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.

Motivated by the need for a better understanding of the melting and stability of floating ice bodies, we experimentally investigated the melting of floating ice cylinders. Experiments were carried out in a tank, with ice cylinders with radii between 5 and 12 cm, floating horizontally with their axis perpendicular to gravity. The water in the tank was at room temperature, with salinities ranging from 0 to 35 g l−1. These conditions correspond to Rayleigh numbers in the range 10$^5\lesssim$ Ra $\lesssim$ 10$^9$. The relative density and thus the floating behaviour was varied by employing ice made of H$_2$O–D$_2$O mixtures. In addition, we explored a two-layer stable stratification. We studied the morphological evolution of the cross-section of the cylinders and interpreted our observations in the context of their interaction with the convective flow. The cylinders only capsize in fresh water but not when the ambient is saline. This behaviour can be explained by the balance between the torques exerted by buoyancy and drag, which change as the cylinder melts and rotates. We modelled the oscillatory motion of the cylinders after a capsize as a damped nonlinear oscillator. The downward plume of the ice cylinders follows the expected scalings for a line-source plume. The plume’s Reynolds number scales with Rayleigh number in two regimes, namely Re $\propto$ Ra$^{1/2}$ for Ra $\lt \mathcal{O}(10^7)$ and Re $\propto$ Ra$^{1/3}$ for Ra $\gt \mathcal{O}(10^7)$, and the heat transfer (non-dimensional as Nusselt number) scales as Nu $\propto$ Ra$^{1/3}$. Although the addition of salt substantially alters the solutal, thermal and momentum boundary layers, these scaling relations hold irrespectively of the initial size or the water salinity. While important differences exist between our experiments and real icebergs, our results can qualitatively be connected to natural phenomena occurring in fjords and around isolated icebergs, especially with regard to the melting and capsizing behaviour in stratified waters.

We examine the circular, self-similar expansion of frictional rupture due to fluid injected at a constant rate. Fluid migrates within a thin permeable layer parallel to and containing the fault plane. When the Poisson ratio $\nu =0$, self-similarity of the fluid pressure implies fault slip also evolves in an axisymmetric, self-similar manner, reducing the three-dimensional problem for the evolution of fault slip to a single self-similar dimension. The rupture radius grows as $\lambda \sqrt {4\alpha _{hy} t}$, where $t$ is time since the start of injection and $\alpha _{hy}$ is the hydraulic diffusivity of the pore fluid pressure. The prefactor $\lambda$ is determined by a single parameter, $T$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range $0\lt \lambda \lt \infty$, the lower and upper limits of which correspond to marginal pressurisation of the fault and critically stressed conditions, in which the fault-resolved shear stress is close to the pre-injection fault strength. In both limits, we derive solutions for slip by perturbation expansion, to arbitrary order. In the marginally pressurised limit ($\lambda \rightarrow 0$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($\lambda \rightarrow \infty$), the perturbation is singular, contains a boundary layer and an outer solution, and the series is divergent. In this case, we provide a composite solution with uniform convergence over the entire rupture using a matched asymptotic expansion. We provide error estimates of the asymptotic expansions in both limits and demonstrate optimal truncation of the singular perturbation in the critically stressed limit.

Wettability quantifies the affinity of a liquid over a substrate and determines whether the surface is repellent or not. When both the liquid and the solid phases are made of the same chemical substance and are at thermal equilibrium, complete wetting is expected in principle, as observed, for instance, with drops of molten metals spreading on their solid counterparts. However, this is not the case for water on ice. Although there is a growing consensus on the partial wetting of water on ice and several estimates available for the value of the associated macroscopic contact angle, the question of whether these values correspond to the contact angle at mechanical and thermal equilibrium is still open. In the present paper, we address this issue experimentally and demonstrate the existence of such a macroscopic contact angle of water on ice, from measurements and theoretical arguments. Indeed, when depositing water droplets on smooth polycrystalline ice layers with accurately controlled surface temperatures, we observe that spreading is unaffected by thermal effects and phase change close enough to the melting point (namely, for undercoolings below 1 K) so that conditions of thermal equilibrium are closely approached. Whereas the short time motion of the contact line is driven by an inertial-capillary balance, the evolution towards mechanical equilibrium is described by a viscous-capillary dynamics and is therefore capillary – and not thermally – related. Moreover, we show that the resulting contact angle remains constant for undercoolings below 1 K. In this way, we show the existence of a non-zero macroscopic contact angle of water on ice under conditions of mechanical and thermal equilibrium, which is very close to $12^\circ$. We anticipate this key finding will significantly improve the understanding of capillary flows in the presence of phase change, which is of special interest in the realm of ice morphogenesis and glaciology, and will also be beneficial with the aim of developing numerical methods for resolving triple-line dynamics.

The inertial migration of hydrogel particles suspended in a Newtonian fluid flowing through a square channel is studied both experimentally and numerically. Experimental results demonstrate significant differences in the focusing positions of the deformable and rigid particles, highlighting the role of particle deformability in inertial migration. At low Reynolds numbers (${Re}$), hydrogel particles migrate towards the centre of the channel cross-section, whereas the rigid spheres exhibit negligible lateral motion. At finite ${Re}$, they focus at four points along the diagonals in the downstream cross-section, in contrast to the rigid particles which focus near the centre of the channel face at similar ${Re}$. Numerical simulations using viscous hyperelastic particles as a model for hydrogel particles reproduced the experimental results for the particle distribution with an appropriate Young’s modulus of the hyperelastic particles. Further numerical simulations over a broader range of ${Re}$ and the capillary number ($Ca$) reveal various focusing patterns of the particles in the channel cross-section. The phase transitions between them are discussed in terms of the inertial lift and the lift due to particle deformation, which would act in the direction towards lower shear. The stability of the channel centre is analysed using an asymptotic expansion approach to the migration force at low ${Re}$ and $Ca$. The theoretical analysis predicts the critical condition for the transition, which is consistent with the direct numerical simulation. These experimental, numerical and theoretical results contribute to a deeper understanding of inertial migration of deformable particles.

In this paper, a freely falling circular cylinder attached by a splitter plate in an infinite fluid domain under gravity is investigated numerically. The kinematic modes and wake patterns are summarised, and their parametric sensitivity with the dimensionless plate length ($L^\ast$), the Galileo number ($Ga$) and the cylindric-fluid density ratio ($\rho ^\ast$) is studied. The kinematic modes of a freely falling circular cylinder with a splitter plate can be classified into six types: the steady falling, the steady oblique falling, the small vibration oblique falling, the zigzag oblique falling, the locked falling and the chaotic falling. In the meantime, the wake patterns can be summarised into five types: the steady wake, the 2S wake, the 2P + nS wake, the 2P + 2S wake, and the chaotic wake. The effect of the length of the splitter plate on the vortex shedding characteristics represented by the Strouhal number is also discussed. Further investigation reveals that the attachment of a splitter plate of different lengths to the rear not only influences the kinematic mode and the vortex shedding of the circular cylinder, but also allows the passive and precise control of its falling posture and trajectory. Finally, through theoretical analysis, scaling laws are proposed to estimate the turn angle $\alpha$ and the drift angle $\beta$. The present study can deepen the understanding of similar natural phenomena, such as gliding birds and falling maple seeds, and provide valuable reference for engineering design of drag-reduction devices or air-dropped objects.

Rotor–stator interactions in turbomachines are characterised by a complex interplay of hydrodynamic instabilities, acoustic pressure waves and receptivity mechanisms, as well as the collision of coherent structures with the blade geometry. An unsteady dual analysis of self-excited instabilities and flow interactions, exemplified by a simple model compressor stage under subsonic conditions, is proposed and presented. Using a low-dissipation sliding-plane implementation, instability-resolving nonlinear-adjoint looping simulations provide detailed sensitivity information that allows for the dissection of the full flow into sub-components linked to distinct flow phenomena. This sensitivity information further links observed flow behaviour to its hydrodynamic or acoustic origin, thereby laying the foundation for a cause-and-effect analysis and for flow control.

When a drop impinges onto a deep liquid pool, it can yield various splashing behaviours, leading to a crown-like structure along the free surface. Under high-speed impact conditions, the upper portion of the thin-walled crown may undergo necking and encapsulate a large bubble, which remains fascinating and is rarely discussed in the literature. In this work, we numerically study this physical process based on the volume-of-fluid and adaptive mesh refinement framework. Our meticulous observations have allowed us to unveil a spectrum of repeatable early-time jet behaviours, vorticity structures and crater evolution, underscoring the rich and complex nature of drop-impact phenomenon. We show that the interplay between aerodynamic pressure and surface tension on the liquid crown could play a significant role in its bending and surface closure. A regime map, incorporating both early-stage jet dynamics and overall bubble-canopy formation, is established across a wide parameter space. This study provides a comprehensive understanding of the diverse splashing regimes, offering insights into the fundamental characteristics of drop-impact phenomenon.

The presence of salt in seawater significantly affects the melt rate and morphological evolution of ice. This study investigates the melting process of a vertical cylinder in saline water using a combination of laboratory experiments and direct numerical simulations. The two-dimensional (2-D) direct numerical simulations and three-dimensional (3-D) experiments achieve thermal Rayleigh numbers up to $\textit{Ra}_{T}= \mathcal{O} (10^{9} )$ and saline Rayleigh numbers up to $\textit{Ra}_{S}=\mathcal{O} (10^{12} )$. Some 3-D simulations of the vertical ice cylinder are conducted at $\textit{Ra}_{T}= \mathcal{O} (10^{5} )$ to confirm that the results in 2-D simulations are qualitatively similar to those in 3-D simulations. The mean melt rate exhibits a non-monotonic relationship with ambient salinity. With increasing salinity, the mean melt rate initially decreases towards the point where thermal and saline effects balance, after which it increases again. Based on the ambient salinity, the flow can be categorised into three regimes: temperature-driven flow, salinity-driven flow and thermal-saline competing flow. In the temperature-driven and competing flow regimes, we find that the mean melt rate follows a $\textit{Ra}_{T_d}^{1/4}$ scaling, where the subscript $d$ denotes a response parameter. In contrast, in the salinity-driven flow regime, we see a transition from a $\textit{Ra}_{T_d}^{1/4}$ to a $\textit{Ra}_{T_d}^{1/3}$ scaling. Additionally, the mean melt rate follows a $\textit{Ra}_{S_d}^{1/3}$ scaling in this regime. The ice cylinder develops distinct morphologies in different flow regimes. In the thermal-saline competing flow regime, distinctive scallop (dimpled) patterns emerge along the ice cylinder due to the competition between thermal buoyancy and saline buoyancy. We observe these scallop patterns to migrate downwards over time, due to local differences in the melt rate, for which we provide a qualitative explanation.

This paper presents an analytical method for modelling the acoustic field radiation from a semi-infinite elliptic duct in the presence of uniform subsonic flow. In contemporary aircraft design, elliptic ducts play crucial roles as inlets for advanced blended wing body configurations owing to their capacity to maximise the pre-compression effect of the fuselage and enhance the stealth performance of aircraft. The method uses Mathieu functions to describe the incident and scattered sound in the elliptic cylindrical coordinates. An analytical Wiener–Hopf technique is developed in this work to derive near- and far-field solutions. Numerical simulations based on a finite element method are conducted to validate the accuracy of the analytical method, revealing a strong correspondence with analytical predictions. A parametric study is conducted to explore the influence of the elliptic cross-section shape on noise directivity. Moreover, we investigate reflections within the duct via an extended derivation of the analytical model. The proposed method can be used to examine the acoustic characteristics of elliptic ducts with inflow mean flows, which holds relevance for noise control and optimisation of turbofan engine inlets and blended wing body applications.

We analyse direct numerical simulations of homogeneous, forced, stably stratified turbulence to study how the pressure–strain and pressure scrambling terms are modified as stability is increased from near neutral to strongly stratified conditions. We decompose the pressure into nonlinear and buoyancy components, and find that the buoyancy part of the pressure–strain correlation changes sign to promote large-scale anisotropy at strong stability, unlike the nonlinear component, which always promotes large-scale isotropy. The buoyancy component of the pressure scrambling term is positive semidefinite and increases monotonically with stability. As its magnitude becomes greater than the nonlinear component (which is negative), the overall scrambling term generates buoyancy flux at very strong stability. We apply quadrant analysis (in the pressure-gradient space) to these correlations to study how contributions from the four quadrants change with stability. Furthermore, we derive exact relationships for the volume-averaged buoyancy components of these correlations which reveal (i) the buoyancy component of the pressure–strain correlation involves a weighted sum of the vertical buoyancy flux cospectrum, so counter-gradient buoyancy fluxes contribute to enhanced anisotropy by transferring vertical kinetic energy into horizontal kinetic energy; (ii) the buoyancy component of the pressure scrambling term involves a weighted sum of the potential energy spectrum; (iii) the weighting factor accentuates contributions from layered motions, which are a prominent feature of strongly stratified flows. These expressions apply generally to all homogeneous stratified flows independent of forcing and across all stability conditions, explaining why these effects have been observed for both forced and sheared stably stratified turbulence simulations.

We introduce a novel approach to derive compressibility corrections for Reynolds-averaged Navier–Stokes (RANS) models. Using this approach, we derive variable-property corrections for wall-bounded flows that take into account the distinct scaling characteristics of the inner and outer layers, extending the earlier work of Otero Rodriguez et al. (Intl J. Heat Fluid Flow, 73, 2018, 114–123). We also propose modifying the eddy viscosity to account for changes in the near-wall damping of turbulence due to intrinsic compressibility effects. The resulting corrections are consistent with our recently proposed velocity transformation (Hasan et al. Phys. Rev. Fluids, 8, 2023, L112601) in the inner layer and the Van Driest velocity transformation in the outer layer. Furthermore, we address some important aspects related to the modelling of the energy equation, primarily focusing on the turbulent Prandtl number and the modelling of the source terms. Compared with the existing state-of-the-art compressibility corrections, the present corrections, combined with accurate modelling of the energy equation, lead to a significant improvement in the results for a wide range of turbulent boundary layers and channel flows. The proposed corrections have the potential to enhance modelling across a range of applications, involving low-speed flows with strong heat transfer, fluids at supercritical pressures, and supersonic and hypersonic flows.

The effect of uniform wall suction on compressible Görtler vortices excited by free stream vortical disturbances is studied via asymptotic and numerical methods. The flow is described by the boundary-region framework, written and solved herein for non-similar boundary layers. The suction, applied downstream of an impermeable region, reduces the amplitude of steady and unsteady Görtler vortices. The vortices are attenuated more when the boundary layer has reached the asymptotic-suction condition than when it is streamwise-dependent. The impact of suction weakens as the free stream Mach number increases. As the boundary layer becomes thinner, the exponential growth of the vortices is prevented because the disturbance spanwise pressure gradient and spanwise viscous diffusion are inhibited. The flow is described by the boundary-layer equations in this case, for which the wall-normal momentum equation is uninfluential at leading order and the curvature effects responsible for the inviscid pressure-centrifugal imbalance are therefore negligible. The influence of unsteadiness weakens as suction intensifies because, in the limit of a thin boundary layer, the boundary-region solution simplifies to a regular-perturbation series whose first terms are described by the steady boundary-layer equations. Suction broadens the stability regions and may favour the presence of oblique Tollmien–Schlichting waves at the expense of more energetic Görtler vortices for relatively high frequencies and moderate Mach numbers.

In compressible turbulent boundary layers (CTBLs), the strong Reynolds analogy (SRA) refers to a set of quantitative relationships between temperature and velocity fluctuations. The essence of the SRA is the linear relationship between these fluctuations in large-scale motions. We investigate the transport processes of the second-order statistical moments associated with temperature and velocity fluctuations to reveal the physical mechanisms underlying this linear correlation. An important finding is that there exists a strong linear mechanism between the turbulent production of velocity and temperature fluctuations. Nonlinear mechanisms, such as the viscous-thermal dissipation, the work contribution, and particularly the pressure term, lead to the failure of the existing SRAs in the outer layer. Based on the above findings, a refined SRA (RSRA) is proposed, which better describes the quantitative relation between the temperature and velocity fluctuation intensities. An approximate expression for the turbulent Prandtl number under different Mach numbers and wall-cooling conditions is derived with the newly proposed RSRA. The relations proposed in this paper are validated through the direct numerical simulation data of flat-plate zero-pressure-gradient CTBLs at different Mach numbers and wall temperatures.

Refraction is the predominant mechanism causing spatially inhomogeneous surface gravity wave fields. However, the complex interplay between depth- and current-induced wave refraction remains poorly understood. Assuming weak currents and slowly varying bathymetry, we derive an analytical approximation to the wave ray curvature, which is validated by an open-source ray tracing framework. The approximation has the form of linear superposition of a current- and a depth-induced component, each depending on the gradients in the ambient fields. This separation enables quantification of their individual and combined contributions to refraction. Through analysis of a few limiting cases, we demonstrate how the sign and magnitude of these components influence the wave refraction, and identify conditions where they either amplify or counteract each other. We also identify which of the two plays a dominant role. These findings provide physically resolved insights into the influence of current and depth gradients on wave propagation, and are relevant for applications related to remote sensing and coastal wave forecasting services.

Sink flow boundary layers on smooth and rough walls were studied experimentally. In all cases a turbulent, zero-pressure-gradient boundary layer was subject to acceleration with K = 3.2 × 10–6, which suppressed the turbulence in the outer region and produced conditions similar to those in turbulent sink flow cases with lower K. In the smooth-wall case, after the momentum thickness Reynolds number had dropped to about 600, the near-wall turbulence then dropped, resulting in relaminarisation. In the rough-wall cases, the near-wall turbulence was sustained in spite of the strong favourable pressure gradient, and relaminarisation did not occur. A temporary equilibrium appears to occur that is similar to that seen with lower K, in spite of the ratio of the boundary-layer thickness to the roughness height dropping to less than 5. Mean velocity and Reynolds stress profiles, quadrant analysis and turbulence spectra are used to show the development of the boundary layer in response to the pressure gradient and the differences between the rough- and smooth-wall cases. This is believed to be the first study to consider the spatial evolution of constant-K rough-wall boundary layers with K large enough to cause relaminarisation in the smooth-wall case.

While flow confinement effects on a shear layer of an one-sided or submerged vegetation array’s interface have been widely studied, turbulent interactions between shear layers in channels with vegetation on both sides remain unclear. This study presents laboratory experiments investigating flow adjustments and turbulent interaction within a symmetrical vegetation–channel–vegetation system, considering varying array widths and densities. In the outer shear layer, the shear stress is primarily balanced by the pressure gradient. As the array extends laterally, the outer penetration of the shear layer reduces from a fully developed thickness to the half-width of the open region, resulting in flow confinement. Flow confinement enhances the pressure gradient, which increases the interior velocity and shear stress at the interface. Despite the time-averaged shear stress being zero at the centreline when the shear layer is confined, the shear instabilities from both sides interact, producing significant turbulent events at the centreline with equal contributions from each side. Furthermore, the two parallel vortex streets self-organised and created a wave response with a $\pi$-radian phase shift , where alternating vortex cores amplify the pressure gradient, intensifying coherent structures and facilitating momentum exchange across the channel centreline. Although the turbulent intensity is enhanced, the decreased residence time for turbulent flow events may limit transport distance. Overall, the shear layer that develops on one interface acts as an additional resistance to shear turbulence on the other interface, leading to a more rapid decline of shear stress in the open region, despite a higher peak at the interface.

In this paper we propose a novel control strategy for modulating nonlinear flapping and symmetry-breaking (S-B) bifurcations of a piezoelectric metamaterial beam behind a circular cylinder subjected to viscous flow. The beam incorporates distributed piezoelectric meta-cells connected via unidirectional circuits to enable self-sensing and adaptive control. A strongly coupled nonlinear fluid-structure-electro-control model within an arbitrary Lagrangian–Eulerian framework is developed for predicting the flapping dynamics of the large deformable piezoelectric metamaterial beam. The system exhibits multiple flow-induced modes, including limit-cycle oscillations, subharmonic responses and S-B deflections. These dynamic regimes arise from nonlinear bifurcations of the system, namely the period-doubling and spontaneous S-B bifurcations. Flapping control and wake topology transition of the system is achieved by suppressing the periodic-doubling bifurcation based on the vibration rebound effect through a self-sensing and adaptive-actuation mechanism of the beam. Floquet stability analysis confirms the effectiveness of control in delaying instability onset and suppressing chaotic transitions. Symmetry modulation of the beam is achieved via the localised perturbations induced from the piezoelectric meta-cells, which reshape the stability of the system. The transition from S-B mode to symmetry-recovery mode reflects a shift from a flow-separation-dominated to vibration-dominated vortex shedding pattern. This symmetry transition reorganises the energy exchange pathways between the flow and the beam. Quantitative analyses of the wake recovery and the energy harvesting efficiency confirm enhanced flow energy conversion under control. These results establish a framework for bifurcation control of slender structures in viscous flow, providing potential applications for underwater energy harvesting and flexible propulsion in unsteady environments.

Turbulent separating and reattaching flows are known to exhibit low-frequency fluctuations manifested in a large-scale contraction and expansion of the reverse-flow region. Previous experimental investigations have been restricted to planar measurements, while the computational cost to resolve the low-frequency spectrum with high-fidelity simulations currently appears to be unaffordable. In this article, we make use of volumetric measurements to reveal the low-frequency dynamics of a turbulent separation bubble (TSB) formed in the fully turbulent flow past a smooth backward-facing ramp. The volumetric velocity field measurements cover the entire separated flow region over a domain with a spanwise extent of $S=0.6\, {\textrm{m}}$. Spectral proper orthogonal decomposition (SPOD) of the velocity fluctuations reveals low-rank low-frequency behaviour at Strouhal numbers ${\textit{St}}\lt 0.05$, which was also observed in previous planar measurements. However, in contrast with the interpretation of a two-dimensional contraction/expansion motion, the low-frequency dynamics is shown to be inherently three-dimensional, and governed by large elongated structures with a spanwise wavelength of approximately $S/2$. A low-order model constructed with the leading SPOD mode confirms substantial changes of the TSB extent in the centre plane, linking it to the modal pattern that is strongly non-uniform in the spanwise direction. The findings presented in this study promote a more complete understanding of the low-frequency dynamics in turbulent separated flows, thereby enabling novel modelling and control approaches.

Based on the assumption of locally quasi-steady behaviour, Duran & Moreau (2013 J. Fluid Mech. 723, 190–231), assumed that, at a critical nozzle throat, the fluctuations of the Mach number vanish for linear perturbations of a quasi-one-dimensional isentropic flow. This appears to be valid only in the quasi-steady-flow limit. Based on the analytical model of Marble & Candel (1977 J. Sound Vib. 55, 225–243) an alternative boundary condition is obtained, which is valid for nozzle geometries with a finite limit of the second spatial derivative of the cross-section on the subsonic side of the throat. When the nozzle geometry does not satisfy this condition, the application of a quasi-one-dimensional theory becomes questionable. The consequences of this for the quasi-one-dimensional modelling of the acoustic response of choked nozzles are discussed for three specific nozzle geometries. Surprisingly, the relative error in the inlet nozzle admittance and acoustic wave transmission coefficient remains below a per cent, when the quasi-steady boundary condition is used at the throat. However, the prediction of the acoustic fluctuations assuming a quasi-steady critical-throat behaviour is incorrect, because the predicted acoustic field is singular at the throat.