A free-Lagrange numerical method is implemented to simulate the axisymmetric jetting collapse of air bubbles in water. This is performed for both lithotripter shock-induced collapses of initially stable bubbles, and for free-running cases where the bubble initially contains an overpressure. The code is validated using two test problems (shock-induced bubble collapse using a step shock, and shock–water column interaction) and the results are compared to numerical and experimental results. For the free-running cases, simulations are conducted for a bubble of initial radius 0.3 mm located near a rigid boundary and near an aluminium layer (planar and notched surfaces). The simulations suggest that the boundary and its distance from the bubble influence the flow dynamics, inducing bubble elongation and jetting. They also indicate stress concentration in the aluminium and the likelihood of aluminium deformation associated with bubble collapse events. For the shock-induced collapse, a lithotripter shock, consisting of 56 MPa compressive and −10 MPa tensile waves, interacts with a bubble of initial radius 0.04 mm located in a free field (case 1) and near a rigid boundary (case 2). The interaction of the shock with the bubble causes it to involute and a liquid jet is formed that achieves a velocity exceeding 1.2 km s−1 for case 1 and 2.6 km s−1 for case 2. The impact of the jet on the downstream wall of the bubble generates a blast wave with peak overpressure exceeding 1 GPa and 1.75 GPa for cases 1 and 2, respectively. The results show that the simulation technique retains sharply resolved gas/liquid interfaces regardless of the degree of geometric deformation, and reveal details of the dynamics of bubble collapse. The effects of compressibility are included for both liquid and gas phases, whereas stress distributions can be predicted within elastic–plastic solid surfaces (both planar and notched) in proximity to cavitation events. There is a movie with the online version of the paper.

Large-eddy simulations of a generic turbulent flow with discrete effusion are reported. The computational domain is periodic in both streamwise and spanwise directions and contains both the injection and the suction sides. The blowing ratio is close to 1.2 while the Reynolds number in the aperture is of order 2600. The numerical results for this fully developed bi-periodic turbulent flow with effusion are compared to available experimental data from a large-scale spatially evolving isothermal configuration. It is shown that many features are shared by the two flow configurations. The main difference is related to the mean streamwise velocity profile, which is more flat for the bi-periodic situation where the cumulative effect of an infinite number of upstream jets is accounted for. The necessity of considering both sides of the plate is also established by analysing the vortical structure of the flow and some differences with the classical jet-in-crossflow case are highlighted. Finally, the numerical results are analysed in terms of wall modelling for full-coverage film cooling. For the operating point considered, it is demonstrated that the streamwise momentum flux is dominated by non-viscous effects, although the area where only the viscous shear stress contributes is very large given the small porosity value (4%).

The large-scale vertical density stratification, represented by buoyancy frequency N, is generally very stable in the upper half of the ocean, and relatively weak in the lower half. However, closer inspection of density profiles demonstrates steps rather than a smooth increase with depth. As is demonstrated here using Richardson number, geostrophic balance and slantwise convective mixing arguments, these layers have a limited set of minimum, weak stratification, N-values Nmin indicating the transition between stably stratified and convective ‘homogeneous’ layers. Adopting the viewpoint that the transition occurs for neutral stability in the direction of Earth's rotation Ω instead of gravity g, three discrete states are hypothesized for mid-latitudes: (i) Nmin = 2fh under linear stability conditions, (ii) Nmin = fh(|ϕ| < 45°) and (iii) Nmin = 4fh, both under nonlinear stability, where horizontal component fh = 2Ω cos ϕ at latitude ϕ. The Nmin are not in terms of inertial frequency f = 2Ω sin ϕ, because the effect of fh is the tilting of vortex tubes away from the local vertical in the direction of Ω. The above explains very well deep-ocean North-Atlantic and Mediterranean observations on transitions in conductivity-temperature with depth profiles, inertial polarization and near-inertial shear. The latter peaks at sub-inertial 0.97f, which is associated with the lower inertio-gravity wave limit for Nmin = 4fh, thereby stressing the importance of fh for the dominant physics associated with mixing in the ocean.

One of the most striking features of rotating turbulence is the inevitable appearance of large-scale columnar structures. Whilst these structures are frequently observed, the processes by which they are created are still poorly understood. In this paper we consider the emergence of these structures from freely decaying, rotating turbulence with Ro ∼ 1. Our study follows the conjecture by Davidson, Staplehurst & Dalziel (J. Fluid Mech., vol. 557, 2006, p. 135) that the structure formation may be due to linear inertial wave propagation, which was shown to be consistent with the growth of columnar eddies in inhomogeneous turbulence. Here we extend that work and consider the case of homogeneous turbulence.

We describe laboratory experiments where homogeneous turbulence is created in a rotating tank. The turbulence is generated with Ro ∼ 1, and as the energy decays, the formation of columnar vortices is observed. The axial growth of these columnar structures is then measured using two-point correlations and in all cases the results are consistent with structure formation via linear inertial wave propagation. In particular, we obtain a self-similar collapse of the two-point correlations when the axial coordinate is normalized by Ωtb, where b is a measure of the integral scale in the horizontal plane and Ω is the rotation rate. Although our results do not exclude the possibility of significant nonlinear dynamics, they are consistent with the conjecture of Davidson et al. (2006) that linear dynamics play a strong guiding hand in structure formation.

A forced two-horizontal-dimension analytical model is developed to investigate the distinguishing physical features of landslide-induced tsunamis generated and propagating on a plane beach. The analytical solution is employed to study the wave field at small times after the landslide motion starts. At larger times, the occurrence of transient edge waves travelling along the shoreline is demonstrated, showing the differences with the transient waves propagating over a bottom of constant depth. Results are satisfactorily compared with available experimental data. Finally, the validity of non-forced numerical models is discussed.

The results of laboratory wall turbulence experiments on a shear-driven rotating boundary layer are presented. The experiments were carried out in the Turin University Laboratory rotating water tank. The flow was generated by changing the rotation speed of the platform and measured by means of particle image velocimetry. In order to analyse the influence of the rotation and of surface roughness, different cases were examined. Several rotation periods were considered. The measurements were performed both over a smooth surface and over a rough-to-smooth transition. Mean flows and the higher-order moments of the velocity probability density function are shown and discussed together with a comparison of the different experimental cases, theory and large-eddy simulations.

Cinematographic stereoscopic particle image velocimetry measurements were performed to resolve small and intermediate scales in the far field of an axisymmetric co-flowing jet. Measurements were performed in a plane normal to the axis of the jet and the time-resolved measurement was converted to quasi-instantaneous three-dimensional data by using Taylor's hypothesis. The quasi-instantaneous three-dimensional data enabled computation of all nine components of the velocity gradient tensor over a volume. The results based on statistical analysis of the data, including computation of joint p.d.f.s and conditional p.d.f.s of the principal strain rates, vorticity and dissipation, are all in agreement with previous numerical and experimental studies, which validates the quality of the quasi-instantaneous data. Instantaneous iso-surfaces of the principal intermediate strain rate (β) show that sheet-forming strain fields (i.e. β > 0) are themselves organized in the form of sheets, whereas line-forming strain fields (β < 0) are organized into smaller spotty structures (not lines). Iso-surfaces of swirling strength (a vortex identification parameter) in the volume reveal that, in agreement with direct numerical simulation results, the intense vortex structures are in the form of elongated ‘worms’ with characteristic diameter of approximately 10η and characteristic length of 60--100η. Iso-surfaces of intense dissipation show that the most dissipative structures are in the form of sheets and are associated with clusters of vortex tubes. Approximately half of the total dissipation occurs in structures that are generally sheet-like, whereas the other half occurs in broad indistinct structures. The largest length scale of dissipation sheets is of order 60η and the characteristic thickness (in a plane normal to the axis of the sheet) is about 10η. The range of scales between 10η (thickness of dissipation sheets, diameter of vortex tubes) to 60η (size of dissipation sheet or length of vortex tubes) is consistent with the bounds for the dissipation range in the energy and dissipation spectrum as inferred from the three-dimensional model energy spectrum.

Fully developed rotating turbulent channel flow has been studied, through direct numerical simulations, for the complete range of rotation numbers for which the flow is turbulent. The present investigation suggests that complete flow laminarization occurs at a rotation number Ro = 2Ωδ/Ub ≤ 3.0, where Ω denotes the system rotation, Ub is the mean bulk velocity and δ is the half-width of the channel. Simulations were performed for ten different rotation numbers in the range 0.98 to 2.49 and complemented with earlier simulations (done in our group) for lower values of Ro. The friction Reynolds number Reτ = uτδ/ν (where uτ is the wall-shear velocity and ν is the kinematic viscosity) was chosen as 180 for these simulations. A striking feature of rotating channel flow is the division into a turbulent (unstable) and an almost laminarized (stable) side. The relatively distinct interface between these two regions was found to be maintained by a balance where negative turbulence production plays an important role. The maximum difference in wall-shear stress between the two sides was found to occur for a rotation number of about 0.5. The bulk flow was found to monotonically increase with increasing rotation number and reach a value (for Reτ = 180) at the laminar limit (Ro = 3.0) four times that of the non-rotating case.

Direct numerical simulation is used to study the temporal development of single vortex rings at various Reynolds numbers and core thicknesses. Qualitative differences between the evolution of thin- and thick-core rings are observed leading to a correction factor to the classical equation for the ring translational velocity. We compare the obtained linear modal growth rates with previous work, highlighting the role of the wake in triply periodic numerical simulations. The transition from a laminar to a turbulent ring is marked by the rearrangement of the outer core vorticity into a clearly defined secondary structure. The onset of the fully turbulent state is associated with shedding of the structure in a series of hairpin vortices. A Lagrangian particle analysis was performed to determine the ring entrainment and detrainment properties and to investigate the possibility of an axial flow being generated around the circumference of the core region prior to the onset of turbulence.

The time development of small three-dimensional disturbances in plane Poiseuille flow of helium II is considered. The study is conducted by considering the interaction of a normal fluid field and a superfluid field. The interaction is caused by a mutual friction forcing between the two flow fields. Specifically, the stability of the normal fluid affected by the mutual forcing is considered. Compared to the ordinary fluid case where the mutual forcing is not present, the presence of the mutual forcing implies a substantial increase of the transient growth of the disturbances. The increase of the transient growth occurs because the mutual forcing reduces the damping of the disturbances. The phase of transient growth becomes thereby more prolonged and higher levels of amplification are reached. There is also a minor effect on the transient growth caused by the modification of the mean flow owing to the mutual forcing. The strongest transient growth occurs for streamwise elongated disturbances, i.e. disturbances with streamwise wavenumber α = 0. When α increases beyond zero, the transient amplification quickly becomes reduced. Striking differences compared to the ordinary fluid case are that the largest transient amplification does not occur when the spanwise wavenumber (β) is close to two and that the peak level of the disturbance energy density amplification does not depend on the square of the Reynolds number.

The study addresses the linear instability of narrow spectra homogeneous seas and its subsequent evolution in time, subject to inhomogeneous disturbances. Specifically, we study unidirectional spectra, where according to the kinetic equation no spectral evolution is expected. In the region of instability, recurrent evolution is discovered. This recurrence is the stochastic counterpart of the Fermi–Pasta–Ulam recurrence obtained for the cubic Schrödinger equation.

The phenomena of self-diffusion and migration of rough spheres in nonlinear shear flows are investigated using a new traction-corrected boundary element method (TC-BEM) in which the near-field asymptotics for the traction solution in the interstitial region between two nearly touching spheres is seamlessly coupled with a traditional direct boundary element method. The TC-BEM is extremely accurate in predicting particle trajectories, and hence can be used to calculate both the particle self-diffusivity and a newly defined migration diffusivity for dilute suspensions. The migration diffusivity is a function of a nonlinearity parameter characterizing the shear flow and arises from the net displacement of the centre of gravity of particle pairs. This net displacement of the centre of gravity of particle pairs does not occur for smooth particles, nor for rough particles in a linear shear flow. An explanation is provided for why two-particle interactions of rough spheres in a nonlinear shear flow result in particle migration.

A device constructed from a filament of paramagnetic beads connected to a human red blood cell will swim when subject to an oscillating magnetic field. Bending waves propagate from the tip of the tail toward the red blood cell in a fashion analogous to flagellum beating, making the artificial swimmer a candidate for studying what has been referred to as ‘flexible oar’ micro-swimming. In this study, we demonstrate that under the influence of a rotating field the artificial swimmer will perform ‘corkscrew’-type swimming. We conduct numerical simulations of the swimmer where the paramagnetic tail is represented as a series of rigid spheres connected by flexible but inextensible links. An optimal range of parameters governing the relative strength of viscous, elastic and magnetic forces is identified for swimming speed. A parameterization of the motion is extracted and examined as a function of the driving frequency. With a continuous elastica/resistive force model, we obtain an expression for the swimming speed in the low-frequency limit. Using this expression we explore further the effects of the applied field, the ratio of the transverse field to the constant field, and the ratio of the radius of the sphere to the length of the filament tail on the resulting dynamics.

The slow downstream (x) variation of a dense and inviscid bottom current (u) in a parabolic strait with a sill at y = 0 is investigated. Vanishing potential vorticity is assumed and the density interface in the 1 1/2-layer model intersects the bottom at y = y1 and y = y2 < y1, where the vanishing layer thickness (h) provides the free dynamical boundary condition. For time-dependent finite-amplitude waves, the nonlinear hyperbolic equations obtained here give the wave velocity and indicate the sense in which lateral wave steepening occurs. The long-wave perturbations of y1(x,t), y2(x,t) are stationary if where g′ is the reduced gravity, μ = ∂2M/∂y2 is the parabolic curvature of the bottom elevation (M), and f is the Coriolis parameter. This controls the upstream–downstream flow, and the downstream nonlinearity generates ‘short’ waves which may initiate lateral mixing with the adjacent (less dense) water mass.

It is also shown that short waves are exponentially amplified with a maximum growth rate (about 1/day) depending only on g′μ/f2. When g′μ/f2 = 1 (a narrow strait) the instability is suppressed, but for small g′μ/f2≪1 the growth rate is comparable to the flat bottom case μ = 0, studied by Griffiths, Killworth & Stern (J. Fluid Mech. Vol. 117, 1982, p. 343.).

Direct numerical simulation is used to study passive scalar transport and mixing in a round turbulent jet, in a laminar crossflow. The ratio of the jet velocity to that of the crossflow is 5.7, the Schmidt number of the scalar is 1.49, and the jet-exit Reynolds number is 5000. The scalar field is used to compute entrainment of the crossflow fluid by the jet. It is shown that the bulk of this entrainment occurs on the downstream side of the jet. Also, the transverse jet entrains more fluid than a regular jet even when the jet has not yet bent into the crossflow. The transverse jet's enhanced entrainment is explained in terms of the pressure field around the jet. The acceleration imposed by the crossflow deforms the jet cross-section on the downstream side, which sets up a pressure gradient that drives downstream crossflow fluid toward the jet. The simulation results are used to comment on the applicability of the gradient–diffusion hypothesis to compute passive scalar mixing in this flow field. Computed values of the eddy diffusivity show significant scatter, and a pronounced anisotropy. The near field also exhibits counter gradient diffusion.

We present direct multipoint velocity measurements of the two-dimensional velocity field in the central region of turbulent Rayleigh–Bénard convection. The local homogeneity and isotropy of the velocity field are tested using a number of criteria and are found to hold to an excellent degree. The properties of velocity circulation Γr are also studied. The results show that the circulation appears to be more effective in capturing the effect of local anisotropy than the velocity field itself. The distribution of Γr is found to depend on the scale r, reflecting strong intermittency. It is further found that the velocity circulation has the same anomalous scaling exponents as the longitudinal and transverse structure functions for low-order moments (p≲5), whereas, for high-order moments (p≳5), the anomalous scaling exponents for circulation are found to be systematically smaller than the scaling exponents of the longitudinal and transverse structure functions.

Reflecting internal gravity waves in a stratified fluid preserve their frequency and thus their angle with the gravitational direction. At boundaries that are neither horizontal nor vertical, this leads to a focusing or defocusing of the waves. Previous theoretical and experimental work has demonstrated how this can lead to internal wave energy being focused onto ‘wave attractors’ in relatively simple geometries. We present new experimental and theoretical results on the dynamics of wave attractors in a nearly two-dimensional trapezoidal basin. In particular, we demonstrate how a basin-scale mode forced by simple mechanical excitation develops an equilibrium spectrum. We find a balance between focusing of the basin-scale internal wave by reflection from a single sloping boundary and viscous dissipation of the waves with higher wavenumbers. Theoretical predictions using a simple ray-tracing technique are found to agree well with direct experimental observations of the waves. With this we explain the observed behaviour of the wave attractor during the initial development, steady forcing, and the surprising increase of wavenumber during the decay of the wave field after the forcing is terminated.

Liu & Chan (J. Fluid Mech. vol. 579, 2007, p. 467, hereinafter referred to as LC) derived analytical solutions for the interactions between shallow water waves and a viscous fluid seabed. In this paper we present a set of new experimental data on the solitary-wave-induced flows in a viscous muddy seabed so as to validate LC's theory and the approximations employed. In the experiments a clear silicone fluid was used as the viscous mud and particle image velocimetry was employed to measure the velocity field inside the viscous mud. The shear stress along the bottom of the mud bed and the displacement of the water--mud interface were also deduced from data. Experimental results showed excellent agreement with the theoretical solutions. Additional analyses were performed to show that the ratio of the muddy seabed thickness to the corresponding bottom boundary-layer thickness, , plays an important role in characterizing mud flow regimes. When , the vertical profile of the horizontal velocity in the mud bed can be parameterized as a parabola. On the other hand, when , the velocity profile appears as a plug flow above a thin viscous layer. When , the flow patterns are more complex than the other two regimes and flow reversal can occur inside the viscous mud bed.

Natural convection in a cubical cavity heated from below with perfectly conducting sidewalls is investigated numerically. A parameter continuation procedure based on a Galerkin spectral method was applied to obtain the bifurcation diagrams for steady flow solutions over the region of Rayleigh numbers Ra ≤ 1.5 × 105 at Prandtl numbers Pr = 0.71 and 130. In both cases, the bifurcation diagrams were more complex than those previously reported for adiabatic sidewalls. Four and nine different convective solutions (without taking into account the solutions obtained by symmetry) that were stable over certain ranges of Ra were respectively identified at Pr = 0.71 and 130. The dependence of the bifurcation diagrams and of the topology of the flow patterns on the Prandtl number were also stronger in the case of conducting sidewalls. Most of the flow patterns investigated evolved to double toroid-like topologies with increasing Rayleigh number. This is especially noticeable at Pr = 130, where all flow patterns adopted double-toroid shapes that were superimposed on the characteristic flow patterns observed at values of Ra slightly above the respective bifurcation points where they originated. At sufficiently high Ra the double-toroid pattern configuration prevailed. This phenomenon, which has not been previously observed in the case of adiabatic lateral walls, is related to the thermal activity of the sidewalls, which locally extract/supply relatively large amounts of heat from/to the fluid. These predictions are consistent with experimental flow transitions and topologies reported in the literature. In addition, a complete bifurcation study in the two-dimensional (Ra, Pr)-plane was carried out for the flow pattern with an initial configuration of four connected half-rolls which was stable at both Pr = 0.71 and 130. Since the surface of Nu over the (Ra, Pr)-plane presented several folds and cusps, different regions were identified as a function of the number of particular realizations of this flow pattern, varying between zero and five. Three different regions of stability were identified for this particular flow pattern in the (Ra, Pr)-plane within the range of parameters investigated, i.e. Ra ≤ 1.5 × 105 and 0.71 ≤ Pr ≤ 130.

Molecular dynamic (MD) simulations are used to probe the ability of Navier–Stokes-order theories to predict each of the constitutive quantities – heat flux, stress tensor and dissipation rate – associated with granular materials. The system under investigation is bounded by two opposite walls of set granular temperature and is characterized by zero mean flow. The comparisons between MD and theory provide evidence of higher-order effects in each of the constitutive quantities. Furthermore, the size of these effects is roughly one order of magnitude greater, on a percentage basis, for heat flux than it is for stress or dissipation rate. For the case of heat flux, these effects are attributed to super-Burnett-order contributions (third order in gradients) or greater, since Burnett-order contributions to the heat flux do not exist. Finally, for the system considered, these higher-order contributions to the heat flux outweigh the first-order contribution arising from a gradient in concentration (i.e. the Dufour effect)