Internal waves shoaling on the continental slope can break and form materially coherent vortices called boluses. These boluses are able to trap and transport material up the continental slope, yet the global extent of bolus transport is unknown. Previous studies of bolus formation primarily focused on systems consisting of two layers of uniform density, which do not account for the presence of ocean pycnoclines of finite thickness. We use hyperbolic tangent profiles to model the density stratification in our simulations and demonstrate the impact of the pycnocline on the bolus. A spectral clustering method is used to objectively identify the bolus as a Lagrangian coherent structure that contains the material advected upslope. The bolus size and displacement upslope are examined as a function of the pycnocline thickness, incoming wave energy, density change across the pycnocline and topographic slope. The dependence of bolus transport on the pycnocline thickness demonstrates that boluses in continuous stratifications tend to be larger and transport material further than in corresponding two-layer stratifications.

The flow-induced vibration (FIV) of an airfoil freely undergoing two-degrees-of-freedom (2-DOF) motions of plunging and pitching is numerically investigated as a function of the reduced velocity and pivot location in a two-dimensional free-stream flow. This investigation covers a wide parameter space spanning the flow reduced velocity range of $0<U^{\ast }=U/(\,f_{n}c)\leqslant 10$ and the pivot location range of $0\leqslant x\leqslant 1$, where $U$ is the free-stream velocity, $f_{n}$ is the natural frequency of the system set equal in the plunge and pitch directions, $c$ is the chord length of the foil and $x$ is the normalised distance of the pivot point from the leading edge. The numerical simulations were performed by employing an immersed boundary method at a low Reynolds number ($Re=Uc/\unicode[STIX]{x1D708}=400$, with $\unicode[STIX]{x1D708}$ the kinematic viscosity of the fluid). Through detailed analyses of the dynamics of the 2-DOF vibrations and wake states, a variety of FIV response regimes are identified, including four regions showing synchronisation or near-synchronisation responses (labelled as S‐I, S‐II, S‐III and S‐IV) and four transition regimes (labelled as T‐I, T‐II, T‐III and T‐IV) that show intermittent, switching or chaotic responses, in the $x{-}U^{\ast }$ space.

Shock buffet on wings encountered in edge-of-the-envelope transonic flight remains an unresolved and disputed flow phenomenon, challenging both fundamental fluid mechanics and applied aircraft aerodynamics. Its dynamics is revealed through the interaction of spanwise shock-wave oscillations and intermittent turbulent boundary-layer separation. Resulting unsteady aerodynamic loads, and their mutual working with the flexible aircraft structure, need to be accounted for in establishing the safe flight envelope. The question of global instability leading to this flow unsteadiness is addressed herein. It is shown for the first time on an industrially relevant configuration that the dynamics of a single unstable oscillatory eigenmode plays a prominent role in near-onset shock buffet on a quasi-rigid wing. Its three-dimensional spatial structure, previously inferred both from experiment and time-marching simulation, describes a spanwise-localised pocket of shear-layer pulsation synchronised with an outboard-propagating shock oscillation. The results also suggest that the concept of a critical global shock-buffet mode commonly reported for two-dimensional aerofoils also applies to three-dimensional finite and swept wings, albeit different modes at play. Specifically, the modern wing design, NASA Common Research Model, with publicly available geometry and experimental data for code validation is studied at a free-stream Mach number of 0.85 with Reynolds number per reference chord of $5.0\times 10^{6}$ and varying angle of attack between 3. 5° and 4. 0° targeting the instability onset. Strouhal number at instability onset just above 3. 7° is approximately 0.39. At the same time, a band of eigenmodes shows reduced decay rate in the Strouhal-number range of 0.3 to 0.7, with additional unstable oscillatory modes appearing beyond onset. Importantly, those emerging modes seem to discretise the continuous band of medium-wavelength modes, as recently reported for infinite swept wings using stability analysis, hence generalising those findings to finite wings. Through conventional time-marching unsteady simulation it is explored how the critical linear eigenmode feeds into the nonlinearly saturated limit-cycle oscillation near instability onset. The established numerical strategy, using an iterative inner–outer Krylov approach with shift-and-invert spectral transformation and sparse iterative linear solver, to solve the arising large-scale eigenvalue problem with an industrial Reynolds-averaged Navier–Stokes flow solver means that such a practical non-canonical test case at a high-Reynolds-number condition can be investigated. The numerical findings can potentially be exploited for more effective unsteady flow analysis in future wing design and inform routes to flow control and model reduction.

A small-disturbance model is presented for the complex dynamics of vortex flows of moist air in a straight, circular pipe with non-equilibrium and homogeneous condensation. The model explores the nonlinear interactions among the vortex near-critical swirl ratio and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. These are the flow inlet swirl ratio $\unicode[STIX]{x1D714}$, the inlet Mach number $Ma_{0}$, the initial humidity $\tilde{\unicode[STIX]{x1D714}}_{0}$, the number of water molecules in a characteristic fluid element $n_{C}$, the inlet centreline super-saturation ratio $S_{0}$ and the ratio of characteristic condensation and flow time scales $K$. Also, the flow field may be described by an ordinary first-order nonlinear differential equation for the flow evolution coupled with a set of four first-order ordinary differential equations along the pipe for the calculation of the condensate mass fraction. An iterative numerical scheme which combines the Runge–Kutta integration technique for the flow dynamics with Simpson’s integration rule for the calculation of the condensation variables is developed. Specifically, equilibrium states are determined, including the possibility of the appearance of multiple states under the same boundary conditions, and the stability characteristics of these states are described. The model is used to study the effects of humidity and of energy supply from nanoscale condensation processes on the large-scale dynamics of vortex flows as well as the effect of flow swirl on condensation processes in swirling flows.

Motivated by planetary-driven applications and experiments in non-spherical geometries, we study compressible fluid modes in rotating rigid ellipsoids. Such modes are also required for modal acoustic velocimetry (MAV), a promising non-invasive method to track the velocity field components in laboratory experiments. To calculate them, we develop a general spectral method in rigid triaxial ellipsoids. The description is based on an expansion onto global polynomial vector elements, satisfying the non-penetration condition on the boundary. Then, we investigate the diffusionless compressible modes in rotating (and magnetised) rigid ellipsoids. The spectral description is successfully benchmarked against three-dimensional finite-element computations and analytical predictions. A spectral convergence is obtained. Our results have direct implications for MAV in experiments, for instance in the ZoRo experiment (gas-filled rigid spheroid). So far, deformation and rotational effects have been theoretically considered separately, as small perturbations of the solutions in non-rotating spheres. We carefully compare the perturbation approach, in this illustrative geometry, to the polynomial solutions. We show that second-order ellipticity effects are often present, even in weakly deformed ellipsoids. Moreover, high-order effects due to rotation and/or ellipticity should be observed for some acoustic modes in experimental conditions. Thus, perturbation theory should be used with care in MAV. Instead, the spectral polynomial method paves the way for future MAV applications in fluid experiments with rigid ellipsoids.

We study the convective patterns that arise in a nearly semicylindrical cavity fed in with hot fluid at the upper boundary, bounded by a cold, porous semicircular boundary at the bottom, and infinitely extended in the third direction. While this configuration is relevant to continuous casting processes that are significantly more complex, we focus on the flow patterns associated with the particular form of mixed convection that arises in it. Linear stability analysis (LSA) and direct numerical simulations (DNS) are conducted, using the spectral-element method to identify observable states. The nature of the bifurcations is determined through Stuart–Landau analysis for completeness. The base flow consists of two counter-rotating rolls driven by the baroclinic imbalance due to the curved isothermal boundary. These are, however, suppressed by the through-flow, which is found to have a stabilising influence as soon as the Reynolds number $Re$ based on the through-flow exceeds 25. For a sufficiently high Rayleigh number, this base flow is linearly unstable to three different modes, depending on $Re$. For $Re\leqslant 75$, the rolls destabilise through a supercritical bifurcation into a travelling wave. For $100\leqslant Re\leqslant 110$, a subcritical bifurcation leads to a standing oscillatory mode, whereas for $Re\geqslant 150$, the unstable mode is non-oscillatory and grows out of a supercritical bifurcation. The DNS confirm that in all cases the dominant mode returned by the LSA precisely matches the topology and evolution of the flow patterns that arise out of the fully nonlinear dynamics.

This paper describes linear stability analysis of the two-dimensional steady motion of periodic deep-water waves with symmetric non-overhanging profiles propagating on a linear shear current, namely a vertically sheared current with constant vorticity. In order to investigate numerically with high accuracy the stability of large-amplitude waves, we adopt a formulation using conformal mapping, in which the time-varying water surface is always mapped onto the real axis of a complex plane. This formulation allows us to apply numerical methods developed for large-amplitude irrotational waves without a shear current directly to the present problem, and reduces the linear stability problem to a generalized eigenvalue problem. Numerical solutions describe both super- and sub-harmonic instabilities of the periodic waves for a wide range of wave amplitudes and clarify how the behaviours of dominant eigenvalues change with the strength of the shear current. In particular, it is shown that, even in the presence of a linear shear current, the steady periodic waves lose stability due to superharmonic disturbances at the wave amplitude where the wave energy attains an extremum, similarly to the case of irrotational waves without a shear current. It is also found that re-stabilization with an increase in wave amplitude characterizes subharmonic instability for weak shear currents, but the re-stabilization disappears for strong shear currents.

The interaction of rarefaction waves and a heavy/light interface is investigated using numerical simulations by solving the compressible Euler equations. An upwind space–time conservation element and solution element (CE/SE) scheme with second-order accuracy in both space and time is adopted. Rarefaction waves are generated by simulating the shock-tube problem. In this work, the SF6/air interface evolution under different conditions is considered. First, the gas physical parameters before and after the rarefaction waves impact the interface are calculated using one-dimensional gas dynamics theory. Then, the interaction between the rarefaction waves and a single-mode perturbation interface is investigated, and both the interface evolution and the wave patterns are obtained. Afterwards, the amplitude growth of the interface over time is compared between cases, considering the effects of the interaction period and the strength of the rarefaction waves. During the interaction of the rarefaction waves with the interface, the Rayleigh–Taylor instability induced by the rarefaction waves is well predicted by modifying the nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61), considering the variable acceleration. After the rarefaction waves leave the interface, the equivalent Richtmyer–Meshkov instability is well depicted by the nonlinear model proposed by Zhang et al. (Phys. Rev. Lett., vol. 121(17), 2018, 174502), considering the growth rate transition from Rayleigh–Taylor instability to Richtmyer–Meshkov instability. The differences in the heavy/light interface amplitude growth under the rarefaction wave condition and the shock wave condition are compared. The interface perturbation is shown to be more unstable under rarefaction waves than under a shock wave.

The aim of this study is to understand the dynamics of Rossby waves induced by a localised and periodic ‘plunger’ forcing – imposed on a background flow – which is intended as an elementary representation of transient mesoscale eddy forcing in the ocean. We consider linearised dynamics and its quasi-nonlinear extension, and focus on the rotating shallow-water model. The plunger induces a spectrum of Rossby waves that drive zonal momentum flux convergence at the forced latitudes. This behaviour has a robust and significant dependence on the background flow, which we treat as zonal and uniform. We systematically analyse this dependence using two methods. First, we use the eddy geometry formulation, in which Reynolds stresses are expressed in terms of eddy elongation and eddy tilt parameters, and consider the relationship between eddy geometry and zonal momentum redistribution. Second, we implement decompositions of flow responses into linear dynamical eigenmodes and compare with expectations from linear Rossby wave theory. Both methods compliment each other and aid the understanding of zonal momentum redistribution and its dependence on uniform background flow. We find that this dependence is determined by two factors: (i) dispersion-constrained resonance with the plunger forcing and (ii) efficiency of nonlinear eddy self-interactions. These results significantly improve our understanding of shallow-water Rossby waves, and may also be applied towards the development of parameterisations of oceanic mesoscale eddies.

The development of laminar thermals and the formation of buoyant vortex rings in thermals are studied by performing direct numerical simulations. The formation number of buoyant vortex rings in thermals is also analysed. We find that the development of thermals can be classified into three modes: the starting vortex ring dominated mode; the mode with the occurrence of a secondary vortex ring with breakup; and the mode with the occurrence of a secondary vortex ring without breakup. For the latter two modes, owing to the stretching of the thermal cap, the fluid at the leading edge rolls up, and a secondary vortex ring occurs, grows and replaces the starting vortex ring. The boundary of non-occurrence and occurrence of the secondary vortex ring is determined in a space of Richardson number (Ri) and injection duration ($t_{i}$). The final mode occurs only in a small region. For $Ri<0.6$, the secondary vortex ring does not occur even for very long injection duration. The effective Rayleigh number ($Ra_{m}$) is proposed to accommodate the cases $Ri>0.7$ and $t_{i}<5$, with $Ra_{m}$ larger than the critical value (approximates to 1. 95 × 105) for the occurrence of the secondary vortex ring. The formation number of buoyant vortex rings in thermals is beyond the universal formation number of 4 for non-buoyant vortex rings, and increases with the increase of the Richardson number and the injection duration. The switching between the thermal modes by changing the Richardson number and the injection duration has no significant effect on the value of the formation number.

The stability of neutrally and non-neutrally buoyant particles immersed in a plane Poiseuille flow of a yield-stress fluid (Bingham fluid) is addressed numerically. Particles being carried by the yield-stress fluid can behave in different ways: they might (i) migrate inside the yielded regions or (ii) be transported without any relative motion inside the unyielded region if the yield stress is large enough compared to the buoyancy stress and the other stresses acting on the particles. Knowing the static stability of particles inside a bath of quiescent yield-stress fluid (Chaparian & Frigaard, J. Fluid Mech., vol. 819, 2017, pp. 311–351), we analyse the latter behaviour when the yield-stress fluid Poiseuille flow is host to two-dimensional particles. Numerical experiments reveal that particles lose their stability (i.e. break the unyielded plug and sediment/migrate) with smaller buoyancy compared to the sedimentation inside a bath of quiescent yield-stress fluid, because of the inherent shear stress in the Poiseuille flow. The key parameter in interpreting the present results is the position of the particle relative to the position of the yield surface in the undisturbed flow (in the absence of any particle): the larger the portion of a particle located inside the undisturbed sheared regions, the more likely is the particle to be unstable. Yet, we find that the core unyielded plug can grow locally to some extent to contain the particles. This picture holds even for neutrally buoyant particles, although they are strictly stable when they are located wholly inside the undisturbed plug. We propose scalings for all cases.

The stability of a uniform flow above an erodible bed composed of a bimodal mixture of sediments is investigated by means of linear analysis. Results show that, for any given set of the flow and sediment parameters, two distinct modes of instability arise, each one characterized by its own wave speed, growth rate and longitudinal wavelength, each one involving spatial variations of both grain size density and bed elevation. Although at a linear level no information on the amplitude of the perturbations is gathered, the analysis of the eigenvectors associated with the two modes of instability allows for an easy classification in terms of the relative amplitudes of the perturbations of bed elevation and size density. One eigenvalue is shown to be associated with the modifications of bed forms induced by the presence of the heterogeneous mixture, such as the local accumulation of finer and coarser material along the unit wavelength, the other with the formation of the low-amplitude sorting waves known as bedload sheets. In the present unidirectional shallow-water framework, only the sorting wave is found to be unstable, since dunes and antidunes, the relevant bed forms for this case, require a more refined rotational flow model in order to become unstable. On the other hand, the simple flow model adopted allows for the formulation of an algebraic eigenvalue problem that can be solved analytically, allowing for a deep insight into the mechanisms that drive both instabilities.

Experiments are performed to investigate the onset of early transition and drag reduction in the flow of polymer (polyacrylamide and polyethylene oxide) solutions through rigid microtubes of diameters in the range 0.49–2.84 mm. We measure friction factor variation with Reynolds number for varying polymer concentrations and tube diameters, and the Reynolds number, $Re_{t}$, at which the experimental data deviate from the laminar value represents the onset of transition. Crucially, owing to the high shear rates encountered in our experiments, we show that it is important to account for shear thinning of the fluid in the theoretical estimation of the friction factor in the laminar regime. We accomplish this using a Carreau model, and show that the use of laminar friction factor calculated without shear thinning leads to an erroneous overestimation of $Re_{t}$. The $Re_{t}$ obtained from friction factor data in the present study is in good agreement with that inferred using micro particle image velocimetry analysis in Chandra et al. (J. Fluid Mech., vol. 844, 2018, pp. 1052–1083). For smaller concentrations of the added polymer, there is a marginal delay in the onset of turbulence, but as the concentration is increased further, the transition Reynolds number decreases much below $2000$, the usual value at which transition occurs in Newtonian pipe flows. Thus, the present study further corroborates the phenomenon of early transition leading to an ‘elasto-inertial’ turbulent state in the flow of polymer solutions. For concentrations such that there is a delay in transition, if $Re$ is maintained above the $Re_{t}$ for Newtonian fluids, the flow is transitional or turbulent in the absence of polymers. At such a fixed $Re$, if the concentration of the polymer is increased gradually, the friction factor decreases and the flow relaminarizes. With further increase in polymer concentration, the flow undergoes a transition due to elasto-inertial instability. The effect of addition of small amounts of polymer on turbulent drag reduction in the flow of water through microtubes is also investigated. Increase in polymer concentration, molecular weight and decrease in tube diameter causes an increase in drag reduction. The friction factor data for different polymer concentrations, molecular weights, tube diameters and $Re$, when plotted with $Wi(1-\unicode[STIX]{x1D6FD})$, show a reasonable collapse, where $Wi$ is the Weissenberg number defined as the product of the longest relaxation time of the polymer solution and the average shear rate in the tube and $\unicode[STIX]{x1D6FD}$ is the ratio of solvent to total solution viscosity. Interestingly, the onset of the maximum drag reduction asymptote, for experiments using varying tube diameters and polymer concentrations, appears to occur at $Wi(1-\unicode[STIX]{x1D6FD})\sim O(1)$.