The propagation at high Reynolds number of a heavy, axisymmetric gravity current of given initial volume over a horizontal boundary is considered in both rotating and non-rotating situations. The investigation combines experiments with theoretical predictions by both shallow-water approximations and numerical solutions of the full axisymmetric equations. Attention is focused on cases when the initial ratio of Coriolis to inertia forces is small. The experiments were performed by quickly releasing a known cylindrical volume of dense salt water of 2 m diameter at the centre of a circular tank of diameter 13 m containing fresh ambient water of typical depth 80 cm. The propagation of the current was recorded for different initial values of the salt concentration, the volume of released fluid, the ratio of the initial height of the current to the ambient depth, and the rate of rotation. A major feature of the rotating currents was the attainment of a maximum radius of propagation. Thereafter a contraction–relaxation motion of the body of fluid and a regular series of outwardly propagating pulses was observed. The frequency of these pulses is slightly higher than inertial, and the amplitude is of the order of magnitude of half the maximum radius. Theoretical predictions of the corresponding gravity currents were also obtained by (i) previously developed shallow-water approximations (Ungarish & Huppert 1998) and (ii) a specially developed finite-difference code based on the full axisymmetric Navier–Stokes equations. The ‘numerical experiments’ provided by this code are needed to capture details of the flow field (such as the non-smooth shape of the interface, the vertical dependence of the velocity field) which are not reproduced by the shallow-water model and are very difficult for, or outside the range of, accurate experimental measurement. The comparisons and discussion provide insight into the flow field and indicate the advantages and limitations of the verified simulation tools.

In this paper the structure of strong transverse waves in two-dimensional numerical simulations of cellular detonations is investigated. Resolution studies are performed and it is shown that much higher resolutions than those generally used are required to ensure that the flow and burning structures are well resolved. Resolutions of less than about 20 numerical points in the characteristic reaction length of the underlying steady detonation give very poor predictions of the shock configurations and burning, with the solution quickly worsening as the resolution drops. It is very difficult and dangerous to attempt to identify the physical structure, evolution and effect on the burning of the transverse waves using such under-resolved calculations. The process of transverse wave and triple point collision and reflection is then examined in a very high-resolution simulation. During the reflection, the slip line and interior triple point associated with the double Mach configuration of strong transverse waves become detached from the front and recede from it, producing a pocket of unburnt gas. The interaction of a forward facing jet of exploding gas with the emerging Mach stem produces a new double Mach configuration. The formation of this new Mach configuration is very similar to that of double Mach reflection of an inert shock wave reflecting from a wedge.

Large-eddy simulations were carried out to simulate laboratory-scale isolated buoyant convection in unstratified water with shelf and slope topography in the presence of rotation and to compare and complement the experimental study of Jacobs & Ivey (1998) under the same conditions. The simulation code developed in this work was a three-dimensional incompressible Navier–Stokes solver and the simulation runs were performed on a distributed memory massively parallel computer, namely the IBM SP2, to study the effects of different applied heat fluxes and system rotation rates. We are able to show for the first time the detailed temporal evolution and spatial structure of the three-dimensional convective flow field. Rayleigh–Bénard instability in the form of circular concentric convective rings is recognized in the initiation process of the convection. The onset of Rayleigh–Bénard instability was investigated and the critical Rayleigh number was found to increase with Taylor number only when the Taylor number is greater than 5 × 103, where both non-dimensional parameters are based on the conductive layer thickness. The horizontally axisymmetric convective rings later break down and evolve into a quasi-two-dimensional vortex field. An azimuthal rim current develops around the periphery of the convective region. Our simulation results confirmed that the rim current velocity scales as Bt1/2/Hf3/2. Here B is the buoyancy flux applied over a bottom circular disk, f is the Coriolis parameter, t is the time and H is the distance between the tank bottom and the shelf. With increasing lateral temperature gradient the rim current undergoes a baroclinic instability. Our study of root-mean-square velocities in the convective region suggests that the transition from the buoyancy-flux-controlled to background-rotation-controlled flow occurred when the natural Rossby number Ro* became smaller than a critical value between 0.015 and 0.044. The simulation results of the convective overturning time, the wavelength of the baroclinic eddies and the density anomaly at steady state are all in reasonable agreement with the experimental data.

One-dimensional turbulence is a stochastic simulation method representing the time evolution of the velocity profile along a notional line of sight through a turbulent flow. In this paper, the velocity is treated as a three-component vector, in contrast to previous formulations involving a single velocity component. This generalization allows the incorporation of pressure-scrambling effects and provides a framework for further extensions of the model. Computed results based on two alternative physical pictures of pressure scrambling are compared to direct numerical simulations of two time-developing planar free shear flows: a mixing layer and a wake. Scrambling based on equipartition of turbulent kinetic energy on an eddy-by-eddy basis yields less accurate results than a scheme that maximizes the intercomponent energy transfer during each eddy, subject to invariance constraints. The latter formulation captures many features of free shear flow structure, energetics, and fluctuation properties, including the spatial variation of the probability density function of a passive advected scalar. These results demonstrate the efficacy of the proposed representation of vector velocity evolution on a one-dimensional domain.

We study by direct numerical simulation the motion of incompressible fluid contained in an ellipsoid of revolution with ellipticity 0.1 or less which rotates about its axis of symmetry and whose rotation axis is executing precessional motion. A solution to this problem for an inviscid fluid given by Poincaré (1910) predicts motion of uniform vorticity. The simulations show how the orientation of the average vorticity of a real fluid is influenced by both pressure and viscous torques exerted by the boundaries. Axisymmetric shear layers appear which agree well with those observed experimentally by Malkus (1968). Shear caused by deviations from a velocity field with uniform vorticity triggers an instability consisting of waves propagating around the average rotation axis of the fluid. The Ekman layers at the boundaries may also become unstable.

This paper analyses a numerical model of a flow around an obstacle bounded in a channel with arbitrary walls. The model is based on the usual two-dimensional model of ideal incompressible weightless fluid. It is validated here by comparison with analytical findings and experimental data, which consisted of the velocity field determined by particle image velocimetry (PIV). Despite the simplicity of our wake model, the numerical data are usually in good agreement with the analytical and experimental data.

The analysis of the self-induced velocity of a single helical vortex (Boersma & Wood 1999) is extended to include equally spaced multiple vortices. This arrangement approximates the tip vortices in the far wake of multi-bladed wind turbines, propellers, or rotors in ascending, descending, or hovering flight. The problem is reduced to finding, from the Biot–Savart law, the additional velocity of a helix due to an identical helix displaced azimuthally. The resulting Biot–Savart integral is further reduced to a Mellin–Barnes integral representation which allows the asymptotic expansions to be determined for small and for large pitch. The Biot–Savart integral is also evaluated numerically for a total of two, three and four vortices over a range of pitch values. The previous finding that the self-induced velocity at small pitch is dominated by a term inversely proportional to the pitch carries over to multiple vortices. It is shown that a far wake dominated by helical tip vortices is consistent with the one-dimensional representation that leads to the Betz limit on the power output of wind turbines. The small-pitch approximation then allows the determination of the blade&s bound vorticity for optimum power extraction. The present analysis is shown to give reasonable estimates for the vortex circulation in experiments using a single hovering rotor and a four-bladed propeller.

Morland, Saffman & Yuen's (1991) study of the stability of a semi-infinite, concave shear flow bounded above by a capillary–gravity wave, for which they obtained numerical solutions of Rayleigh's equation, is revisited. A variational formulation is used to construct an analytical description of the unstable modes for the exponential velocity profile U = U0 exp(y/d), −∞ < y [les ] 0. The assumption of slow waves ([mid ]c[mid ] [Lt ] U0) yields an approximation that agrees with the numerical results of Morland et al. The assumption of short waves (kd [Gt ] 1) yields Shrira's (1993) asymptotic approximation.

Direct numerical simulations (DNS) and optimal control theory are used in a predictive control setting to determine controls that effectively reduce the turbulent kinetic energy and drag of a turbulent flow in a plane channel at Reτ = 100 and Reτ = 180. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The algorithm used for the control optimization is based solely on the control objective and the nonlinear partial differential equation governing the flow, with no ad hoc assumptions other than the finite prediction horizon, T, over which the control is optimized.

Flow relaminarization, accompanied by a drag reduction of over 50%, is obtained in some of the control cases with the predictive control approach in direct numerical simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (using this type of actuation) via adaptive strategies such as neural networks, intuition-based strategies such as opposition control, and the so-called ‘suboptimal’ strategies, which involve optimizations over a vanishingly small prediction horizon T+ → 0. To achieve flow relaminarization in the predictive control approach, it is shown that it is necessary to optimize the controls over a sufficiently long prediction horizon T+ [gsim ] 25. Implications of this result are discussed.

The predictive control algorithm requires full flow field information and is computationally expensive, involving iterative direct numerical simulations. It is, therefore, impossible to implement this algorithm directly in a practical setting. However, these calculations allow us to quantify the best possible system performance given a certain class of flow actuation and to qualify how optimized controls correlate with the near-wall coherent structures believed to dominate the process of turbulence production in wall-bounded flows. Further, various approaches have been proposed to distil practical feedback schemes from the predictive control approach without the suboptimal approximation, which is shown in the present work to restrict severely the effectiveness of the resulting control algorithm. The present work thus represents a further step towards the determination of optimally effective yet implementable control strategies for the mitigation or enhancement of the consequential effects of turbulence.

We investigate the formation of small-scale three-dimensional bedforms due to interactions of an erodible bed with a sea wave that obliquely approaches the coast, being partially reflected at the beach. In this case the trajectories of fluid particles at the top of the bottom boundary layer are ellipses in the horizontal plane, the axes of which depend on the angle of wave incidence and the distance from the shore. A weakly nonlinear stability analysis of an initially flat, cohesionless, sandy bottom is performed. We focus on the resonant interaction of three perturbation components. The results show that these elliptical forcing conditions are responsible for the formation of both brick-pattern ripples and tile ripples. In particular tile ripples are associated with a flow at the top of the bottom boundary layer which is near-circular (ellipticity close to one), whereas brick-pattern ripples are related to a unidirectional oscillatory flow (zero ellipticity).

We study the instability of a laminar vortex column (in an external orthogonal strain field) to an axisymmetric core size perturbation, and the resulting transition to fine-scale turbulence. The perturbation, which evolves as a standing wave oscillation (i.e. core dynamics, CD), is inviscidly amplified by the external strain. Analysis of a weakly strained Rankine vortex explains the physical mechanism of instability: resonant interaction between the perturbation – the azimuthal wavenumber m = 0 wave – and m = ±2 waves. The CD instability (CDI) – a type of elliptic instability – experiences the fastest growth when the CD oscillation frequency equals vortex column's fluid angular velocity, such matching occurring only at specific discrete values of the axial wavenumber k. At this resonant frequency, the net effect of the swirl-induced tilting of perturbation vorticity and the CD-induced tilting of base flow vorticity is such that perturbation vorticity is continually aligned with the stretching direction of the external strain. Such strain–vorticity locking occurs for all m; hence all waves are unstable, the instability oscillation frequency being dependent on m. In a viscous Gaussian-like vortex, CDI has low-strain, low-Re and high-k cutoffs – consequences of the competing effects of inviscid amplification and viscous damping. Direct numerical simulation reveals two physical-space mechanisms of transition: (i) formation of a thin annular vortex sheath surrounding a low-enstrophy ‘bubble’ (similar to axisymmetric vortex breakdown) and the sheath's subsequent roll-up into smaller ‘vortexlets’; and (ii) folding and reconnection of core vortex filaments giving rise to additional fine-grained random vorticity within the bubble – both mechanisms caused by CD-induced intense axial flow within the vortex column. The resulting finer tubular vortices (similar to ‘worms’) have in turn their own CD, and thus this transition scenario suggests a physical-space cascade process in developed turbulence (as well as a concomitant anti-cascade process during the bubble's collapse phase). Additionally, we show that bending waves, in spite of their faster growth, effect surprisingly much slower transfer of energy into fine scales than CDI does, and hence are less effective than CDI in vortex transition and in turbulence cascade.

Standing gravity waves forced beyond the maximum height for perfect periodicity can produce vertical jets with sharp-pointed tips. In this paper, canonical forms for the wave crests are derived which display sharp cusps in the limit as the time t tends to infinity. The theoretical profile is in general quartic in the space coordinates, and can describe the smooth transition of a fairly low wave crest to a cusped form. There is no singularity as the surface slope passes through 45°.

The motion and shape evolution of viscous drops made from a dilute suspension of tiny, spherical glass beads sedimenting in an otherwise quiescent liquid is investigated both experimentally and theoretically for conditions of low Reynolds number. In the (presumed) absence of any significant interfacial tension, the Bond number [Bscr ] = (Δρ)gR2/σ is effectively infinite. The key stages of deformation of single drops and pairs of interacting drops are identified. Of particular interest are (i) the coalescence of two trailing drops, (ii) the subsequent formation of a torus, and (iii) the breakup of the torus into two or more droplets in a repeating cascade. To overcome limitations of the boundary-integral method in tracking highly deformed interfaces and coalescing and dividing drops, we develop a formal analogy between drops of homogeneous liquid and a dilute, uniformly distributed swarm of sedimenting particles, for which only the 1/r far-field hydrodynamic interactions are important. Simple, robust numerical simulations using only swarms of Stokeslets reproduce the main phenomena observed in the classical experiments and in our flow-visualization studies. Detailed particle image velocimetry (PIV) for axisymmetric configurations enable a mechanistic analysis and confirm the theoretical results. We expose the crucial importance of the initial condition – why a single spherical drop does not deform substantially, but a pair of spherical drops, or a bell-shaped drop similar to what is actually formed in the laboratory, does undergo the torus/breakup transformation. The extreme sensitivity of the streamlines to the shape of the ring-like swarm explains why the ring that initially forms in the experiments does not behave like the slender open torus analysed asymptotically by Kojima, Hinch & Acrivos (1984). Essentially all of the phenomena described above can be explained within the realm of Stokes flow, without resort to interfacial tension or inertial effects.

A fast computational method for fully nonlinear non-overturning water waves is derived in two and three dimensions. A corresponding time-stepping scheme is developed in the two-dimensional case. The essential part of the method is a fast converging iterative solution procedure of the Laplace equation. One part of the solution is obtained by fast Fourier transform, while another part is highly nonlinear and consists of integrals with kernels that decay quickly in space. The number of operations required is asymptotically O(N log N), where N is the number of nodes at the free surface. While any accuracy of the computations is achieved by a continued iteration of the equations, one iteration is found to be sufficient for practical computations, while maintaining high accuracy. The resulting explicit approximation of the scheme is tested in two versions. Simulations of nonlinear wave fields with wave slope even up to about unity compare very well with reference computations. The numerical scheme is formulated in such a way that aliasing terms are partially or completely avoided.

Internal hydraulic theory is often used to describe idealized bi-directional exchange flow through a constricted channel. This approach is formally applicable to layered flows in which velocity and density are represented by discontinuous functions that are constant within discrete layers. The theory relies on the determination of flow conditions at points of hydraulic control, where long interfacial waves have zero phase speed. In this paper, we consider hydraulic control in continuously stratified exchange flows. Such flows occur, for example, in channels connecting stratified reservoirs and between homogeneous basins when interfacial mixing is significant. Our focus here is on the propagation characteristics of the gravest vertical-mode internal waves within a laterally contracting channel.

Two approaches are used to determine the behaviour of waves propagating through a steady, continuously sheared and stratified exchange flow. In the first, waves are mechanically excited at discrete locations within a numerically simulated bi-directional exchange flow and allowed to evolve under linear dynamics. These waves are then tracked in space and time to determine propagation speeds. A second approach, based on the stability theory of parallel shear flows and examination of solutions to a sixth-order eigenvalue problem, is used to interpret the direct excitation experiments. Two types of gravest mode eigensolutions are identified: vorticity modes, with eigenfunction maxima centred above and below the region of maximum density gradient, and density modes with maxima centred on the strongly stratified layer. Density modes have phase speeds that change sign within the channel and are analogous to the interfacial waves in hydraulic theory. Vorticity modes have finite propagation speed throughout the channel but undergo a transition in form: upwind of the transition point the vorticity mode is trapped in one layer. It is argued that modes trapped in one layer are not capable of communicating interfacial information, and therefore that the transition points are analogous to control points. The location of transition points are identified and used to generalize the notion of hydraulic control in continuously stratified flows.

We investigate the miscible Rayleigh–Taylor (RT) instability in both two and three dimensions using direct numerical simulations, where the working fluid is assumed incompressible under the Boussinesq approximation. We first consider the case of randomly perturbed interfaces. With a variety of diagnostics, we develop a physical picture for the detailed temporal development of the mixed layer: we identify three distinct evolutionary phases in this development, which can be related to detailed variations in the growth of the mixing zone. Our analysis provides an explanation for the observed differences between two- and three-dimensional RT instability; the analysis also leads us to concentrate on the RT models which (i) work equally well for both laminar and turbulent flows, and (ii) do not depend on turbulent scaling within the mixing layer between fluids. These candidate RT models are based on point sources within bubbles (or plumes) and their interaction with each other (or the background flow). With this motivation, we examine the evolution of single plumes, and relate our numerical results (for single plumes) to a simple analytical model for plume evolution.