Results are presented from a computational study of the flow over a forward-facing step in a plane channel. The aim of the study is to gain better insight into the three-dimensionality that is typically observed in the separation region of flows over steps and ribs, and in similar configurations. We consider laminar flow at a Reynolds number of 330, based on step height and bulk velocity of the oncoming flow, and the step is assumed to be infinitely extended in the spanwise direction. High-resolution simulations are undertaken using a mixed spectral/spectral-element code. Moreover, a linear stability study of the flow at the step is performed. The results show that, in the case considered, the three-dimensionality is not related to some absolute instability of the separation bubble in front of the step; rather, it is a sensitive reaction of the flow to three-dimensional perturbations present in the oncoming stream. It is demonstrated that disturbance amplitudes of less than 1% of the mean flow (at, say, 10 step heights ahead of the step) already suffice to produce a visibly three-dimensional structure of the separation zone. If the disturbance level is systematically decreased, the three-dimensional state evolves to an almost two-dimensional recirculation. Here, the key finding is that the intensity of the flow response is proportionate to the amplitude of the inflow disturbance, meaning that the breakup of the flow in the step region is a linear (i.e. small) perturbation of the two-dimensional base flow. A comparison of the present simulation results with experimental data shows close agreement concerning, for example, the flow topology in the step region, and the spanwise spacing of the characteristic streaks that form further downstream.

We examine the settling of monodisperse heavy particles released into a fluid when the resulting motion is sufficiently vigorous that the particle cloud initially assumes the form of a turbulent thermal. A laboratory study is complemented by numerical simulations of particle cloud dynamics in both homogeneous and stratified ambients. In the homogeneous ambient, the cloud generated by a total buoyancy excess $Q= g'N_pV_p$, where $g'$ is the reduced gravity of the $N_p$ spherical particles of volume $V_p\,{=}\,{4\upi a^3}/3$, evolves in a manner consistent with a classical fluid thermal. The cloud grows through turbulent entrainment and decelerates until its speed is exceeded by that of the individual particles $w_s$, at which point the particles rain out as individuals. For particle Reynolds numbers $\hbox{\it Re}_p\,{=}\,w_s a/\nu$ in the range of 0.1–300, the fallout height $Z_f$ is found to be $Z_f/a\,{=}\,(11 \pm 2) (Q^{1/2}/(w_sa))^{0.83}$. For high $\hbox{\it Re}_p$ particles, the fallout height assumes the simple form: $Z_f/a\,{=}\,(9 \pm 2) N_p^{1/2}$. Following fallout, the particles sink at their individual settling speeds in the form of a bowl-shaped swarm. In a stratified environment characterized by a constant Brunt–Väisälä frequency $N$, the mode of fallout depends explicitly on the stratified cloud number, $N_s\,{=}\,w_s Q^{-1/4} N^{-1/2}$. For $N_s\,{<}\,1$, the cloud overshoots, rebounds past, then intrudes at the neutral height, $Z_N$, of the equivalent fluid thermal. The particles fall out between the depth of maximum penetration and the spreading neutral cloud, and may be distributed over a relatively broad area. For $N_s\,{>}\,1$, the particles fall out in the form of a bowl-shaped swarm at a height $Z_f\,{<}\,Z_N$, thus giving rise to a relatively localized deposit. For $N_s\,{>}\,4$, the fallout height is largely uninfluenced by the stratification and is adequately described by the homogeneous result. Regardless of $N_s$, following particle fallout in a stratified ambient, the fluid entrained by the thermal ascends and intrudes at a rebound height given to leading order by ${3} Z_f/4$. Criteria for three distinct modes of particle deposition in a stratified ambient are developed.

Nano-bubbles have recently been observed experimentally on smooth hydrophobic surfaces; cracks on a surface can likewise be the site of bubbles when partially wetting fluids are used. Because these bubbles may provide a zero shear stress boundary condition and modify considerably the friction generated by the solid boundary, it is of interest to quantify their influence on pressure-driven flow, with particular attention given to small geometries. We investigate two simple configurations of steady pressure-driven Stokes flow in a circular pipe whose surface contains periodically distributed regions of zero surface shear stress. In the spirit of experimental studies probing slip at solid surfaces, the effective slip length of the resulting flow is evaluated as a function of the degrees of freedom describing the surface heterogeneities, namely the relative width of the no-slip and no-shear stress regions and their distribution along the pipe. Comparison of the model with experimental studies of pressure-driven flow in capillaries and microchannels reporting slip is made and a possible interpretation of the experimental results is offered which is consistent with a large number of distributed slip domains such as nano-size and micron-size nearly flat bubbles coating the solid surface. Further, the possibility is suggested of a shear-dependent effective slip length, and an explanation is proposed for the seemingly paradoxical behaviour of the measured slip length increasing with system size, which is consistent with experimental results to date.

The selection of the axial wavelength in axisymmetric Taylor-vortex flow was studied by numerical experiments where the inner cylinder speed was linearly increased from subcritical to supercritical values over a finite ramp time to values not far above $\hbox{\it Re}_c$. For impulsive increases of the inner cylinder speed (zero ramp time) the preferred axial wavelength was less than the critical wavelength. As the ramp time was increased, the preferred axial wavelength increased and approached the critical wavelength, so that for very slow increases of the inner cylinder speed the preferred axial wavelength was equal to the critical wavelength. A linear model was developed which revealed that a linearly increased inner cylinder speed resulted in a delayed growth for each of the amplitudes of the modes. When the ramp time was sufficiently large, the amplitude of the mode with the critical wavelength was delayed the least from growing to high amplitudes. This mode then self-interacted and saturated resulting in steady Taylor-vortex flow. Finally, nonlinear effects and state selection are discussed from the point of view of nonlinear dynamics.

Two-dimensional potential flows due to progressive surface waves in deep water are considered. For periodic waves, only gravity is included in the dynamic boundary condition, but both gravity and surface tension are taken into account for solitary waves. The validity of the steady first-order cnoidal wave approximation, i.e. the periodic solution of KdV, is extended to infinite depth by renormalizations. This renormalized cnoidal wave (RCW) solution is expressed as a Fourier–Padé approximation. It is analytically simpler and more accurate than fifth-order Stokes approximations. It is also capable of describing the recently discovered sharp-crested wave. A sharp-crested wave is obtained when the fluid velocity at the crest is larger than the phase speed. When the wavelength is infinite, RCW yields an algebraic solitary wave. Depending on the surface tension, the solitary wave involves one or two interfaces: a wave of depression; a wave of depression with a pocket of air; a wave of elevation with a pocket of air. Solitary waves are found for all values of the surface tension. However, this does not necessarily mean that these waves are solutions of the exact equations. Moreover, RCW approximate solitary waves always present a dipole singularity. It is also shown that a cnoidal wave in deep water can be rewritten as a periodic distribution of dipoles, each dipole representing an algebraic solitary wave. This provides a new paradigm for descriptions of water wave phenomena.

The unsteady velocity field generated by an underexpanded jet has been investigated using stereoscopic particle image velocimetry (PIV). A 4:1 aspect ratio converging–diverging rectangular nozzle designed to operate at a fully expanded condition of $M=1.44$ was used. The nozzle was operated at off-design conditions to generate imperfectly expanded jets with intense screech tones. Phase-locked PIV measurements show the spatial and temporal evolution of the three-dimensional jet with high fidelity. In addition to the globally averaged mean and turbulence velocity field data, the phase-averaged data for the velocity and vorticity fields were also obtained. The turbulence quantities were resolved into contributions from the periodic and random motions. The deformation of the periodic spanwise structures results in the formation of strong streamwise vortices that appear to govern the mixing of the jet. It is shown that the presence of coherent vorticity of significant strength, in addition to the shock cell strength, is largely responsible for determining the screech intensity.

A soft elastohydrodynamic lubrication model is formulated for deformable roll coating involving two contra-rotating rolls, one rigid and the other covered with a compliant layer. Included is a finite-strip model (FSM) for the deformation of the layer and a lubrication model with suitable boundary conditions for the motion of the fluid. The scope of the analysis is restricted to Newtonian fluids, linear elasticity/viscoelasticity and equal roll speeds, with application to the industrially relevant highly loaded or ‘negative gap’ regime. Predictions are presented for coated film thickness, inter-roll thickness, meniscus location, pressure and layer deformation as the control parameters – load (gap), elasticity, layer thickness and capillary number, $\hbox{\it Ca}$ – are varied. There are four main results:

1. Hookean spring models are shown to be unable to model effectively the deformation of a compliant layer when Poisson's ratio $\nu\rightarrow 0.5$. In particular, they fail to predict the swelling of the layer at the edge of the contact region which increases as $\nu\rightarrow 0.5$; they also fail to locate accurately the position of the meniscus, $X_M$, and to identify the presence, close to the meniscus, of a ‘nib’ (constriction in gap thickness) and associated magnification of the sub-ambient pressure loop.

2. Scaling arguments suggest that layer thickness and elasticity may have similar effects on the field variables. It is shown that for positive gaps this is true, whereas for negative gaps they have similar effects on the pressure profile and flow rate yet quite different effects on layer swelling (deformation at the edge of the contact region) and different effects on $X_M$.

3. For negative gaps and $\hbox{\it Ca}\,{\sim}\,O(1)$, the effect of varying either viscosity or speed and hence $\hbox{\it Ca}$ is to significantly alter both the coating thickness and $X_M$. This is contrary to the case of fixed-gap rigid roll coating.

4. Comparison between theoretical predictions and experimental data shows quantitive agreement in the case of $X_M$ and qualitive agreement for flow rate. It is shown that this difference in the latter case may be due to viscoelastic effects in the compliant layer.

A quantitative theory of the average features of turbulent flow in a channel is described without the introduction of empirical parameters. The qualitative problem consists of maximizing the dissipation rate of the mean flow subject to the Rayleigh condition that the mean flow has no inflections. The quantitative features result from a boundary stability study which determines a smallest scale of motion in the transport of momentum. The velocity fields satisfying these conditions, the averaged equations and the boundary conditions uniquely determine an entire mean velocity profile at all Reynolds numbers within ten per cent of the data. The maximizing condition for the reproducibility of averages emerges from the Navier–Stokes equations as a consequence of a novel definition of nonlinear instability. The smallest scale of motion results from a theory for a time-dependent re-stabilization of the boundary layer following a disruptive instability. Computer reassessment of the several asymptotic estimates of the critical boundary eigenstructure can establish the limits of validity of the quantitative results.

Conditions determining the existence of localized steadily translating two-layer vortices (modons) of arbitrary symmetric form on the $\beta$-plane are considered. A numerical method for direct construction of modon solutions is suggested and its accuracy is analysed in relation to the parameters of the computational procedure and the geometrical and physical parameters of the modon sought. Using this method, several non-circular baroclinic solutions are constructed marked by nonlinearity of the dependence of the potential vorticity (PV) on the streamfunction in the trapped-fluid area of the modon, i.e. where the streamlines are closed. The linearity of this dependence and the circularity of the trapped-fluid area are shown to be equivalent properties of a modon. Special attention is given to elliptical modons – extended both in the direction of the modon propagation and in the orthogonal direction, the baroclinic PV component being assumed continuous. The differences between the two types of elliptical modons are discussed. The simplest vortical couples and shielded modons are considered. In the context of the continuity of the baroclinic PV field, the stability of modons is discussed based on numerical simulations.

We present the results of a combined theoretical and experimental investigation of the influence of surface tension $\sigma$ on the laminar circular hydraulic jump. An expression is deduced for the magnitude of the radial curvature force per unit length along a circular jump, $F_c\,{=}\,{-}\sigma ( s - \uDelta R )/R_j$, where $R_j$ is the jump radius, and $s$ and $\uDelta R$ are, respectively, the arclength along the jump surface and radial distance between the nearest points at the nose and tail of the jump at which the surface is horizontal. This curvature force is dynamically significant when $2\sigma/(\rho g R_j \uDelta H)$ becomes appreciable, where $\uDelta H$ is the jump height, $\rho$ the fluid density and $g$ the acceleration due to gravity. The theory of viscous hydraulic jumps (Watson 1964) is extended through inclusion of the curvature force, and yields a new prediction for the radius of circular hydraulic jumps. Our experimental investigation demonstrates that the surface tension correction is generally small in laboratory settings, but appreciable for jumps of small radius and height.

The transport of scalars in the coastal ocean is considered through the analysis of a vertically constrained plume which disperses laterally. Observations of the plume are made using an autonomous underwater vehicle (AUV) operating in two modes: (i) repeated transects of the plume at a fixed distance from the source; and (ii) a large-scale mapping of the plume development. Together, these measurements define both the variability in the plume centreline (i.e. the meandering) and the growth of the plume around the centreline (i.e. the relative dispersion). The analysis of the measurements suggests that the meandering is well-described by a spatially uniform but temporally variable velocity field, indicating that large-scale flow structures dominate the centreline variability. The growth of the plume downstream is seen to follow a scale-dependent dispersion law, most likely of a compound structure: a 4/3-law in the near field, and a scale-squared law in the far field. This transition between dispersion laws is consistent with the transition from three-dimensional turbulence structures to two-dimensional eddies, which is due to the constraints imposed on the vertical dimension at the site. Comparing the two dispersion processes, the effective dispersion created by meandering is found to be comparable with or larger than the relative dispersion in the near field; but in the far field, the relative dispersion is found to dominate considerations of overall dispersion.

Results of an experimental study on the solidification and convection of a ternary solution of H$_2$O–CuSO$_4$–Na$_2$SO$_4$ when cooled from a vertical side boundary are presented. Cooling and crystallization at a sidewall result in the formation of horizontal gradients of composition and temperature which are restricted to thin boundary layers adjacent to the solid. The dynamics of the buoyancy-induced flows in the compositional and thermal boundary layers is shown to depend critically on the morphology of the growing solid as well as on the density changes in the fluid released on solidification. In a ternary system, the possibility of a density reversal in the residual fluid as a second component solidifies leads to a wide range of possible behaviours. Three broad regimes are identified and the flows which arise in two of these are studied in detail; the behaviour in the third regime can be inferred from these results. When the thermal and compositional boundary layers are both heavy and flow to the base of the tank (regime I), the evolution of the interior fluid is shown to be initially quantitatively similar to that arising from thermal convection only. Slow compositional changes in the interior only later result in a number of double-diffusive effects. When the compositional and thermal buoyancies in the boundary layers are opposed (regimes II and III), a unidirectional downflow, counterflow and upflow are all observed at various stages of the experiment. The onset and duration of these different flows is shown to depend on the ratio of compositional to thermal buoyancies, the relative fluxes in the boundary layers, and the crystal morphology. The effect of surface roughness for the particular ternary system investigated is quantified by measuring its effect on the fluxes in the boundary layers, which allows the modification of existing theories to predict the conditions under which the different boundary layer dynamics arise.

The effective contribution to the lateral heat transport in a rotating differentially heated annulus attributable to fully developed baroclinic eddies is determined by the combined use of laboratory measurements and numerical simulations. The total heat transport is determined in the laboratory by real-time calorimetry to a precision of around $\pm 2.5/%$ over a wide range of parameters sampling a wide cross-section of the regular baroclinic wave regime accessible in the rotating annulus up to the transition to geostrophic turbulence. High-resolution numerical simulations of steady axisymmetric flow in the rotating annulus were carried out under comparable parametric conditions to the laboratory experiments, to determine the contribution to total heat transport due to the axisymmetric boundary-layer circulation in the system. The difference between the Nusselt or Péclet numbers determined in these two ways enables the heat transport attributable to the presence of the baroclinic eddies to be determined unambiguously. The variation of the resulting excess Péclet number with external parameters appears to be consistent with predictions from a weakly nonlinear model of baroclinic instability within the regular baroclinic wave regime, at least for weak–moderate supercriticality, whereas at higher rotation rates a parametrization based on the linear instability approach of Green (1970) may be more appropriate. This approach seems to offer an accurate and incisive means of evaluating schemes proposed to parametrize the transport properties of baroclinic eddies in a variety of models used in geophysical and engineering applications.

Results of experiments on decaying turbulence in shallow water flow bounded by a solid bottom and a free surface are presented. The evolution of the turbulence structures generated by a grid, with horizontal mesh dimensions larger than the water depth, is measured using laser Doppler velocimetry and particle image velocimetry (PIV). The vertical confinement of the flow forces the large turbulence structures to move in the horizontal plane, thus attaining strongly two-dimensional features. This two-dimensionality and its consequences for the intensities and length scales of the large structures are analysed. It is shown that the decay of the vortices that are shed from the grid is determined by the characteristic size of the grid elements rather than the grid spacing. Furthermore, during the decay process the merging of vortices is observed in combination with a $-3$ slope in the energy density spectrum of the velocity fluctuations. Using the PIV data, spatial properties like divergence and enstrophy can be derived for the velocity field near the free surface. The distinct effect of water depth that is found in the velocity fluctuations is almost insignificant in the enstrophy decay.

A new shallow-water theory valid for arbitrary bottom slope, due to Bouchut et al. (2003), is derived systematically by a scaling method. The fact that the pressure is hydrostatic, and the form of the velocity parallel to the bottom, are consequences of the scaling method, and need not be assumed.

Observations show that the near-sill dynamics of dense abyssal overflows is variable and is governed, to a significant extent, by a balance between rotation, bottom friction and downslope acceleration due to gravity. Numerical simulations indicate that the near-sill downslope velocities are comparable to the phase/group velocities of long internal gravity waves. This suggests the possibility that overflows can become supercritical and destabilized by bottom friction. A theory is presented for the frictional destabilization of rotating abyssal overflows and the accompanying baroclinic coupling with the overlying ocean. This mode of transition allows for the formation of downslope and alongslope propagating periodic bores or pulses in the overflow and the generation of amplifying long internal gravity waves in the overlying ocean, and may help to explain aspects of the observed variability which seem unrelated to purely inertial baroclinic instability.