Laboratory experiments on barotropic vortices in a rotating fluid revealed that the instability behaviour of cyclonic and anticyclonic vortices is remarkably different. Depending on its initial vorticity distribution, the cyclonic vortex has in a number of experiments been observed to be unstable to wavenumber-2 perturbations, leading to the gradual formation of a stable tripolar vortex structure. This tripole consists of an elongated cyclonic core vortex adjoined by two anticyclonic satellite vortices.

In contrast, the anticyclonic vortex shows a rather explosive instability behaviour, in the sense that it is observed to immediately split up into two dipoles. Under somewhat different circumstances the higher-order mode-3 instability is observed, in which the anticyclonic core has a triangular shape, with three smaller cyclonic satellite vortices at its sides.

A modified version of Rayleigh's instability criterion offers a qualitative explanation for this apparent difference between unstable cyclonic and anticyclonic vortices.

A liquid film flowing down an inclined heated plane subject to surface wave and thermocapillary instabilities is studied. Three mechanisms exist by which energy can be transferred to the disturbance. Two of these mechanisms are associated with the thermocapillary forces and one with the shear stress of the basic flow at the deformed free surface. Depending on which mechanism is dominant, the instability can assume the form of either long transverse waves or short longitudinal rolls.

An earlier derivation (Miles 1990a) of the partial differential equation for the complex amplitude of a gravity–capillary wave in a shallow, viscous liquid of variable depth and fixed contact line is extended to accommodate a meniscus with a moving contact line at which the slope of the meniscus is assumed to be proportional to (but not necessarily in phase with) the velocity. The motion of the contact line implies capillary dissipation, which is absent for a fixed contact line. The results are applied to the normal reflection of a wave incident from a region of uniform depth on a beach of uniform slope. The reflection coefficient has the form R = R1RνRc, where R1 is the coefficient for an ideal fluid, and Rν and Rc comprise the respective effects of viscosity and capillarity.

The motion of the free surface of a viscous droplet is investigated. By using lubrication theory a model is developed for the motion of the free surface which includes both the effect of slip and the dependence of the contact angle on the slip velocity. We solve the resulting nonlinear partial differential equation in several ways. First we investigate the initial motion of the drop at a non-equilibrium contact angle using the method of matched asymptotics. Then we develop a pseudo-spectral method to numerically solve the full nonlinear system. The dependence of the spreading rate of the drop on the various physical parameters and for different slip models is determined.

The splitting of a smooth gradient into mixed layers separated by thin interfaces is studied in a constant stable solute gradient heated uniformly from below by raising the temperature at its bottom to a prescribed value above the ambient. The phenomenon under study is characterized by the development of a first mixed layer from the bottom up to a certain height; it is shown that its instantaneous thickness is proportional to the thermal expansion coefficient and the instantaneous temperature difference between the bottom heated plate and the undisturbed far fluid and is inversely proportional to the solute gradient. A second layer starts its formation during the growth of the first one and persists above the bottom layer notwithstanding the advancing interface below. Further layer formation occurs, depending on the prescribed Rayleigh number of the system (which is based on the vertical temperature difference and the initial solute gradient). The secondary layers grow with time and the final thickness of all the layers (except the bottom one) is almost equal. Their average final thickness is found to depend mostly on the solute gradient and the fluid properties. A final quasi-steady state is reached in which the depth of the whole convective region is proportional to the expansion coefficient and the vertical temperature difference and is inversely proportional to the initial solute gradient

The structure of the far-field turbulent region of an incompressible free jet developing downstream of an axisymmetric nozzle is studied by means of the Reynolds time-averaged equations. The analysis employs the method of matched asymptotic expansions, with limit-process expansions developed in the limit of large Reynolds number. The analysis reveals the existence, far downstream of the nozzle exit, of a turbulent core region, an irrotational exterior region, and a distinguished intermediate region. Self-similar formulations are sought for all three regions in terms of appropriate independent and dependent variables. The stress- and pressure-function solutions for the exterior region, unlike the mean-velocity solutions, represent new information on the far-field flow behaviour. The analytical results of the centreline decay of the mean axial velocity and those of the radial distributions of the axial and radial mean-velocity components and the shear- and normal-stress components are compared with available experimental data.

Simulated upwelling fronts have been generated around the outer edge of a cylindrical tank filled with a two-layer fluid system and driven by a surface stress. Initially, an axisymmetric front was observed which subsequently became unstable to small baroclinic eddies. These eddies continued to grow until they reached an equilibrium size. Under some circumstances, cyclonic eddies pinched-off from the fully developed front and moved away from the mean position of the front into the fluid interior. Streak photographs of the fully developed flow field were digitized to generate a velocity field interpolated on to a regular grid. A direct two-dimensional Fourier transform was performed on the turbulent kinetic energy field deduced from such images and one-dimensional energy E(k) spectra were extracted. Consistent $k^{-\frac{5}{3}}$ energy spectra were found at lower wavenumber, k and approximately k−5.5 spectra at higher k. In any given experiment, the two spectral slopes meet close to a wavenumber kw = 2π/λw (where λw is the mean diameter of a frontal eddy and kw is the associated wavenumber). According to classical theories, kw would be the input wavenumber, and the range of k with a $k^{-\frac{5}{3}}$ spectrum would correspond to an inverse energy cascade range; this yielded a Kolmogorov constant (C) that varied within the limits 2.8 [les ] C [les ] 3.8. The approximately k−5.5 range, which is much steeper than that predicted by the original statistical theories, is nevertheless consistent with those found frequently in numerical experiments.

The spectral slopes inferred from particle dispersion methods and from one-dimensional Fourier transforms of the longitudinal velocity correlations were compared with the results obtained above and in previous laboratory experiments. In general, the global energy spectra are consistent with an interpretation of the fluid dynamics as being that of two-dimensional turbulence. This in turn implies that known properties of such flows may be invoked to explain the appearance of a number of naturally occurring phenomena in coastal upwelling fronts.

A new approach for investigating two-layer hydraulic exchange flows in channels is introduced. The approach is based on the functional formalism of Gill (1977) and applied to the flow through a contraction in width and to flow over a simple sill. The sill geometry is an extension of that looked at by earlier workers, in particular Farmer & Armi (1986) who used a Froude-number-plane approach. In the present paper a simple relationship between the composite Froude number and the hydraulic functional is derived, though the functional approach may also be applied to channels where a Froude number is not readily defined. The ability to trace roots of this functional from one reservoir to the other is a prerequisite for the flow to be realizable. Two hydraulic transitions are required for the flow to be fully controlled and the exchange flow rate to be maximal. If only one hydraulic transition is present, the flow is governed by the conditions in one of the reservoirs and the exchange flow rate is found to be submaximal. The flow along a channel is found to be very sensitive to small departures from symmetry about a horizontal plane. The response of the interface to the introduction of a net (barotropic) flow is found to be a discontinuous function of the strength of the forcing for some range of sill heights.

The effect of rotation on mixing across a density interface is studied experimentally in a two-layer stratified fluid. Mixing is caused by turbulence produced in one of the layers by an oscillating grid. The flow depends on the Richardson number Ri = g′l/u2 and the Rossby number Ro = u/2Ωl. The most important result is the observed decrease of the entrainment rate E in the presence of rotation, when compared with non-rotating experiments. In a certain range of the two parameters, a general entrainment law in the form E = 0.5RoRi−1 is established, whereas the entrainment law in non-rotating conditions is $E = 1.6 Ri^{-\frac{3}{2}}$. Additional information concerning the dynamics of the interface in rotating conditions is provided by interface displacement spectra, showing that rotation favours low-frequency oscillations of the interface, whereas high-frequency oscillations are not modified by rotation. Finally, the role of inertial waves is discussed on the basis of velocity measurements in the non-stirred layer.

A new theoretical approach is presented, which determines the entrainment laws with and without background rotation in a two-layer stratified fluid with one layer stirred by turbulent motions. The model gives the entrainment coefficient E as a function of the local turbulent Richardson number Ri = g′l/u2, the Rossby number Ro = u/fl and the Péelet number Pe = ul/κ. The following entrainment laws are obtained: (i) without rotation and for a high Péclet number: $E \propto Ri^{-\frac{3}{2}}$; (ii) without rotation and for a low to moderate Péclet number: $E \propto Pe^{-\frac{1}{3}}Ri^{-1}$ (iii) with rotation and for a high Péclet number: E ∝ RoRi−l. These entrainment laws are consistent with experiments for the three different cases. The model relies to a great extent on the spectral distribution of interface oscillations measured in experiments. Comparison is made with experiments and with earlier models of entrainment.

The influence of small topographic features on the dynamic responses of an inviscid, stratified, f-plane ocean to a time-dependent upstream flow is studied. Based on the quasi-geostrophic vorticity equation we analyse a quasi-nonlinear approach, in which the horizontal currents in the advection terms are replaced by the upstream flow. It appears that the quasi-nonlinear theory is susceptible to analytical treatment and contains the essential dynamics to give a sufficient description of the response scenario. As examples, infinitely long ridges and a right-circular cylinder are considered. In the case of an infinitely long top-hat ridge closed-form expressions can be derived. In the case of a cylindrical obstable the theory gives explicit indications of which term corresponds to which of the processes involved, such as topographic waves, which are generated in the starting phase and move clockwise round the obstacle, vortex shedding, and formation of a vortex over the cylinder in the final steady state.

This paper is a report on the mechanism of turbulent momentum transport normal to the wall in a turbulent wall-bounded flow. The objective of this study is to examine the ‘background’ turbulent flow field as a first step toward understanding suspended sediment transport. Specifically, the hypothesis that fine grain particles can be kept in suspension through a net upward momentum flux in a turbulent boundary layer is examined. The net momentum flux can arise if the probability density distribution of the fluctuating v–signal is positively skewed; i.e. the positive v–fluctuations are predominantly of large amplitude and short duration while the negative v-fluctuations are of small amplitude and long duration. High-resolution, two-component laser-doppler anemometer measurements of the v-velocity component in a fully developed turbulent water channel flow were examined spanning a Reynolds number range of 3000 to 23000. Averages of these signals demonstrate that, for very small particles, there is net upward momentum flux in the range y+ > 30, while there is a net downward momentum flux in the range 10 [les ] y+ [les ] 30. Preliminary results which categorize the normal velocity according to quadrants of motion are also included.

This paper presents experimental results on the nonlinear phase locking present in the resonant growth of unstable modes in the shear layer of an axisymmetric jet. The initial instability modes scale with the exiting shear layer and grow convectively with downstream distance. Because of the special condition at the exit lip of the jet, the initial growth of modes is very sensitive to local unsteady pressure fields. A part of the unsteady field is stochastic in nature. To a larger extent, the pressure field at the lip of the jet contains the imprint of the downstream-developing instability modes, in particular the first unstable axisymmetric mode and its subharmonic. These are felt at the lip of the jet as a result of the energetic processes of the first vortex rollup and vortex pairing. As a result, a resonant feedback exists which under special conditions makes the initial region of this flow in some sense absolutely unstable. The features of this process are brought out by the normalized crossbispectrum or cross-bicoherence between the instantaneous unsteady pressure at the lip of the jet and velocity time series measured at the same azimuthal position for different downstream locations. These give a measure of the nonlinear phase locking between the principle modes and their sum and difference modes. Analysis of these show a perfect nonlinear phase locking at the fundamental axisymmetric and subharmonic frequencies between the pressure field at the lip and the velocity field at the downstream locations corresponding to the energy saturations of the fundamental and subharmonic modes. This resonance process can be suppressed or enhanced by low-amplitude axisymmetric mode forcing at the natural preferred frequency of slightly detuned cases. Contrasted to this is the behaviour of the fundamental m = ± 1 helical mode. This mode was found to have the same spatial growth rate as the axisymmetric mode and a streamwise frequency approximately 20 % higher, in agreement with theoretical predictions. However, short-time spectral estimates showed that these two fundamental modes do not exist at the same time or space. This suggests that each is a basin of attraction which suppresses the existence of the other. The apparent non-deterministic switching observed between these modes is probably the result of the response of the jet to stochastic input of axisymmetric or non-axisymmetric disturbances. This scenario may lead to a low-dimensional temporal model based on the interaction between these two modes which captures most of the early random nature seen in our experiments.

A thin fluid drop is at rest on a plane vertical surface, supported against gravity by surface tension. The perimeter of the drop is required to lie on a given closed curve, upon which the contact angle is arbitrary. If the drop is of sufficient volume, it can wet the whole area interior to this curve. However, for any given curve, there is a certain critical volume below which this fully wetted configuration is not physically acceptable, the formal solution having negative thickness. It is suggested here as an alternative that the upper portion of the drop, above a free boundary to be determined, must drain completely. Some time-dependent computations in two dimensions are presented to illustrate this draining property. In three dimensions, the static free boundary has zero contact angle, and must be determined as part of the solution. An example solved here is that where the original boundary is a circle, and the free boundary is a non-trivial curve lying inside it, whose shape is found by numerical methods. This problem also has relevance to the shape of a raindrop on a windowpane where surface contamination prevents contact-line motion, and the drop may again be considered to be confined within a prescribed boundary.

The present study provides a detailed description of the dissipative structure of shock waves propagating in dense gases which have relatively large specific heats. The flows of interest are governed by the usual Navier–Stokes equations supplemented by realistic equations of state and realistic models for the density dependence of the viscosity and thermal conductivity. New results include the first computation of the structure of finite-amplitude expansion shocks and examples of shock waves in which the thickness increases, rather than decreases, with strength. A new phenomenon, referred to as impending shock splitting, is also reported.

An equation, derived by Phillips (1984) to describe the variation of a wind-wave spectrum in the presence of a current, is shown to have an exact solution. For certain choices of current, the results are shown to simplify even further.

Sound generation by the Lighthill quadrupole is an important mechanism in the noise of supersonic and transonic propellers. Full numerical calculation of the quadrupole radiation must, however, require knowledge of the flow at all points exterior to the blades (involving transonic aerodynamics) and the evaluation of special functions. We describe how these difficulties may be largely avoided, using an asymptotic approximation that the number of blades, B, is large, and prove that to leading order the problem of the radiation in a given direction reduces to one of determining the (two-dimensional) flow field at just one radial station, legitimately achieved by linearized supersonic analysis. Simple formulae are derived for the far-field acoustic pressure generated by unswept blades, from which absolute level predictions can be made accurately and quickly. These formulae predict a significantly greater intensity, over broad angular ranges, than is predicted by the linear theory for thickness noise sources.

Two-dimensional cusped interfaces are line singularities of curvature. We create such cusps by rotating a cylinder half immersed in liquid. A liquid film is dragged out of the reservoir on one side and is plunged in at the other, where it forms a cusp at finite speeds, if the conditions are right. Both Newtonian and non-Newtonian fluids form cusps, but the transition from a rounded interface to a cusp is gradual in Newtonian liquids and sudden in non-Newtonian liquids. We present an asymptotic analysis near the cusp tip for the case of zero surface tension, and we make some remarks about the effects of a small surface tension. We also present the results of numerical simulations showing the development of a cusp. In those simulations, the fluid is filling an initially rectangular domain with a free surface on top. The fluid enters from both sides and is sucked out through a hole in the bottom.

The steady-state and time-dependent flow transitions observed in a well-characterized viscoelastic fluid flowing through an abrupt axisymmetric contraction are characterized in terms of the Deborah number and contraction ratio by laser-Doppler velocimetry and flow visualization measurements. A sequence of flow transitions are identified that lead to time-periodic, quasi-periodic and aperiodic dynamics near the lip of the contraction and to the formation of an elastic vortex at the lip entrance. This lip vortex increases in intensity and expands outwards into the upstream tube as the Deborah number is increased, until a further flow instability leads to unsteady oscillations of the large elastic vortex. The values of the critical Deborah number for the onset of each of these transitions depends on the contraction ratio β, defined as the ratio of the radii of the large and small tubes. Time-dependent, three-dimensional flow near the contraction lip is observed only for contraction ratios 2 [les ] β [les ] 5, and the flow remains steady for higher contraction ratios. Rounding the corner of the 4:1 abrupt contraction leads to increased values of Deborah number for the onset of these flow transitions, but does not change the general structure of the transitions.

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, Lapwood convection in a two-dimensional saturated porous cavity heated from below. The focus of the study concerns the destabilization, through symmetry-preserving Hopf bifurcations, of the various stable convective flow patterns that can form in a rectangular cavity. We show how the limits of stability of steady convection in a porous medium can be determined by bifurcation techniques that locate Hopf bifurcations, and we predict a surprisingly complex evolution of the Hopf bifurcation along the unicellular branch as the aspect ratio varies. The continuation methods that we adopt reveal interactions of Hopf bifurcations with limit points that signal complicated dynamical behaviour for certain container sizes. The study demonstrates the role of Hopf bifurcation in destabilizing completely the unicellular flow at aspect ratios greater than 2.691. A simple relationship between symmetry-preserving Hopf bifurcations from the alternative steady flows is also derived, and used to define upper limits on the stability of the alternative steady flows and the thresholds for oscillatory convection as a function of aspect ratio.