We study the dynamics of a cavitation bubble near beds of sand of different grain sizes. We use high-speed imaging to observe the motion of the bubble and the sand for different values of the stand-off parameter $\gamma$ (dimensionless bubble-boundary distance) between 0.3 and 5.3. Compared with a rigid boundary, we find that a granular boundary leads to bubbles with shorter lifetimes and reduced centroid displacements. Above $\gamma \approx 1.3$, the behaviour of the bubble is almost independent of the granularity of the sand. When the stand-off parameter lies between $0.6$ and $1.3$, a mound of sand develops beneath the bubble, which can force the latter to assume a conical shape as it collapses. For $\gamma \lesssim 0.6$, the bubble develops a bell-shaped form, leading to the formation of thin and surprisingly fast micro-jets ($v_{jet} > 1000\,{\rm m}\,{\rm s}^{-1}$). Moreover, between $\gamma \approx 1.3$ and $\gamma \approx 0.3$, granular jets erupt from the sand surface following the bubble collapse. We additionally develop a simple numerical model, based on the boundary integral method, to predict the dynamics of a cavitation bubble near a bed of sand, which we replace by an equivalent liquid. The simulations are remarkably consistent with experimental observations for values of $\gamma$ down to $1.3$. We also show how the anisotropy parameter $\zeta$, a dimensionless version of the Kelvin impulse, can be adapted for the case of a nearby bed of sand and predict the displacement of the bubble centroid for $\zeta \lesssim 0.08$.

The finite-amplitude instability of the buoyancy-driven boundary layer is considered on a vertical plate immersed in a thermally stratified ambient medium, where the wall and surrounding fluid have different temperature gradients. Although the linear stability in this configuration has been investigated, the finite-amplitude solution arising from the critical instability has been studied only for specific parameter values. We extend this by using the amplitude expansion method. The primary bifurcations to the two-dimensional least unstable mode for different temperature gradient ratios ($0 \leqslant \lambda \leqslant 10$) and Prandtl numbers ($10^{-1} \leqslant Pr \leqslant 10^{4}$) are investigated. Only supercritical bifurcations are found to occur when $0 \leqslant \lambda < 2$ and $Pr \leqslant 2800$, while subcritical bifurcations are also found for larger values of temperature gradient ratio and Prandtl number. Analysis of the contribution of the nonlinear terms in the Landau coefficient reveals that the interaction of the modification of the mean flow and second harmonic for velocity with the fundamental mode for temperature plays an important role in subcritical bifurcation. Based on the Landau equation, the threshold amplitude of the nonlinear equilibrium solution is discussed as well. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.

The computation of invariant solutions and the visualisation of the associated state space have played a key role in the understanding of transition and the self-sustaining process in wall-bounded shear flows. In this study, an extension of this approach is sought for a turbulent flow which explicitly exhibits multi-scale behaviour. The minimal unit of multi-scale near-wall turbulence, which resolves two adjacent spanwise integral length scales of motion, is considered using a shear stress-driven flow model (Doohan, Willis & Hwang J. Fluid Mech., vol. 913, 2021, A8). The edge state, 26 travelling waves and two periodic orbits are computed, which represent either the large- or small-scale self-sustaining processes. Given that the spanwise length scales are not widely separated here, it could be envisaged that turbulent trajectories visit these solutions in the state space. Considering the intra- and inter-scale dynamics of the flow, numerous phase portraits are examined, but the turbulent state is not found to approach any of these solutions. A detailed analysis reveals that this is due to the lack of scale interaction processes captured by the invariant solutions, including the mean–fluctuation interaction, the energy cascade in the streamwise wavenumber space and the cascade-driven energy production discovered recently. There is a single solution that resembles turbulence much more than the others, which captures two-scale energetics and a scale interaction process involving energy feeding from small to large spanwise scales through the subharmonic sinuous streak instability mode. Based on these observations, it is conjectured that the state space view of turbulent trajectories wandering between solutions would need suitable refinement to model multi-scale turbulence, when each solution does not represent multi-scale processes of turbulence. In particular, invariant solutions that are inherently multi-scale would be required.

The two-dimensional (2-D) unsteady shock reflection over a single wedge is studied theoretically and numerically, and the emphasis is placed on the trajectory of the triple point (TP). Skews’ relation and the three-shock theory are, respectively, used for determining the trajectory angles of the corner-generated disturbance and the TP, and, subsequently, a model capable of predicting the TP trajectory is established for 2-D unsteady shock reflections over a single wedge. Then, a systematically numerical study is carried out on the 2-D unsteady shock reflection over a single wedge, including five types of shock reflection with the wedge angle increased and five types of shock reflection with the wedge angle decreased. It is found that the Mach stem is always slightly concavely (convexly) curved for reflections with the wedge angle increased (decreased), which should be caused by corner-generated compression (rarefaction) waves propagating along the Mach stem. The new model reasonably predicts the TP trajectory in all the 2-D unsteady shock reflections, and its performance is related not only to the variation trend of the wedge angle (increase or decrease), but also to the type of shock and the initial wedge angle. Specifically, for the shock reflection with the wedge angle increased (decreased), the model generally provides a slightly better (worse) prediction if the TP trajectory angle calculated by the three-shock theory is compulsively modified. The shock-shock/expansion polar analysis presented can partially explain the model performance for predicting the TP trajectory.

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and four application cases highlight common non-normal mechanisms in hydrodynamics: the streamwise convective non-normal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.

The flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically with a gas–liquid Navier–Stokes solver. The solver relies on the volume-of-fluid method to account for different phases, and the interface tracking is carried out by using novel schemes based on a tailored total-variation-diminishing limiter. The numerical solver is proved to be characterized by a low numerical dissipation, thanks to the use of a scheme that guarantees energy conservation in the discrete form. Both two- and three-dimensional simulations have been performed, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. To this purpose, coherent vortical structures, such as vortex tubes and vortex sheets, are identified, and the different behaviours of the vortex sheets and tubes at various Reynolds numbers are highlighted. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is identified primarily by the vortex sheets, whereas the vortex tubes govern mainly the intermittency.

In this paper, the interaction between an upper-layer vortex and a bottom topography is investigated using an $f$-plane two-layer quasi-geostrophic model with a point vortex and step-like topography. The contour dynamics method is used to formulate the model. A steadily propagating linear solution along the topography, known as the pseudoimage solution, is derived analytically for a weak point vortex, and the nonlinear solution is obtained numerically. Numerical experiments show that the nonlinear pseudoimage solution collapses with time. Saddle-node points in the velocity field are critical in this collapse. Even after the collapse, the point vortex propagates along the topography similarly to in the steadily propagating solution. Numerical experiments with various initial conditions show that the point vortex has two types of motion in this system: motion along the topography and motion away from the topography. In the latter case, the point vortex and lower-layer potential vorticity anomaly form a heton-like dipole structure. The motion classification results show that an anticyclonic (cyclonic) point vortex on the deeper (shallower) side is more likely to form a dipole structure than a cyclonic (anticyclonic) vortex on the deeper (shallower) side when their initial distance from the topography is the same.

The acoustic analogy is adopted to characterise the signature of a seven-bladed submarine propeller, relying on a high-fidelity large-eddy simulation, performed on a computational grid consisting of 840 million points. Results demonstrate that the nonlinear terms of the Ffowcs-Williams and Hawkings equation quickly become dominant moving away from the propeller along the direction of its wake development. While the linear terms experience a decay moving downstream, the nonlinear terms grow in the near wake, as a result of the development of wake instability. In particular, this growth affects frequencies lower than the blade frequency. Therefore, the acoustic signature of the propeller is mainly tonal in the near field only, due to the thickness and loading components of noise from the surface of the propeller and the periodic perturbation caused by its tip vortices. They develop instability at a faster rate, compared with the hub vortex, triggering the process of energy cascade towards higher frequencies and contributing in this way to broadband noise.