An oscillating body floating at the water surface produces a field of self-generated waves. When the oscillation induces a difference in fore–aft wave amplitude squared, these self-generated waves can be used as a mechanism to propel the body horizontally across the surface (Longuet-Higgins 1977 Proc. R. Soc. Lond. A, vol. 352, no. 1671, pp. 463–480). The optimisation of this wave-driven propulsion is the interest of this work. To study the conditions necessary to produce optimal thrust we will consider a shallow water set-up where a periodically oscillating pressure source acts as the body. In this framework, an expression for the thrust is derived by relation to the difference in fore–aft amplitude squared. The conditions on the source for maximal thrust are explored both analytically and numerically in two optimal control problems. The first case is where a bound is imposed on the norm of the control function to regularise it. Secondly, a more physically motivated case is studied where the power injected by the source is bounded. The body is permitted to have a drift velocity $U$. When scaled with the wave speed $c$, the dimensionless velocity $v=U/c$ divides the study into subcritical, critical and supercritical regimes and the optimal conditions are presented for each. The result in the bounded power case is then used to demonstrate how the modulation of power injected can slowly change the cruising velocity from rest to supercritical velocities.

Six types of shock wave interference resulting from the impingement of an incident shock on a bow shock are revisited by examining the sub-types that arise between the canonical types. Several new sub-types are predicted based on the theories of weak shock reflection and double-wedge shock interference, and verified via numerical simulations. Two additional types, Type IIw and Type IIs, are identified between Type II and Type III, whereas a Type Vw emerges between Type IV and Type V. These types originate from the transformation of the Mach reflection at the triple point, which evolves through weak shock reflections (von Neumann reflection, Vasilev reflection, Guderley reflection) before disappearing. The transition from Type III to Type IV is further shown to mirror sequences of double-wedge shock interference. Two additional types (Type IIIb and Type IVt) are found. Meanwhile, it is found that under large incoming flow Mach number ($M_0$ = 5), Types III, IV and their sub-types dominate, whereas Type II is absent; under small incoming flow Mach number ($M_0$ = 2.5), Types III and IV vanish and a modified Type Va emerges. This study adds seven new sub-types to the existing six types of shock interference. These work extend the classical six types of shock interferences into six-plus shock interference, a picture that shed new insight into shock interference.

The linear stability of nanofluid boundary-layer flow over a flat plate is investigated using a two-phase formulation that incorporates the Brinkman (1952 J. Chem. Phys., vol. 20, pp. 571–581) model for viscosity along with Brownian motion (BM) and thermophoresis (TP), building upon the earlier work of Buongiorno (2006 J. Heat Transfer, vol. 128, pp. 240–250). Solutions to the steady boundary-layer equations reveal a thin nanoparticle concentration layer near the plate surface, with a characteristic thickness of $O({\textit{Re}}^{-1/2}{\textit{Sc}}^{-1/3})$, for a Reynolds number ${\textit{Re}}$ and Schmidt number ${\textit{Sc}}$. When BM and TP are neglected, the governing equations reduce to the standard Blasius formulation for a single-phase fluid, and the nanoparticle concentration layer disappears, resulting in a uniform concentration across the boundary layer. Neutral stability curves and critical conditions for the onset of the Tollmien–Schlichting (TS) wave are computed for a range of nanoparticle materials and volume concentrations. Results indicate that while the effects of BM and TP are negligible, the impact of nanoparticle density is significant. Denser nanoparticles, such as silver and copper, destabilise the TS wave, whereas lighter nanoparticles, like aluminium and silicon, establish a small stabilising effect. Additionally, the viscosity model plays a crucial role, with alternative formulations leading to different stability behaviour. Finally, a high Reynolds number asymptotic analysis is undertaken for the lower branch of the neutral stability curve.

We present a mathematical model for tsunami and induced magnetic anomalies originating from a time-dependent seabed deformation in an otherwise quiescent ocean over a conductive seafloor. The deformation is assumed to be a slender fault, whose lateral extension is much larger than the longitudinal scale. Using a perturbative method with multiple time scales and Green’s function approach, we examine the slow evolution of the wave field and induced magnetic anomaly over transoceanic distances from the fault. The model is validated against deep-ocean observations from the 2011 Tōhoku-oki tsunami. Our study reveals that lateral propagation in two horizontal dimensions decreases the period of both the surface wave and induced magnetic signal compared with one-horizontal-dimension scenarios. Over time, initially longitudinal wave propagation alters as wave fronts bend and stretch, affecting the magnetic signal accordingly. Interestingly, the magnetic anomaly gradually separates from the leading tsunami wave and travels ahead of the tsunami by a distance proportional to the fault’s longitudinal scale. We show that increased lateral propagation reduces the detectability of magnetic anomalies. Finally, we derive an asymptotic formula valid for the long leading wave that travels ahead of the dispersive group over transoceanic distances. This formula holds promise for the rapid assessment of tsunami risk. These findings advance fundamental understanding and may inform the development of future tsunami early warning systems relying on magnetic field detection.

We study transverse profiles and time fluctuations of turbulence dissipation rate, turbulence kinetic energy and integral length scales by means of high-speed stereoscopic particle image velocimetry in the turbulent wake of a 6 : 1 prolate spheroid that has its principal axis aligned with the incoming non-turbulent flow. This turbulent wake of a slender body differs from turbulent bluff body wakes in terms of transverse non-homogeneity of turbulence dissipation rate and because it is not axisymmetric even though it nominally is. Even so, both transverse profiles and time fluctuations of turbulence dissipation rate coefficients (inverse ratio between the rate with which the large scales lose energy and the rate with which the small scales dissipate energy) and of the Taylor length-based Reynolds number (ratio between the turbulent kinetic energy mostly in the large scales and the turbulent kinetic energy at the smallest scales) obey self-regulating non-equilibrium, as previously found in various other turbulent flows. However, the power law relating the transverse variations and the time fluctuations of these two ratios differs from previously reported self-regulating non-equilibrium power law scalings in other turbulent flows.

The coupling between Rayleigh–Taylor (R–T) and Saffman–Taylor (S–T) instabilities, when a gas displaces a high-viscosity liquid, remains challenging to elucidate due to the unclear roles of density and viscosity contrasts. Counterintuitively, our radial Hele-Shaw cell experiments revealed that viscosity contrast – typically considered a damping factor – serves as the primary driver of instability. We observed that the glycerin–air interface, despite its higher viscosity, exhibits significantly greater instability than the water–air interface. This anomalous behaviour arises from the S–T mechanism, which accelerates the onset of nonlinearity and induces an early transition to fingering. We applied a unified model to decouple the competing influences of surface tension oscillation and viscous damping on R–T instability and the S–T destabilisation. Moreover, we proposed criteria for either mostly enhancing or completely freezing the instability. These findings offer valuable insights into manipulating hydrodynamic instabilities in contracting/expanding geometries through surface tension and viscosity.

High-intensity focused ultrasound (HIFU) is a non-invasive alternative to traditional surgery for detection and treatment. When HIFU targets a specific area, ultrasonic cavitation occurs with mechanical stress, causing tissue damage, a process that is significantly influenced by the surroundings. This paper presents a numerical study on the cavitation initiation and evolution mechanisms under focused ultrasonic waves considering the influence of a solid surface. Firstly, the dynamic property of focused ultrasonic waves and the generation of diffraction waves is explained based on the Huygens–Fresnel principle, and the prefocused phenomenon is analysed. Notably, the scenario considering the existence of a solid wall is discussed, with the corresponding cavitation clouds in a ‘tree-like’ pattern that can be generally divided into three or four subregions. The different initiation mechanisms of the near-wall cavitation clouds under a different relative distance between the theoretical focal point and the solid wall are discussed in detail. Finally, by considering the effects of the incident waves, scattered waves and their reflected waves on the solid wall, a wave superposition model is established that can clearly explain the distribution characteristics of the near-wall cavitation clouds with different modes. The understanding of the ultrasonic cavitation mechanism may support precise control in future HIFU applications.

This paper presents the first experimental measurement of the Prandtl–Meyer function in the non-ideal compressible flow regime. Planar contoured nozzle profiles expand the flow to the supersonic regime, providing a uniform parallel flow of siloxane MM (hexamethyldisiloxane, $\textrm{C}_{6}\textrm{H}_{18}\textrm{OSi}_{2}$). Prandtl–Meyer expansions are then generated at sharp convex corners, for discrete flow deflection angles from 5$^\circ$ to 30$^\circ$. Stagnation pressures and temperatures are measured in the settling chamber, immediately upstream of the test section, to estimate the level of non-ideality of the investigated flows, ranging from mild non-ideal conditions to dilute ideal-gas states. Non-ideal thermodynamic effects through the expansions are characterised by means of independent measurements of Mach number by schlieren visualisations, and static pressure. Experimental comparisons across different thermodynamic states confirm the role of the compressibility factor evaluated at total conditions as a similarity parameter for moderately high non-ideal flows. To extract values of the Prandtl–Meyer function from the measurements, a simplified analytical model for the Prandtl–Meyer function dependency on the Mach number is formulated. The recovered values agree with Prandtl–Meyer theory, complemented with state-of-the-art thermodynamic models, for all the examined operating conditions.

The effect of a smooth surface hump on laminar–turbulent transition over a swept wing is investigated using direct numerical simulation (DNS), and results are compared with wind tunnel measurements. When the amplitude of incoming crossflow (CF) perturbation is relatively low, transition in the reference (without hump) case occurs near $53\,\%$ chord, triggered by the breakdown of type I secondary instability. Under the same conditions, no transition is observed in the hump case within the DNS domain, which extends to $69\,\%$ chord. The analysis reveals a reversal in the CF velocity component downstream of the hump’s apex. Within this region, the structure and orientation of CF perturbations are linearly altered, particularly near the wall. These perturbations gradually recover their original state further downstream. During this recovery phase, the lift-up mechanism is weakened, reducing linear production, which stabilises the stationary CF perturbations and weakens spanwise gradients. Consequently, the neutral point of high-frequency secondary CF instability modes shifts downstream relative to the reference case, leading to laminar–turbulent transition delay in the presence of the surface hump. In contrast, when the amplitude of the incoming CF perturbation is relatively high, a pair of stationary counter-rotating vortices forms downstream of the hump. These vortices locally deform the boundary layer and generate regions of elevated spanwise shear. The growth of secondary instabilities in these high-shear regions leads to a rapid advancement of transition towards the hump, in agreement with experimental observations.