Highly resolved three-dimensional and two-dimensional simulations of gravity currents in planar and cylindrical configurations are presented. The volume of release of the heavy fluid is varied and the different phases of spreading, namely acceleration, slumping, inertial and viscous phases, are studied. The incompressible Navier–Stokes equations are solved assuming that the Boussinesq approximation is valid for small density difference. The simulations are performed for three different Reynolds numbers (Re): 895, 3450 and 8950 (this particular choice corresponds to values of Grashof number: 105, 1.5 × 106 and 107, respectively). Following their sudden release, the gravity currents are observed to go through an acceleration phase in which the maximum front velocity is reached. As the interface of the current rolls up, the front velocity slightly decreases from the maximum and levels off to a nearly constant value. At higher Re, three-dimensional disturbances grow rapidly and the currents become strongly turbulent. In contrast, in two-dimensional simulations, the rolled-up vortices remain coherent and very strong. Depending on the initial Re of the flow and on the size of the release, the current may transition from the slumping to the inertial phase, or directly to the viscous phase without an inertial phase. New criteria for the critical Re are introduced for the development of the inertial phase. Once the flow transitions to the inertial or viscous phase, it becomes fully three-dimensional. During these phases of spreading, two-dimensional approximations underpredict the front location and velocity. The enhanced vortex coherence of the two-dimensional simulations leads to strong vortex interaction and results in spurious strong time variations of the front velocity. The structure and dynamics of the three-dimensional currents are in good agreement with previously reported numerical and experimental observations.

This paper considers a paradigm large-Prandtl-number, large-Rayleigh-number forced convection problem suggested by the batch melting process in the glass industry. Although the fluid is heated from above, non-uniform heating in the horizontal direction induces thermal boundary layers in which colder liquid is driven over hotter liquid. This leads to an interesting selection problem in the boundary layer analysis, whose resolution is suggested by a combination of analytical and numerical evidence.

The stretching rate, normalized by the reciprocal of the Kolmogorov time, of sufficiently extended material lines and surfaces in statistically stationary homogeneous isotropic turbulence depends on the Reynolds number, in contrast to the conventional picture that the statistics of material object deformation are determined solely by the Kolmogorov-scale eddies. This Reynolds-number dependence of the stretching rate of sufficiently extended material objects is numerically verified both in two- and three-dimensional turbulence, although the normalized stretching rate of infinitesimal material objects is confirmed to be independent of the Reynolds number. These numerical results can be understood from the following three facts. First, the exponentially rapid stretching brings about rapid multiple folding of finite-sized material objects, but no folding takes place for infinitesimal objects. Secondly, since the local degree of folding is positively correlated with the local stretching rate and it is non-uniformly distributed over finite-sized objects, the folding enhances the stretching rate of the finite-sized objects. Thirdly, the stretching of infinitesimal fractions of material objects is governed by the Kolmogorov-scale eddies, whereas the folding of a finite-sized material object is governed by all eddies smaller than the spatial extent of the objects. In other words, the time scale of stretching of infinitesimal fractions of material objects is proportional to the Kolmogorov time, whereas that of folding of sufficiently extended material objects can be as long as the turnover time of the largest eddies. The combination of the short time scale of stretching of infinitesimal fractions and the long time scale of folding of the whole object yields the Reynolds-number dependence. Movies are available with the online version of the paper.

We perform numerical box simulations of strongly stratified turbulence. The equations solved are the Boussinesq equations with constant Brunt–Väisälä frequency and forcing either in rotational or divergent modes, or, with another terminology, in vortical or wave modes. In both cases, we observe a forward energy cascade and inertial-range scaling of the horizontal kinetic and potential energy spectra. With forcing in rotational modes, there is approximate equipartition of kinetic energy between rotational and divergent modes in the inertial range. With forcing in divergent modes the results are sensitive to the vertical forcing wavenumber kfv. If kfv is sufficiently large the dynamics is very similar to the dynamics of the simulations which are forced in rotational modes, with approximate equipartition of kinetic energy in rotational and divergent modes in the inertial range. Frequency spectra of rotational, divergent and potential energy are calculated for individual Fourier modes. Waves are present at low horizontal wavenumbers corresponding to the largest scales in the boxes. In the inertial range, the frequency spectra exhibit no distinctive peaks in the internal wave frequency. In modes for which the vertical wavenumber is considerably larger than the horizontal wavenumber, the frequency spectra of rotational and divergent modes fall on top of each other. The simulation results indicate that the dynamics of rotational and divergent modes develop on the same time scale in stratified turbulence. We discuss the relevance of our results to atmospheric and oceanic dynamics. In particular, we review a number of observational reports indicating that stratified turbulence may be a prevalent dynamic process in the ocean at horizontal scales of the order of 10 or 100 m up to several kilometres.

Inviscid gravity currents released from a finite-length lock are known to propagate at a constant speed to a predicted finite distance before decelerating. By extension this should occur in a two-layer fluid with equal upper- and lower-layer depths for an intrusion having the average density of the ambient. The experiments presented here show this is not necessarily the case. The finite-depth thickness of the interface non-negligibly influences the evolution of the intrusion so that it propagates well beyond the predicted constant-speed limit; it propagates without decelerating beyond 22 lock lengths in a rectilinear geometry and beyond 6 lock radii in an axisymmetric geometry. Experiments and numerical simulations demonstrate that the intrusion speed decreases to half the two-layer speed in the circumstance in which the interface spans the domain. The corresponding long mode-2 interfacial wave speed increases rapidly with interfacial thickness, becoming comparable with the intrusion speed when the interfacial thickness is approximately one-quarter the domain height. For somewhat thinner interfacial thicknesses, the intrusion excites solitary waves that move faster than the long-wave speed. The coupling between intrusions and the waves they excite, together with reduced mixing of the current head, result in constant-speed propagation for longer times.

Turbidity currents act to sculpt the submarine environment through sediment erosion and deposition. A sufficiently swift turbidity current on a steep slope can be expected to be supercritical in the sense of the bulk Richardson number; a sufficiently tranquil turbidity current on a mild slope can be expected to be subcritical. The transition from supercritical to subcritical flow is accomplished through an internal hydraulic jump. Consider a steady turbidity current flowing from a steep canyon onto a milder fan, and then exiting the fan down another steep canyon. The flow might be expected to undergo a hydraulic jump to subcritical flow near the canyon–fan break, and then accelerate again to critical flow at the fan–canyon break downstream. The problem of locating the hydraulic jump is here termed the ‘jump problem’. Experiments with fine-grained sediment have confirmed the expected behaviour outlined above. Similar experiments with coarse-grained sediment suggest that if the deposition rate is sufficiently high, this ‘jump problem’ may have no solution with the expected behaviour, and in particular no solution with a hydraulic jump. In such cases, the flow either transits the length of the low-slope fan as a supercritical flow and shoots off the fan–canyon break without responding to it, or dissipates as a supercritical flow before exiting the fan. The analysis presented below confirms the existence of a range associated with rapid sediment deposition where no solution to the ‘jump problem’ can be found. The criterion for this range is stated in terms of an order-one dimensionless parameter involving the fall velocity of the sediment. The criterion is tested and confirmed against the experiments mentioned above. A sample field application is presented.

A hybrid method for the problem of transient shock-induced filtration of the gas flow through granular media is developed. The hybrid method combines a controlled shock-tube test and Morrison's simplified approach to the problem of gas filtration. It is demonstrated that most pressure traces that have been recorded in various laboratories with a large variety of granular material samples and under different conditions are limited to situations in which the pressure losses in the flow are dominated by the Forchheimer mechanism. The hybrid method enables these results to be described by a single, universal pressure curve, and specification of the Forchheimer coefficient which is one of the two key parameters responsible for the correct simulation of the filtration-flow behaviour. The second key parameter, the Darcy coefficient, cannot be evaluated by the available experimental results. To overcome this shortcoming, a new controlled laboratory test that provides a wider range of the flow conditions, from the Forchheimer to the developed mixed flow, was conducted. In turn, a comprehensive gas dynamic analysis of the transient flow inside the shock tube enables us to define, from the single controlled laboratory test, the two coefficients of the Forchheimer resistance law, a and b.

The tip-leakage flow in a turbomachinery cascade is studied using large-eddy simulation with particular emphasis on understanding the underlying mechanisms for viscous losses in the vicinity of the tip gap. Systematic and detailed analysis of the mean flow field and turbulence statistics has been made in a linear cascade with a moving endwall. Gross features of the tip-leakage vortex, tip-separation vortices, and blade wake have been revealed by investigating their revolutionary trajectories and mean velocity fields. The tip-leakage vortex is identified by regions of significant streamwise velocity deficit and high streamwise and pitchwise vorticity magnitudes. The tip-leakage vortex and the tip-leakage jet which is generated by the pressure difference between the pressure and suction sides of the blade tip are found to produce significant mean velocity gradients along the spanwise direction, leading to the production of vorticity and turbulent kinetic energy. The velocity gradients are the major causes for viscous losses in the cascade endwall region. The present analysis suggests that the endwall viscous losses can be alleviated by changing the direction of the tip-leakage flow such that the associated spanwise derivatives of the mean streamwise and pitchwise velocity components are reduced.

Recent experiments have shown intriguing regions of intense luminescence or ‘hotspots’ in the vicinity of triple-point shear layers in propagating gaseous detonation waves. Localized explosions have also been observed to develop in these fronts. These features were observed in higher effective activation energy mixtures, but not in lower effective activation energy mixtures. The increased lead shock oscillation through a cell cycle in higher activation energy mixtures may result in a significantly increased disparity in the induction time on either side of the triple-point shear layer, and thus an enhanced mixing between reacted and non-reacted streams supported by Kelvin–Helmholtz instability. The relation between the shear-layer instability and the mixture effective activation energy is analysed by carrying out a spatial linear stability study for three mixtures with different activation energies and injection conditions that correspond to the experimental conditions. The role of vortical structures associated with Kelvin–Helmholtz instability in the formation of localized ignition is investigated by performing two-dimensional Navier–Stokes simulations with detailed chemical kinetics and transport. In the low activation energy mixture, large-scale vortical structures are observed to occur downstream of the induction distance; these structures do not have a noticeable effect on the reaction. In higher effective activation energy mixtures, a thin transverse ignition front develops near the interface between the two gas streams and results in a combustion structure decoupled from the entrainment region. The decoupling leads to attenuation of the instability growth rate when compared to frozen calculations, and a reduced heat release in the high vorticity region. The analysis indicates the instability plays a modest role in ignition events for high activation energy mixtures. The formation of localized explosions observed in high activation energy systems is instead linked to the impossibility of a one-dimensional reactive combustion wave supported by the injection conditions. In the absence of curvature effects and stream-tube divergence, a system of shock waves is formed which spreads the ignition to the cold gas stream.

Solving the Boltzmann–BGK equation, we investigate a flow generated by an infinite plate oscillating with frequency ω. The geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation 0 ≤ ωτ ≤ ∞, where τ is a properly defined relaxation time. A transition from viscoelastic behaviour of a Newtonian fluid (ωτ → 0) to purely elastic dynamics in the limit ωτ → ∞ is discovered. The relation of the derived solutions to nanofluidics is demonstrated on a solvable example of a ‘plane oscillator’. The results from the derived formulae compare well with experimental data on various nanoresonators operating in a wide range of both frequency and pressure variation.

A set of complete two- and three-dimensional direct numerical simulations (DNS) in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is presented in this paper. Although the physical phenomenon is three-dimensional, owing to its prohibitive computational costs the majority of the previous DNS of turbulent and transition natural convection flows in enclosed cavities assumed a two-dimensional behaviour. The configurations selected here (Rayleigh number based on the cavity height 6.4 × 108, 2 × 109 and 1010, Pr = 0.71) are an extension to three dimensions of previous two-dimensional problems.

An overview of the numerical algorithm and the methodology used to verify the code and the simulations is presented. The main features of the flow, including the time-averaged flow structure, the power spectra and probability density distributions of a set of selected monitoring points, the turbulent statistics, the global kinetic energy balances and the internal waves motion phenomenon are described and discussed.

As expected, significant differences are observed between two- and three-dimensional results. For two-dimensional simulations the oscillations at the downstream part of the vertical boundary layer are clearly stronger, ejecting large eddies to the cavity core. In the three-dimensional simulations these large eddies do not persist and their energy is rapidly passed down to smaller scales of motion. It yields on a reduction of the large-scale mixing effect at the hot upper and cold lower regions and consequently the cavity core still remains almost motionless even for the highest Rayleigh number. The boundary layers remain laminar in their upstream parts up to the point where these eddies are ejected. The point where this phenomenon occurs clearly moves upstream for the three-dimensional simulations. It is also shown that, even for the three-dimensional simulations, these eddies are large enough to permanently excite an internal wave motion in the stratified core region. All these differences become more marked for the highest Rayleigh number.

Three-dimensional spatially growing perturbations in a two-dimensional compressible boundary layer are considered within the scope of linearized Navier–Stokes equations. The Cauchy problem is solved under the assumption of a finite growth rate of the disturbances. It is shown that the solution can be presented as an expansion into a biorthogonal eigenfunction system. The result can be used in a decomposition of flow fields derived from computational studies when pressure, temperature, and all the velocity components, together with some of their derivatives, are available. The method can also be used if partial data are available when a priori information may be utilized in the decomposition algorithm.

We present both analytical and numerical solutions describing seepage flows in an unsaturated permeable seabed induced by transient long waves. The effects of compressibility of pore water in the seabed due to a small degree of unsaturation are considered in the investigation. To make the problem tractable analytically, we first focus our attention on situations where the horizontal scale of the seepage flow is much larger than the vertical scale. With this simplification the pore-water pressure in the soil column is governed by a one-dimensional diffusion equation with a specified pressure at the water–seabed interface and the no-flux condition at the bottom of the seabed. Analytical solutions for pore-water pressure and velocity are obtained for arbitrary transient waves. Special cases are studied for periodic waves, cnoidal waves, solitary waves and bores. Numerical solutions are also obtained by simultaneously solving the Navier–Stokes equations for water wave motions and the exact two-dimensional diffusion equation for seepage flows in the seabed. The analytical solutions are used to check the accuracy of the numerical methods. On the other hand, numerical solutions extend the applicability of the analytical solutions. The liquefaction potential in a permeable bed as well as the energy dissipation under various wave conditions are then discussed.

This paper considers two-dimensional flow generated in a stably stratified porous medium by monotonic differential heating of the upper surface. For a rectangular cavity with thermally insulated sides and a constant-temperature base, the flow near the upper surface in the high-Darcy–Rayleigh-number limit is shown to consist of a double horizontal boundary layer structure with descending motion confined to the vicinity of the colder sidewall. Here there is a vertical boundary layer structure that terminates at a finite depth on the scale of the outer horizontal layer. Below the horizontal boundary layers the motion consists of a series of weak, uniformly stratified counter-rotating convection cells. Asymptotic results are compared with numerical solutions for the cavity flow at finite values of the Darcy–Rayleigh number.

Two-dimensional local wall suction is applied to a fully developed turbulent boundary layer such that near-wall turbulence structures are completely sucked out, but most of the turbulent vortices in the original outer layer can survive the suction and cause the resulting laminar flow to undergo re-transition. This enables us to observe and clarify the whole process by which the suction-surviving strong vortical motions give rise to near-wall low-speed streaks and eventually generate wall turbulence. Hot-wire and particle image velocimetry (PIV) measurements show that low-frequency velocity fluctuations, which are markedly suppressed near the wall by the local wall suction, soon start to grow downstream of the suction. The growth of low-frequency fluctuations is algebraic. This characterizes the streak growth caused by the suction-surviving turbulent vortices. The low-speed streaks obtain almost the same spanwise spacing as that of the original turbulent boundary layer without the suction even in the initial stage of the streak development. This indicates that the suction-surviving turbulent vortices are efficient in exciting the necessary ingredients for the wall turbulence, namely, low-speed streaks of the correct scale. After attaining near-saturation, the low-speed streaks soon undergo sinuous instability to lead to re-transition. Flow visualization shows that the streak instability and its subsequent breakdown occur at random in space and time in spite of the spanwise arrangement of streaks being almost periodic. Even under the high-intensity turbulence conditions, the sinuous instability amplifies disturbances of almost the same wavelength as predicted from the linear stability theory, though the actual growth is in the form of a wave packet with not more than two waves. It should be emphasized that the mean velocity develops the log-law profile as the streak breakdown proceeds. The transient growth and eventual breakdown of low-speed streaks are also discussed in connection with the critical condition for the wall-turbulence generation.

The interaction and merger of a pair of co-rotating vortices has been studied experimentally in a wind tunnel. The vortices are generated using two separate wings, allowing both initial circulation ratio and initial separation to be varied. At the nominal test speed of 30 ms−1, the circulation-based Reynolds number is typically 6.4 × 104. Mean crossflow velocities at stations downstream are obtained with a traverse-mounted yaw meter. The vortex interaction is characterized by fitting Lamb–Oseen profiles to the measured, azimuthally averaged, tangential velocity profiles.

It is found that the merger process differs in several important respects from lower-Reynolds-number studies. First, the separation of the vortices decreases continually throughout the initial, ‘viscous growth’, phase, instead of remaining constant. Second, the vortex core growth in this phase appears to be greater than can be accounted for by turbulent diffusivity, even after correcting for the effects of wandering. Finally, the time to merger lies well below that predicted by expressions based on the lower-Reynolds-number observations, and is further reduced when the circulation ratio departs from unity. We conclude that the enhanced core growth is probably due to the short-wavelength ‘elliptic’ instability that has already been observed in some high-Reynolds-number experiments. The mechanism behind the decrease in separation, which is a crucial factor in the reduced merger time, is three-dimensional, but, beyond this, remains unknown.

Lubrication theory is used to derive a coupled pair of strongly nonlinear partial differential equations governing the evolution of interfaces separating a thin film of a pure melt from its crystalline phase and from a gas. The free melt–gas (MG) interface deforms in response to the local state of stress and the crystal–melt (CM) interface can deform by freezing and melting only. A linear stability analysis of a static, uniform film subject to the effects of MG interface capillary forces, thermocapillary forces, the latent heat of fusion, van der Waals attraction, heat transfer and solidification volume change effects, reveals stationary and oscillatory instabilities. The effect of a temperature gradient (by increasing the gas phase temperature) is to stabilize a film. As the temperature gradient is reduced, the onset of instability is oscillatory and is at a unique, finite wavenumber. Instability is oscillatory for all marginally stable, non-isothermal cases. Crystals with higher density than the melt are more stable, whereas crystals with lower density are less stable in the presence of an applied temperature gradient. Fully nonlinear numerical solutions show that oscillatory instabilities lead to rupture by growth of standing or travelling waves. Rupture times and the number of oscillations to rupture increase as the temperature gradient is increased. For stationary linearly unstable initial conditions, the CM interface retreats by melting away from the tip region of the encroaching MG interface due to a rise in the heat flux there as the film thins and nears rupture. Larger amplitude disturbances increase the maximum allowable temperature for instability, at a given wavenumber, and decrease the time to rupture at fixed temperature and wavenumber.

We have measured the trajectory and visualized the wake of a single sphere falling in a fluid confined between two closely spaced glass plates (a Hele-Shaw cell). The position of a sedimenting sphere was measured to better than 0.001d, where d is the sphere diameter, for Reynolds numbers (based on the terminal velocity) between 20 and 330, for gaps between the plates ranging from 1.014d to 1.4d. For gaps in the range 1.01d–1.05d, the behaviour of the sedimenting sphere is found to be qualitatively similar to that of an unconfined cylinder in a uniform flow, but our sedimenting sphere begins to oscillate and shed von Kármán vortices for Re>200, which is far greater than the Re = 49 for the onset of vortex shedding behind cylinders in an open flow. When the gap is increased to 1.10d–1.40d, the vortices behind the sphere are different – they are qualitatively similar to those behind a sphere sedimenting in the absence of confining walls. Our precision measurements of the velocity of a sedimenting sphere and the amplitude and frequency of the oscillations provide a benchmark for numerical simulations of the sedimentation of particles in fluids.

We consider model problems for the tear film over multiple blink cycles in limits that yield a single equation for the tear film; the single nonlinear partial differential equation that governs the film thickness arises from lubrication theory. The two models arise from considering the absence of naturally occurring surfactant and the case when the surfactant strongly affects the surface tension. The film is considered on a sinusoidally varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. A main contribution of this article is computation of solutions for multiple complete blink cycles; the results of these non-trivial computations show a distinct similarity to quantitative in vivo observations of the tear film under partial blink conditions. A transition between periodic and non-periodic solutions has been estimated and this may be a criterion for what is effectively a full blink according to fluid dynamic considerations.

Laminar flow beneath a finite heated horizontal plate in a rapidly rotating system is considered in both axisymmetric and planar geometries. In particular, we examine the case where the Ekman layer is confined well within a much deeper (yet still thin) thermal boundary layer. This situation corresponds to the regime E−3/2 ≪ Ra ≪ E−5/2, where E and Ra are the natural Ekman and Rayleigh numbers for the system (equation (2.6)). The outward flux of buoyant fluid from beneath the plate occurs primarily in the Ekman layer, while outward flow in the thicker thermal boundary layer is inhibited by a dominant thermal-wind balance. The O(Ra−1/2E−3/4 thickness of the thermal boundary layer is determined by a balance between Ekman suction and diffusion. There are several possible asymptotic regimes near the outer edge of the plate, differing only by logarithmic factors, but in all cases the edge corresponds to a simple boundary condition on the interior flow. With a uniform plate temperature, the dimensionless heat transfer (equation (7.6)) is given by a Nusselt number . The solution for a uniform plate heat flux is also presented.