Aerothermal issues in hypersonic transitional swept shock wave/boundary-layer interactions (SBLIs) are critical for the structural safety of high-speed vehicles but remain poorly understood. In this work, previously scarce, high-resolution heat transfer distributions of the hypersonic transitional swept SBLIs, are obtained from fast-responding temperature-sensitive paint (fast TSP) measurements. A series of $34^\circ$ compression ramps with sweep angles ranging from $0^\circ$ to $45^\circ$ are tested in a Mach 12.1 shock tunnel, with a unit Reynolds number of 3.0 $\times$ 10$^{6}$ m$^{-1}$. The fast TSP provides a global view of the three-dimensional aerothermal effects on the ramps, allowing in-depth analysis on the sweep effects and the symmetry of heat transfer. The time-averaged results reveal that the heat flux peak near reattachment shifts upstream with decreasing amplitude as the sweep angle increases, and a second peak emerges in the $45^\circ$ swept ramp due to a type V shock–shock interaction. Downstream of reattachment, the heat flux streaks induced by Görtler-like vortices weaken with increasing sweep angle, whereas their dominant projected wavelengths show little dependence on sweep angle or spanwise location. Away from the ramp’s leading side, the transition onset of the reattached boundary layer gradually approaches the reattachment point. Finally, a general quasi-conical aerothermal symmetry is identified upstream of reattachment, although spanwise variations in transition onset, shock–shock interaction and heat flux streaks are found to disrupt this symmetry downstream of reattachment with varying degrees.

This study presents an analytical advancement in predicting the growth rate of perturbation amplitude in two-dimensional non-standard Richtmyer–Meshkov instability (RMI), driven by the interaction of a first-phase rippled shock wave at moderate Mach number with a heavy–light interface. We extend the irrotational model to encompass non-standard RMI scenarios, establishing a generalised framework validated through numerical simulations. Distinct from previous models, our model is free of empirical coefficients, and demonstrates superior accuracy across diverse perturbation configurations and Mach numbers. The analyses reveal the fundamental disparity of non-standard RMI from classical RMI: the vorticity deposition mechanism in non-standard RMI arises not only from normal pressure gradients at the shock front but crucially from tangential pressure gradients behind the shock wave. The asymptotic circulations are also well predicted by our model. Moreover, the relationship of the amplitudes between sinusoidal shock and perturbed interface is derived based on the model to realise the freeze-out of interface amplitude. The initial fundamental mode’s amplitude growth is frozen well, and the mixing width is greatly suppressed.

Hysteresis in the transition between regular reflection (RR) and Mach reflection (MR) has been predicted theoretically and numerically for decades, yet successful experimental demonstrations have remained limited to wedge-angle-variation-induced hysteresis. This work presents the first successful experimental demonstration of Mach-number-variation-induced hysteresis. Utilising a newly developed continuously variable Mach 5–8 wind-tunnel nozzle, Mach-sweep experiments were conducted on a pair of wedges at three different angles ($25^{\circ }$, $27^{\circ }$ and $28^{\circ }$). A stable RR was first established at Mach 7 within the dual-solution domain for each angle, and then the Mach number was decreased to 5. For the $27^{\circ }$ and $28^{\circ }$ cases, transition from RR to MR was observed at Mach 5.3 and 5.9, respectively, during the downward Mach sweep, and the MR state persisted throughout the upward sweep back to Mach 7. During the $25^{\circ }$ case, a stable RR was maintained throughout the entire Mach sweep, prompting further experiments into the effect of free-stream disturbances on the stability of the RR state. Preliminary results revealed a free-stream-disturbance-induced hysteresis and that the RR state is metastable with potential stochastic behaviour.

The Richtmyer–Meshkov instability (RMI) develops when a planar shock front hits a rippled contact surface separating two different fluids. After the incident shock refraction, a transmitted shock is always formed and another shock or a rarefaction is reflected back. The pressure/entropy/vorticity fields generated by the rippled wavefronts are responsible of the generation of hydrodynamic perturbations in both fluids. In linear theory, the contact surface ripple reaches an asymptotic normal velocity which is dependent on the incident shock Mach number, fluid density ratio and compressibilities. In this work we only deal with the situations in which a shock is reflected. Our main goal is to show an explicit, closed form expression of the asymptotic linear velocity of the corrugation at the contact surface, valid for arbitrary Mach number, fluid compressibilities and pre-shock density ratio. An explicit analytical formula (closed form expression) is presented that works quite well in both limits: weak and strong incident shocks. The new formula is obtained by approximating the contact surface by a rigid piston. This work is a natural continuation of J. G. Wouchuk (2001 Phys. Rev. E vol. 63, p. 056303) and J. G. Wouchuk (2025 Phys. Rev. E vol. 111, p. 035102). It is shown here that a rigid piston approximation (RPA) works quite well in the general case, giving reasonable agreement with existing simulations, previous analytical models and experiments. An estimate of the relative error incurred because of the RPA is shown as a function of the incident shock Mach number $M_i$ and ratio of $\gamma $ values at the contact surface. The limits of validity of this approximation are also discussed. The calculations shown here have been done with the scientific software Mathematica. The files used to do these calculations can be retrieved as Supplemental Files to this article.

The propagation of detonations in a non-uniform mixture exhibits notable distinctions from that in a uniform mixture. This study first delves into the analytical analysis of the one-dimensional shock transmission problem and the two-dimensional shock propagation in a mixture with temperature non-uniformity. Additionally, the research extends to the numerical simulation of the propagation of shocks and detonations, building upon the insights garnered from the analytical analysis. The numerical results indicate that introducing a temperature interface in a non-uniform gas creates a discrete flow field and wavefront, resulting in oblique shocks that connect hot and cold layers. A competitive mechanism between the transverse waves and non-uniformity is responsible for the detonation propagation. The temperature amplitude tends to inhibit the propagation of transverse waves. In contrast, the wavelengths primarily affect the spacing and strength of these transverse waves, especially during the early stages of propagation. In a Zel’Dovich–von Neumann–Döring detonation, the non-uniformities distort the detonation front, creating transverse wave spacings comparable to the wavelength and reducing the front velocity. However, the detonation can recover its Chapman–Jouguet velocity and approach a steady states as intrinsic instabilities come into play. In the steady state, the cell sizes are found to be determined by the temperature amplitude. When the temperature amplitude is sufficiently high, the detonation cells effectively disappear.

For hypersonic inlets, buzz is a self-sustained oscillatory flow characterised by strong nonlinear and unsteady behaviour. Our recent study shows that, unlike conventional alterations in flow conditions at the inlet entrance or exit, flexible lip deformation is a newly identified trigger for buzz. However, the mechanism by which this fluid–structure interaction (FSI) behaviour induces buzz remains unclear. To clarify how FSI acts as a dominant factor in triggering flow instability leading to buzz, this study investigates a more general flexible plate model within the inlet. The results show that the plate FSI introduces a prolonged instability accumulation process for buzz evolution, resulting in a ‘gradual-onset’ characteristic differing from previous studies. During this process, plate FSI amplifies downstream flow oscillations while accumulating unstable energy. Eventually, the excessive unstable energy causes the shock train to destabilise and be disgorged from the inlet, initiating a complete instability process dominated by buzz. Notably, buzz induced by plate FSI exhibits unsteady characteristics similar to those observed in rigid inlets. Therefore, as an internal self-excited disturbance source, plate FSI produces relatively weaker disturbances than conventional flow modifications, but exhibits highly persistent accumulation effects and distinct multistage characteristics. This study reveals the buzz evolution mechanism under plate FSI, providing new insights into flow instability in hypersonic inlets.

Transonic buffet presents time-dependent aerodynamic characteristics associated with shock, turbulent boundary layer and their interactions. Despite strong nonlinearities and a large degree of freedom, there exists a dominant dynamic pattern of a buffet cycle, suggesting the low dimensionality of transonic buffet phenomena. This study seeks a low-dimensional representation of transonic airfoil buffet at a high Reynolds number with machine learning. Wall-modelled large-eddy simulations of flow over the OAT15A supercritical airfoil at two Mach numbers, $M_\infty = 0.715$ and 0.730, respectively producing non-buffet and buffet conditions, at a chord-based Reynolds number of ${Re} = 3\times 10^6$ are performed to generate the present datasets. We find that the low-dimensional nature of transonic airfoil buffet can be extracted as a sole three-dimensional latent representation through lift-augmented autoencoder compression. The current low-order representation not only describes the shock movement but also captures the moment when the separation occurs near the trailing edge in a low-order manner. We further show that it is possible to perform sensor-based reconstruction through the present low-dimensional expression while identifying the sensitivity with respect to aerodynamic responses. The present model trained at ${Re} = 3\times 10^6$ is lastly evaluated at the level of a real aircraft operation of ${Re} = 3\times 10^7$, exhibiting that the phase dynamics of lift is reasonably estimated from sparse sensors. The current study may provide a foundation towards data-driven real-time analysis of transonic buffet conditions under aircraft operation.

We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

This study quantitatively investigates the two-dimensional pseudosteady shock refraction at an inclined air–water interface, referred to as the water wedge, in the weak and strong incident shock strength groups. Numerical simulations are employed to validate the predicted refraction sequences from a previous study (Anbu Serene Raj et al. 2024 J. Fluid Mech. 998, A49). A distinctive irregular refraction pattern, referred to as the bound precursor refraction with a Mach reflection, is numerically validated in the weak shock group. Based on the numerical simulations, an enhanced formulation is proposed to determine the sonic line of the incident flow Mach number ($M_b$) in water, thereby providing an appropriate transition condition for an irregular refraction with a Mach reflection to a free precursor refraction with a Mach reflection transition. Furthermore, comparative studies on solid and water wedges of wedge angle $20^\circ$ reveal discernible differences in the shock reflection patterns. The interplay of the energy dissipation due to the transmitted shock wave and the Richtmyer–Meshkov instability at the air–water interface results in the variation of the triple-point trajectory and transition angles between single Mach reflection (SMR) to transitional Mach reflection (TMR) occurring in air.

Turbulence amplification is crucial in shock-wave/turbulent boundary layer interaction (SWTBLI). To examine the impact of interaction intensity on turbulence amplification and inter-component energy transfer, direct numerical simulations of impinging oblique shock reflections at strong ($37^\circ$) and weak ($33.2^\circ$) incident angles are conducted. The results indicate that strong interaction generates a larger permanent separation zone, featuring the unique ‘oblique platform’ in Reynolds stress peaks and ‘secondary turbulence amplification’ downstream. Reynolds stress budget and spanwise spectral analyses reveal that $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ amplify primarily by production terms. $u''$, $v''$ and $w''$ represent the streamwise, wall-normal and spanwise velocity fluctuations. At the investigated Reynolds number, deceleration effect dominates the initial amplification of $\widetilde {u^{\prime \prime}u^{\prime \prime}}$, influencing multi-scale wall-bounded turbulence structures, while shear effect remains active along the shear layer and may primarily affects streaky structures. The initial amplification of $-\!\widetilde{\ u^{\prime\prime}v^{\prime\prime}}$ is driven by the adverse pressure gradient, which reshapes the velocity profile and affects the wall-normal velocity. The primary energy for $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ and $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ amplification originates from $\widetilde{ u^{\prime \prime}u^{\prime \prime}}$ via the pressure-strain term. The delayed amplification of $\!\widetilde{\ v^{\prime\prime}v^{\prime\prime}}$ is influenced by its production term and energy redistribution, with $\widetilde {w^{\prime \prime}w^{\prime \prime}}$ exhibiting higher spectral consistency with $\widetilde {u^{\prime \prime}u^{\prime \prime}}$ and receiving more energy. In strong interaction, the ‘oblique platform’ serves as a stable dissipation region, formed by increased separation–incident shock distance, characterised by progressively concentrated stress spectra and the transition to large-scale streaks. The downstream ‘secondary amplification’ process resembles the initial amplification near the separation shock foot, driven by intermittent compression waves that strengthen shear instabilities and the deceleration effect. These findings detail the streamwise stress evolution, providing a more comprehensive turbulence amplification mechanism in SWTBLI.

This study examines the reflection of a rightward-moving shock (RMS) over expansion waves, dividing the reflection structure into three components. The first component analyses the pre- and post-interaction parts of the expansion waves, categorising primary flow patterns into four types with defined transition criteria, visualised through Mach contours. The second component investigates the curved perturbed shock. Through numerical simulations, the influence of increasing shock strength on the flow structures is displayed. A triple point forms for an RMS of the first family, and the Mach stem height increases with the increase of shock strength. When the RMS is strong enough, a vortex forms in the near-wall region, which acts like a wedge to distort the near-foot part of the RMS. The third component, the near-foot region, is analysed using a one-dimensional Riemann problem approach. The calculated wave speeds are used to mark waves in Mach contours for eight cases. The position of the waves indicates that the left-going shock for an RMS of the first family or the right-going shock for an RMS of the second family corresponds to the foot of the RMS. This can explain the finding that the right-hand side of an RMS of the first family or the left-hand side of an RMS of the second family is disturbed. The regions to have different wave patterns solved from the one-dimensional Riemann problem are displayed in the original Mach number–shock speed Mach number plane.

A model for galloping detonations is conceived as a sequence of very fast re-ignitions followed by long periods of evolution with quenched reactions. Numerical simulations of the one-dimensional Euler equations are conducted in this limit. While the mean speed and structure is found in reasonable agreement with Chapman–Jouguet theory, very strong pulsations of the lead shock appear, along with a train of rear-facing N-waves. These dynamics are analysed using characteristics. A closed-form solution for the lead shock dynamics is formulated, which is found in excellent agreement with numerics. The model relies on the presence of a single time scale of the process, the pulsation period, which controls the shock dynamics via the shock change equations and establishes a shock decay with a single time constant. These long periods of shock decay with known dynamics are punctuated by energy release events, with ‘kicks’ in the shocked speed controlled by the pressure increase and resulting lead shock amplification. Model predictions are found in excellent agreement with previous numerical results of pulsating detonations far from the stability limit.

The flow instabilities in shock-wave–boundary-layer interactions at Mach 6 are comprehensively investigated through compression corner and incident shock cases. The boundary of global stability and the characteristics of globally unstable modes are determined by global stability analysis. In resolvent analysis, cases are categorized into flat plate, no separation, small separation and large separation flows. The optimal response shifts from the first mode in the flat plate case to streaks after the amplification in the interaction region. The amplification of streaks and the first mode (oblique mode) are both attributed to the Görtler instability. Meanwhile, the second mode exhibits minimal growth and higher Mack’s modes appear within the separation bubble. Rounded corner case and linear stability analysis are utilized to further validate the amplification mechanism of the oblique mode.

Large-scale spanwise motions in shock wave–turbulent boundary-layer interactions over a $ 25^{\circ }$ compression ramp at Mach 2.95 are investigated using large-eddy simulations. Spectral proper orthogonal decomposition (SPOD) identifies coherent structures characterised by low-frequency features and a large-scale spanwise wavelength of $ O(15\delta _{0})$, where $ \delta _{0}$ is the incoming boundary-layer thickness. The dominant frequency is at least one order of magnitude lower than that of the shock motions. These large-scale spanwise structures are excited near the shock foot and are sustained along the separation shock. Global stability analysis (GSA) is then employed to investigate the potential mechanisms driving these structures. The GSA identifies a stationary three-dimensional (3-D) mode at a wavelength of $ 15\delta _{0}$ with a similar perturbation field, particularly near the separation shock. Good agreement is achieved between the leading SPOD mode and the 3-D GSA mode both qualitatively and quantitatively, which indicates that global instability is primarily responsible for the large-scale spanwise structures surrounding the shock. The reconstructed turbulent separation bubble (TSB) using the 3-D global mode manifests as spanwise undulations, which directly induce the spanwise rippling of the separation shock. Furthermore, the coupled TSB motions in the streamwise and spanwise directions are examined. The TSB oscillates in the streamwise direction while simultaneously exhibiting spanwise undulations. The filtered wall-pressure signals indicate the dominant role of the streamwise motions.

The dynamics of self-excited shock train oscillations in a back pressured axisymmetric duct was investigated to deepen the understanding of the isolator/combustor coupling in high-speed propulsion systems. The test article consisted of an internal compression inlet followed by a constant area isolator, both having a circular cross-section. A systematic back pressure variation was implemented by using a combination of aerodynamic and physical blockages at the isolator exit. High bandwidth two-dimensional pressure field imaging was performed at $8\,{\rm kHz}$ repetition rate within the isolator for different back pressure settings. The acquisition rate was considerably higher than the dominant frequency of the shock train oscillations across the different back pressure settings. The power spectral density of the pressure fluctuations beneath the leading shock foot exhibited broadband low frequency oscillations across all back pressures that resembled the motions of canonical shock–boundary layer interaction units. A node in the vicinity of reattachment location that originated the pressure perturbations within the separation shock was also identified, which further ascertained that the leading shock low frequency motions were driven by the separation bubble pulsations. Above a threshold back pressure, additional peaks appeared at distinct higher frequencies that resembled the acoustic modes within the duct. However, none of the earlier expressions of the resonance acoustic frequency within a straight duct agreed with the experimentally observed value. Cross-spectral analyses suggested that these modes were caused by the shock interactions with upstream propagating acoustic waves that emanate from the reattachment location, originally proposed for transonic diffusers by Robinet & Casalis (2001) Phys. Fluids 13, 1047–1059. Feedback interactions described using one-dimensional stability analysis of the shock perturbations by obliquely travelling acoustic waves (Robinet & Casalis 2001 Phys. Fluids 13, 1047–1059) made favourable comparisons on the back pressure threshold that emanated the acoustic modes as well as the acoustic mode frequencies.

This work presents detailed 3D modelling and simulation of the mechanical effects induced by lightning strikes in protected carbon fibre-reinforced polymer laminates. Firstly, physically based models that represent the mechanical overpressure that results from a lightning strike are revisited. In particular, this paper compares the implementation of an analytical strong shock wave approximation with the solutions obtained from computational fluid dynamics (CFD), considering different equations of state, to represent the supersonic expansion of the hot plasma channel when simulating the mechanical damage induced by lightning strikes. The assessment of the pressure profiles, the numerical predictions of the displacement and velocity fields and the analysis of the predicted damage maps show that, for two lightning protection layers, the effects of the supersonic plasma expansion loads obtained from the strong shock wave approximation compare reasonably well with those obtained from CFD, independently of the equation of state solved numerically. Subsequently, the predictions of the 3D modelling strategy of the mechanical response of composite laminates subjected to lightning strike employing the strong shock wave approximation are compared with mechanical deformation measurements obtained from lab-scale lightning test results. Accurate deflection and out-of-plane velocity fields are predicted, validating the 3D modelling strategy. Moreover, the predicted damage maps correlate well with the (bulk) damage identified by C-scan (considering only the damaged area below the second ply).

Over-expansion flow can generate asymmetric shock wave interactions, which lead to significant lateral forces on a nozzle. However, there is still a lack of a suitable theory to explain the phenomenon of asymmetry. The current work carefully investigates the configurations of shock wave interactions in a planar nozzle, and proposes a theoretical method to analyse the asymmetry of over-expansion flows. First, various possible flow patterns of over-expansion flows are discussed, including regular and Mach reflections. Second, the free interaction theory and the minimum entropy production principle are used to analyse the boundary layer flow and main shock wave interactions, establish the relationship between the separation shock strength and separation position, and predict asymmetric configurations. Finally, experiments are conducted to validate the theoretical method, and similar experiments from other studies are discussed to demonstrate the effectiveness of the proposed method. Results demonstrate that the direction of asymmetric over-expansion flow is random, and the separated flow strives to adopt a pattern with minimal total pressure loss. Asymmetric interaction is a mechanism through which the flow can achieve a more efficient thermodynamic balance by minimising entropy production.

Shock interactions on a V-shaped blunt leading edge (VBLE) that are commonly encountered at the cowl lip of an inward-turning inlet are investigated at freestream Mach numbers ($ M_\infty$) 3–6. The swept blunt leading edges of the VBLE generate a pair of detached shocks with varying shapes due to the changes in $ M_\infty$ and $L/r$ (i.e. the ratio of the leading-edge length $L$ to the leading-edge blunt radius $r$), which causes intriguing shock interactions at the crotch of the VBLE. Three subtypes of regular reflection (RR) and a Mach reflection (MR) are produced successively with increasing $ M_\infty$ for a given $L/r$, which appear in the opposite order to those with increasing $L/r$ for a given $ M_\infty$. These shock interactions identified in numerical simulations are verified in supersonic and hypersonic wind tunnel experiments. It is demonstrated that the relative position of the shocks is crucial in determining the transitions of shock interactions by varying either $L/r$ or $ M_\infty$. Transition criteria between subtypes of RR and from RR to MR are theoretically established in the parameter space $(M_\infty,L/r)$ by analysing the shock structures, showing good agreement with the numerical and experimental results. Interactions between either immature or fully developed detached shocks are embedded in these criteria. Specifically, the transition criteria asymptotically approach the corresponding critical $ M_\infty$ when $L/r$ is sufficiently large. These transition criteria provide guidelines for improving the design of the cowl lip of an inward-turning inlet in supersonic and hypersonic regimes.

Compressible jets impinging on a perpendicular surface can produce high-intensity, discrete-frequency tones. The character of these tones is a function of nozzle shape, jet Mach number, impingement-plate geometry, and the distance between nozzle and plate. Though it has long been recognised that these tones are associated with a resonance cycle, the exact mechanism by which they are generated has remained a topic of some debate. In this work, we present evidence for a number of distinct tone-generation mechanisms, reconciling some of the different findings of prior authors. We demonstrate that the upstream-propagating waves that close resonance can be confined within the jet, or external to it. These waves can be either weak and relatively linear, or strong and nonlinear from their inception. The waves can undergo coalescence or merging, and in some configurations, pairs of waves rather than singletons appear. We discuss both historical and new evidence for multiple distinct processes by which upstream-propagating waves are produced: direct vortex sound, shock leakage, wall-jet-boundary fluctuations, and wall-jet shocklets. We link these various mechanisms to the disparate collection of upstream-propagating waves observed in the data. We also demonstrate that multiple mechanisms can be provoked by a single vortex, providing an explanation as to why sometimes pairs of waves or merging waves are observed. Through this body of work, we demonstrate that rather than being in opposition, the various pieces of past research on this topic were simply identifying different mechanisms that can support resonance.

The Richtmyer–Meshkov instability at gas interfaces with controllable initial perturbation spectra under reshock conditions is investigated both experimentally and theoretically. A soap-film method is adopted to generate well-defined single-, dual- and triple-mode air/SF$_6$ interfaces. By inserting an acrylic block into the test section, a reflected shock with controllable reshock timing is created. The results reveal a complex relationship between the post-reshock perturbation growth rate and the pre-reshock interface morphology. For single-mode interfaces, the post-reshock growth rate exhibits a strong dependence on pre-reshock conditions. In contrast, for multi-mode interfaces, this dependence weakens significantly due to mode-coupling effects. It is found that, following reshock, each fundamental mode develops independently and later is significantly influenced by mode-coupling effects. Based on this finding, we propose an empirical model that matches the initial linear growth rate and the asymptotic growth rate, accurately predicting the evolution of fundamental modes from early to late stages across all three configurations. Furthermore, a theoretical formula is derived, linking the empirical coefficient in the model of Charakhch’An (2020 J. Appl. Mech. Tech. Phys. vol. 41, no. 1, pp. 23–31) to the initial perturbation. This provides a unified framework to explain the varying dependence of post-reshock growth rates on pre-reshock morphology observed in previous experiments.