The interaction of linear waves with a planar shock is analysed using a spatial framework. Unlike conventional temporal analyses which often overlook post-shock evanescent waves, our spatial framework intrinsically captures all wave modes. By prescribing the frequency and tangential wavenumbers, the shock-normal wavenumbers are solved simultaneously on both sides, yielding a complete wavevector correspondence across the shock. Based on this relationship, a linear system is established that links the amplitudes of all six incident waves (from either side) to five downstream responses: the four generated post-shock waves plus shock deformation. This system enables the solution of both the forward problem (predicting downstream responses from given incident waves) and the inverse problem. Specifically, by imposing no incident downstream acoustic wave, the inverse solution uniquely determines the required upstream waves to produce any desired downstream wave field. We further demonstrate that while the tangential base flow velocity alters the group velocity and frequency via the Doppler effect, it leaves the disturbance wavenumbers and amplitudes unchanged. Consequently, oblique shock interactions can be reduced to equivalent normal shock problems. As an application, we present a complete solution map for a Mach 2 normal shock flow. The complex amplitudes of evanescent waves reveal characteristic phase shifts relative to incident waves. Notably, we present a scenario where shock oscillations occur without any transmitted waves downstream, demonstrating that the shock can act as a perfect energy absorber. Finally, direct numerical simulations for three representative cases provide quantitative validation of these theoretical predictions.

Shock-tube experiments are conducted to examine the initial-amplitude dependence of strong-shock-driven interfacial instability in reflected-shock configurations at Mach numbers ($M_s$) exceeding 3.0. Qualitative observations reveal a marked suppression (enhancement) of bubble (spike) growth under strong shocks, with stronger effects at higher dimensionless initial amplitudes ($ka_0$). The initial-amplitude effect on linear growth exhibits a non-monotonic dependence on $M_s$, contrary to the previously anticipated saturation behaviour, arising from the $M_s$ dependence of compressibility and vorticity-deposition contributions. Compressibility suppresses bubble growth via a shock-proximity mechanism while promoting spike evolution through a newly identified Mach-reflection mechanism, with their competition yielding a saturation dependence on $M_s$. In contrast, the effectiveness of vorticity deposition is modulated by the Mach-reflection structure, with reduced contributions at high $M_s$ due to geometric separation. Based on these insights, an empirical factor is proposed to quantify the initial-amplitude effect across a wide range of $ka_0$ and $M_s$. Dimensionless nonlinear amplitude evolution collapses across diverse $ka_0$ conditions. However, flow evolution depends strongly on $ka_0$, indicating that amplitude alone is insufficient to characterise the dynamics. Reliable prediction of nonlinear bubble (spike) evolution requires accurate representation of the shock-proximity and secondary-compression mechanisms (Mach-reflection and spike-acceleration mechanisms). However, existing nonlinear models fail to satisfy this requirement. The present study provides a foundation for future investigations of interfacial instability under extreme conditions relevant to inertial confinement fusion and astrophysical flows.

This study considers two key components of the mechanisms in screeching jets: the triadic interaction between the Kelvin–Helmholtz wavepacket and the shocks, and the deformation of the mean flow by these wavepackets. Intermittency in the resonance loop leads to multiple ‘mean-flow’ states; the existence of short-time mean flows associated with different manifestations of jet screech allows closer analysis of the relationship between screech and a given mean flow. Frequency–time analysis of high-speed schlieren data reveals a strong correlation between screech modes and mean-flow distortion, with each mode modifying the mean flow distinctly. Bispectral mode decomposition (BMD) is applied to elucidate the nonlinear interactions responsible for mean-flow distortion. The BMD results show that all dominant triad interactions arise from the direct difference self-interaction of the screech modes. Bispectral interaction maps further identify regions of the flow where triadic interaction between the screech mode and shocks occurs. Prior work hypothesised that local quasi-periodicity underpins resonance, and the BMD technique explicitly identifies these regions. This approach links spectral and spatial domains: instead of defining quasi-periodicity only via an effective wavenumber, the interaction maps localise it in space. Across a wide range of operating conditions, the wavenumber extracted from BMD maps matches the expected wavenumber from the unifying theory model (Edgington-Mitchell et al., J. Fluid Mech., 2022, vol. 945, p. A8) more closely than that from time-averaged shock structures. While highly consistent for the dominant screech frequency, the method is less so for weaker, intermittent tones. Critically, the results confirm that localised quasi-periodic regions underpin the triadic interactions that close the screech resonance loop.

The viscous shock tube is a canonical test case for assessing Navier–Stokes (NS) solvers in the continuum-flow regime, widely used to validate numerical accuracy and probe flow physics. It features a rich set of interacting structures – shock and rarefaction waves, contact discontinuities, boundary layers and their couplings – spanning multiple spatial and temporal scales. However, NS-based modelling, which presumes near-equilibrium behaviour, may fail to capture important non-equilibrium effects even in nominally continuum conditions. This study investigates the viscous shock tube at low Reynolds numbers and demonstrates the presence of non-equilibrium phenomena within the conventional continuum regime. To obtain physically consistent solutions across scales, we employ the unified gas-kinetic scheme (UGKS) and compare its results with NS solutions computed using the gas-kinetic scheme (GKS). Discrepancies between UGKS and GKS solutions reveal pronounced non-equilibrium effects in regions where shock waves interact with boundary layers. For continuum flows at high Mach and low Reynolds numbers, such multiscale non-equilibrium transport becomes important, underscoring the need for multiscale methods in analysis and prediction.

The physical mechanism of low-frequency unsteadiness in shock-wave/turbulent boundary layer interaction (STBLI) is studied using direct numerical simulations at Mach 2.9 with momentum-thickness Reynolds number ${\textit{Re}}_\theta \approx 2400$ over three ramp configurations with varying corner curvature. The configurations include a sharp $24^\circ$ compression ramp (R24) and two curved ramps with radii of 7 (C7) and 14 (C14) times the boundary layer thickness. By varying the ramp curvature, the extent of separation is controlled while maintaining sufficient streamline concavity to ensure that the Görtler number remains above its critical threshold. Cases R24 and C7 exhibit a mean separation with a detached shear layer accompanied by pronounced low-frequency unsteadiness, whereas C14 shows only incipient separation with markedly attenuated low-frequency unsteadiness. Sparsity-promoting dynamic mode decomposition reveals the emergence and downstream growth of counter-rotating structures in both low- and mid-frequency bands in all cases. In R24 and C7, these structures undergo growth and merging within the detached shear layer, giving rise to large-scale low-frequency motion, while in C14, their growth remains gradual and largely frequency-independent. The interactions between the counter-rotating structures and detached shear layer redistribute energy toward lower wavenumbers with a reduction at mid-wavenumbers. During this phase, the streamwise enstrophy dominates the enstrophy magnitude. Enstrophy transport analysis shows that the wall-normal-to-streamwise enstrophy tilting dominates the initial generation of streamwise enstrophy, whereas stretching is the primary source of the subsequent downstream growth and alignment. The results suggest that the interaction between the detached shear layer and counter-rotating structures develops into large-scale structures that govern the low-frequency unsteadiness in STBLI.

According to the state of incoming boundary layer, shock wave/boundary layer interaction can be classified into three groups: shock wave/laminar boundary layer interaction (SLBLI), shock wave/transitional boundary layer interaction (STrBLI) and shock wave/turbulent boundary layer interaction (STBLI). Although STBLI has been comprehensively studied, the difference between SLBLI and STrBLI is still not fully understood, especially in the framework of laminar–turbulent transition. In this work, a compression-ramp flow at Mach 6.5 is studied using experiment, direct numerical simulations (DNS) and stability analysis. Two distinct transition scenarios are revealed. On the one hand, without introducing external disturbances in the DNS, transition to turbulence is observed downstream of reattachment. This transition process is excited by the global instability in the separation bubble. On the other hand, the experiment captures a much smaller separation bubble under the same flow conditions. When realistic disturbances are superimposed on the freestream of DNS, a good agreement between the DNS and the experiment is achieved. The first mode is shown to be responsible for the transition. It is found that in the first transition scenario, the transition zone is located downstream of the reattachment region, and as a result, the size of the separation bubble is nearly not affected by the transition. This identifies an SLBLI. In the second transition scenario, however, the reattachment region is fully covered by the transition zone, accompanied by the shrinking of the separation bubble. This features an STrBLI. Therefore, the relative position between the reattachment region and the transition zone can be used to identify whether a flow belongs to SLBLI or STrBLI.

This work experimentally investigates Richtmyer–Meshkov (RM) turbulence driven by a cylindrical converging shock using time-resolved planar laser-induced fluorescence. An automatically retractable plate is developed to generate membraneless yet sharp interfaces characterised by random short-wavelength perturbations and controllable long-wavelength components. Five initial interfaces are examined: one with random short-wavelength perturbations, and four with these random perturbations superimposed with single-mode long-wavelength perturbations of controlled amplitude and wavenumber. The shocked interface in the converging configuration exhibits strongly unsteady motion: it initially moves inwards uniformly, then decelerates, moves outwards following reshock, and finally decelerates again with noticeable oscillations. The interface initially develops with a well-defined morphology, but quickly transitions to turbulent mixing after reshock. Both reshock and Rayleigh–Taylor instability, unique to converging RM flows, significantly influence the mixing layer evolution. The mixing width exhibits significantly higher post-reshock growth rates compared with the planar counterpart. Scalar mixedness analysis indicates that the random-perturbation case achieves significantly higher homogenisation efficiency than single-mode cases, whereas the mean scalar dissipation rate follows a similar temporal pattern regardless of initial morphology. The initial perturbation spectrum strongly influences the timing of turbulent transition: the high-wavenumber single-mode case reaches the empirical threshold earliest, followed by the random-perturbation case, while the low-wavenumber single-mode cases exhibit substantially delayed or incomplete transition. Scalar energy spectra further highlight that initial perturbations critically affect the efficiency of turbulent cascade and the small-scale energy distribution.

Richtmyer–Meshkov instability (RMI) at quasi-single-mode interfaces subjected to strong shocks with Mach number exceeding 3.0 is investigated through shock-tube experiments. To reveal the influence of higher-order initial modes, one single-mode and four quasi-single-mode interfaces with varying modal compositions are examined. The Richtmyer theory is experimentally verified for the first time to accurately capture the single-mode interface evolution from start-up to the linear stage. In contrast, it fails for quasi-single-mode scenarios, highlighting the significant effect of higher-order modes. By incorporating the influence of higher-order modes via linear superposition, a linear model for quasi-single-mode interface evolution is proposed, whose validation indicates negligible modal coupling in early evolution. For the nonlinear period, a representative model for weak-shock-driven RMI at a single-mode interface performs poorly, as it does not consider the influence of higher-order modes and key mechanisms such as shock proximity, secondary compression and spike acceleration. Based on the present findings, an empirical nonlinear model with favourable predictive capability is developed. Furthermore, by matching the new linear and nonlinear models, a complete description of the amplitude evolution of single-mode and quasi-single-mode interfaces from start-up to nonlinear stages is achieved, offering new insight into modelling strong-shock-driven RMI.

Building on classical oblique jump theory, we develop a one-dimensional (1-D) analytical framework that incorporates non-Newtonian rheology to predict the onset of hydraulic jumps, their internal structure and the associated Mach-front geometry. Source terms representing bed slope and wall friction are included, and the resulting formulation is systematically assessed against laboratory experiments, two-dimensional (2-D) shallow-water simulations and fully three-dimensional (3-D) computational fluid dynamics. Experiments with Newtonian, shear-thinning and shear-thickening fluids on converging sidewalls demonstrate a good match with the 1-D formulation. For Newtonian and shear-thinning fluids on mild slopes, the 1-D formulation with source terms closely reproduces the measured shock-front geometry and the 2-D simulation results. The analysis shows that upstream flow deceleration governs the reduction of the Mach angle and the resulting curvature. By contrast, in tests with shear-thickening fluids and steeper slopes, gravitational contributions produce detachment and strong front curvature that are not captured by the 1-D model. Comparisons of the transverse front position confirm that 1-D models lose validity when the upstream Froude number decreases sharply along the front. Fully 3-D simulations reveal concave front deformation driven by shear, strong dominance of tangential over normal velocities and flow features absent in depth-averaged models. The results demonstrate that 2-D shallow-water models capture the key dynamics for mild slopes and shear-thinning conditions, while accurate prediction for shear-thickening fluids requires 3-D approaches, motivating future hybrid strategies.

This study investigates a self-similar solution as intermediate asymptotics describing cylindrical shock waves driven by a piston in van der Waals gas under solid-body rotation. The solution is obtained for the exponential variations in the ambient density and the shock radius. The viscous stress follows Newton’s law of viscosity, while heat flux obeys Fourier’s law of heat conduction. The viscosity and thermal conductivity coefficients follow power-law dependencies on temperature and density. The solutions exist with pressure correction for increasing ambient density, while with volume correction for constant ambient density. The viscosity and volume corrections tend to weaken the shock, whereas the pressure correction and the specific heat ratio enhance it. Shock-induced compression intensifies with increasing viscosity, pressure correction and specific heat ratio, but decreases with increasing volume correction. The temperature and density exponents in the viscosity coefficient significantly affect shock compression, shock strength and the distribution of flow variables. Reduced density and radial velocity decrease with viscosity and heat conduction. Viscosity enhances tangential velocity while heat flux affects the normal and tangential stresses. The total energy behind the shock scales as the sixth power of the shock radius for pressure correction and the fourth power for volume correction. Solid body rotation is coupled with shock Mach number and ratio of specific heats. The reduced density, radial velocity and heat flux decrease, while pressure and normal viscous stress increase with pressure correction. Volume correction leads to decreases in density and pressure, but increases in tangential velocity and heat flux.

The evolution of main shock waves generated by multiple finite-energy blast sources with time-delayed energy release is investigated. We demonstrate that a merged shock wave forms when the individual blasts interact within a critical interval of delay times. This interval is determined principally by the initial sound speed ratio between the compressed gas of the blasts and the ambient medium. Numerical simulations confirm both the existence and the boundaries of this merging regime. Extending the analysis to the strong-shock stage, we derive an asymptotic solution that incorporates an equivalent energy release and a temporal scaling for the dual-source system, thereby describing the propagation of the merged shock front. These results provide a foundational model for multiple blast–shock processes, with potential applications to astrophysical phenomena, intense laser–matter interactions, and blast-wave dynamics in general.

This paper investigates asymmetric shock reflection in a supersonic overexpanded jet. A theoretical model that can predict the size and shape of flow field structures such as shocks, expansion waves and sliplines is established. For symmetric Mach reflection, the current model exhibits better agreement with numerical simulation results. For asymmetric Mach reflection, the current model also shows good agreement with numerical simulation results in predicting Mach stem height. The research indicates that the Mach stem height decreases with increasing nozzle pressure ratio and nozzle exit length difference, and decreases as the nozzle exit Mach number decreases. In addition, the critical geometric condition for complete misalignment of the upper and lower slipline interference segments (i.e. when the interference between one side of the expansion wave and the slipline ends, the other side of the expansion wave has not yet begun to interfere with the slipline) is given, which increases approximately linearly with nozzle pressure ratio and decreases as the nozzle exit Mach number increases. This study provides important theoretical support for the engineering application of asymmetric shock reflection in supersonic overexpanded jet.

The hypersonic flow over 30$^{\circ }$–50$^{\circ }$ double-cone configurations with three nose bluntness levels was experimentally investigated at Mach 6. High-speed schlieren photography, pressure sensors and pressure-sensitive paint were used to examine both global flow patterns and unsteady dynamics at a transitional Reynolds number. The experimental results indicate that the size of the separation region at the cone junction increases with increasing nose bluntness. Type V shock–shock interactions were observed in all three configurations, while the shock wave structures in the region below the triple point exhibited two patterns: Mach shock wave reflection in the sharp and small-blunt-nose cases, and regular shock wave reflection in the large-blunt-nose case. Spectral analysis of high-speed schlieren sequences revealed two types of unsteadiness across all cases: low-frequency shock oscillations and high-frequency unsteady structures along the boundary of supersonic jet on the second cone. For the low-frequency unsteadiness, shock oscillations displayed a broadband nature in the sharp and small-blunt-nose configurations, while a dominant frequency of approximately 2 kHz was observed in the large-blunt-nose case, characterised by shock motion and bubble breathing – an observation not experimentally reported before. Additionally, spectral analysis of wall pressure contours indicated that the low-frequency unsteadiness was primarily characterised by axisymmetric modes for all configurations. Global stability analysis and resolvent analysis further demonstrated noise-amplifier behaviour in all configurations, and the dominant low-frequency unsteadiness in the large-blunt-nose case is attributed to modal resonance induced by environmental noise.

The explosive dispersal of granular media, exemplified by the rapid radial expansion of a dense particle ring driven by internal pressurised gases, serves as a paradigmatic system for investigating multiphase blast dynamics. Despite the ubiquity of jetting and clustering phenomena in explosive dispersal scenarios, their governing mechanisms remain poorly resolved. In this work, we combine compressible computational fluid dynamics–discrete parcel method simulations, and theoretical modelling to elucidate the multiscale physics underlying explosion-induced particle jetting. We reveal a hierarchy of jetting structures, comprising non-jetting, suppressed jetting and prominent jetting, which are governed by the interplay between microscale particle force-chain evolution, mesoscale gas–particle coupling and macroscale ring dynamics. Jetting initiation emerges from the transient competition between shock-induced particle compaction and gas filtration during the early expansion phase, whereas sustained jet development requires subsequent ring implosion driven by adverse pressure gradients. By unifying this multiscale dynamics, we reduce the system’s complexity into two dimensionless parameters: one characterising mesoscale gas–particle interactions and another quantifying macroscale implosion intensity. A phase diagram for jetting morphology under weak-shock conditions is established in this dimensionless parameter space, delineating two necessary criteria for jet formation. Systems failing either criterion exhibit no jetting, resolving long-standing ambiguities in the prediction of explosive dispersal structures.

The interaction between cylindrically converging shock waves (SWs) in a water–gelatine solution and a coaxial cylindrical air bubble is studied experimentally and numerically. Two configurations are considered: (i) an azimuthally symmetric, cylindrically converging SW of Mach 1.35 impinging on a coaxial cylindrical bubble, and (ii) a semicylindrical converging SW of Mach 1.45 (corresponding to half of the cylindrical front), interacting with the same target. Shock waves are generated by exploding wire arrays driven by a high-voltage pulsed power system at beamline ID19 of the European Synchrotron Radiation Facility, delivering currents up to $130\,\text{kA}$ with rise times of $0.35$ and $0.55\,\unicode{x03BC} \text{s}$ to the cylindrical and semicylindrical wire loads, respectively. X-ray radiography is conducted at a pulse repetition rate of 5.68 MHz using two synchronised high-speed cameras. Numerical hydrodynamic simulations are performed using a compressible multiphase Navier–Stokes solver. A Gilmore-type model for compressible cylindrical bubble pulsation provides an independent analytical estimate of the interface evolution. In the cylindrical SW configuration, the bubble collapse in experiments exhibits Richtmyer–Meshkov instability spikes. The cylindrically converging shock is analysed with Guderley’s solution and Whitham’s approximation using a real-gas equation of state, predicting Mach 14.1 near the focus. In the semicylindrical configuration, momentum focuses into a single supersonic jet with a speed of 885 $\pm$ 30 m s−1, producing localised high-pressure regions, coherent vortices and complex internal Mach reflections. Experiments, simulations and theory are consistent in collapse time, interface motion and overall flow dynamics.

An experimental study was conducted to investigate the characteristics of unsteady oblique shock trains in a constant-area rectangular duct under an asymmetric incoming boundary layer. High-speed schlieren techniques and high-frequency pressure measurements were utilised in this study. The results indicate that the oblique shock train mainstream leans significantly towards the thin-boundary-layer side. Under downstream periodic excitation, the shock train moves periodically, and its shape changes during the movement. This phenomenon occurs to match the downstream pressure by altering the relative Mach number in front of the shock train, with the average pressure rise slope along the thick-boundary-layer side changing periodically. Additionally, unlike a normal shock train, the pressure rise distribution along the thick-boundary-layer side is nearly linear, and the correlation coefficients between the transducers on this side and the most downstream transducer are higher than those on the thin-boundary-layer side. Due to differences in flow structure and pressure rise distribution, the existing amplitude prediction model proposed by Xiong et al. (J. Fluid Mech. vol. 846, 2018, pp. 240–262) for the unsteady normal shock train is no longer applicable to the unsteady oblique shock train. Therefore, a new prediction model is derived and verified by experiments. Moreover, it is found that using only the downstream pressure transducer information on the thick-boundary-layer side can effectively predict the amplitude of the shock-train motion. Combined with the prediction model, a novel method is proposed to estimate the amplitude of the shock-train motion conveniently.

Supersonic diamond airfoils operating in ground effect exhibit choking phenomena, where slight variations in free-stream Mach number can induce significant alterations in the ground effect flow structure and consequently affect the aerodynamic loading on the airfoil. However, existing models for predicting the choking limit Mach number demonstrate systematic discrepancies. This study establishes a novel predictive model by analysing the steady inviscid supersonic flow field around a two-dimensional diamond airfoil in ground effect. Benchmarking against numerical simulations demonstrates that the prediction errors for the choking limit Mach number across various diamond airfoil geometries are all below 3.5 %. These results affirm the high accuracy of the proposed predictive model. Under critical choking conditions, the ground effect flow field manifests multiple shock structures, including regular reflection, curved reflection and strong Mach reflection. Crucially, all of these configurations share the characteristic feature of the reflected shock impinging on the lower vertex of the airfoil. Consequently, the problem of predicting the choking limit is reformulated as determining the free-stream Mach number at which the reflected shock strikes the lower vertex of the airfoil. To circumvent complications from the reflected shock curvature inherent to critical choking, the model solves mass and momentum conservation equations for a strategically defined control volume. This approach eliminates curvature-induced errors, enabling precise prediction of the choking limit Mach number for supersonic diamond airfoils in ground effect.

Shock tube experiments are essential in understanding the environment encountered by hypersonic vehicles. Such experiments provide information used to determine rate constants of chemical, relaxation and radiative processes taking place in non-equilibrium plasmas. These constants are significant drivers of uncertainty in surface heat flux predictions. Recent work has shown that flow non-uniformities in real shock tube experiments can be misinterpreted as a need to alter these parameters; however, no comprehensive model exists to decouple the effects. We show that there is a rigorous method to achieve this by using experimental measurements as boundary conditions and including their effects via reverse time integration. This method improves over previous implementations by rigorously enforcing conservation laws, incorporating two-temperature, non-equilibrium thermochemistry and explicitly modelling both forward- and backward-running sound waves in the shock tube test slug through a method of characteristics formulation. This approach allowed the effect of shock speed variation in highly non-equilibrium tests, specifically those relevant to Titan entry, to be studied for the first time. A validation study showed that properties predicted by the method were found to agree with results from a viscous, two-dimensional axisymmetric Navier–Stokes solver within 1.5 %. When applied to shock tube test cases from the EAST and T6 facilities for simulation of lunar return and Titan entry representative conditions, the method offered improved agreement with experimentally measured oxygen 777 nm and 240–440 nm radiance, respectively, when compared with previous implementations, particularly towards the rear of the test slug where forward-running sound waves from the driver become influential.

Richtmyer–Meshkov instability (RMI) at a single-mode interface separating an inert gas (N$_2$) and a reactive gas mixture (H$_2$/O$_2$/Xe) under reshock conditions is numerically investigated using a newly developed compressible reactive Navier–Stokes solver. The solver employs the Kéromnès mechanism (10 species, 21 reactions) for combustion modelling and a dual-flux algorithm to suppress numerical oscillations at material interfaces, demonstrating high accuracy across a wide range of benchmark tests. By systematically varying incident shock Mach numbers, we identify four distinct evolution regimes: an inert regime (${\textit{Ma}} \lt 1.80$), characterised by negligible combustion effects on interface evolution; a deflagration regime ($1.80 \lt Ma \lt 1.86$), marked by strong coupling between interface dynamics and combustion through sustained interactions; a detonation regime ($1.86 \lt Ma \lt 2.50$), where rapid transition to detonation leads to moderate coupling; and an immediate detonation regime (${\textit{Ma}} \gt 2.50$), where detonation occurs directly after incident shock impact, modifying interface evolution from the outset through intense heat release and pressure waves. Mixing width and mixing level are most significantly enhanced in the deflagration regime due to prolonged combustion-flow interactions, while cases with higher Mach numbers show reduced mixing due to rapid combustion completion. Heat release and enstrophy also display clear regime-dependent evolution behaviour: maximum heat release occurs in the detonation regime, while peak enstrophy is observed in the deflagration regime. A clear correlation is observed between the Damköhler number ($Da$), which represents the ratio of hydrodynamic to chemical time scales, and the flow regimes: for ${\textit{Ma}} \lt 1.80$, $Da \lt 1$ indicates negligible coupling; at ${\textit{Ma}} = 1.83$, $Da \approx 1$ reflects sustained coupling; and for ${\textit{Ma}} \gt 2.00$, $Da \gt 1$ denotes strong early coupling. This correlation provides a theoretical basis for interpreting the distinct regimes and guiding the modulation of reactive RMI.

Fluid mixture models are essential for describing a wide range of physical phenomena, including wave dynamics and spinodal decomposition. However, there is a lack of consensus in the modelling of compressible mixtures, with limited connections between different classes of models. On the one hand, existing compressible two-phase flow models accurately describe wave dynamics, but do not incorporate phase separation mechanisms. On the other hand, phase-field technology in fluid dynamics consists of models incorporating spinodal decomposition; however, a general phase-field theory for compressible mixtures remains largely undeveloped. In this paper we take an initial step toward bridging the gap between compressible two-phase flow models and phase-field models by developing a theory for compressible, isothermal N-phase mixtures. Our theory establishes a system of reduced complexity by formulating N mass balance laws alongside a single momentum balance law, thereby naturally extending the Navier–Stokes Korteweg model to N phases and providing the Navier–Stokes Cahn–Hilliard/Allen–Cahn model for compressible mixtures. Key aspects of the framework include its grounding in continuum mixture theory and its preservation of thermodynamic consistency despite its reduced complexity.