A series of laboratory experiments was performed to investigate the overall mixing characteristics of oscillatory stratified flow past an isolated topography. The experiments were conducted by oscillating a right-circular cylinder in an otherwise quiescent linearly stratified fluid contained in a rectangular basin. The mixing was largely confined to the turbulent ‘core’ region around the cylinder. This mixed fluid was then injected into the fluid interior of the basin by numerous intrusive tongues. These intrusions were accompanied by return currents of unmixed stratified fluid into the turbulent core. The overall effect of this mixing process was to increase the potential energy of the fluid in the basin. An expression is derived to relate the rate of change of potential energy of the system to the basin-averaged buoyancy flux. This formula was then used to calculate the mean buoyancy flux from measurements of the rate of change of potential energy of the fluid system. Basin-averaged diapycnal eddy diffusivities for the experiments were evaluated and the results were found to be in good agreement with the predictions of a heuristic model based on the energetics of the mixing. Observations on the spreading of intrusions and the evolution of the density field are also presented.

Numerical experiments are described to ascertain how the steady flow past a circular cylinder loses stability as the Reynolds number is increased. A novel feature of the present study is that the cylinder is confined between parallel planes, allowing a more definitive specification of the flow, both experimentally and computationally, than is possible for the unbounded case. Since the structure of the bifurcation is unclear from the extant literature, with the experimental and computational evidence not in good agreement, a critical appraisal of both sets of evidence is presented.

A study has been made of the formation of the steady vortex pair behind the cylinder, and it has been determined that the first appearance of the vortices is not associated with a bifurcation of the full dynamical problem but instead it is probably associated with a bifurcation of a restricted kinematical problem.

A set of numerical experiments has been made in which the steady flow past the cylinder was perturbed slightly and the ensuing time-dependent motions were computed. These experiments revealed that, for a given blockage ratio, the perturbation would die away at small Reynolds numbers but that, above a critical Reynolds number, the disturbance would be amplified and the flow would eventually settle down to a new state comprising a time-periodic motion.

Experiments were also carried out to determine the bifurcation point numerically by considering an eigenvalue problem based on a linearization about the computed steady flow past the cylinder. The calculations showed that stability is lost through a symmetry-breaking Hopf bifurcation and that, for a given blockage ratio, the critical Reynolds number was in very good agreement with that estimated from the time-dependent computations.

Steady flow with constant circulation into a vertical drain is considered. The precise details of the outflow are simplified by assuming that the drain is equivalent to a distributed volume sink, into which the fluid flows with uniform downward speed. It is shown that a maximum outflow rate exists, corresponding to no fluid circulation and vertical entry into the drain hole. Numerical solutions to the full nonlinear problem are computed, using the method of fundamental solutions. An approximate analysis, based on the use of the shallow-water equations, is presented for flows in which the free surface enters the drain. There is, in addition, a second type of solution, having a stagnation point at the free surface and no fluid circulation. These flows are also computed numerically, and results are presented.

Steady planar flow of a liquid layer over an obstacle is studied for favourable slopes. First, half-plane Poiseuille flow is found to be a non-unique solution on a uniformly sloping surface since eigensolutions exist which are initially exponentially small far upstream. These have their origin in a viscous–inviscid interaction between the retarding action of viscosity and the hydrostatic pressure from the free surface. The cross-stream pressure gradient caused by the curvature of the streamlines also comes into play as the slope increases. As the interaction becomes nonlinear, separation of the liquid layer can occur, of a breakaway type if the slope is sufficiently large. The breakaway represents a hydraulic jump in the sense of a localized relatively short-scaled increase in layer thickness, e.g. far upstream of a large obstacle. The solution properties give predictions for the shape and structure of hydraulic jumps on various slopes. Secondly, the possibility of standing waves downstream of the jump is addressed for various slope magnitudes. A limiting case of small gradient, governed by lubrication theory, allows the downstream boundary condition to be included explicitly. Numerical solutions showing the free-surface flow over an obstacle confirm the analytical conclusions. In addition the predictions are compared with the experimental and computational results of Pritchard et al. (1992), yielding good qualitative and quantitative agreement. The effects of surface tension on the jump are also discussed and in particular the free interaction on small slopes is examined for large Bond numbers.

This work reports the first part of a series of numerical simulations carried out in order to improve knowledge of the forces acting on a sphere embedded in accelerated flows at finite Reynolds number, Re. Among these forces added mass and history effects are particularly important in order to determine accurately particle and bubble trajectories in real flows. To compute these hydrodynamic forces and more generally to study spatially or temporally accelerated flows around a sphere, the full Navier–Stokes equations expressed in velocity–pressure variables are solved by using a finite-volume approach. Computations are carried out over the range 0.1 ≤ Re ≤ 300 for flows around both a rigid sphere and an inviscid spherical bubble, and a systematic comparison of the flows around these two kinds of bodies is presented.

Steady uniform flow is first considered in order to test the accuracy of the simulations and to serve as a reference case for comparing with accelerated situations. Axisymmetric straining flow which constitutes the simplest spatially accelerated flow in which a sphere can be embedded is then studied. It is shown that owing to the viscous boundary condition on the body as well as to vorticity transport properties, the presence of the strain modifies deeply the distribution of vorticity around the sphere. This modification has spectacular consequences in the case of a rigid sphere because it influences strongly the conditions under which separation occurs as well as the characteristics of the separated region. Another very original feature of the axisymmetric straining flow lies in the vortex-stretching mechanism existing in this situation. In a converging flow this mechanism acts to reduce vorticity in the wake of the sphere. In contrast when the flow is divergent, vorticity produced at the surface of the sphere tends to grow indefinitely as it is transported downstream. It is shown that in the case where such a diverging flow extends to infinity a Kelvin–Helmholtz instability may occur in the wake.

Computations of the hydrodynamic force show that the effects of the strain increase rapidly with the Reynolds number. At high Reynolds numbers the total drag is dramatically modified and the evaluation of the pressure contribution shows that the sphere undergoes an added mass force whose coefficient remains the same as in inviscid flow or in creeping flow, i.e. CM = ½, whatever the Reynolds number. Changes found in vorticity distribution around the rigid sphere also affect the viscous drag, which is markedly increased (resp. decreased) in converging (resp. diverging) flows at high Reynolds numbers.

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coefficient of the highest derivative in the corresponding boundary-value problem vanishes at the point where the depth profile has an extremum. Although this singularity is similar to a critical level, it cannot be regularized by the simplistic introduction of infinitesimal viscosity through the assumption that the phase speed of the disturbance is complex. In order to regularize the singularity properly, one should consider the problem within the framework of the original ageostrophic viscous equations and, having obtained the boundary-value problem for harmonic disturbance, take the limit Rossby number → 0, viscosity → 0.

The results obtained analytically and (for special cases) numerically indicate that the stability of flows with non-monotonic profiles strongly depends on the depth of the upper layer. If the upper layer is ‘thick’ (i.e. if the average depth H1 of the upper layer is of the order of the total depth of the fluid H0), the stability boundary-value problem does not have any solutions at all, which means stability (however, this stability is structurally unstable, and the flow, generally speaking, can be made weakly unstable by any small effect such as external forcing, viscosity, or ageostrophic corrections). In the case of ‘thin’ upper layer (H1/H0 [lsim ] Ro), the order of the singularity changes and all non-monotonic flows are unstable regardless of their profiles. It is also demonstrated that thin-upper-layer flows do not have to be non-monotonic to be unstable: if u–βR20 (where u is the zonal velocity, β is the β-parameter, and R0 is the deformation radius) changes sign somewhere in the flow, the stability boundary-value problem has another singular point which leads to instability.

A non-similar boundary layer theory for air blowing over a water layer on a flat plate is formulated and studied as a two-fluid problem in which the position of the interface is unknown. The problem is considered at large Reynolds number (based on x), away from the leading edge. We derive a simple non-similar analytic solution of the problem for which the interface height is proportional to x1/4 and the water and air flow satisfy the Blasius boundary layer equations, with a linear profile in the water and a Blasius profile in the air. Numerical studies of the initial value problem suggest that this asymptotic non-similar air–water boundary layer solution is a global attractor for all initial conditions.

Experiments were conducted in a two-stream planar mixing layer at convective Mach numbers, Mc, of 0.28, 0.42, 0.50, 0.62 and 0.79. Planar laser Mie scattering (PLMS) from a condensed alcohol fog and planar laser-induced fluorescence (PLIF) of nitric oxide were used for flow visualization in the side, plan and end views. The PLIF signals were also used to characterize the turbulent mixture fraction fluctuations.

Visualizations using PLMS indicate a transition in the turbulent structure from quasi-two-dimensionality at low convective Mach number, to more random three-dimensionality for $M_c\geqslant 0.62$. A transition is also observed in the core and braid regions of the spanwise rollers as the convective Mach number increases from 0.28 to 0.62. A change in the entrainment mechanism with increasing compressibility is also indicated by signal intensity profiles and perspective views of the PLMS and PLIF images. These show that at Mc = 0.28 the instantaneous mixture fraction field typically exhibits a gradient in the streamwise direction, but is more uniform in the cross-stream direction. At Mc = 0.62 and 0.79, however, the mixture fraction field is more streamwise uniform and with a gradient in the cross-stream direction. This change in the composition of the structures is indicative of different entrainment motions at the different compressibility conditions. The statistical results are consistent with the qualitative observations and suggest that compressibility acts to reduce the magnitude of the mixture fraction fluctuations, particularly on the high-speed edge of the layer.

Green & Gerrard (1993) have presented in a recent paper the results of experiments to measure the distribution of vorticity in the near wake of a circular cylinder at low Reynolds numbers (up to Re = 220). They also compared the various definitions of the vortex formation region length which have been proposed by Gerrard (1966), Griffin (1974), and others for both high and low Reynolds numbers. The purpose of this note is to expand the work of Green & Gerrard, and to further their proposition that the end of the vortex formation region at all Reynolds numbers mark both the initial position of the fully shed vortex and the location at which its strength is a maximum. The agreement discussed here between several definitions for the formation region length will allow further understanding to be gained from investigations of the vortex wakes of stationary bluff bodies, and the wakes of oscillating bodies as well.

The unsteady nonlinear potential flow induced by a submerged line source or sink is studied by a vortex sheet method, both to trace the free surface evolution and to explore the possible existence of steady-state solutions. Only steady-state flows have been considered by other investigators, and these flows have been insensitive to whether they are generated by a source or sink, except with respect to the flow direction along the streamlines. The time-dependent solution permits an assessment of the stability of previously found steady solutions, and also reveals differences between source and sink flows: for the infinite-depth case, steady stagnation-point-type solutions are found both for source flows and sink flows, above the critical value reported by other investigators; finally, it is shown that streamline patterns of steady stagnation-point flows are identical for source and sink flows only in the limiting case of infinite depth.

In contrast to the well-known columnar convection mode in rapidly rotating spherical fluid systems, the viscous dissipation of the preferred convection mode at sufficiently small Prandtl number Pr takes place only in the Ekman boundary layer. It follows that different types of velocity boundary condition lead to totally different forms of the asymptotic relationship between the Rayleigh number R and the Ekman number E for the onset of convection. We extend both perturbation and numerical analyses with the stress-free boundary condition (Zhang 1994) in rapidly rotating spherical systems to those with the non-slip boundary condition. Complete analytical solutions – the critical parameters for the onset of convection and the corresponding flow and temperature structure – are obtained and a new asymptotic relation between R and E is derived. While an explicit solution of the Ekman boundary-layer problem can be avoided by constructing a proper surface integral in the case of the stress-free boundary problem, an explicit solution of the spherical Ekman boundary layer is required and then obtained to derive the solvability condition for the present problem. In the corresponding numerical analysis, velocity and temperature are expanded in terms of spherical harmonics and Chebychev functions. Accurate numerical solutions are obtained in the asymptotic regime of small E and Pr, and comparison between the analytical and numerical solutions is then made to demonstrate that a satisfactory quantitative agreement between the analytical and numerical analyses is reached.

An experimental and theoretical investigation of low Reynolds number, high subsonic Mach number, compressible gas flow in channels is presented. Nitrogen, helium, and argon gases were used. The channels were microfabricated on silicon wafers and were typically 100 μm wide, 104 μm long, and ranged in depth from 0.5 to 20 μm. The Knudsen number ranged from 10-3 to 0.4. The measured friction factor was in good agreement with theoretical predictions assuming isothermal, locally fully developed, first-order, slip flow.

A new asymptotic analysis of slender vortices in three dimensions, based solely on the vorticity transport equation and the non-local vorticity–velocity relation gives new insight into the structure of slender vortex filaments. The approach is quite different from earlier analyses using matched asymptotic solutions for the velocity field and it yields additional information. This insight is used to derive three different modifications of the thin-tube version of a numerical vortex element method. Our modifications remove an O(1) error from the node velocities of the standard thin-tube model and allow us to properly account for any prescribed physical vortex core structure independent of the numerical vorticity smoothing function. We demonstrate the performance of the improved models by comparison with asymptotic solutions for slender vortex rings and for perturbed slender vortex filaments in the Klein–Majda regime, in which the filament geometry is characterized by small-amplitude–short-wavelength displacements from a straight line. These comparisons represent a stringent mutual test for both the proposed modified thin-tube schemes and for the Klein–Majda theory. Importantly, we find a convincing agreement of numerical and asymptotic predictions for values of the Klein–Majda expansion parameter ε as large as ½. Thus, our results support their findings in earlier publications for realistic physical vortex core sizes.

Nonlinear kink instabilities of high-Reynolds-number supersonic shear layers have been studied using high-resolution computer simulations with the piecewise-parabolic-method (PPM). The transition region between the two fluids of the shear layer is spread out over many computational zones to avoid numerical effects introduced on the smallest lengthscales. Mach number, density contrast, and perturbation speed and amplitude were varied to study their effects on the growth of the kink instabilities. In response to a perturbing sound wave, a travelling kink mode grows in amplitude until enough of a disturbance on the shear layer has been created for it to roll up and rapidly grow in thickness. The time it takes for this rapid growth to be initiated is proportional to the initial shear-layer thickness and increases for increasing Mach number or decreasing perturbation amplitude. For equal density, Mach 4 shear layers, perturbed by a sound wave with a 2% amplitude at the travelling mode velocity, the growth time is τg = (546 ± 24) δ/c, where c is the sound speed and δ the half-width of the shear layer.

The nonlinear dynamics of nonlinear modulated cross-waves of resonant frequency ω1 and carrier frequency ω ≈ ω1 is investigated. In a long channel of width b, that contains fluid of depth d and which is subjected to a vertical oscillation of frequency 2ω, the wave can appear in solitary form. As has been shown previously, the solitary wave is only stable in a certain parameter regime; depending on damping and driving amplitudes the wave becomes unstable. The nonlinear development of the instabilities of solitary waves is the central problem of this paper. It is shown how instabilities are saturated following generic routes to chaos in time with spatially coherent structures. Finally, the case of time-modulated driving amplitudes is also considered. In most cases it appears that nonlinear waves of simple spatial structures take part in the nonlinear dynamics, but a few cases of spatial chaos are also reported.

The trapping of surface water waves by a thin plate in deep water is reduced to finding non-trivial solutions of a homogeneous, hypersingular integral equation for the discontinuity in velocity potential across the plate. The integral equation is discretized using an expansion-collocation method, involving Chebyshev polynomials of the second kind. A non-trivial solution to the problem is given by the vanishing of the determinant inherent in such a method. Results are given for inclined flat plates, and for curved plates that are symmetric with respect to a line drawn vertically through their centre. Comparisons with published results for horizontal flat plates (in water of finite depth) and for circular cylinders are made.

This paper is concerned with the effect of a weak spanwise-variable mean-flow distortion on the growth of oblique instability waves in a Blasius boundary layer. The streamwise component of the distortion velocity initially grows linearly with increasing streamwise distance, reaches a maximum, and eventually decays through the action of viscosity. This decay occurs slowly and allows the distortion to destabilize the Blasius flow over a relatively large streamwise region. It is shown that even relatively weak distortions can cause certain oblique Rayleigh instability waves to grow much faster than the usual two-dimensional Tollmien–Schlichting waves that would be the dominant instability modes in the absence of the distortion. The oblique instability waves can then become large enough to interact nonlinearly within a common critical layer. It is shown that the common amplitude of the interacting oblique waves is governed by the amplitude evolution equation derived in Goldstein & Choi (1989). The implications of these results for Klebanoff-type transition are discussed.