We investigate the linear instability of flows that are stable according to Rayleigh’s criterion for rotating fluids. Using Taylor–Couette flow as a primary test case, we develop large-Reynolds-number-matched asymptotic expansion theories. Our theoretical results not only aid in detecting instabilities previously reported by Deguchi (Phys. Rev. E, vol 95, 2017, p. 021102(R)) across a wide parameter range, but also clarify the physical mechanisms behind this counterintuitive phenomenon. Instability arises from the interaction between large-scale inviscid vortices and the viscous flow structure near the wall, which is analogous to Tollmien–Schlichting waves. Furthermore, our asymptotic theories and numerical computations reveal that similar instability mechanisms occur in boundary layer flows over convex walls.

In this work, the stability and transition to turbulence over blunt flat plates with different leading-edge radii are investigated computationally. The benchmark experimental work for comparative studies is conducted by Borovoy et al. (AIAA J., vol. 60, 2022, pp. 497–507). The freestream Mach number is 5, the unit Reynolds number is $6\times 10^7$ m$^{-1}$, and the maximum nose-tip radius 3 mm exceeds the experimental reversal value. High-resolution numerical simulation and stability analysis are performed. Three-dimensional broadband perturbation is added on the far field boundary to initiate the transition. The highlight of this work is that the complete physical process is considered, including the three-dimensional receptivity, linear and nonlinear instabilities, and transition. The experimental reversal phenomenon is reproduced favourably in the numerical simulation for the first time. Linear stability analysis shows that unstable first and second modes are absent in the blunt-plate flows owing to the presence of the entropy layer, although these modes are evident in the sharp-leading-edge case. Therefore, the transition on the blunt plate is due to non-modal instabilities. Numerical results for all the blunt-plate cases reveal the formation of streamwise streaky structures downstream of the nose (stage I) and then the presence of intermittent turbulent spots in the transitional region (stage II). In stage I, a preferential spanwise wavelength approximately 0.9 mm is selected for all the nose-tip radii, and low-frequency components are dominant. In stage II, high-frequency secondary instabilities appear to grow, which participate in the eventual breakdown. By contrast, leading-edge streaks are not remarkable in the sharp-leading-edge case, where transition is induced by oblique first and Mack second modes. The transition reversal beyond the critical nose-tip radius arises from an increasing magnitude of the streaky response in the early stage, while the transition mechanism stays similar qualitatively.

Green water loads on prismatic obstacles (representing topside structures) mounted on the raised deck of a simplified vessel are investigated using computational fluid dynamics simulations and physical model testing with emphasis on examining different structure shapes, orientation angles and relative structure size. For each scenario investigated, several flow features are identified that characterize the green water interaction with the structure and influence loads, namely delayed flow diversion, formation of a vertical jet, scattered wave formation and the development of complex wake patterns. Comparing across structures, these interactions are more pronounced for blunt objects, and the associated force impulse is larger. For example, a cube with flow at normal incidence is found to experience approximately twice the force impulse of a circular cylinder of the same projected area. Equally, rotation of the cube leads to reduced run-up height and streamwise force on the structure. To explain these trends, a theoretical model based on Newtonian flow theory is adopted. This model provides an estimate of the streamwise force exerted on obstacles in high-Froude-number flows and shows good agreement with the numerical results when the flow is supercritical, shallow (small water depth relative to structure width) and the structure is tall (large structure height relative to water depth). Despite some limitations, the model should provide an efficient force prediction tool for practical use in design.

The nonlinear stability of two-dimensional (2-D) plane Couette flow subject to a constant throughflow is analysed at finite and asymptotically large Reynolds numbers $\textit {Re}$. The speed of this throughflow is quantified by the non-dimensional throughflow number $\eta$. The base flow exhibits a linear instability provided $\eta \gtrsim 3.35$, with multi-deck upper and lower branch structures developing in the limit $1\ll \eta \ll \mathit {O}(\textit {Re})$. This instability provides a springboard for the computation of nonlinear travelling waves which bifurcate subcritically from the linear neutral curve, allowing us to map out a neutral surface at different values of $\eta$. Using strongly nonlinear critical layer theory, we investigate the waves that bifurcate from the upper branch at asymptotically large $\textit {Re}$. This asymptotic structure exists provided the throughflow number is larger than the critical value of $\eta _c\approx 1.20$ and is shown to give quantitatively similar results to the numerical solutions at Reynolds numbers of $\mathit {O}(10^5)$.

The problem of axisymmetric supersonic laminar flow separation over a compression corner has not been considered within the framework of triple-deck theory for several decades, despite significant advances in both theoretical methods and numerical techniques. In this study, we revisit the problem considered by Gittler & Kluwick (J. Fluid Mech., vol. 179, 1987, pp. 469–487), using the numerical method of Ruban (Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol. 18, issue 5, 1978, pp. 1253–1265) and Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285), termed the Ruban–Cassel method (RCM). The solution shows good agreement with the results of Gittler & Kluwick (J. Fluid Mech., vol. 179, 1987, pp. 469–487) for a scale external radius of 1 and scale angles from 1 to 6. However, for scale angles above 6.8, a wave packet appears. This wave packet is similar to that reported by Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285) for two-dimensional supersonic flow. As the external scale radius increases (from 1 to 10), the axisymmetric solution converges towards the two-dimensional solution for equivalent scale angle values. For a scale external radius of 10, the wave packet appears at a scale angle of 3.8, compared with the value of 3.9 by Cassel et al. (J. Fluid Mech., vol. 300, 1995, pp. 265–285). Inspection of the velocity profiles reveals that inflection points, while ubiquitous in shear flow, do not seem to play a relevant role in the appearance of the wave packet for the axisymmetric flow. Axisymmetric effects become more important as the scale external radius decreases below 0.5. A larger scale angle is necessary to produce a flow structure equivalent to that of the two-dimensional case. For scale external radius 0.1, the pressure gradient is substantially diminished and the solution is devoid of a second shear-stress minimum.

We study the generation, nonlinear development and secondary instability of unsteady Görtler vortices and streaks in compressible boundary layers exposed to free-stream vortical disturbances and evolving over concave, flat and convex walls. The formation and evolution of the disturbances are governed by the compressible nonlinear boundary-region equations, supplemented by initial and boundary conditions that characterise the impact of the free-stream disturbances on the boundary layer. Computations are performed for parameters typical of flows over high-pressure turbine blades, where the Görtler number, a measure of the curvature effects, and the disturbance Reynolds number, a measure of the nonlinear effects, are order-one quantities. At moderate intensities of the free-stream disturbances, increasing the Görtler number renders the boundary layer more unstable, while increasing the Mach number or the frequency stabilises the flow. As the free-stream disturbances become more intense, vortices over concave surfaces no longer develop into the characteristic mushroom-shaped structures, while the flow over convex surfaces is destabilised. An occurrence map identifies Görtler vortices or streaks for different levels of free-stream disturbances and Görtler numbers. Our calculations capture well the experimental measurements of the enhanced skin friction and wall-heat transfer over turbine-blade pressure surfaces. The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances. Nonlinearly saturated disturbances are unstable to secondary high-frequency modes, whose growth rate increases with the Görtler number. A new varicose even mode is reported, which may promote transition to turbulence at the stem of nonlinear streaks.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

In transonic flow conditions, buffeting associated with finite-amplitude lift fluctuations can limit the operational envelope of an aircraft. For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation. We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behaviour of the oscillatory buffet-onset instability on airfoils. The flow is governed by the unsteady Reynolds-averaged Navier–Stokes equations, with a basic state provided by a steady-state solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation. The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states. The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil. Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behaviour needed to estimate the buffet-onset boundary. The stability analysis is shown to capture the envelope of lift oscillations during a continuous pitch of an airfoil, from pre-buffet through post-buffet lift levels.

The spatio-temporal scales, as well as a comprehensive self-sustained mechanism of the reattachment unsteadiness in shock wave/boundary layer interaction, are investigated in this study. Direct numerical simulations reveal that the reattachment unsteadiness of a Mach 7.7 laminar inflow causes over $26\,\%$ variation in wall friction and up to $20\,\%$ fluctuation in heat flux at the reattachment of the separation bubble. A statistical approach, based on the local reattachment upstream movement, is proposed to identify the spanwise and temporal scales of reattachment unsteadiness. It is found that two different types, i.e. self-induced and random processes, dominate different regions of reattachment. A self-sustained mechanism is proposed to comprehend the reattachment unsteadiness in the self-induced region. The intrinsic instability of the separation bubble transports vorticity downstream, resulting in an inhomogeneous reattachment line, which gives rise to baroclinic production of quasi-streamwise vortices. The pairing of these vortices initiates high-speed streaks and shifts the reattachment line upstream. Ultimately, viscosity dissipates the vortices, triggering instability and a new cycle of reattachment unsteadiness. The temporal scale and maximum vorticity are estimated with the self-sustained mechanism via order-of-magnitude analysis of the enstrophy. The advection speed of friction, derived from the assumption of coherent structures advecting with a Blasius-type boundary layer, aligns with the numerical findings.

Supersonic shear layers experience instabilities that generate significant adverse effects; in complex configurations, these instabilities have global impacts as they foster compounding complications with other independent flow features. We consider the flow near the exit of a dual-stream rectangular nozzle. The supersonic core and sonic bypass streams mix downstream of a splitter plate trailing edge (SPTE) just above an adjacent deck. We perform two-dimensional direct numerical simulations with laminar inflow conditions to parametrically explore the influence of active flow control, considering various actuation angles and locations. The goal is to alleviate the prominent tone associated with vortices shed at the SPTE; these vortices initiate an unsteady shock system that affects the entire flow field through a shock-induced separation and the downstream evolution of plume shear layers. Resolvent analysis is performed on the baseline flow. The identified optimal response guides the placement of steady-blowing actuators. Since the resolvent analysis fundamentally investigates the input–output dynamics of a system, it is also utilized to uncover actuation-induced changes in the forcing–response dynamics. Spectral analysis shows that the baseline flow fluctuating energy is concentrated in the shedding instability. Actuating at optimal angles based on location disperses this energy into various flow features; this affects the shedding itself, and the structure and unsteadiness of the shock system, and thus the response of the deck and nozzle wall boundary layers and the plume. The resolvent analysis indicates, and Navier–Stokes solutions confirm, that favourable control is obtained by either indirectly or directly mitigating the baseline instability.

A previous paper of the authors (Duck & Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

We present a Mach 15 air flow over a blunt two-dimensional wedge simulated using the direct molecular simulation method. As electronically excited states are not modelled, the resulting air mixture around the wedge contains the electronic ground states only, namely ${\rm N}_2(\text {X}^1 \varSigma _g^{+})$, ${\rm O}_2(\text {X}^3 \varSigma _g^{-})$, ${\rm NO}(\text {X}^2\varPi _r)$, ${\rm N}(^4{\rm S})$ and ${\rm O}(^3{\rm P})$. All the potential energy surfaces (PESs) that are used to model the various interactions between air particles are ab initio, with two notable exceptions, namely ${\rm N}_2+{\rm NO}$ and ${\rm O}_2+{\rm NO}$. At the selected free-stream conditions, strong vibrational non-equilibrium is observed in the shock layer. The flow is characterized by significant chemical activity, with near-complete oxygen dissociation, considerable formation of NO and minimal molecular nitrogen dissociation. Complex mass diffusion kinetics, driven by composition, temperature and pressure gradients, are identified in the shock layer. All these physical phenomena are directly coupled to, and responsible for, the mechanics of the gas flow and are all solely traceable to the PESs’ inputs, without the need for any thermochemical models, mixing rules or constitutive laws for transport properties. Because the flow is entirely at near-continuum conditions, it is a gas-phase thermophysics benchmark that is useful to enhance the fidelity of continuum models used in computational fluid dynamics of hypersonic flows.

In the present study, we perform direct numerical simulations of compressible turbulent boundary layers at free stream Mach numbers $2\unicode{x2013}6$ laden with dilute phase of spherical particles to investigate the Mach number effects on particle transport and dynamics. Most of the phenomena observed and well-recognized for inertia particles in incompressible wall-bounded turbulent flows – such as near-wall preferential accumulation and clustering beneath low-speed streaks, flatter mean velocity profiles, and trend variation of the particle velocity fluctuations – are identified in the compressible turbulent boundary layer as well. However, we find that the compressibility effects are significant for large inertia particles. As the Mach number increases, the near-wall accumulation and the small-scale clustering are alleviated, which is probably caused by the variations of the fluid density and viscosity that are crucial to particle dynamics. This can be affected by the fact that the forces acting on the particles with viscous Stokes number greater than 500 are modulated by the comparatively high particle Mach numbers in the near-wall region. This is also the reason for the abatement of the streamwise particle velocity fluctuation intensities with the Mach numbers.

Coherent small-amplitude unsteadiness of the shock wave and the separation region over a canonical double cone flow, termed in literature as oscillation-type unsteadiness, is experimentally studied at Mach 6. The double cone model is defined by three non-dimensional geometric parameters: fore- and aft-cone angles ($\theta _1$ and $\theta _2$), and ratio of the conical slant lengths ($\varLambda$). Previous studies of oscillations have been qualitative in nature, and mostly restricted to a special case of the cone model with fixed $\theta _1 = 0^\circ$ and $\theta _2 = 90^\circ$ (referred to as the spike-cylinder model), where $\varLambda$ becomes the sole governing parameter. In the present effort we investigate the self-sustained flow oscillations in the $\theta _1$-$\varLambda$ parameter space for fixed $\theta _2 = 90^\circ$ using high-speed schlieren visualisation. The experiments reveal two distinct subtypes of oscillations, characterised by the motion (or lack thereof) of the separation point on the fore-cone surface. The global time scale associated with flow oscillation is extracted using spectral proper orthogonal decomposition. The non-dimensional frequency (Strouhal number) of oscillation is seen to exhibit distinct scaling for the two oscillation subtypes. The relationship observed between the local flow properties, instability of the shear layer, and geometric constraints on the flow suggests that an aeroacoustic feedback mechanism sustains the oscillations. Based on this understanding, a simple model with no empiricism is developed for the Strouhal number. The model predictions are found to match well with experimental measurements. The model provides helpful physical insight into the nature of the self-sustained flow oscillations over a double cone at high speeds.

Cowl-induced incident shock wave/boundary layer interactions (ISWBLIs) under the influence of shoulder expansion represent one of the dominant phenomena in supersonic inlets. To provide a more comprehensive understanding of how an expansion corner affects the ISWBLI, a detailed experimental and analytical study is performed in a Mach 2.73 flow in this work. Pressure measurement, schlieren photography and surface oil-flow visualisation are used to record flow features, including the pressure distribution, separation extent and surface-flow topological structures. Our results reveal three types of ISWBLIs influenced by the expansion corner. When the shock intensity is weak, the separation is small scale with the expansion waves emanating from the expansion corner. This is the first type of expansion-corner-affected ISWBLI (EC-ISWBLI). When the incident shock wave is strong, large-scale separation occurs, accompanied by the disappearance of expansion waves, forming the second type of EC-SWBLI. The expansion corner induces a ‘lock-in’ effect in which the separation onset is consistently locked near the expansion corner regardless of the incident shock intensity and impingement position. The third type of EC-ISWBLI occurs when the shock is sufficiently strong and the impingement point is close to the expansion corner. In this interaction, the ‘lock-in’ effect ceases to manifest. Moreover, a shock polar-incorporating inviscid model is employed to elucidate the shock patterns. Two criteria are established by combining free interaction theory with this model. The first criterion provides valuable insights into the evolution of separations with a minimal overall pressure rise and the second criterion determines the threshold for the occurrence of the ‘lock-in’ effect.

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2$^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

The geometrical properties of streamlines, such as the curvatures, directions and positions, are studied in steady inviscid compressible flow fields via differential geometry theories and conservation laws. The influences of the streamline geometries on the flow speeds and pressures are also identified and discussed. By transforming the streamlines to fill the domain and satisfy the boundary conditions, a unified geometry-based solver, the streamline transformation method, is proposed for both subsonic and supersonic regions. The governing equations and boundary conditions along streamlines and shock waves are also derived. This method is verified by numerical results of three typical flow fields, including the subsonic channel flow, the supersonic downstream of attached shock waves and especially the subsonic/supersonic downstream of detached bow shock waves. Both two-dimensional planar and axisymmetric flow fields are considered. Compared with the results from computational fluid dynamics, good agreements are achieved by this method, while fewer computational resources, by an order of magnitude, are consumed. Features of these flow fields are also analysed from a geometrical perspective, such as flow speeds and pressures deviated by the wall curvatures, and three-dimensional effects in the after-shock flow fields. For a hyperbolic-shaped bow shock wave, the stand-off distances and the transitions from subsonic to supersonic regions are also discussed. As indicated by the accuracy, efficiency and applicability in a wide range of flow speeds, the streamline transformation method would be a potential candidate for the theoretical analysis and inverse design of high-speed flow fields, especially where the subsonic regions exist downstream of strong shock waves.

This work is a numerical and experimental study of a rectangular thin plate undergoing stall flutter at Mach 0.8. This constitutes one of the first studies of this kind where three-dimensionality is fully implemented in a numerical simulation including the test-section effects characterizing wind-tunnel experiments. In order to break down the fluid–structure interaction to its main driving phenomena, an aerodynamic model is proposed that is based on computationally inexpensive steady-state simulations. Two types of dynamic instability are observed in the numerical simulations; Flutter by mode coalescence is promoted at zero flow incidence, however, high bending precludes this from happening for higher values of angle of attack. Stall flutter is instead a nonlinear one-degree type of instability. Both of these instability mechanisms can be explained in terms of hysteretic behaviour of the pressure distribution, which becomes more pronounced at high angles of attack, when a large separation region is formed. Tests were conducted employing titanium alloy plates in order to survive the aerodynamic loads characterizing the wind-tunnel initial transient. However, due to wall interference, high bending was promoted so that the internal stress exceeded the yield values before flutter could be measured. Numerical simulations were in general agreement with the experiment in terms of both amplitude and oscillation frequency.

This paper investigates the effect of curvature on curved detonation and its reflections. Specifically, the study focuses on two aspects: the effect of curvature on the postwave parameters and their gradients, and the stabilization of Mach reflection. Relationships are established between the curvature and the gradients of the postwave parameters, thus providing a basis for examining detonation reflections and obtaining a comprehensive understanding of curved detonation. In particular, these relationships offer a valuable analytical tool to predict the postwave gradients, as well as providing a fresh perspective to understand the transformation from Mach reflection to regular reflection in curved detonation. The validity of these relationships is confirmed by comparison with simulation results. Two mechanisms by which curvature influences the stationarity of Mach reflection are identified. An increase in wave angle and interference between wave systems leading to the generation and integration of subsonic zones are the reasons for the non-stationarity of the Mach reflection in curved detonation. Besides, the effect mechanisms of choked flow which is considered to be the root cause are analysed in detail. On the basis of a theoretical model, the development of a quantitative criterion for the stability of detonation reflection is proposed, and its validity is confirmed by simulations. This criterion is used in a comprehensive investigation of the primary factors affecting the stability of detonation wave reflections, providing insights that will be of great value for the further development of detonation engines.