We consider a horizontal three-dimensional saturated porous layer, confined in an upright cubic box, heated from below and cooled from above. In the absence of a controller, the fluid maintains a no-motion state for subcritical Rayleigh numbers R < Rc, where Rc depends on the box's aspect ratio. Once this critical number is exceeded, fluid motion ensues. We demonstrate that, with the use of feedback control strategies which suppress flow instabilities, one can maintain a stable no-motion state for Rayleigh numbers far exceeding the classical critical one for the onset of convection. To preserve the equilibrium no-motion state of the classical problem, the controller alters the system's dynamics so as to stabilize an otherwise non-stable state.

Experiments are presented for three turbulent boundary layers generated by laterally converging, laterally diverging and parallel flow on a flat plate. A converging potential flow field outside the boundary layer was generated by superposing a parallel flow in the x-direction, a row of equally spaced line sources in the wall-normal (y) direction and an analogous row of sinks in the transversal (z) direction. This arrangement resulted in a velocity that was constant far upstream, far downstream and along the x-axis. The convergence – ∂W/∂z has its maximum in the plane of the source and sink rows. This flow field was realized with the test section shown in figure 1, based on streamlines intersecting a rectangular cross-section far upstream. The diverging flow was generated by reversing the flow direction through the test section.

The tests were conducted at about 42 m/s leading to a unit Reynolds number of 2.5 × 106/m and to a Reynolds number based on the momentum thickness of 4000 to 4700 at the inlet of the test sections, increasing up to 25000 at the outlet. In all three cases the velocity distribution near the wall agreed very well with the logarithmic law of the wall. The wake contribution in the outer layer was considerably increased by convergence and decreased by divergence. The Reynolds stresses, measured with crossed hot-wire probes, and the wall shear stress, measured with a floating-element balance, were generally increased by divergence and decreased by convergence and the same holds true for the mixing length and the turbulent viscosity.

A finite-difference boundary-layer code using a simple turbulence model was used to predict the experimental results. The comparison showed good agreement for the two-dimensional flow, reasonable agreement for the diverging flow and poor agreement for the converging one. Use of the experimentally determined turbulent viscosity as input into the computation did not systematically improve the agreement but excellent agreement was found if it was combined with anisotropy of the turbulent viscosity. It was much more difficult to predict the converging flow as small errors in the crossflow had a large effect on the flow in the plane of symmetry (z = 0).

We discuss a series of numerical experiments on the dispersion of neutrally buoyant particles in two-dimensional turbulent flows. The topology of two-dimensional turbulence is parametrized in terms of the relative dominance of deformation or rotation; this leads to a segmentation of the turbulent field into hyperbolic and elliptic domains. We show that some of the characteristic structural domains of two-dimensional turbulent flows, namely coherent structures and circulation cells, generate particle traps and peculiar accelerations which induce several complex properties of the particle dispersion processes at intermediate times. In general, passive particles are progressively pushed from the coherent structures and tend to concentrate in highly hyperbolic regions in the proximity of the isolines of zero vorticity. For large dispersion times, the background turbulent field is a privileged domain of particle richness; there is however a permanent particle exchange between the background field and the energetic circulation cells which surround the coherent structures. At intermediate times, an anomalous dispersion regime may appear, depending upon the relative weight of the different topological domains active in two-dimensional turbulence. The use of appropriate conditional averages allows the basic topology of two-dimensional turbulence to be characterized from a Lagrangian point of view. In particular, an intermediate $t^{\frac{5}{4}}$ anomalous dispersion law is shown to be associated with the action of hyperbolic regions where deformation dominates rotation; the motion of the advected particles in strongly elliptic regions where rotation dominates over deformation is shown to be associated with a $<e1>t<sup>t^{\frac{5}{3}}$ dispersion law. Because neutral particles concentrate on average in hyperbolic regions, the $<e1>t^{\frac{5}{4}}$ dispersion law is quite robust and it can be observed under very general circumstances.

A theoretical examination is made of convection, induced by applied thermal and solutal gradients inclined to the vertical, in a shallow horizontal layer of a saturated porous medium. The horizontal components of these gradients induce a Hadley circulation, which becomes unstable when the vertical components are sufficiently large. A linear stability analysis is carried out, and calculations are made using a low-order Galerkin approximation for the various modes of instability. The orientation of the preferred mode and the other critical quantities are determined for representative parameter values.

The motion of an elliptical cylindrical particle immersed in an incompressible Newtonian fluid in a narrow channel is examined numerically in the zero-Reynolds-number limit. It is assumed that no external forces or torques act on the elliptical cylinder, and the effects of inertia forces on the motion of the fluid and the particle are neglected. The Stokes equations are solved by a finite-element method for various positions and orientations of the cylinder, yielding the instantaneous velocities of the particle that satisfy the conditions of zero force and zero torque on the particle. Using the computed longitudinal, lateral, and angular velocities of the particle, the evolution of the particle's position and orientation is determined for various initial configurations. An elliptical cylinder is found to either tumble or oscillate in rotation, depending on the particle-channel size ratio, the axis ratio of the elliptical cylinder, and the initial conditions. In the first case, the particle rotates continuously in one direction that is well approximated by Jeffery's solution for an elliptical cylinder in unbounded shear flow with a so-called equivalent axis ratio; in the second case, the particle changes its direction of rotation during part of each period. In both cases, the particle translates with a periodically varying longitudinal velocity, accompanied by a considerable side drift due to the walls. The oscillatory motion is more likely to occur when the particle-channel size ratio or axis ratio is increased. The tumbling motion is inhibited for elliptic cylinders whose size ratios are larger than threshold values that depend on the axis ratio.

Arnol'd's second stability theorem is proved for arbitrary perturbations of the potential vorticity field δq and the circulation(s) δγ. The formal stability condition is essentially the same as that for δγ ≡ 0, which is much easier to obtain. Similarly, the condition obtained assuming $\overline{\delta q} \equiv 0$ (the overbar denoting a horizontal average) is found to be also valid for $\overline{\delta q} \not\equiv 0$. It is argued that a Lyapunov functional that is extreme only on the sheet of constant Casimirs (and other integrals of motion) also proves stability for perturbations off the sheet, even though its second variation may not be sign definite for general perturbations. This conjecture is illustrated by means of a very simple mechanical problem: a point particle subject to the action of a central force. For the case of Phillips’ problem in a periodic channel, formal stability conditions on the isovortical sheet coincide with the criteria obtained from normal modes analysis.

When an alternating electric field is applied to a colloid the particles oscillate at a velocity proportional to the applied field strength. The complex constant of proportionality is termed the dynamic mobility (O'Brien 1988). Although this quantity can now be determined from electroacoustic measurements in suspensions of arbitrary concentration (O'Brien 1990), the theory for interpreting these measurements in terms of the size and charge of the particles is limited to dilute suspensions.

In this paper we derive an expression for the O(ϕ) correction to the dynamic mobility in a random suspension of uniform spheres with volume fraction ϕ. It is assumed that the particle radius is much greater than the double layer thickness but much smaller than the sound wavelength. The mobility is calculated using O'Brien's 1979 macroscopic boundary integral technique. This method ensures a correct mathematical formulation of the problem, and yields an absolutely convergent expression for the average particle velocity. The evaluation of this expression to O(ϕ) involves the determination of the velocities of an isolated pair of particles at various separations and frequencies of oscillation. These velocities are computed using the collocation technique and the O(ϕ) correction to the dynamic mobility is then obtained by numerically integrating over all particle separations.

We show that a recent publication by Liron & Barta (1992) concerning a single-layer boundary integral equation for the tractions is mathematically equivalent to Karrila & Kim's (1989) Riesz method. In actual computational schemes, the second viewpoint is preferable since the integral operator has a spectral radius less than one and even large problems can be solved by fast iterative methods.

The effect of rapid mean compression on compressible turbulence at a range of turbulent Mach numbers is investigated. Rapid distortion theory (RDT) and direct numerical simulation results for the case of axial (one-dimensional) compression are used to illustrate the existence of two distinct rapid compression regimes. These regimes – the nearly solenoidal and the ‘pressure-released’ – are defined by a single parameter involving the timescales of the mean distortion, the turbulence, and the speed of sound. A general RDT formulation is developed and is proposed as a means of improving turbulence models for compressible flows. In contrast to the well-documented observation that ‘compressibility’ (measured, for example, by the turbulent Mach number) is often associated with a decrease in the growth rate of turbulent kinetic energy, we find that under rapid distortion compressibility can produce an amplification of the kinetic energy growth rate. We also find that as the compressibility increases, the magnitude of the pressure–dilation correlation increases, in absolute terms, but its relative importance decreases compared to the magnitude of the kinetic energy production.

An integral method is developed to solve the two-dimensional Stokes problem with Neumann boundary conditions for multiply connected domains in which the inside hole area can shrink and disappear. The method is applied to simulate viscous sintering. In particular the sintering of glasses can be modelled as such, i.e. a viscous incompressible Newtonian volume flow driven solely by surface tension. A Boundary Element Method is applied to solve the integral equations of Stokes flow involved, and the time integration is carried out by a variable-step, variable-order Backward Differences Formulae method. The derived numerical algorithm is demonstrated for several arbitrarily shaped multiply connected sintering domains. In particular some cylindrical packings are considered. The latter simulations provide a justification for the use of ‘unit problems’ in the theory of sintering.

The nonlinear interactions between a free surface and a shed vortex shear layer in the inviscid wake of a vertical surface-piercing plate are studied numerically using a mixed-Eulerian-Lagrangian method. For a plate with initial submergence d starting abruptly from rest to constant horizontal velocity U, the problem is governed by a single parameter, the Froude number Fn = U/(gd)½, where g is the gravitational acceleration. Depending on Fn, three classes of interaction dynamics (subcritical, transcritical and supercritical) are identified. For subcritical Fn ([lsim ] 0.7), the free surface plunges on both the forward and lee sides of the plate before significant interactions with the vortex sheet occur. For transcritical and supercritical Fn, interactions between the free surface and the starting vortex result in a stretching of the vortex sheet which eventually rolls up into double-branched spirals as a result of Kelvin-Helmholtz instability. In the transcritical range (Fn ∼ 0.7–1.0), the effect of the free surface on the double-branched spirals remains weak, while for supercritical Fn ([gsim ] 1.0), strong interactions lead to entrainment of the double-branched spiral into the free surface resulting in prominent surface features.