This work is a numerical study of a transitional shock wave boundary layer interaction (SWBLI). The main goal is to improve our understanding of the well-known low-frequency SWBLI unsteadiness and especially the suspected role of triadic interactions in the underlying physical mechanism. To this end, a direct numerical simulation is performed using a high-order finite-volume scheme equipped with a suitable shock capturing procedure. The resulting database is then extensively post-processed in order to extract the main dynamical features of the interaction zone dynamics (involved characteristic frequencies, characteristics of the vortical structures, etc.). The dynamical organisation and space–time evolution of the flow at dominant frequencies are then further characterised by mean of an spectral proper orthogonal decomposition analysis. In order to study the role of triadic interactions occurring in the interaction region, a bispectral mode decomposition analysis is applied to the database. It allows us to extract the significant triadic interactions, their location and the resulting physical spatial modes. Strong triadic interactions are detected in the downstream part of the separation bubble whose role on the low-frequency unsteadiness is characterised. All the results of the various analyses are then discussed and integrated to formulate a possible mechanism fuelling low-frequency SWBLI unsteadiness.

Attenuation of shock waves through dense granular media with varying macro-scale and micro-scale parameters has been numerically studied in this work by a coupled Eulerian–Lagrangian approach. The results elucidate the correlation between the attenuation mechanism and the nature of shock-induced unsteady flows inside the granular media. As the shock transmission becomes trivial relative to the establishment of unsteady interpore flows, giving way to the gas filtration, the shock attenuation mechanism transitions from the shock dynamics and deduction of propagation area associated with the shock transmission, to the drag-related friction dissipation alongside the gas filtration. The ratio between the maximum shock transmission length and the thickness of the particle layer is found to be a proper indicator of the nature of shock-induced flows. More importantly, it is this ratio that successfully collapses the upstream and downstream pressures of shock impacted particle layers with widely ranging thickness and volume fraction, leading to a universal scaling law for the shock attenuation effect. We further propose a correlation between the structure of particle layer and the corresponding maximum shock transmission length, guaranteeing adequate theoretical predictions of the upstream and downstream pressures. These predictions are also necessary for an accurate estimation of the spread rate of shock dispersed particle bed through a pressure-gradient-based scaling method.

The reflection of multiple incident shock waves that converge to a single point on the reflecting surface is studied in this paper. The number of the incident shocks, denoted $K$, is arbitrary. The interaction between the reflected shock of one incident shock and the other incident shocks may produce various possible configurations, such as type-I, type-II and type-IV shock interferences. The number of possible reflection configurations is shown to be an exponential function of ($K-1$) with base 2. The possibility of pre-, middle- and post-Mach reflections, which means Mach reflection occurs for the first, middle and last incident shock, is revealed through numerical simulation for $K=3$. For the particular case where the incident shocks are produced by equal variation of wedge surface deflection, the conventional von Neumann condition and detachment condition for the $k\mathrm{th}$ incident shock to have Mach reflection are derived. It is shown that the von Neumann condition for regular reflection is lowered and the detachment condition for Mach reflection is elevated as $k$ increases. The shock reflection patterns for $ K=1,2,\ldots ,10$ are obtained by numerical simulations. We observe a shock interaction train structure, where we have pre-Mach reflection followed by ($K-1$) type-I or type-II shock interferences. We also observe that the Mach stem height decreases with $K$ well above the von Neumann condition and becomes non-monotonic near the von Neumann condition.

Reflection of a rightward-moving shock over a steady oblique shock, equivalent to a shock-on-shock interaction, is typically studied with post-formed shock waves. Law, Felthun and Skews (Shock Waves, vol. 13, 2003, pp. 381–394) reported post-formed expansion fan (PFEF) reflection for second-family incident shock. Here, we show that PFEF reflection also exists for first-family incident shock. We derive the critical condition for PFEF reflection in the shock speed Mach number and incident shock angle plane. Our findings indicate PFEF emergence near type post-I region. Numerical simulations reveal that PFEF with rising incident angle can intersect the incident shock, triple point or Mach stem, echoing the Hillier (J. Fluid Mech., vol. 575, 2007, pp. 399–424) three-type classification of shock–expansion fan interactions. The complex shock reflection pattern is essentially composed of an upstream structure linked to the moving shock wave, and a downstream structure linked to the unperturbed oblique shock wave. Under the conditions investigated, the upstream structure is characterized by a Mach reflection of the incident shock over the wall, potentially featuring a triple point formed within the Mach stem. Below this triple point, there is a curved segment of the Mach stem that is close to the wall. As the inclined angle increases, the curved shock may expand and interact with the incident shock, leading to an asymmetric regular reflection, which is a phenomenon that has not been observed previously. The downstream structure is a double $\lambda$ shock structure, with the lower shock resulting from the generation of recompression shock waves.

This work reports high-fidelity shock-tube experiments on the convergent Richtmyer–Meshkov (RM) instability at a heavy gas layer. The convergent shock tube is designed based on shock dynamics theory, significantly mitigating interface deceleration and reflected shock. As a result, long-term observation of instability growth up to nonlinear stage, free of interface deceleration and reshock, is achieved. Various types of SF$_6$ layers surrounded by air with controllable thicknesses and shapes, created using a soap film technique, are examined. For thick layers, the evolutions of the outer and inner interfaces are nearly decoupled regardless of the layer shape. The weakly nonlinear model of Wang (Phys. Plasmas,vol. 22, 2015, p. 082702), designed for cylindrical RM instability at a single interface, provides a reasonable prediction of perturbation growth at the inner interface, while slightly underestimating instability growth at the outer interface, as it neglects the effects of rarefaction wave. For thin layers, perturbation growth is fastest at either interface when both interfaces initially possess in-phase perturbations, moderate when only one interface is initially perturbed and slowest when the two interfaces have anti-phase perturbations. This variation in growth rates is due to the fact that the evolution of a thin layer is influenced by both reverberating waves and interface coupling, with each factor being highly sensitive to the layer shape. The original vortex method is extended to address the convergent RM instability by incorporating the influences of unsteady background flow, interface coupling and reverberating waves into the transport of a vortex sheet. This extended vortex method enables accurate prediction of convergent RM instability at a gas layer, covering the full range from early linear to late nonlinear stages.

The dependence of the Richtmyer–Meshkov instability (RMI) on post-shock Atwood number ($A_1$) is experimentally investigated for a heavy–light single-mode interface. We create initial interfaces with density ratios of heavy to light gases ranging from 1.73 to 34.07, and achieve the highest $|A_1|$ value reported to date for gaseous-interface experiments (0.95). For the first time, spike acceleration is observed in experiments with a heavy–light configuration. The models for the start-up, linear and weakly nonlinear evolution stages are evaluated over a wide range of $A_1$ conditions. Specifically, the models proposed by Li et al. (Phys. Fluids, vol. 36, 2024, 056104) and Wouchuk & Nishihara (Phys. Plasmas, vol. 4, 1997, 1028–1038) effectively describe the start-up and linear stages, respectively, across all cases. None of the considered nonlinear models is valid under all $A_1$ conditions. Based on the dependence of spike and bubble evolutions on $A_1$ provided by the present work and previous study (Chen et al., J. Fluid Mech., vol. 975, 2023, A29), the SEA model (Sadot et al., Phys. Rev. Lett., vol. 80, 1998, pp. 1654–1657), whose expression has clear physical meanings, is modified by revising the coefficient that governs its prediction for early-time evolution. The modified model applies to prediction of the weakly nonlinear evolution of RMI with $A_1$ ranging from −0.95 to −0.35 and from 0.30 to 0.86. Based on this model, an approximation of the critical $A_1$ for the occurrence of spike acceleration is obtained.

The aerodynamic deformation and breakup of wall-attached droplets in axisymmetric stagnation flow are investigated experimentally. A vertical shock tube is used to generate the shock wave accompanying the post-wave airflow, and the axisymmetric stagnation flow is formed through the impingement of an air stream on a solid wall. For the wall-attached droplets with initially hemispherical profile, four typical droplet deformation and breakup modes can be identified with the continuous increase of the droplet local Weber number, which are the vibrating mode, the compressing mode, the sheet thinning mode and the shear-induced entrainment mode. Quantitative analyses of droplet evolution dynamics are also conducted for the compressing mode and the sheet thinning mode, and the significant differences of air flow separation at the droplet lateral surface between these two modes are revealed. The potential flow model and the energy conservation model are further developed to predict the entire droplet deformation processes. The vibrating frequency and amplitude of droplets under the vibrating mode are predicted by a spring-mass model, and the surface perturbation wavelengths of droplets under the shear-induced entrainment mode are estimated based on the dispersion relation of Kelvin–Helmholtz instability. This work is proposed to give potential guidance for regulating the aerodynamic fragmentation of wall-attached droplets in practical engineering applications.

Wave propagation in channels with area changes is a topic of significant practical interest that involves a rich set of coupled physics. While the acoustic wave problem has been studied extensively, the shock propagation problem has received less attention. In addition to its practical significance, this problem also introduces deep fundamental issues associated with how energy in propagating large-amplitude disturbances is redistributed upon interaction with inhomogeneities. This paper presents a study of shock scattering and entropy and vorticity coupling for shock wave propagation through discrete area changes. It compares results from computational fluid dynamics to those of one-dimensional quasi-steady calculations. The solution space is naturally divided into five ‘regimes’ based upon the incident shock strength and area ratio. This paper also presents perturbation methods to quantify the dimensionless scaling of physical effects associated with wave reflection/transmission and energy transfer to other disturbances. Finally, it presents an analysis of the ‘energetics’ of the interaction, quantifying how energy that initially resides in dilatational disturbances and propagates at the shock speed is redistributed into finite-amplitude reflected and transmitted waves as well as convecting vortical and entropy disturbances.

Turbulent transonic buffet is an aerodynamic instability causing periodic (albeit, often irregular) oscillations of lift/drag in aerospace applications. Involving complex coupling between inviscid and viscous effects, buffet is characterised by shock wave oscillations and flow separation/reattachment. Previous studies have identified both two-dimensional (2-D) chordwise shock-oscillation and three-dimensional (3-D) buffet-/stall-cell modes. While the 2-D instability has been studied extensively, investigations of 3-D buffet have been limited to only low-fidelity simulations or experiments. Due to computational cost, almost all high-fidelity studies to date have been limited to narrow span-widths around 5 % of aerofoil chord length (aspect ratio, ), which is insufficiently wide to observe large-scale three-dimensionality. In this work, high-fidelity simulations are performed up to , on an infinite unswept NASA Common Research Model (CRM) wing profile at $Re=5\times 10^{5}$. At , intermittent 3-D separation bubbles are observed at buffet conditions. While previous Reynolds-averaged Navier–Stokes (RANS)/stability-based studies predict quasi-simultaneous onset of 2-D- and 3-D-buffet, a case that remains essentially 2-D is identified here. Strongest three-dimensionality was observed near low-lift phases of the buffet cycle at maximum flow separation, reverting to essentially 2-D behaviour during high-lift phases. Buffet was found to become 3-D when extensive mean flow separation was present. At , multiple 3-D separation bubbles form in a spanwise wavelength range $\lambda =1c$ to $1.5c$. Spectral proper orthogonal decomposition (SPOD) was applied to analyse the spatio/temporal structure of 3-D buffet-cells. In addition to the 2-D chordwise shock-oscillation mode (Strouhal number $St \approx 0.07-0.1$), 3-D modal structures were observed at the shock wave/boundary layer interaction at $St \approx 0.002-0.004$.

We report the first measurement of turbulent mixing developing from the convergent Richtmyer–Meshkov (RM) instability using time-resolved planar laser-induced fluorescence in a semi-annular convergent shock tube. A membraneless yet sharp interface with random short-wavelength perturbations, but controllable long-wavelength perturbations, is created by an automatically retractable plate, enhancing the reproducibility and reliability of RM turbulence experiments. The cylindrical air/SF$_6$ interface formed is first subjected to a convergent shock, then to its reflected shock and subsequently transits to turbulent mixing. It is found that the mixing width after reshock has a linear growth rate more than twice the rate in planar geometry. Also, the mixing width does not present power-law growth at late stages as in a planar geometry. However, the scalar spectrum and transition criterion obtained are similar to their planar counterparts. These findings indicate that the geometric constraint greatly affects the large scales of the flow, while having a weaker effect on the small scales. It is also found that the reflected shock significantly increases both scale separation and Reynolds number, explaining the rapid transition to turbulence following reshock.

In this paper, the reflection of shock waves with downstream expansion fan interference in two-dimensional, inviscid flow is investigated, including steady Mach reflection (MR) and the unsteady transition process from regular reflection (RR) to MR. A threshold for the configuration based on non-dimensional wedge length is proposed. The analytical model for the steady MR and RR$\rightarrow$MR transition process is established based on the classical shock and expansion wave relations, whose prediction agrees well with results obtained through inviscid numerical simulation. It is found that the expansion fan interference significantly influences the steady flow patterns, especially the height of the Mach stem and the shape of the slip line. The interaction accelerates the formation of the sonic throat, stabilizing the flow structure rapidly, and results in generally small Mach stem heights. The exposure of the triple point to the expansion fan eliminates the inflection point on the slip line, whose slope increases smoothly. The interaction further affects the time evolution of the Mach stem during the multiple-interaction stage of the RR$\rightarrow$MR transition process. It appears that the modifications come from the curvature of the incident shock brought by the wave interference. During the multiple-interaction stage, the triple point moves upstream along the curved incident shock, where the incident shock angle changes according to the curvature, resulting in the variation of the evolution velocity.

We report the first shock-tube experiments on Richtmyer–Meshkov instability at a single-mode light–heavy interface accelerated by a strong shock wave with Mach number higher than 3.0. Under the proximity effect of the transmitted shock and its induced secondary compression effect, the interface profile is markedly different from that in weakly compressible flows. For the first time, the validity of the compressible linear theory and the failure of the impulsive model in predicting the linear amplitude evolution in highly compressible flows are verified through experiments. Existing nonlinear and modal models fail to accurately describe the perturbation evolution, as they do not account for the shock proximity and secondary compression effects on interface evolution. The shock proximity effect manifests mainly in the early stages when the transmitted shock remains close to the interface, while the effect of secondary compression manifests primarily at the period when interactions of transverse shocks occur at the bubble tips. Based on these findings, we propose an empirical model capable of predicting the bubble evolution in highly compressible flows.

The interaction between planar incident shocks and cylindrical boundary layers is prevalent in missiles equipped with inverted inlets, which typically leads to substantial three-dimensional flow separation and the formation of vortical flow. This study utilizes wind-tunnel experiments and theoretical analysis to elucidate the shock structure, surface topology and pressure distributions induced by a planar shock with finite width impinging on a cylinder wall at Mach 2.0. In the central region, a refraction phenomenon occurs as the transmitted shock bends within the boundary layer, generating a series of compression waves that coalesce into a shock, forming a ‘shock triangle’ structure. As the incident shock propagates backward along both sides, it gradually evolves into a Mach stem, where the transmitted shock refracts the expansion wave. The incident shock interacts with the boundary layer, resulting in the formation of a highly swept separation region that yields a pair of counter-rotating horseshoe-like vortices above the separation lines. These vortices facilitate the accumulation of low-energy fluid on both sides. Although the interaction of the symmetry plane aligns with free-interaction-theory, the separation shock angle away from the centre significantly deviates from the predicted value owing to the accumulation of low-energy fluids. The primary separation line and pressure distribution jointly exhibit an elliptical similarity on the cylindrical surface. Furthermore, the potential unsteady behaviour is assessed, and the Strouhal number of the low-frequency oscillation is found to be 0.0094, which is insufficient to trigger significant alterations in the flow field structure.

This study identifies two previously unrecognised screech modes in non-axisymmetric jets. Spectral proper orthogonal decomposition (SPOD) of ultra-high-speed schlieren images reveals a bi-axial flapping mode in a rectangular jet and a quasi-helical mode in an elliptical jet. To educe the complex three-dimensional structure of these new modes, SPOD is performed on datasets from different viewing perspectives, produced by rotating the nozzle with respect to the schlieren path to an azimuthal angle $\theta$. The bi-axial flapping mode is strongly antisymmetric from any perspective. However, the SPOD eigenvalue at the screech frequency ($\lambda _s$) varies with $\theta$ and the axial distance of the SPOD domain from the nozzle lip. This mode most closely resembles a flapping mode in the minor-axis plane close to the nozzle lip and a wagging mode in the major-axis plane further downstream. This transition from flapping to wagging at the same frequency correlates with the axis switching defined by the shock-cell structure in the mean flow. The quasi-helical mode in the elliptical jet is characterised by an antisymmetric structure present in the SPOD spatial modes whose eigenvalue $\lambda _s$ is insensitive to both $\theta$ and the axial domain. These findings indicate that the spatial evolution of the mean flow in non-axisymmetric jets may allow them to support a range of additional screech modes that differ significantly from those supported by the original three-dimensional shape of the jet.

In this paper, we study the rapid transition in Richtmyer–Meshkov instability (RMI) with reshock through three-dimensional double-layer swirling vortex rings. The rapid transition in RMI with reshock has an essential influence on the evolution of supernovas and the ignition of inertial confinement fusion, which has been confirmed in numerical simulations and experiments in shock-tube and high-energy-density facilities over the past few years. Vortex evolution has been confirmed to dominate the late-time nonlinear development of the perturbed interface. However, few studies have investigated the three-dimensional characteristics and nonlinear interactions among vortex structures during the transition to turbulent flows. The coexistence of co-rotating and counter-rotating vortices is hypothesized to induce successive large-scale strain fields, which are the main driving sources for rapid development. The three-dimensional effect is reflected in the presence of local swirling motion in the azimuthal direction, and it decreases the translation velocity of a vortex ring. Large-, middle- and small-scale strain fields are employed to describe the development process of RMI with reshock, e.g. vorticity deposited by the reshock, formation of the coexistence of the co-rotating and counter-rotating vortices, iterative cascade under the amplification of the strain fields and viscous dissipation to internal energy. This provides theoretical suggestions for designing practical applications, such as the estimation of the hydrodynamic instability and mixing during the late-time acceleration phase of the inertial confinement fusion.

The manipulation of the Richtmyer–Meshkov instability growth at a heavy–light interface via successive shocks is theoretically analysed and experimentally realized in a specific shock-tube facility. An analytical model is developed to forecast the interface evolution before and after the second shock impact, and the possibilities for the amplitude evolution pattern are systematically discussed. Based on the model, the parameter conditions for each scenario are designed, and all possibilities are experimentally realized by altering the time interval between two shock impacts. These findings may enhance the understanding of how successive shocks influence hydrodynamic instabilities in practical applications.

Fluid dynamics systems driven by dominant, near-periodic dynamics are common across wakes, jets, rotating machinery and high-speed flows. Traditional modal decomposition techniques have been used to gain insight into these flows, but can require many modes to represent key physical processes. With the aim of generating modes that intuitively convey the underlying physical mechanisms, we propose an intrinsic phase-based proper orthogonal decomposition (IPhaB POD) method, which creates energetically ranked modes that evolve along a characteristic cycle of the dominant near-periodic dynamics. Our proposed formulation is set in the time domain, which is particularly useful in cases where the cyclical content is imperfectly periodic. We formally derive IPhaB POD within a POD framework that therefore inherits the energetically ranked decomposition and optimal low-rank representation inherent to POD. As part of this derivation, a dynamical systems representation is utilized, facilitating a definition of phase within the system's near-periodic cycle in the time domain. An expectation operator and inner product are also constructed relative to this definition of phase in a manner that allows for the various cycles within the data to demonstrate imperfect periodicity. The formulation is tested on two sample problems: a simple, low Reynolds number aerofoil wake, and a complex, high-speed pulsating shock wave problem. The method is compared to space-only POD, spectral POD (SPOD) and cyclostationary SPOD. The method is shown to better isolate the dominant, near-periodic global dynamics in a time-varying IPhaB mean, and isolate the tethered, local-in-phase dynamics in a series of time-varying modes.

This study investigates the heating issue associated with a V-shaped blunt leading edge (VBLE) in a hypersonic flow. The heat flux generation on the VBLE is highly correlated with the shock interaction configurations in the crotch region, determined by the relative position of the triple point T and the curved shock (CS). The primary Mach reflection (MR), accompanied by a series of secondary shock–shock interactions and shock wave–boundary layer interactions, can produce extremely high heating peaks on the crotch. To configure the shock wave structures and reduce the heat flux, a shock-controllable design approach is developed based on the simplified continuity method. The strategy involves the inverse design of the crotch sweep path, according to the location of the triple point and the contour of the CS. The comparisons between the pre-designed shock configurations and the numerical results demonstrate the reliability of the design approach across various free stream Mach numbers ranging from 6 to 10. A VBLE model designed with the shock configuration of regular reflection from the same family (sRR) at a free stream Mach number of 8 is examined. Under the design conditions, the outermost heat flux peak is reduced by 80 % compared with the baseline case. The heating reduction capabilities of the model under varying free stream Mach numbers and sideslip angles are also evaluated, confirming its robustness under undesigned operating scenarios.

An asymptotic matching modal model is established based on the singular perturbation method for predicting mode evolution in single- and dual-mode interfaces accelerated by a shock wave. The startup process is incorporated into the model to provide a complete description of the mode evolution after the shock impact. Through considering the feedback from high-order harmonic to the third-order harmonic, the model accuracy is improved and the model divergence is prevented. In addition, the model can evaluate the mutual-coupling effect on the amplitude variations of high-order harmonics besides the ‘beat modes’. To validate the model, experiments on both light–heavy and heavy–light interfaces subject to a shock wave are conducted, and both single- and dual-mode interfaces formed by the soap-film technique are involved. The interface profiles extracted from mode decomposition and predicted by the model show high consistency with the experimental counterparts. Good agreement of the mode amplitude growths between the experiments and theoretical predictions shows the superiority of the model, especially for the heavy–light interface.

In this study, direct numerical simulation of the particle dispersion and turbulence modulation in a sonic transverse jet injected into a supersonic cross-flow with a Mach number of 2 was carried out with the Eulerian–Lagrangian point-particle method. One single-phase case and two particle-laden cases with different particle diameters were simulated. The jet and particle trajectories, the dispersion characteristics of particles, and the modulation effect of particles on the flow were investigated systematically. It was found that large particles primarily accumulate around shear layer structures situated on the windward side of the jet trajectory. In contrast, small particles exhibit radial transport, accessing both upstream and downstream recirculation zones. Moreover, small particles disperse extensively within the boundary layer and large-scale shear layers, evidently influenced by the streamwise vortices. The particles increase the mean wall-normal velocity near the wall in the wake region of the transverse jet, while reducing the mean streamwise and wall-normal velocities in outer regions. Particles significantly alter the flow velocity adjacent to shock fronts. In particular, the turbulent fluctuations near the windward barrel shock and bow shock are reduced, while those around the leeward barrel shock are increased. An upward displacement of the bow shock in the wall-normal direction is also observed due to particles. In the regions away from the shocks, small particles tend to amplify the Reynolds stress, while large particles attenuate the turbulent kinetic energy.