The flow of a river in a channel of slowly varying width is investigated using an asymptotic approach. The work was motivated by a recent experimental investigation of this problem. The river transports sediment as bedload and is susceptible to an instability which causes variations in the depth of the river. The asymptotic theory is, in the first instance, used to describe steady-state flows in channels of varying width and it is found to give excellent agreement with experimental observations on this flow. The theory shows conclusively that a river of slowly increasing width will form central bars. Secondly, the approach is used to investigate the instability of the flow. The major result obtained is that in a symmetric channel which diverges from a width where the flow is unstable to one where instability is possible, then the preferred mode of instability is likely to be a central bar rather than an alternating bar as is the case in straight channels.

We present the results of an experimental investigation of the flows generated by monodisperse particles settling at low Reynolds number in a stably stratified ambient with an inclined sidewall. In this configuration, upwelling beneath the inclined wall associated with the Boycott effect is opposed by the ambient density stratification. The evolution of the system is determined by the relative magnitudes of the container depth, $h$, and the neutral buoyancy height, $h_n\,{=}\,c_0(\rho_p-\rho_f)/|{\rm d}\rho/{\rm d}z|$, where $c_0$ is the particle concentration, $\rho_p$ the particle density, $\rho_f$ the mean fluid density and ${\rm d}\rho/{\rm d}z\,{<}\,0$ the stable ambient stratification. For sufficiently weak stratification, $h\,{<}\,h_n$, the Boycott layer transports dense fluid from the bottom to the top of the system; subsequently, the upper clear layer of dense saline fluid is mixed by convection. For sufficiently strong stratification, $h\,{>}\,h_n$, layering occurs. The lowermost layer is created by clear fluid transported from the base to its neutral buoyancy height, and has a vertical extent $h_n$; subsequently, smaller overlying layers develop. Within each layer, convection erodes the initially linear density gradient, generating a step-like density profile throughout the system that persists after all the particles have settled. Particles are transported across the discrete density jumps between layers by plumes of particle-laden fluid.

The steady two-dimensional mixed-convection flow past a heated or cooled horizontal plate of finite length is analysed for large Péclet numbers and weak buoyancy effects. The plate is assumed to be aligned with the free stream. It is shown that the hydrostatic pressure jump across the wake, combined with the Kutta condition, gives rise to a potential flow that can be determined by distributing vortices in the wake as well as in the plate. Solutions in closed form are obtained for two cases, i.e. laminar flow of a fluid with very small Prandtl number and turbulent flow. As a result, a lift force opposite to the buoyancy force and a tangential force opposite to the viscous drag force are found. For the case of laminar flow of a fluid with very small Prandtl number, closed-form solutions are also obtained for the temperature distribution and the heat transfer rate, respectively.

The role of mesoscale oceanic eddies in driving large-scale currents is studied in an eddy-resolving midlatitude double-gyre ocean model. The reference solution is decomposed into large-scale and eddy components in a way which is dynamically consistent with a non-eddy-resolving ocean model. That is, the non-eddy-resolving solution driven by this eddy-forcing history, calculated on the basis of this decomposition, correctly approximates the original flow. The main effect of the eddy forcing on the large-scale flow is to enhance the eastward-jet extension of the subtropical western boundary current. This is an anti-diffusive process, which cannot be represented in terms of turbulent diffusion. It is shown that the eddy-forcing history can be approximated as a space–time correlated, random-forcing process in such a way that the non-eddy-resolving solution correctly approximates the reference solution. Thus, the random-forcing model can potentially replace the diffusion model, which is commonly used to parameterize eddy effects on the large-scale currents. The eddy-forcing statistics are treated as spatially inhomogeneous but stationary, and the dynamical roles of space–time correlations and spatial inhomogeneities are systematically explored. The integral correlation time, oscillations of the space correlations, and inhomogeneity of the variance are found to be particularly important for the flow response.

We investigate theoretically and numerically the inverse energy cascade in statistically steady two-dimensional turbulence. We perform a numerical testing of the analytical results proposed by Lindborg (J. Fluid Mech. vol. 338, 1999, p. 259), and we show that these predictions are quantitatively verified in the inverse energy cascade provided that the quantities involved in the spatial average computation are also averaged over all directions. Then, we define a simple measurable criterion based on the kinetic energy induced by coherent vortices in physical space. Using this criterion, we introduce more selective analyses of the energy cascade that reveal spatial properties of energy transfers which are concealed by global spatial averages. We conclude that there exist convective fluxes in both physical and scale space that feed the energy cascade processes in strongly energetic regions. In two dimensions, these regions are mostly localized around coherent structures. In the turbulent background, this mechanism manifests itself as a deficit of the kinetic energy and weaker inverse energy transfers.

An investigation of detonation diffraction through an abrupt area change has been carried out via a set of two-dimensional numerical simulations parameterized by the activation energy of the reactant. Our analysis is specialized to a reactive mixture with a perfect gas equation of state and a single-step reaction in the Arrhenius form. Lagrangian particles are injected into the flow as a diagnostic tool for identifying the dominant terms in the equation that describes the temperature rate of change of a fluid element, expressed in the shock-based reference system. When simplified, this equation provides insight into the competition between the energy release rate and the expansion rate behind the diffracting front. The mechanism of spontaneous generation of transverse waves along the diffracting front is carefully analysed and related to the sensitivity of the reaction rate to temperature. We study in detail three highly resolved cases of detonation diffraction that illustrate different types of behaviour, super-, sub- and near-critical diffraction.

The leading-edge contamination (LEC) problem of an infinite swept wing is shown here as vortex-induced instability. The governing equation for receptivity is presented for LEC in terms of disturbance energy based on the Navier–Stokes equation. The unperturbed shear layer given by the swept Hiemenz boundary-layer solution is two-dimensional and an exact solution of incompressible the Navier–Stokes equation. Thus, the LEC problem is solved numerically by solving the full two-dimensional Navier–Stokes equation. The contamination at the attachment-line is shown by solving a receptivity to a convecting vortex moving outside the attachment-line boundary layer, which triggers subcritical spatio-temporal instability.

The mechanism of LEC is shown to be due essentially to a convecting counter-clockwise rotating vortex, whereas a clockwise rotating vortex displays much weaker receptivity. These results are consistent with experimental results for the bypass mechanism.

The role of linear and nonlinear mechanisms in the contamination problem is discussed as interactions between vorticity and velocity terms of the developed receptivity equation. The computed temporal growth rates reveal pattern formation during such instabilities. Proper orthogonal decomposition (POD) of the numerical solution shows the structure of the leading eigenvector as the coherent eddy excited during the bypass transition.

The flow from line and point sources through an inclined fibrous sheet is studied experimentally and theoretically for wicking from a saturated region and flow from a constant-flux source. Wicking from a saturated line generates a wetted region whose length grows diffusively, linearly or tends to a constant, depending on whether the sheet is horizontal or inclined downwards or upwards. A constant-flux line source generates a wetted region which ultimately grows linearly with time, and is characterized by a capillary fringe whose thickness depends on the relative strength of the source, gravitational and capillary forces. Good quantitative agreement is observed between experiments and similarity solutions.

Capillary-driven and constant-flux source flows issuing from a point on a horizontal sheet generate a wetted patch whose radius grows diffusively in time. The flow is characterized by the relative strength of the source and spreading induced by the action of capillary forces, $\gamma$. As $\gamma$ increases, the fraction of the wetted region which is saturated increases. Wicking from a saturated point corresponds to $\gamma\,{=}\, \gamma_c$, and spreads at a slower rate than from a line source. For $\gamma \,{<}\,\gamma_c$, the flow is partially saturated everywhere. Good agreement is observed between measured moisture profiles, rates of spreading, and similarity solutions.

Numerical solutions are developed for point sources on inclined sheets. The moisture profile is characterized by a steady region circumscribed by a narrow boundary layer across which the moisture content rapidly changes. An approximate analytical solution describes the increase in the size of the wetted region with time and source strength; these conclusions are confirmed by numerical calculations. Experimental measurements of the downslope length are observed to be slightly in excess of theoretical predictions, though the dependence on time, inclination and flow rate obtained theoretically is confirmed. Experimental measurements of cross-slope width are in agreement with numerical results and solutions for short and long times. The effect of a percolation threshold is observed to ultimately arrest cross-slope transport, placing a limitation on the long-time analysis.

The dynamics of linear internal waves in the ocean is analysed without adopting the ‘traditional approximation’, i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm–Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm–Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.

In the near-inertial range, non-traditional effects act as a singular perturbation; this is seen from the sub-inertial short-wave limit, which is present whenever the ‘non-traditional’ terms are there, but disappears under the traditional approximation. In the dispersion relation the sub-inertial modes represent a smooth continuation of the super-inertial ones. The combined effect of the horizontal component of rotation and a vertical inhomogeneity in the stratification is found to play a crucial role in the dynamics of sub-inertial waves. They are trapped in waveguides localized around minima of the buoyancy frequency. The presence of horizontal inhomogeneities in the effective Coriolis parameter (such as shear currents or beta effect) are shown to enable a transition from super-inertial to sub-inertial waves (and thus effectively an irreversible transformation of large-scale into small-scale motions). It is suggested that this transformation provides a mechanism for mixing in the deep ocean.

The notion of critical reflection of internal waves at a sloping bottom is also modified by non-traditional effects, and they strongly increase the probability of critical reflection in the near-inertial to tidal range.

The influence of inertia and elasticity on the onset and stability of Rayleigh–Bénard thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The Galerkin projection method is used to obtain the departure from the conduction state. The solution is capable of displaying complex dynamical behaviour for viscoelastic fluids in the elastic and inertio-elastic ranges, which correspond to ${\it Ra} \,{<}\, {\it Ra}_c^s$ and ${\it Ra} \,{>}\, {\it Ra}_c^s $, respectively, ${\it Ra}_c^s $ being the critical Rayleigh number at which stationary thermal convection emerges. This behaviour is reminiscent of that observed experimentally for viscoelastic Taylor–Couette flow. For a given ${\it Ra}$ in the pre-critical range, finite-amplitude periodic oscillatory convection emerges when the elasticity number, $E$, exceeds a threshold. Periodicity is lost as $E$ increases, leading to a $T^{2}$ quasi-periodic behaviour, and the breakup of the torus as $E$ increases further. Although no experimental data are available for direct comparison, this scenario is reminiscent of the flow sequence observed by Muller et al. (1993) in the Taylor–Couette flow of a Boger fluid. Stationary thermal convection emerges, via a supercritical bifurcation, when ${\it Ra}$ exceeds ${\it Ra}_c^s $. The amplitude of motion is found to be little influenced by fluid elasticity or retardation time, especially as the Rayleigh number increases. However, the range of stability of the stationary thermal convection narrows considerably for viscoelastic fluids. In this case, oscillatory thermal convection is favoured. The onset and the stability of other steady convective patterns, namely hexagons and squares, are studied in the inertio-elastic range by using an amplitude equation approach. The range of stability of each pattern is examined, simultaneously allowing the validation of the two-dimensional picture.

This paper addresses shear-induced self-diffusion in a monodisperse suspension of non-Brownian particles in Couette flow by two-dimensional computer simulations following the lattice-Boltzmann method. This method is suited for the study of (many-particle) particulate suspensions and can not only be applied for Stokes flow, but also for flow with finite Reynolds number. At relatively low shear particle Reynolds numbers (up to 0.023), shear-induced diffusivity exhibited a linear dependence on the shear rate, as expected from theoretical considerations. Simulations at shear particle Reynolds numbers between 0.023 and 0.35, however, revealed that in this regime, shear-induced diffusivity did not show this linear dependence anymore. Instead, the diffusivity was found to increase more than linearly with the shear rate, an effect that was most pronounced at lower area fractions of 0.10 and 0.25. In the same shear regime, major changes were found in the flow trajectories of two interacting particles in shear flow (longer and closer approach) and in the viscosity of the suspension (shear thickening). Moreover, the suspended particles exhibited particle clustering. The increase of shear-induced diffusivity is shown to be directly correlated with this particle clustering. As for shear-induced diffusivity, the effect of increasing shear rates on particle clustering was the most intensive at low area fractions of 0.10 and 0.25, where the radius of the clusters increased from about 4 to about 7 particle radii with an increase of the shear Reynolds number from 0.023 to 0.35. The importance of particle clustering to shear-induced diffusion might also indicate the importance of other factors that can induce particle clustering, such as, for example, colloidal instability.

Kinematics of nominally periodic surf zone waves have been measured in the laboratory using LDA (laser Doppler anemometry), above trough level as well as below, for weakly plunging breakers transforming into bores in shallower water. The aim was to determine, through phase- or ensemble-averaging, periodic flow structures in a two-dimensional vertical plane, from large-scale down to small-scale vortical structures. Coherent multiple vortical structures were evident at the initiation of breaking, becoming elongated along the surface during bore propagation. The initial region is likely to become more extensive as waves become more strongly plunging and could explain the difference in turbulence characteristics between plunging and spilling breakers observed elsewhere. Comparison of vorticity magnitudes with hydraulic-jump measurements showed some similarities during the initial stages of breaking, but these quickly grew less as breaking progressed into shallower water. Period-averaged kinematics and vorticity were also obtained showing shoreward mass transport above trough level and undertow below, with a thick layer of vorticity at trough level and a thin layer of vorticity of opposite rotation at the bed. There were also concentrated regions of mean vorticity near the end of the plunging region. Residual turbulence of relatively high frequency was presented as Reynolds stresses, showing marked anisotrophy. Dynamic pressure (pressure minus its hydrostatic component) was determined from the kinematics. The magnitudes of different effects were evaluated through the depth-integrated Reynolds-averaged Navier–Stokes (RANS) equations, which may be reduced to nine terms (the standard inviscid terms of the shallow-water equations conserving mass and momentum with hydrostatic pressure, and six additional terms), assuming that the complex, often aerated, free surface is treated as a simple interface. All terms were evaluated, assuming that a space/time transformation was justified with a slowly varying phase speed, and the net balance was always small in relation to the maxima of the larger terms. Terms due to dynamic pressure and vertical dispersion (due to the vertical variation of velocity) were as significant as the three terms in the inviscid shallow-water equations; terms involving residual turbulence were insignificant. The r.m.s. (root mean square) variation of each along the slope is highly irregular, with the inertia term due to (Eulerian) acceleration always greatest. This is consistent with complex, though repetitive, coherent structures. Modelling the flow with the shallow-water equations, using the surface elevation variation at the break point as input, nevertheless gave a good prediction of the wave height variation up the slope.

The unsteady development toward breaking of the wavy flow generated by a submerged hydrofoil starting from rest is investigated numerically by solving the two-dimensional Navier–Stokes equations for the two-phase flow of air and water and by using a level-set technique to capture the interface. The study, carried out for hydrofoils with various chord lengths, shows that, when passing from longer to shorter scales, the role played by surface tension becomes more and more relevant and different flow regimes are recovered ranging from intense plunging jet, eventually leading to a large amount of entrapped air, down to microscale breakers, in which the jet is replaced by a bulge growing on the wave crest and the breaking event takes place without air entrapment. The ‘toe’ of the bulge slides down upon the forward face of the wave and large downstream-propagating surface fluctuations are observed. Wavenumber and frequency spectra of the computed free-surface profiles, aimed at understanding the downstream motion of free-surface fluctuations, are evaluated and found in good agreement with similar investigations carried out experimentally and available in literature.

A careful inspection of the instantaneous vorticity field under the microbreakers reveals the presence of an intense shear flow originating at the toe, instability of which eventually gives rise to coherent vortex-structures. These structures are convected downstream by the flow at a growing speed, remaining confined in a thin layer just beneath the free surface. This continuous interaction is responsible for the generation of the free-surface fluctuations experimentally found.