Direct numerical simulations were used to study the dynamics of a vortex ring impacting a wall at normal incidence. The boundary layer formed as the ring approaches the wall undergoes separation and roll-up to form a secondary vortex ring. The secondary ring can develop azimuthal instabilities which grow rapidly owing to vortex stretching and tilting in the presence of the mean strain field generated by the primary vortex ring. The stability of the secondary ring was investigated through complementary numerical experiments and stability analysis. Both perturbed and unperturbed evolutions of the secondary ring were simulated at a Reynolds number of about 645, based on the initial primary-ring propagation velocity and ring diameter. The linear evolution of the secondary vortex-ring instability was modelled analytically by making use of a quasi-steady approximation. This allowed a localized stability analysis following Widnall & Sullivan's (1973) earlier treatment of an isolated vortex ring. Amplitude evolution and growth-rate predictions from this analysis are in good agreement with the simulation results. The analysis shows that the secondary vortex ring is unstable to long-wavelength perturbations, even though an isolated ring having similar characteristics would be stable.

Irrotational flows caused by a body moving with a constant velocity in an unbounded homentropic compressible fluid at rest at infinity are considered. Provided the (steady) flow relative to the body is everywhere subsonic, it is shown that the momentum mass is always equal to the drift mass, and the kinetic-energy mass is equal to the drift mass under certain conditions.

Accurate numerical solutions to the problem of finding the location of the interface between two unconfined regions of fluid of different density during the withdrawal process are presented. Supercritical flows are considered, in which the interface is drawn directly into the sink. As the flow rate is reduced, the interface enters the sink more steeply, until the solution method breaks down just before the interface enters the sink vertically from above, and becomes flow from the lower layer only. This lower bound on supercritical flow is compared with the upper bound on single-layer (free surface) flow with good agreement.

The evolution of two-dimensional regular flows laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle-particle interactions. Flows with large Reynolds and Froude numbers are considered, when effects related to viscous dissipation and gravity are negligible. A Cauchy problem is solved for an initially uniform distribution of particles with Stokes (St) and Reynolds (Rep) numbers of order unity in several types of flows representing steady solutions of the two-dimensional Euler equations. We consider flows in the vicinity of the hyperbolic stagnation point (with a uniform strain and zero vorticity) and the elliptic stagnation point (where vorticity is uniform), a circular vortex (with vorticity depending on the radius) and Stuart vortex flow. Analytical solutions are obtained, for the case of sufficiently small St, describing the accumulation of particles and corresponding modification of the fluid flow. Solutions derived show that the concentration of particles, although remaining uniform, decreases at the elliptic stagnation point and grows at the hyperbolic point. Owing to the coupling between the particulate and fluid dynamics, the flow vorticity is reduced at the elliptic point, while flow strain rate is enhanced at the hyperbolic point. Solutions obtained for the circular vortex show that the accumulation of particles proceeds in the form of a travelling wave. The concentration grows locally, forming the crest of the wave which propagates away from the vortex centre. Owing to the influence of the particulate on the carrier flow, the vorticity is reduced in the vortex centre. At the location of the crest the gradient of the flow grows owing to the drag forces between the fluid and particles and a vorticity peak is generated. Analytical solutions are also obtained for a chain of particle-laden Stuart vortices. Owing to the coupling effects, the concentration is diminished and the vorticity is reduced at the centres of the vortices. A sheet of increased concentration and vorticity is formed extending from the braid region to the periphery of the vortices, and the flow strain in the braid region is enhanced. Results of numerical simulations performed for St = 0.5 show good agreement with analytical solutions.

The nonlinear development of stationary Görtler vortices leads to a highly distorted mean flow field where the streamwise velocity depends strongly not only on the wall-normal but also on the spanwise coordinates. In this paper, the inviscid instability of this flow field is analysed by solving the two-dimensional eigenvalue problem associated with the governing partial differential equation. It is found that the flow field is subject to the fundamental odd and even (with respect to the Görtler vortex) unstable modes. The odd mode, which was also found by Hall & Horseman (1991), is initially more unstable. However, there exists an even mode which has higher growth rate further downstream. It is shown that the relative significance of these two modes depends upon the Görtler vortex wavelength such that the even mode is stronger for large wavelengths while the odd mode is stronger for short wavelengths. Our analysis also shows the existence of new subharmonic (both odd and even) modes of secondary instability. The nonlinear development of the fundamental secondary instability modes is studied by solving the (viscous) partial differential equations under a parabolizing approximation. The odd mode leads to the well-known sinuous mode of break down while the even mode leads to the horseshoe-type vortex structure. This helps explain experimental observations that Görtler vortices break down sometimes by sinuous motion and sometimes by developing a horseshoe vortex structure. The details of these break down mechanisms are presented.

Subcritical transition in a flat-plate boundary layer is examined experimentally through observing its nonlinear response to energetic hairpin eddies acoustically excited at the leading edge of the boundary-layer plate. When disturbed by the hairpin eddies convecting from the leading edge, the near-wall flow develops local three-dimensional wall shear layers with streamwise vortices. Such local wall shear layers also evolve into hairpin eddies in succession to lead to the subcritical transition beyond the x-Reynolds number Rx = 3.9 × 104, where the momentum thickness Reynolds number Rθ is 127 for laminar Blasius flow without excitation, and is about 150 under the excitation of energetic hairpin eddies. It is found that in terms of u- and v-fluctuations, the intensity of the near-wall activity at this critical station is of almost the same order as or slightly less than that of the developed wall turbulence. The development of wall turbulence structure in this transition is also examined.

The transient deformation of liquid capsules enclosed by elastic membranes subject to simple shear flow is studied numerically using a new implementation of the boundary element method. The numerical results for capsules with spherical unstressed shapes and varying degrees of surface elasticity are compared with the predictions of an asymptotic theory for small deformations due to Barthès-Biesel and coworkers, and the significance of nonlinear effects due to finite deformation is assessed. It is found that the capsules exhibit continuous elongation when the dimensionless shear rate becomes larger than a critical threshold, in agreement with recent experimental observations of capsules with polymerized interfaces. Membrane failure at large deformations is discussed with respect to membrane thinning and development of excessive elastic tensions, and it is argued that the location where the membrane is likely to rupture due to continued deformation is insensitive to the precise mechanism of rupture. The numerical results suggest that a dilute suspension of capsules behaves like shear-thinning medium with some elastic properties. Results of oblate spheroidal capsules suggest that the points of maximum membrane thinning and tension coincide but their location depends upon the unstressed capsule shape.

Thermohaline convection in a salt water loop is discussed. Fluid temperature is affected by relaxation on the loop surface and fluid salinity by a freshwater flux through the loop surface. In addition, other boundary conditions on salinity, such as the equivalent virtual salt flux or salinity relaxation condition, are examined and the dynamic role of diffusion in thermohaline convection is analysed.

Both analytical and numerical analyses indicate that the system behaviour depends sensitively on the nature of the salinity boundary condition. For the saline-only loop model, analysis indicates that perturbations are advected by the mean flow, and flow stability is independent of the strength of the boundary forcing. In the full thermohaline loop problem, the virtual salt flux formulation accurately mirrors the freshwater flux results when the system is in the thermal mode. However, these formulations can differ substantially when the system is in the haline mode, especially in the strongly forced, weakly diffusive limit.

For both types of loop configuration, salinity profiles governed by freshwater flux have scales determined by the internal parameters, while virtual salt flux profiles necessarily reflect the lengthscales of applied boundary conditions. Negative salinities can also appear under virtual salt flux owing to the inaccuracies inherent in the approximation, while freshwater flux ensures positive-definite salinity values.

Our analysis supports the use of the physically more accurate freshwater flux boundary conditions when simulating thermohaline circulation.

A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.

In this paper we investigate numerically the periodic, axisymmetric response of a liquid annulus enclosed in a revolving cylinder and subject to a periodic axial excitation within the range of the natural frequencies. We use a description which employs as solution variables the transverse component of the vorticity vector, the scalar stream function and the transverse velocity component, as well as the position of the free surface. The acceleration due to gravity is neglected, whereas friction and surface tension are taken into account. At the fixed walls the no-slip condition is fulfilled except at points on the contact line. At the contact line the slip condition is applied. The solution of this problem is achieved using the spectral method in the time direction and finite differences in the space direction, whereby surface-adapted coordinates are utilized. Far away from the walls the computational results show good agreement with results obtained from a linearized theory assuming an inviscid liquid, while at the walls boundary layers are generated.

Effects of viscous damping on mass transport velocity in a two-layer fluid system are studied. A temporally decaying small-amplitude interfacial wave is assumed to propagate in the fluids. The establishment and the decay of mean motions are considered as an initial-boundary-value problem. This transient problem is solved by using a Laplace transform with a numerical inversion. It is found that thin ‘second boundary layers’ are formed adjacent to the interfacial Stokes boundary layers. The thickness of these second boundary layers is of O(ε1/2) in the non-dimensional form, where ε is the dimensionless Stokes boundary layer thickness defined as $\epsilon = \hat{k}\hat{\delta}=\hat{k}(2\hat{v}/\hat{\sigma})^{1/2}$ for an interfacial wave with wave amplitude â, wavenumber $\hat{k}$ and frequency $\hat{\sigma}$ in a fluid with viscosity $\hat{v}$. Inside the second boundary layers there exists a strong steady streaming of O(α2ε−1/2), where $\alpha = \hat{k}\hat{a}$ is the surface wave slope. The mass transport velocity near the interface is much larger than that in a single-layer system, which is O(α2) (e.g. Longuet-Higgins 1953; Craik 1982). In the core regions outside the thin second boundary layers, the mass transport velocity is enhanced by the diffusion of the mean interfacial velocity and vorticity. Because of vertical diffusion and viscous damping of the mean interfacial vorticity, the ‘interfacial second boundary layers’ diminish as time increases. The mean motions eventually die out owing to viscous attenuation. The mass transport velocity profiles are very different from those obtained by Dore (1970, 1973) which ignored viscous attenuation. When a temporally decaying small-amplitude surface progressive wave is propagating in the system, the mean motions are found to be much less significant, O(α2).

Results of numerical computations are presented of time-dependent three-dimensional convection flows in a horizontal layer heated from below which evolve from the oscillatory blob instability of steady two-dimensional rolls. It is shown that the heat transport is typically increased in the transition to blob convection. Oscillatory blob convection exists in the forms of standing or travelling blob convection. The latter type of solution represents the stable form bifurcating supercritically at the Rayleigh number RII for the onset of the oscillatory blob instability. In contrast to standing blob convection travelling blob convection exhibits a mean flow.

In analogy with Kolmogorov's refined similarity hypotheses for the velocity field, two hypotheses are stated for passive scalar fields mixed by high-Reynolds-number turbulence. A ‘refined’ Yaglom equation is derived under the new assumption of local isotropy in pure ensembles, which is stronger than the usual assumption of local isotropy but weaker than the isotropy of the large scale. The new theoretical result is shown to be consistent with the hypotheses of refined similarity for passive scalars. These hypotheses are approximately verified by experimental data on temperature fluctuations obtained (in air) at moderate Reynolds numbers in the wake of a heated cylinder. The fact that the refined similarity hypotheses are stated for high Reynolds (and Péclet) numbers, but verified at moderate Reynolds (and Péclet) numbers suggests that these hypotheses are not sufficiently sensitive tests of universality. It is conjectured that possible departures from universality are hidden by the process of taking conditional expectations.

The stability of the flow of interstitial liquid in a mushy layer driven by expansion or contraction upon solidification is analysed. The full perturbation equations are reduced in a particular aymptotic limit that allows the principal mechanisms controlling instability to be identified. Comparisons are made with the acid-etching instabilities in porous rocks. The full equations are then solved to determine the parametric dependences of the instability. It is found that, though the potential for instability exists, it is unlikely to occur in practice.

Numerical evidence of the existence of edge waves travelling along a periodic coastline consisting of a straight and vertical cliff face from which protrudes an infinite number of identical rectangular barriers, each extending throughout the water depth, is given based on a Galerkin approximation to an integral representation of the problem derived using the linear theory of water waves.

An analytical method somewhat analogous to finite wing theory has been developed which enables the flow induced by a linearized propeller actuator disk with variable radial distribution of load to be solved in closed form for the first time. Analytical solutions are given for various load distributions including the case of an arbitrary polynomial loading. As in finite wing theory, the case of elliptic loading is exceptionally simple and the induced velocities and stream function are simple expressions of elementary functions. Results are also given for a practical propeller load distribution with finite hub. The method can also be used to solve a wide range of analogous electromagnetic problems.

The stability of the flow in a half-zone configuration is analysed with the aid of direct numerical simulation. The work is concentrated on the small Prandtl numbers relevant for typical semiconductor melts. The axisymmetric thermocapillary flow is found to be unstable to a steady non-axisymmetric state with azimuthal wavenumber 2, for a zone with aspect ratio 1. The critical Reynolds number for this bifurcation is 1960. This three dimensional steady solution loses stability to an oscillatory state at a Reynolds number of 6250. For small Prandtl numbers, both bifurcations are seen to be quite insensitive to changes in the Prandtl number, and are thus hydrodynamic in nature. An analogy to the instability of thin vortex rings is made. This analogy suggests a physical mechanism behind the instability and also gives an explanation of how the azimuthal wavenumber of the bifurcated solution is selected. The implications of this for the floating-zone crystal growth process are discussed.

We consider the transport of a tracer substance through a system consisting of a tube containing flowing fluid surrounded by a wall layer in which the tracer is soluble. The fluid moves with either a Poiseuille or a uniform flow profile, and the outer boundary of the wall layer is either impermeable to tracer or absorbs it perfectly. The development of dispersive transport following the injection of tracer is described in terms of three time-dependent effective transport coefficients, viz. the fraction of tracer remaining in the system, the apparent convection velocity and the dispersion coefficient; the last two are defined in terms of the rates of change of the mean and variance of the axial tracer distribution. We assume that the timescale for tracer diffusion across the wall layer is much larger than that for diffusion across the flowing phase, and derive an asymptotic approximation corresponding to each timescale. Numerical results are given to illustrate sensitivity to the physical parameters of the system. It is shown that if the coefficients are based on tracer concentration in the fluid phase alone, as in previous work, paradoxical behaviour, such as negative apparent convection velocities, can result; we therefore base our results on averages of concentration over both phases. On the shorter timescale (the same timescale over which Taylor dispersion develops) at leading order it is found that the influence of the wall layer can be characterized by a single dimensionless parameter, and that conditions at the outer boundary have no effect. In many cases transport is also rather insensitive to the form of the flow profile. On the longer timescale, at leading order the influence of the wall layer is characterized by another dimensionless parameter, and unless uptake is very small diffusion within the layer is the rate-determining process; consequently transport is independent of the form of the flow profile. A further important conclusion is that the usual effective convection and dispersion coefficients, based on spatial moments, are of little use in predicting the time-varying concentration at a fixed position, because the spatial concentration profile becomes Gaussian only over the longer timescale.