The present paper is devoted to an analysis of tip vortex cavitation under confined situations. The tip vortex is generated by a three-dimensional foil of elliptical planform, and the confinement is achieved by flat plates set perpendicular to the span, at an adjustable distance from the tip. In the range of variation of the boundary-layer thickness investigated, no significant interaction was observed between the tip vortex and the boundary layer which develops on the confinement plate. In particular, the cavitation inception index for tip vortex cavitation does not depend significantly upon the length of the plate upstream of the foil. On the contrary, tip clearance has a strong influence on the non-cavitating structure of the tip vortex and consequently on the inception of cavitation in its core. The tangential velocity profiles measured by a laser-Doppler velocimetry (LDV) technique through the vortex, between the suction and the pressure sides of the foil, are strongly asymmetric near the tip. They become more and more symmetric downstream and the confinement speeds up the symmetrization process. When the tip clearance is reduced to a few millimetres, the two extrema of the velocity profiles increase. This increase results in a decrease of the minimum pressure in the vortex centre and accounts for the smaller resistance to cavitation observed when tip clearance is reduced. For smaller values of tip clearance, a reduction of tip clearance induces on the contrary a significant reduction in the maxima of the tangential velocity together with a significant increase in the size of the vortex core estimated along the confinement plate. Hence, the resistance to cavitation is much higher for such small values of tip clearance and in practice, no tip vortex cavitation is observed for tip clearances below 1.5 mm. The cavitation number for the inception of tip vortex cavitation does not correlate satisfactorily with the lift coefficient, contrary to classical results obtained without any confinement. Owing to the specificity introduced by the confinement, the usual procedure developed in an infinite medium to estimate the vortex strength from LDV measurements is not applicable here. Hence, a new quantity homogeneous to a circulation had to be defined on the basis of the maximum tangential velocity and the core size, which proved to be better correlated to the cavitation inception data.

The reflection of asymmetric shock waves in steady flows is studied both theoretically and experimentally. While the analytical model was two-dimensional, three-dimensional edge effects influenced the experiments. In addition to regular and Mach reflection wave configurations, an inverse-Mach reflection wave configuration, which has been observed so far only in unsteady flows (e.g. shock wave reflection over concave surfaces or over double wedges) has been recorded. A hysteresis phenomenon similar to the one that exists in the reflection of symmetric shock waves has been found to also exist in the reflection of asymmetric shock waves. The domains and transition boundaries of the various types of overall reflection wave configurations are analytically predicted.

A linear stability analysis is made of a family of natural convection flows in an arbitrarily inclined rectangular enclosure. The flow is driven by prescribed heat or mass fluxes along two opposing walls. The analysis allows for perturbations in arbitrary directions; however, the purely longitudinal or transverse modes are numerically found to be the most unstable. For the numerical treatment, a finite difference method with automatically calculated differencing molecules, variable order of accuracy, and accurate boundary treatment is developed. In cases with boundary layers, a special scaling is applied.

For base solutions with natural (bottom heavy) stratification, critical conditions are solved for as a function of the Rayleigh number, Ra, and the angle of inclination to the bottom-heated case, α, for different Prandtl numbers (Pr), with complete results for Pr=0.025, 0.1, 0.7, 7, 1000, and Pr→∞. The uniform flux case is found to be much more stable than that of Hart (1971) with fixed wall temperatures, a fact which is attributed to the much larger stratification which occurs in the base solution. As could be expected, instabilities tend to be favoured by a decrease in Pr, an increase in Ra, and a decrease in α; however, exceptions to all these rules could be found.

Cases in which the wavenumber is zero, or approaches zero in different ways, are studied analytically. Integral conditions, derived from the unresolved end regions, are applied in the analysis. The results show that all the base solutions with unnatural (top heavy) stratification are unstable to large-wavelength stationary rolls whose axes are parallel with the base flow.

Real-valued perturbations are constructed and visualized for some of the modes considered.

Particle-driven gravity currents, as exemplified by either turbidity currents in the ocean or ignimbrite flows in the atmosphere, are buoyancy-driven flows due to a suspension of dense particles in an ambient fluid. We present a theoretical study on the dynamics of and deposition from a turbulent current flowing down a uniform planar slope from a constant-flux point source of particle-laden fluid. The flow is modelled using the shallow-water equations, including the effects of bottom friction and entrainment of ambient fluid, coupled to an equation for the transport and settling of the particles. Two flow regimes are identified. Near the source and for mild slopes, the flow is dominated by a balance between buoyancy and bottom friction. Further downstream and for steeper slopes, entrainment also affects the behaviour of the current. Similarity solutions are also developed for the simple cases of homogeneous gravity currents with no settling of particles in the friction-dominated and entrainment-dominated regimes. Estimates of the width and length of the deposit from a monodisperse particle-driven gravity current with settling are derived from scaling analysis for each regime, and the contours of the depositional patterns are determined from numerical solution of the governing equations.

We present measurements of the density and velocity fields produced when an oscillating circular cylinder excites internal gravity waves in a stratified fluid. These measurements are obtained using a novel, non-intrusive optical technique suitable for determining the density fluctuation field in temporally evolving flows which are nominally two-dimensional. Although using the same basic principles as conventional methods, the technique uses digital image processing in lieu of large and expensive parabolic mirrors, thus allowing more flexibility and providing high sensitivity: perturbations of the order of 1% of the ambient density gradient may be detected. From the density gradient field and its time derivative it is possible to construct the perturbation fields of density and horizontal and vertical velocity. Thus, in principle, momentum and energy fluxes can be determined.

In this paper we examine the structure and amplitude of internal gravity waves generated by a cylinder oscillating vertically at different frequencies and amplitudes, paying particular attention to the role of viscosity in determining the evolution of the waves. In qualitative agreement with theory, it is found that wave motions characterized by a bimodal displacement distribution close to the source are attenuated by viscosity and eventually undergo a transition to a unimodal displacement distribution further from the source. Close quantitative agreement is found when comparing our results with the theoretical ones of Hurley & Keady (1997). This demonstrates that the new experimental technique is capable of making accurate measurements and also lends support to analytic theories. However, theory predicts that the wave beams are narrower than observed, and the amplitude is significantly under-predicted for low-frequency waves. The discrepancy occurs in part because the theory neglects the presence of the viscous boundary layers surrounding the cylinder, and because it does not take into account the effects of wave attenuation resulting from nonlinear wave–wave interactions between the upward and downward propagating waves near the source.

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.

The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.

Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

Large-eddy simulation (LES) has been used to study the flow in a planar asymmetric diffuser. The wide range of spatial and temporal scales, the presence of an adverse pressure gradient, and the formation of an unsteady separation bubble in the rear part of the diffuser make this flow a challenging test case for assessing the predictive capability of LES. Simulation results for mean flow, pressure recovery and skin friction are in excellent agreement with data from two recent experiments. The inflow consists of a fully developed turbulent channel flow at a Reynolds number based on shear velocity, Reτ=500. It is found that accurate representation of the in flow velocity field is critical for accurate prediction of the flow in the diffuser. Although the simulation in the diffuser is well resolved, the subgrid-scale model plays a significant role for both mean momentum and turbulent kinetic energy balances. Subgrid-scale stresses contribute a maximum of 8% to the local value of the total shear stress with the maximum values found in the inlet duct and along the flat wall where the flow remains attached. The subgrid-scale model adapts to the enhanced turbulence levels in the rear part of the diffuser by providing more than 80% of the dissipation rate for turbulent kinetic energy. The unsteady separation excites large scales of motion which extend over the major part of the duct cross-section and penetrate deeply into the core of the flow. Instantaneous flow reversal is observed along both walls immediately behind the diffuser throat which is far upstream of the location of main separation. While the mean flow profile changes gradually as the flow enters the expansion, turbulent stresses undergo rapid changes over a short streamwise distance along the deflected wall. An explanation is offered which considers the strain field as well as the influence of geometry changes. The effect of grid resolution and spanwise domain size on the flow field prediction has been documented and this allows an assessment of the computational requirements for carrying out such simulations.

The Rossby adjustment problem for a homogeneous fluid in a channel is solved for large values of the initial depth discontinuity. We begin by analysing the classical dam break problem in which the depth on one side of the discontinuity is zero. An approximate solution for this case can be constructed by assuming semigeostrophic dynamics and using the method of characteristics. This theory is supplemented by numerical solutions to the full shallow water equations. The development of the flow and the final, equilibrium volume transport are governed by the ratio of the Rossby radius of deformation to the channel width, the only non-dimensional parameter. After the dam is destroyed the rotating fluid spills down the dry section of the channel forming a rarefying intrusion which, for northern hemisphere rotation, is banked against the right-hand wall (facing downstream). As the channel width is increased the speed of the leading edge (along the right-hand wall) exceeds the intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid separates from the sidewall at a point whose speed decreases to zero as the channel width approaches infinity. Numerical computations of the evolving flow show good agreement with the semigeostrophic theory for widths less than about a deformation radius. For larger widths cross-channel accelerations, absent in the semigeostrophic approximation, reduce the agreement. The final equilibrium transport down the channel is determined from the semigeostrophic theory and found to depart from the non-rotating result for channels widths greater than about one deformation radius. Rotation limits the transport to a constant maximum value for channel widths greater than about four deformation radii.

The case in which the initial fluid depth downstream of the dam is non-zero is then examined numerically. The leading rarefying intrusion is now replaced by a Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion speed. The shock is either straight across the channel or attached only to the right-hand wall depending on the channel width and the additional parameter, the initial depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed downstream depth, increase above the non-rotating values with increasing channel width. However, rotation reduces the speed of a shock of given amplitude below the non-rotating case. We also find evidence of resonant generation of Poincaré waves by the bore. Shock characteristics are compared to theories of rotating shocks and qualitative agreement is found except for the change in potential vorticity across the shock, which is very sensitive to the model dissipation. Behind the leading shock the flow evolves in much the same way as described by linear theory except for the generation of strongly nonlinear transverse oscillations and rapid advection down the right-hand channel wall of fluid originally upstream of the dam. Final steady-state transports decrease from the zero upstream depth case as the initial depth difference is decreased.

Stably stratified flows past three-dimensional orography have been investigated using a stratified towing tank. Flows past idealized axisymmetric orography in which the Froude number, Fh=U/Nh (where U is the towing speed, N is the buoyancy frequency and h is the height of the obstacle) is less than unity have been studied. The orography considered consists of two sizes of hemisphere and two cones of different slope. For all the obstacles measurements show that as Fh decreases, the drag coefficient increases, reaching between 2.8 and 5.4 times the value in neutral flow (depending on obstacle shape) for Fh[les ]0.25. Local maxima and minima in the drag also occur. These are due to the finite depth of the tank and can be explained by linear gravity-wave theory. Flow visualization reveals a lee wave train downstream in which the wave amplitude is O(Fhh), the smallest wave amplitude occurring for the steepest cone. Measurements show that for all the obstacles, the dividing-streamline height, zs, is described reasonably well by the formula zs/h=1−Fh. Flow visualization and acoustic Doppler velocimeter measurements in the wake of the obstacles show that vortex shedding occurs when Fh[les ]0.4 and that the period of the vortex shedding is independent of height. Based on velocity measurements in the wake of both sizes of hemisphere (plus two additional smaller hemispheres), it is shown that a blockage-corrected Strouhal number, S2c =fL2/Uc, collapses onto a single curve when plotted against the effective Froude number, Fhc=Uc/Nh. Here, Uc is the blockage-corrected free-stream speed based on mass-flux considerations, f is the vortex shedding frequency and L2 is the obstacle width at a height zs/2. Collapse of the data is also obtained for the two different shapes of cone and for additional measurements made in the wake of triangular and rectangular at plates. Indeed, the values of S2c for all these obstacles are similar and this suggests that despite the fact that the obstacle widths vary with height, a single length scale determines the vortex-street dynamics. Experiments conducted using a splitter plate indicate that the shedding mechanism provides a major contribution to the total drag (∼25%). The addition of an upstream pointing ‘verge region’ to a hemisphere is also shown to increase the drag significantly in strongly stratified flow. Possible mechanisms for this are discussed.

We study theoretically the adsorption of surfactant onto the interface of gas bubbles in creeping flow rising steadily in an infinite liquid phase containing surface-active agents. When a bubble rises in the fluid, surfactant adsorbs onto the surface at the leading edge, is convected by the surface flow to the trailing edge and accumulates and desorbs off the back end. This transport creates a surfactant concentration gradient on the surface that causes the surface tension at the back end to be lower than that at the front end, thus retarding the bubble velocity by the creation of a Marangoni force. In this paper, we demonstrate numerically that the mobility of the surfactant-retarded bubble interface can be increased by raising the bulk concentration of surfactant. At high bulk concentrations, the interface saturates with surfactant, and this saturation acts against the convective partitioning to decrease the surface surfactant gradient. We show that as the Péclet number (which scales the convective effect) increases, larger concentrations are necessary to remobilize the surface completely. These results lead to a technologically useful paradigm where the drag and interfacial mobility of a bubble can be controlled by the level of the bulk concentration of surfactant.

Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this scale reduction, the stratification is ‘overturned’ and the Miles–Howard condition for stratified shear flow stability is violated. However, for all slope angles, the ‘overturning’ occurs before the Miles–Howard stability condition is violated and so we argue that the first instability is convective.

Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987).

One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

A number of experiments have been performed on the properties of propane diffusion flames at relatively low fuel flow rates and using a variety of burner types. Optical methods were used to observe the flame and plume above it. We have studied the transition from a steady flame, with a plume exhibiting a helical instability at low flow rates, to an axisymmetric instability of the whole flame/plume, that grows from the base of the flame and oscillates at a well-defined and robust frequency, at higher flow rates. These results and the observation of the effects of various external modifications, e.g. the generation of an annular counter flow, a pressure perturbation, etc. are consistent with the view that the transition to the axisymmetric state, at which the flame flickers, is one to a globally excited oscillation forced by a region of absolutely unstable flow at or near the base of the burners used in this study.

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

The problem of second-order water wave diffraction of an incident monochromatic wave field by an array of bottom-mounted circular cylinders is solved by a semi-analytical approach. The solution for the second-order potential is obtained by combining eigenfunction expansions with an integral representation. Unlike the indirect approach for second-order forces (Lighthill 1979; Molin 1979), this approach gives complete information about local flow characteristics (pressure, velocities, wave elevation, etc.) thus providing a basis for solving the third-order problem. The results obtained are compared with other published data, and new detailed results, useful for benchmarking purposes, are given. Finally the influences of wave incidence, cylinder radius and cylinder configuration are considered. This leads to the suggestion that there exists a near-trapping phenomenon for the second-order wave in an array of cylinders, at half the wave frequency at which the corresponding linear near-trapped mode occurs.