\text{RR}\\rightarrow \\text{MR}$transition on a convex wedge.

We present computational results on the evolution of the shock-accelerated heavy bubbles surrounded by nitrogen with the Atwood number At = 0.497-0.677 and the emphasis is on the jet phenomenon caused by the shock focusing. The multi-fluid Eulerian equation is solved by a finite volume method based on MUSCL-Hancock approach. Based on the numerical schlieren and the distributions of density and pressure, it is found that there are three typical jet structures （outward jet, no jet, inward jet） for different combinations of gas mixture inside the bubble which determine the position of shock focusing relative to the downstream pole of the heavy bubble （upstream of the pole, at the pole, downstream the pole）. Compared with the inward jet, the velocity of outward jet is obviously larger. As At increases, the moment of jet formation is postponed, and the maximal values and magnifications of pressure and density increase distinctly. Therefore, the energy convergence effects are heavily enhanced with the increase of bubble gas density.

The two-dimensional (2-D) unsteady shock reflection over a single wedge is studied theoretically and numerically, and the emphasis is placed on the trajectory of the triple point (TP). Skews’ relation and the three-shock theory are, respectively, used for determining the trajectory angles of the corner-generated disturbance and the TP, and, subsequently, a model capable of predicting the TP trajectory is established for 2-D unsteady shock reflections over a single wedge. Then, a systematically numerical study is carried out on the 2-D unsteady shock reflection over a single wedge, including five types of shock reflection with the wedge angle increased and five types of shock reflection with the wedge angle decreased. It is found that the Mach stem is always slightly concavely (convexly) curved for reflections with the wedge angle increased (decreased), which should be caused by corner-generated compression (rarefaction) waves propagating along the Mach stem. The new model reasonably predicts the TP trajectory in all the 2-D unsteady shock reflections, and its performance is related not only to the variation trend of the wedge angle (increase or decrease), but also to the type of shock and the initial wedge angle. Specifically, for the shock reflection with the wedge angle increased (decreased), the model generally provides a slightly better (worse) prediction if the TP trajectory angle calculated by the three-shock theory is compulsively modified. The shock-shock/expansion polar analysis presented can partially explain the model performance for predicting the TP trajectory.

Richtmyer–Meshkov instability (RMI) at a light–heavy single-mode interface over a wide range of post-shock Atwood numbers $A_1$ is studied systematically through elaborate experiments. The interface generation and $A_1$ variation are achieved by the soap-film technology and gas-layer scheme, respectively. Qualitatively, the nonlinear interface evolution features, including spike, bubble and roll-up structures, are more significant in RMI with higher $A_1$. Quantitatively, both the impulsive model and an analytical linear model perform well in predicting the linear growth rate under a wide range of $A_1$ conditions. For the weakly nonlinear stage, the significant spike acceleration occurring when $A_1$ is high, which is observed experimentally for the first time, results in the evolution law of RMI with high $A_1$ being different from the counterpart with low or intermediate $A_1$. None of the considered nonlinear models is found to be applicable for RMI under all $A_1$ conditions, and the predictive capabilities of these models are analysed and summarized. Based on the present experimental results, an empirical nonlinear model is proposed for RMI over a wide range of $A_1$. Further, modal analysis shows that in RMI with high (low or intermediate) $A_1$, high-order harmonics evolve rapidly (slowly) and cannot (can) be ignored. Accordingly, for RMI with high (low or intermediate) $A_1$, the modal model proposed by Zhang & Sohn (Phys. Fluids, vol. 9, 1997, pp. 1106–1124) is less (more) accurate than the one proposed by Vandenboomgaerde et al. (Phys. Fluids, vol. 14, 2002, pp. 1111–1122), since the former ignores perturbation solutions higher than fourth order (the latter retains only terms with the highest power in time).