Richtmyer–Meshkov (RM) instability at a single-mode interface impacted by a cylindrical divergent shock with low to moderate Mach numbers is investigated experimentally. The motion of an unperturbed interface is first examined to obtain the background flow. The shocked interface moves uniformly at the early stage, but later decelerates. The stronger the incident shock, the larger the interface deceleration, which is reasonably predicted by a one-dimensional model considering the effect of postshock non-uniformity. Such a deceleration greatly inhibits the growths of harmonics of an initially perturbed interface and, consequently, the divergent RM instability presents very weak nonlinearity from early to late stages. Particularly, higher-Mach-number cases present weaker nonlinearity due to larger deceleration there. This abnormal linear growth regime is reported for the first time. Benefiting from this, the incompressible linear model holds validity at all stages of divergent RM instability. It is also found that compressibility inhibits the initial growth rate, but produces a weak influence on the subsequent instability growth.

Shock-tube experiments and theoretical studies have been performed to highlight mode-coupling in an air–SF$_6$–air fluid layer. Initially, the two interfaces of the layer are designed as single mode with different basic modes. It is found that as the two perturbed interfaces become closer, interface coupling induces a different mode from the basic mode on each interface. Then mode coupling further generates new modes. Based on the linear model (Jacobs et al., J. Fluid Mech., vol. 295, 1995, pp. 23–42), a modified model is established by considering the different accelerations of two interfaces and the waves’ effects in the layer, and provides good predictions to the linear growth rates of the basic modes and the modes generated by interface coupling. It is observed that interface coupling behaves differently to the nonlinear growth of the basic modes, which can be characterized generally by the existing or modified nonlinear model. Moreover, a new modal model is established to quantify the mode-coupling effect in the layer. The mode-coupling effect on the amplitude growth is negligible for the basic modes, but is significant for the interface-coupling modes when the initial wavenumber of one interface is twice the wavenumber of the other interface. Finally, amplitude freeze-out of the second single-mode interface is achieved theoretically and experimentally through interface coupling. These findings may be helpful for designing the target in inertial confinement fusion to suppress the hydrodynamic instabilities.

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier–Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier–Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

We report the first experiments on hydrodynamic instabilities of a single-mode light/heavy interface driven by co-directional rarefaction and shock waves. The experiments are conducted in a specially designed rarefaction-shock tube that enables the decoupling of interfacial instabilities caused by these co-directional waves. After the impacts of rarefaction and shock waves, the interface evolution transitions into Richtmyer–Meshkov unstable states from Rayleigh–Taylor (RT) stable states, which is different from the finding in the previous case with counter-directional rarefaction and shock waves. A scaling method is proposed, which effectively collapses the RT stable perturbation growths. An analytical theory for predicting the time-dependent acceleration and density induced by rarefaction waves is established. Based on the analytical theory, the model proposed by Mikaelian (Phys. Fluids, vol. 21, 2009, p. 024103) is revised to provide a good description of the dimensionless RT stable behaviour. Before the shock arrival, the unequal interface velocities, caused by rarefaction-induced uneven vorticity, result in a V-shape-like interface. The linear growth rate of the amplitude is insensitive to the pre-shock interface shape, and can be well predicted by the linear superposition of growth rates induced by rarefaction and shock waves. The nonlinear growth rate is higher than that of a pure single-mode case, which can be predicted by the nonlinear models (Sadot et al., Phys. Rev. Lett., vol. 80, 1998, pp. 1654–1657; Dimonte & Ramaprabhu, Phys. Fluids, vol. 22, 2010, p. 014104).

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).

Shock-tube experiments on Richtmyer–Meshkov (RM) instability at a perturbed SF$_6$ layer surrounded by air, induced by a cylindrical divergent shock, are reported. To explore the effects of reverberating waves and interface coupling on instability growth, gas layers with various shapes are created: unperturbed inner interface and sinusoidal outer interface (case US); sinusoidal inner and outer interfaces that have identical phase (case IP); sinusoidal inner and outer interfaces that have opposite phase (case AP). For each case, three thicknesses are considered. Results show that reverberating waves inside the layer dominate the early-stage instability growth, while interface coupling dominates the late-stage growth. The influences of waves on divergent RM instability are more pronounced than the planar and convergent counterparts, which are estimated accurately based on gas dynamics theory. Both the wave influence and interface coupling depend heavily on the layer shape, leading to diverse growth rates: the quickest growth for case AP, medium growth for case US, the slowest growth for case IP. In particular, for the IP case, there exists a critical thickness below which the instability growth is suppressed by both the reverberating waves and interface coupling. This provides an efficient way to modulate the growth of divergent RM instability. It is found that divergent RM instability involves weak nonlinearity and strong interface coupling such that the linear theory of Mikaelian (Phys. Fluids, vol. 17, 2005, 094105) can well reproduce the instability growth at late stages for all cases. This constitutes the first experimental confirmation of the Mikaelian theory.

The scaling law of the structure function of Richtmyer–Meshkov (RM) turbulence is investigated both numerically and theoretically. High-fidelity simulations with a minimum-dispersion, adaptive-dissipation scheme are first performed. Results show that the mixing width experiences an exponential growth and the turbulent kinetic energy has a visible $-3/2$ spectrum. The scalar field exhibits a greater degree of intermittency than the velocity field, and also the small-scale statistics suffer a larger influence of large scales. Visible differences in the scaling law of the structure function among the RM turbulence and other types of turbulence are observed, which reveal the unique characteristic of RM turbulence. A phenomenological theory, which gives the spatial and temporal scaling laws of the structure functions of velocity and scalar of RM turbulence, is developed for the first time by introducing an external agent. The spatial scaling exponents of structure functions from simulation deviate from the Kolmogorov exponents, but are quite close to the RM-modified anomalous exponents. This demonstrates the validity of the present phenomenological theory. The temporal scaling exponents of structure functions first meet the RM-modified anomalous exponents, and then approach the Kolmogorov–Obukhov–Corrsin non-intermittent ones.

Attenuation and even freeze-out (amplitude growth stagnation) of the perturbation amplitude growth of a shocked SF$_6$–air interface are first realized in shock-tube experiments through reflected rarefaction waves, which produce reverse baroclinic vorticity offsetting the vorticity deposited by the shock. A theoretical model is constructed to predict the perturbation growth after the impact of rarefaction waves, and seven possibilities of amplitude growth are analysed. Experimentally, a planar air–helium interface is used to produce reflected rarefaction waves. Through changing the perturbation wavelength and the time interval of two impacts, five experiments with specific initial conditions are carried out, and three different possibilities of perturbation growth attenuation are realized.

The reshocked Richtmyer–Meshkov instability (RMI) is examined in three different configurations via shock-tube experiments: RMI at a single-mode interface with a planar reshock (configuration I); RMI at a flat interface with a sinusoidal reshock (configuration II); RMI at a single-mode interface with a sinusoidal reshock (configuration III). The sinusoidal reshock is created by an incident shock reflecting off a sine-shaped wall surface. For all three configurations, the initial conditions of the experiment are specially set such that the interface evolution is at the linear stage when the reshock arrives. It is found that the amplitude of the reshocked interface increases linearly with time for all three configurations. For configuration I, the post-reshock perturbation growth depends heavily on the pre-reshock amplitude and growth rate, which can be predicted by a modified Mikaelian model (Phys. Rev. A, vol. 31, 1985, pp. 410–419). For configuration II, velocity perturbation associated with the non-uniform rippled reshock plays an important role in the instability growth. For configuration III, the post-reshock instability growth is much quicker (lower) than in configuration I when the sinusoidal reshock is in phase (out of phase) with the interface. A major reason is that for the in-phase (anti-phase) case, the velocity perturbation gives rise to an instability growth with an identical (opposite) direction to the pressure perturbation. A linear theory is developed that takes velocity perturbation, pressure perturbation and pre-reshock growth rate into account, which gives a reasonable prediction of the growth of the reshocked RMI in configurations II and III.

We report the first shock-tube experiments on two-dimensional dual-mode air–SF$_6$ interfaces with different initial spectra subjected to a convergent shock wave. The convergent shock tube is specially designed with a tail opening to highlight the Bell–Plesset (BP) and mode-coupling effects on amplitude development of fundamental mode (FM). The results show that the BP effect promotes the occurrence of mode coupling, and the feedback of high-order modes to the FM also arises earlier in convergent geometry than that in its planar counterpart. Relatively, the amplitude growth of the FM with a higher mode number is inhibited by the feedback, and saturates earlier. The FM with a lower mode number is affected more heavily by the BP effect, and finally dominates the flow. A new model is proposed to well predict the amplitude growths of the FM and high-order modes in convergent geometry. In particular, for FM that reaches its saturation amplitude, the post-saturation relation is introduced in the model to achieve a better prediction.

Shock-tube experiments are performed on the convergent Richtmyer–Meshkov (RM) instability of a multimode interface. The temporal growth of each Fourier mode perturbation is measured. The hydrodynamic instabilities, including the RM instability and the additional Rayleigh–Taylor (RT) effect, imposed by the convergent shock wave on the dual-mode interface, are investigated. The mode-coupling effect on the convergent RM instability coupled with the RT effect is quantified. It is evident that the amplitude growths of all first-order modes and second-order harmonics and their couplings depend on the variance of the interface radius, and are influenced by the mode-coupling from the very beginning. It is confirmed that the mode-coupling mechanism is closely related to the initial spectrum, including azimuthal wavenumbers, relative phases and initial amplitudes of the constituent modes. Different from the conclusion in previous studies on the convergent single-mode RM instability that the additional RT effect always suppresses the perturbation growth, the mode-coupling might result in the additional RT effect promoting the instability of the constituent Fourier mode. By considering the geometry convergence, the mode-coupling effect and other physical mechanisms, second-order nonlinear solutions are established to predict the RM instability and the additional RT effect in the cylindrical geometry, reasonably quantifying the amplitude growths of each mode, harmonic and coupling. The nonlinear solutions are further validated by simulations considering various initial spectra. Last, the temporal evolutions of the mixed mass and normalized mixed mass of a shocked multimode interface are calculated numerically to quantify the mixing of two fluids in the cylindrical geometry.

Experiments on divergent Richtmyer–Meshkov (RM) instability at a heavy gas layer are performed in a specially designed shock tube. A novel soap-film technique is extended to generate gas layers with controllable thicknesses and shapes. An unperturbed gas layer is first examined and its two interfaces are found to move uniformly at the early stage and be decelerated later. A general one-dimensional theory applicable to an arbitrary-thickness layer is established, which gives a good prediction of the layer motion in divergent geometry. Then, six kinds of perturbed SF$_6$ layers with various thicknesses and shapes surrounded by air are examined. At the early stage, the amplitude growths of the inner interface for various-thickness layers collapse quite well and also can be predicted by the Bell model for cylindrical RM instability at a single interface, which indicates a negligible interface coupling effect. Later, a rarefaction wave accelerates the inner interface, causing a dramatic rise in the growth rate. It is found that a thicker gas layer will result in a larger extent that the rarefaction wave can promote the instability growth. A modified Bell model accounting for both Rayleigh–Taylor (RT) instability and interface stretching caused by a rarefaction wave is established, which well reproduces the quick instability growth. At late stages, reverberating waves inside the layer are negligibly weak such that the inner interface growth is dominated by RM instability and RT stability. The major factors driving the outer interface development are a compression wave and interface coupling. A new interface coupling phenomenon existing uniquely in divergent geometry caused by the gradual thinning of the gas layer is observed and also modelled.

‘Freeze-out’ of amplitude growth, i.e. the amplitude growth stagnation of a shocked helium–air interface, is realized through a reflected shock, which produces baroclinic vorticity of the opposite sign to that deposited by the first shock. Theoretically, a model is constructed to calculate the relations among the initial parameters for achieving freeze-out. In particular, if the amplitude growth is within the linear regime at the arrival of the reflected shock, the time interval between the impacts of two shock waves is linearly related to the initial perturbation wavelength, and is independent of the initial perturbation amplitude. Experimentally, an air–SF$_6$ (or air–argon) plane interface is adopted to produce a weak reflected shock. Seven experimental runs with specific initial conditions are examined. For all cases, freeze-out is achieved after the reflected shock impact under the designed conditions.

Shock-induced instability developments of two successive interfaces have attracted much attention, but remain a difficult problem to solve. The feedthrough and reverberating waves between two successive interfaces significantly influence the hydrodynamic instabilities of the two interfaces. The evolutions of two successive slow/fast interfaces driven by a weak shock wave are examined experimentally and numerically. First, a general one-dimensional theory is established to describe the movements of the two interfaces by studying the rarefaction waves reflected between the two interfaces. Second, an analytical, linear model is established by considering the arbitrary wavenumber and phase combinations and compressibility to quantify the feedthrough effect on the Richtmyer–Meshkov instability (RMI) of two successive slow/fast interfaces. The feedthrough significantly influences the RMI of the two interfaces, and even leads to abnormal RMI (i.e. phase reversal of a shocked slow/fast interface is inhibited) which is the first observational evidence of the abnormal RMI provided by the present study. Moreover, the stretching effect and short-time Rayleigh–Taylor instability or Rayleigh–Taylor stabilisation imposed by the rarefaction waves on the two interfaces are quantified considering the two interfaces’ phase reversal. The conditions and outcomes of the freeze-out and abnormal RMI caused by the feedthrough are summarised based on the theoretical model and numerical simulation. A specific requirement for the simultaneously freeze-out of the instability of the two interfaces is proposed, which can potentially be used in the applications to suppress the hydrodynamic instabilities.

Richtmyer–Meshkov instability induced by two successive shock waves is experimentally studied in a specific shock tube. To create two successive shock waves synchronously, a driver section is added between the driver and driven sections of the standard shock tube, and an electronically controlled membrane rupture equipment is adopted. The shock-tube flow after the membranes rupture is well described by combining the shock relations, isentropic wave relations with compatibility relations across the contact surface (region). The new shock tube is capable of generating two successive shock waves with controllable strengths and time interval, and provides a relatively ‘clean’ wave system. Then the developments of single-mode light–heavy interfaces with different initial conditions induced by two successive shock waves are investigated. The initial amplitudes are all small enough such that the first-shocked interface is within the linear growth regime at the arrival of the second shock. The results show that if the pre-second-shock perturbation amplitude is small, the linear, nonlinear and modal evolutions of the double-shocked interface can be reasonably predicted by the existing models proposed for predicting the perturbation growth induced by a single shock. For the double-shocked interface, the second shock provides an additional perturbation velocity field to the original one introduced by the first shock impact. The validity of the linear superposition model indicates that the linear superposition of these two perturbation velocity fields is satisfied. Therefore, a double-shocked interface evolves similarly to a single-shocked interface provided that their postshock amplitudes and linear growth rates are the same.

Shock-tube experiments on various two-bubble and two-spike interfaces are performed to examine the dependence of bubble competition and spike competition on the initial spectra and density ratio of the interface. The differences in the influences of bubble competition and spike competition on the Richtmyer–Meshkov instability are highlighted for the first time. The bubble-competition effect is mainly dependent on the initial spectra of the two-bubble configuration. In contrast, the spike-competition effect is determined by both the initial spectra and density ratio. The extended buoyancy–drag model is adopted to explain the variation of the drag force imposed on the long-wavelength and short-wavelength structures as the initial conditions change. Based on the spectrum analysis, it is found that the constituent modes of two-bubble and two-spike interfaces have different contributions to the long-wavelength and short-wavelength perturbation growths. A generalised, nonlinear, analytical model is then established to quantify the bubble-competition effect and spike-competition effect considering arbitrary initial spectra and density ratio. The bubble-competition effect is believed to be stronger than the spike-competition effect at a high density ratio because it suppresses the high-frequency perturbation growth more evidently.

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.

The Richtmyer–Meshkov instability of heavy/light single-mode (SM), trapezoid (TR) and sawtooth (ST) interfaces is studied experimentally by considering the reshock. The TR and ST interfaces can be expanded into Fourier series with a dominant fundamental mode and more high-order modes, recognized as quasi-single-mode ones. In experiments, the distorted interfaces at the time of first reshock arrival develop in the weakly nonlinear stage, ensuring an approximate single-scale function of evolving interface. The results show an evident memory of initial interface shapes: the bubbles and spikes of ST interface after reshock mainly develop in the streamwise direction with sharp heads, while the counterparts of TR interface tend to grow in the spanwise direction. The influences of high-order modes are amplified by the reshock, resulting in significant interface shape dependence of mixing width growths. The amplitude superposition of major odd-order modes promotes the growth rates of mixing widths for the SM and ST cases, different from the TR one. The ST interface has larger mixing width growth rates in comparison with the SM interface, since high-order modes play a great role in promoting the increase of ST amplitudes, while the TR interface has a relatively small one. The linear and nonlinear mixing width growths of SM, TR and ST interfaces before and after reshock are further analysed theoretically, indicating that the fundamental mode still has a predominant influence on the interface evolution after reshock, and the growth behaviours exhibit strong similarities to those for the singly shocked cases.

The shock-induced evolution of a gas layer with two fast/slow interfaces is investigated theoretically and experimentally. Specifically, the gas layer is located between a lighter gas and a heavier gas, forming a light/medium/heavy (LMH) configuration. Linear stability analysis is utilised to derive a new analytical model to quantify the Richtmyer–Meshkov instability (RMI) on the two interfaces of such an A/B/C-type fluid layer. Three quasi-one-dimensional (1-D) LMH fluid layers with different initial layer thicknesses are generated to study the wave patterns and interface motions. A general 1-D theory is established to describe the motions of the shock/rarefaction waves reverberating inside the fluid layer and the displacements of the two interfaces. Six quasi-2-D LMH fluid layers with diverse initial layer-thickness and amplitude combinations are created to explore the hydrodynamic instabilities of the two interfaces. It is found that the interface coupling significantly influences the interface evolution, even resulting in an abnormal phase reversal of a shocked fast/slow interface if the two interfaces are anti-phase and the initial layer is very thin. The specific conditions for the abnormal phase reversal and the instability freeze out are deduced. Moreover, the additional RMI (or Rayleigh–Taylor stabilisation) imposed by the shock (or rarefaction waves) reverberating inside an LMH fluid layer on the first (or second) interface is quantified. It is proved that the reverberating waves inside an LMH fluid layer stabilise the two interfaces. Finally, a nonlinear model is obtained by incorporating the nonlinearity effect into the linear model, which well describes the perturbation growths of the two interfaces in a later regime.

Shock-induced light-fluid-layer evolution is firstly investigated experimentally and theoretically. Specifically, three quasi-one-dimensional helium gas layers with different layer thicknesses are generated to study the wave patterns and interface motions. Six quasi-two-dimensional helium gas layers with diverse layer thicknesses and amplitude combinations are created to explore the Richtmyer–Meshkov instability of a light-fluid layer. Due to the multiple reflected shocks reverberating inside a light-fluid layer, the speeds of the two interfaces gradually converge, and the layer thickness saturates eventually. A general one-dimensional theory is adopted to describe the two interfaces’ motions and the layer thickness variations. It is found that, for the first interface, the end time of its phase reversal determines the influence of the reflected shocks on it. However, the reverberated shocks indeed lead to the second interface being more unstable. When the two interfaces are initially in phase, and the initial fluid layer is very thin, the two interfaces’ spike heads collide and stabilise the two interfaces. Linear and nonlinear models are successfully adopted by considering the interface-coupling effect and the reverberated shocks to predict the two interfaces’ perturbation growths in all regimes. The interfacial instability of a light-fluid layer is quantitatively compared with that of a heavy-fluid layer. It is concluded that the kind of waves reverberating inside a fluid layer significantly affects the fluid-layer evolution.