We investigate the three-dimensional characteristics of a general class of resonant temporal instabilities of internal gravity waves, in which the disturbance comprises two infinitesimal internal gravity waves satisfying the conditions $\omega_I + \omega_{II} = n\tilde{\omega}$, with growth rates of order $(\vert \tilde{u}\vert \vert \tilde{k}\vert /N)^n$, assuming small dimensional primary-wave velocity amplitude $\vert \tilde{u} \vert$. We derive simple equations for their wave-numbers, frequency, growth rate, and energy budget. Interactions involving more than two disturbance components are shown not to represent distinct families of solutions, but rather to comprise order transitions linking together two or more ‘two-disturbance-component’ solutions of different order n. Unlike n = 1 resonant instabilities, those with n [ges ] 3 can align with the wave shear flow. We calculate the peak growth rate, spanwise wavenumber, and energy budget of shear-aligned resonance, as functions of wave frequency; they extract energy from both the wave shear and buoyancy fields.

Two-particle Lagrangian models in turbulence are used to consider dispersion in decaying isotropic homogeneous turbulence (i.e. approximating grid-generated wind-tunnel flows). Thomson's formulation is used, and his model is extended to incorporate molecular diffusivity and viscosity, i.e. the range of scales modelled includes the dissipation subranges as well as the intertial subrange. New terms are proposed which consistently provide well-mixed models and the impact on Saffman's well-known small-time results is considered. The new model is ideal for comparison with recent concentration-fluctuation measurements in decaying wind-tunnel turbulence and the results are encouraging. In particular, the fluctuation intensities, both along and across the wind tunnel, are well described by the new model. In addition, small source-size effects are far better explained when we include the molecular effects. A surprising results is the persistence in time of both source-geometry and molecular effects upon concentration fluctuations. Both of the effects are negligible at large times for mean concentrations, but persist for significant times for the fluctuations, indicating an important role for small-scale dynamics.

We consider three-dimensional hydraulic jumps (shocks) propagating along boundaries in rotating fluids. This study is motivated by earlier work (Fedorov & Melville 1995), which dealt with the evolution to breaking of nonlinear Kelvin waves. We obtain the jump relations and derive an evolution equation for the jump as it propagates along the boundary. It is shown that after some initial adjustment the Kelvin-type jump assumes a permanent form and propagates with a constant velocity along the boundary or the coast. At some distance offshore the jump becomes oblique to the coastline, and the final shape of the jump and its speed depend only on the jump strength. The jump gives rise to a moderate mass transport offshore. The potential vorticity remains almost constant across the jump. The energy loss in the jump is proportional to the third power of the jump amplitude, which is similar to classical two-dimensional hydraulic jumps in non-rotating fluids. Jump properties are discussed for both weak and strong nonlinearity, and the role of a boundary layer region behind the leading edge of the jump is considered.

Thin-layer Navier–Stokes solutions are obtained for an incompressible laminar flow over a hemisphere–cylinder at 10°, 30° and 50° incidence to exhibit some three-dimensional separated flow characteristics. Some of the results are compared with a previous water-tunnel investigation for the same body geometry. Good agreement is found, even for some detailed features. Although the geometry is relatively simple, the separated flow surprisingly embraces a number of intricate structural characteristics unique to three-dimensional flows. Particularly noteworthy are the separation sequence at increasing incidence, tornado-like vortices, outward-spiralling vortices, limit cycles, coaxial counter-spiralling patterns and horseshoe vortex systems. Physical insights to these new features are offered.

The flow evolution of water in a completely filled rectangular container, impulsively rotated from rest to a steady angular speed, is investigated experimentally and numerically. The pathlines of the fluid from rest to solid-body rotation primarily follow one of two possible configurations that have been described previously in the literature. The first, consisting of two cyclones that form following the separation and roll-up of the sidewall boundary layers and an anticyclone that forms subsequently, results in a pattern on the path to spin-up of cyclonic–anticyclonic–cyclonic vorticity. In the second configuration the cyclones migrate into the interior of the container and merge, resulting in a pattern on the path to spin-up of anticyclonic–cyclonic–anticyclonic vorticity. The experiments provide a parameterization of the possible evolutionary configurations as a function of horizontal and vertical aspect ratios and Reynolds numbers. Critical Reynolds numbers for vortex merger are determined experimentally. Evolutionary configurations in addition to the primary two are observed; in particular symmetry breaking occurs at high Reynolds numbers causing complicated patterns of flow evolution. For some flow conditions at high Reynolds numbers, more than one evolutionary pattern is observed for the same external parameters. The experiments are conducted with a rigid lid showing that a free surface is not required for vortex merger. Numerical integrations of the two-dimensional Navier–Stokes equations (a situation corresponding to the limiting case of a container of infinite depth, where there are no effects from the top and bottom and all flow is horizontal) reproduce qualitatively many of the features of the experimental observations, in particular the merger events. The numerical results show that neither vertical flow due to Ekman boundary layers at the top and bottom nor a free surface are necessary for the observed vortex merger.

Thermoacoustic refrigeration occurs in periodic flow in a duct with heat transfer within the fluid and to the tube. This study considers the periodic limit cycle with large pressure oscillations that is obtained in a tube when prescribed, phase-shifted, periodic velocities at the tube ends, at frequencies lower than acoustic eigenmodes, sweep a length comparable to the tube length. The temperature differences between the two ends are of arbitrary magnitude, heat transfer in the transverse direction within the fluid is assumed to be very effective and the thermal mass of the wall is large. The geometry is two-dimensional, axisymmetric, and conduction is accounted for, not only in the fluid, but also with and within the tube wall. A perturbation solution valid in a local near-isothermal limit determines the equilibrium longitudinal temperature profile that is reached at the periodic regime, the pressure field including longitudinal gradients, and the longitudinal enthalpy flux. Results are presented for tubes open at both ends and also with one end closed. In the latter case, a singularity occurs in the temperature at the closed end, with behaviour identical to Rott's result for acoustic flow with small pressure amplitude. Other new results obtained for tubes open at both ends show that when velocities at both ends are in opposite phase, internal singularities in the temperature profiles may occur.

The governing equations of the flow field which is obtained when a thermoelastic rigid porous medium is struck head-one by a shock wave are developed using the multiphase approach. The one-dimensional version of these equations is solved numerically using a TVD-based numerical code. The numerical predictions are compared to experimental results and good to excellent agreements are obtained for different porous materials and a wide range of initial conditions.

The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in circular Couette flow has been investigated. Numerical solution of the governing integro-differential equations for different initial conditions shows that the equilibrium states of Taylor-vortex, wavy-vortex or spiralvortex flows are not unique, but depend on the initial disturbance. The presence of multiple solutions at a fixed Reynolds number for a given Taylor–Couette geometry has been known since Coles’ seminal contribution in 1965. The current study indicates that the equilibrium state of flows on a stable bifurcation branch is a natural consequence of nonlinear wave resonance and is dependent on the initial conditions. The resulting wavenumber can take any value within an accessible finite band. Since such multiple solutions have also been found numerically for mixed-convection flows and experimentally for several other flows, there is evidence to support the conclusion that a non-uniqueness in the sense of Coles is a generic property for all fluid flows.

This paper presents an experimental study of the behaviour of single particles and suspensions in polymer solutions in a torsional flow. Four issues are investigated in detail: the radial migration of a shperical particle; the rotation and migration of a cylindrical particle; the particle-particle interaction and microstructures in a suspension of spheres; and the microstructures in a suspension of rods. Newtonian fluids are also tested under similar flow conditions for comparison. A spherical particle migrates outward at constant velocity unless the polymer solution is very dilute. A rod in a viscoelastic fluid has two modes of motion depending on its initial orientation, aspect ratio, the local shear rate and the magnitude of normal stresses in the fluid. In the first mode, the rod rotates along a Jeffery-like orbit around the local vorticity axis. It also migrates slowly inward. The second mode of motion has the rod aligned with the local stream at all times; the radial migration is outward. A hypothesis proposed by Highgate & Whorlow (1968) on the radial force on a particle in a cone-and-plate geometry is generalized to explain the variation of migration speed in torsional flows. Spheres form chains and aggregates when the suspension is sheared. The chains are along the flow direction and may connect to form circular rings; these rings migrate outward at a velocity much higher than that of a single sphere. Rods interact with one another and aggregate in much the same way, but to a lesser extent than spheres. It is argued that the particle interaction and aggregation are not direct results of the shear flow field. Two fundamental mechanisms discovered in sedimentation are applied to explain the formation of chains and aggregates. Finally, the competition between inertia and elasticity is discussed. A change of type is not observed in steady shear, but may happen in small-amplitude oscillatory shear.

When a suspension of the bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere, complex bioconvection patterns form. These arise because the cells (a) are denser than water, and (b) swim upwards on average so that the density of an initially uniform suspension becomes greater at the top than at the bottom. When the vertical density gradient becomes large enough an overturning instability occurs which evolves ultimately into the observed patterns. The cells swim upwards because they are oxytactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell and oxygen concentrations, which, for the pre-instability stage of the pattern formation process, have been solved in a previous paper (Hillesdon, Pedley & Kessler 1995). In this paper we carry out a linear instability analysis of the steady-state cell and oxygen concentration distributions. There are intrinsic differences between the shallow-and deep-chamber cell concentration distributions, with the consequence that the instability is non-oscillatory in shallow chambers, but must be oscillatory in deep chambers whenever the critical wavenumber is non-zero. We investigate how the critical Rayleigh number for the suspension varies with the three independent parameters of the problem and discuss the most appropriate definition of the Rayleigh number. Several qualitative aspects of the solution of the linear instability problem agree with experimental observation.

A thin layer of liquid advancing over a dry, heated, inclined plate is studied. A lubrication model with contact line motion is derived. The plate is at constant temperature, and the surface Biot number is specified. The steady-state solution is obtained numerically. In addition, the steady-state solution is studied analytically in the neighbourhood of the contact line. A linear stability analysis about the steady state is then performed. The effects of gravity, thermocapillarity and contact line motion are discussed. In particular, we determine a band of unstable wavenumbers, and the maximum growth rate as a function of these parameters.

The motion of a rigid ellipsoidal particle freely suspended in a Poiseuille flow of an incompressible Newtonian fluid through a narrow tube is studied numerically in the zero-Reynolds-number limit. It is assumed that the effect of inertia forces on the motion of the particle and the fluid can be neglected and that no forces or torques act on the particle. The Stokes equation is solved by a finite element method for various positions and orientations of the particle to yield the instantaneous velocity of the particle as well as the flow field around it, and the particle trajectories are determined for different initial configurations. A prolate spheroid is found to either tumble or oscillate in rotation, depending on the particle–tube size ratio, the axis ratio of the particle, and the initial conditions. A large oblate spheroid may approach asymptotically a steady, stable configuration, at which it is located close to the tube centreline, with its major axis slightly tilted from the undisturbed flow direction. The motion of non-axisymmetric ellipsoids is also illustrated and discussed with emphasis on the effect of the particle shape and size.

Sound wave structures, downstream of moving incident shocks reflecting from straight compressive wedges, are analysed for both weak and strong Mach reflections (MR) using existing experiments. It is shown that the reflected waves can be well described by using the acoustic criterion or the weak oblique shock approximation, when the classical three-shock theory gives forward-facing reflected shock solutions. The predicted triple-point trajectory angles are found to be in close agreement with the experiments. The distinction between the applicabilities of the two methods is given by an analytically defined ‘smallness’ for the angle of reflecting wedges. The physics of the success of the two methods is discussed. It is concluded that forward-facing reflected shock solutions of pseudo-steady MR should be ruled out physically because sound waves cannot coalesce into Mach waves that propagate upstream of the triple point. In their place, MR-like phenomena occur with the reflected waves being normal Mach waves or finite compression waves for ‘small’ or ‘not-small’ reflecting wedge angles, respectively, and they are classified as the first- or second-king von Neumann reflections, respectively. Boundaries separating regimes between the first and second kinds of von Neumann reflections, and backward-facing MR are determined.

We develop a theoretical analysis of the displacement of inviscid fluid particles and material surfaces caused by the unsteady flow around a solid body that is moving away from a wall. The body starts at position hs from the wall, and the material surface is initially parallel to the wall and at distance hL from it. A volume of fluid Df+ is displaced away from the wall and a volume Df- towards the wall. Df+ and Df- are found to be sensitive to the ratio hL/hs. The results of our specific calculations for a sphere can be extended in general to other shapes of bodies.

When the sphere moves perpendicular to the wall the fluid displacement and drift volume Df+ are calculated numerically by computing the flow around the sphere. These numerical results are compared with analytical expressions calculated by approximating the flow around the sphere as a dipole moving away from the wall. The two methods agree well because displacement is an integrated effect of the fluid flow and the largest contribution to displacement is produced when the sphere is more than two radii away from the wall, i.e. when the dipole approximation adequately describes the flow. Analytic expressions for fluid displacement are used to calculate Df+ when the sphere moves at an acute angle α away from the wall.

In general the presence of the wall reduces the volume displaced forward and this effect is still significant when the sphere starts 100 radii from the wall. A sphere travelling perpendicular to the wall, α = 0, displaces forward a volume Df+(0) = 4πa3hL/33/2hS when the marked surface starts downstream, or behind the sphere, and displaces a volume Df+(0) ∼ 2πa3/3 forward when it is marked upstream or in front of the body. A sphere travelling at an acute angle away from the wall displaces a volume Df+(α) ∼ Df+(0) cos α forward when the surface starts downstream of the sphere. When the marked surface is initially upstream of the sphere, there are two separate regions displaced forward and a simple cosine dependence on α is not found.

These results can all be generalized to calculate material surfaces when the sphere moves at variable speed, displacements no longer being expressed in terms of time, but in relation to the distance travelled by the sphere.

Two-dimensional steady laminar flows past multiple thin blades positioned in near or exact sequence are examined for large Reynolds numbers. Symmetric configurations require solution of the boundary-layer equations alone, in parabolic fashion, over the successive blades. Non-symmetric configurations in contrast yield a new global inner–outer interaction in which the boundary layers, the wakes and the potential flow outside have to be determined together, to satisfy pressure-continuity conditions along each successive gap or wake. A robust computational scheme is used to obtain numerical solutions in direct or design mode, followed by analysis. Among other extremes, many-blade analysis shows a double viscous structure downstream with two streamwise length scales operating there. Lift and drag are also considered. Another new global interaction is found further downstream. All the interactions involved seem peculiar to multi-blade flows.

Most steady flows with constant vorticity and elliptical streamlines are known to be unstable. These, and certain axisymmetric time-periodic flows, can be analysed by Floquet theory. However, Floquet theory is inapplicable to other time-periodic flows that yield disturbance equations containing a quasi-periodic, rather than periodic, function. A practical method for surmounting this difficulty was recently given by Bayly, Holm & Lifschitz. Employing their method, we determine the stability of a clas of three-dimensional time-periodic flows: namely, those unbounded flows with fixed ellipsoidal stream surfaces and spatially uniform but time-periodic strain rates. Corresponding, but bounded, flows are those within a fixed ellipsoid with three different principal axes. This is perhaps the first exact stability analysis of non-reducibly three-dimensional and time-dependent flows. Though the model has some artificial features, the results are likely to shed light on more complex systems of practical interest.

Symmetric suction flows are computed. The flows are free-surface flows with two stagnation points. The configuration is related to the modelling of wave breaking at the bow of a ship. It is shown that there is a countably infinite number of solutions and that the free-surface profiles are characterized by waves.