We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.

Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.

When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

We consider the motion of small-amplitude surface gravity waves over variable bathymetry. Although the governing equations of motion are linear, for general bathymetric variations they are non-separable and cannot be solved exactly. For slowly varying bathymetry, however, approximate solutions based on geometric (ray) techniques may be used. The ray equations are a set of coupled nonlinear ordinary differential equations with Hamiltonian form. It is argued that for general bathymetric variations, solutions to these equations - ray trajectories - should exhibit chaotic motion, i.e. extreme sensitivity to initial and environmental conditions. These ideas are illustrated using a simple model of bottom bathymetry, h(x,y) = h0(1 + εcos (2πx/L) cos (2πy/L)). The expectation of chaotic ray trajectories is confirmed via the construction of Poincaré sections and the calculation of Lyapunov exponents. The complexity of chaotic geometric wavefields is illustrated by considering the temporal evolution of (mostly) chaotic wavecrests. Some practical implications of chaotic ray trajectories are discussed.

A vortex pair in a viscous, incompressible fluid rises vertically toward a deformable free surface. The mathematical, description of this flow situation is a time-dependent nonlinear free-surface problem that has been solved numerically for a two-dimensional laminar flow with the aid of the Navier-Stokes equations by using boundary-fitted coordinates. For a number of selected flow parameters, results are presented on the decay of the primary vortices and their paths, the generation of surface vorticity and secondary vortices, the development and final stage of the disturbed free surface, and the influence of surface tension. High and low Froude numbers represent the two extremes of free-surface yielding and stiffness, respectively. For an intermediate Froude number, a special rebounding due to the presence of secondary vortices has been observed: the path of the primary vortex centre portrays a complete loop.

We analyse the stability of horizontally periodic, two-dimensional, finite-amplitude Kelvin-Helmholtz billows with respect to infinitesimal three-dimensional perturbations having the same streamwise wavelength for several different levels of the initial density stratification. A complete analysis of the energy budget for this class of secondary instabilities establishes that the contribution to their growth from shear conversion of the basic-state kinetic energy is relatively insensitive to the strength of the stratification over the range of values considered, suggesting that dynamical shear instability constitutes the basic underlying mechanism. Indeed, during the initial stages of their growth, secondary instabilities derive their energy predominantly from shear conversion. However, for initial Richardson numbers between 0.065 and 0.13, the primary source of kinetic energy for secondary instabilities at the time the parent wave climaxes is in fact the conversion of potential energy by convective overturning in the cores of the individual billows. A comparison between the secondary instability properties of unstratified Kelvin-Helmholtz billows and Stuart vortices is made, as the latter have often been assumed to provide an adequate approximation to the former. Our analyses suggest that the Stuart vortex model has several shortcomings in this regard.

The instantaneous velocity distribution in trailing vortices generated by lifting hydrofoils has been measured in the Low Turbulence Water Tunnel at the California Institute of Technology. Two different rectangular planform hydrofoils with small aspect ratios were tested. Double-pulsed holography of injected microbubbles, which act much as Lagrangian flow tracers, was used to determine instantaneous axial and tangential velocities. Measurements were made at various free-stream velocities, angles of attack, and downstream distances. The vortex core mean axial velocity is consistently greater than the free-stream velocity near the hydrofoil trailing edge, and decreases with downstream distance. The mean axial velocity is strongly Reynolds-number dependent.

Axial flow in the trailing vortex is highly unsteady for all the flow conditions studied; peak-to-peak fluctuations on the centreline as large as the free-stream velocity have been observed. The amplitude of these fluctuations falls rapidly with increasing distance from the centreline. For an angle of attack of 10° the fluctuations consist of both ‘fast’ and ‘slow’ components, whereas for α = 5° only ‘fast’ fluctuations have been observed. Peak decelerations of the centreline fluid occur with amplitude comparable to the maximum centripetal acceleration around the centreline. Certain unusual structures of the vortex core - regions in which the flow direction quickly diverges from the free-stream direction, and then equally quickly recovers - have been labelled ‘vortex kinks.’

A boundary-integral method is developed for computing first-order and mean second-order wave forces on floating bodies with small forward speed in three dimensions. The method is based on applying Green's theorem and linearizing the Green function and velocity potential in the forward speed. The velocity potential on the wetted body surface is then given as the solution of two sets of integral equations with unknowns only on the body. The equations contain no water-line integral, and the free-surface integral decays rapidly. The Timman-Newman symmetry relations for the added mass and damping coefficients are extended to the case when the double-body flow around the body is included in the free-surface condition. The linear wave exciting forces are found both by pressure integration and by a generalized far-field form of the Haskind relations. The mean drift force is found by far-field analysis. All the derivations are made for an arbitrary wave heading. A boundary-element program utilizing the new method has been developed. Numerical results and convergence tests are presented for several body geometries. It is found that the wave exciting forces and the mean drift forces are most influenced by a small forward speed. Values of the wave drift damping coefficient are computed. It is found that interference phenomena may lead to negative wave drift damping for bodies of complicated shape.

The motion of subharmonic resonant modes of surface waves in a rectangular container subjected to vertical periodic oscillation is studied based on the weakly nonlinear model equations derived by both the average Lagrangian and the two-timescale method. Explicit estimates of the nonlinearity of some specific modes are given. The bifurcations of stationary states including a Hopf bifurcation are examined. Numerical calculations of the dissipative dynamical equations show periodic and chaotic attractors. Theoretical parameter-space diagrams and numerical results are compared in detail with Simonelli & Gollub's (1989) surface-wave modecompetition experiments. It is shown that the average Hamiltonian system for the present 2:1:1 external-internal resonance with suitable coefficients has homoclinic chaos, which was mathematically proven by Holmes (1986) for the specific case of 2:1:2 external-internal resonance.

Visual observations are made on the flow around a sphere which is half-submerged in still water and forced to oscillate up and down. It is found that three kinds of steady flows are generated in the water: the surface flow, the undersurface flow, and the vertical jet flow. The surface flow is produced in the very thin layer on the water surface. The undersurface flow is induced in the thick layer under the water surface. The vertical jet flow is ejected vertically from the bottom of the sphere. The surface flow and the undersurface flow are produced only when petal-shaped waves are generated on the water surface. The velocity of the undersurface flow is much smaller than that of the surface flow.

We study the mixing of fluid in a viscously decaying row of point vortices. To this end, we employ a simplified model based on Stuart's (1967) one-parameter family of solutions to the steady Euler equations. Our approach relates the free parameter to a vortex core size, which grows in time according to the exact solution of the Navier-Stokes equations for an isolated vortex. In this way, we approach an exact solution for small values of t/Re. We investigate how the growing core size leads to a shrinking of the cat's eye and hence to fluid leaking out of the trapped region into the free streams. In particular, we observe that particles initially located close to each other in neighbouring intervals along the streamwise direction escape from the cat's eye near opposite ends. The size of these intervals scales with the inverse square root of the Reynolds number. We furthermore examine the particle escape times and observe a self-similar blow-up for the particles near the border between two adjacent intervals. This can be explained on the basis of a simple stagnation-point flow. An investigation of interface generation shows that viscosity leads to an additional factor proportional to time in the growth rates. Numerical simulations confirm the above results and give a detailed picture of the underlying mixing processes.

The modelling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - appears to yield improved results over the Launder, Reece & Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in five independent homogeneous turbulent flows. However, some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modelling of the pressure–strain correlation. Possible alternative approaches are discussed briefly.

This paper treats a liquid-metal flow through a sharp elbow connecting two constant-area, rectangular ducts with thin metal walls. There is a uniform, strong magnetic field in the plane of the ducts’ centrelines, and the velocity component normal to the magnetic field is in opposite directions upstream and downstream of the elbow. The magnetic field is sufficiently strong that inertial effects are negligible everywhere and viscous effects are confined to boundary layers and to an interior layer lying along the magnetic field lines through the inside corner of the elbow. The interior layer involves large velocities parallel to the magnetic field and carries roughly half of the flow between the upstream and downstream ducts for the case considered.

A simple, phenomenological model is proposed for the formation of spanwise cells behind slender bodies of revolution in crosswise, uniform or non-uniform oncoming flow. The model yields estimates for the position of the cells, their frequencies, their amplitudes of oscillation along the span, and the local shedding angle. The qualitative features of the solutions of this theory agree well with experiments. A quantitative comparison with experiments for a slender cone is presented.

A spatially developing countercurrent mixing layer was established experimentally by applying suction to the periphery of an axisymmetric jet. A laminar mixing region was studied in detail for a velocity ratio R = ΔU/2U between 1 and 1.5, where ΔU describes the intensity of the shear across the layer and U is the average speed of the two streams. Above a critical velocity ratio Rr = 1.32 the shear layer displays energetic oscillations at a discrete frequency which are the result of very organized axisymmetric vortex structures in the mixing layer. The spatial order of the primary vortices inhibits the pairing process and dramatically alters the spatial development of the shear layer downstream. Consequently, the turbulence level in the jet core is significantly reduced, as is the decay rate of the mean velocity on the jet centreline. The response of the shear layer to controlled external forcing indicates that the shear layer oscillations at supercritical velocity ratios are self-excited. The experimentally determined critical velocity ratio of 1.32, established for very thin axisymmetric shear layers, compares favourably with the theoretically predicted value of 1.315 for the transition from convective to absolute instability in plane mixing layers (Huerre & Monkewitz 1985).

This paper examines the sound waves generated when a spherical water drop impacts upon an initially quiescent water surface. The pressure fluctuations and the acoustic energy radiated by the initial impact are calculated analytically. It is shown that the rapid momentum exchange between the fluid in the falling drop and that in the main water body causes the radiation of compressive waves. These waves are radiated in the form of a wave packet with a densely packed edge which is heard in the far field as a noisy shock-like pulse followed by a quickly decreasing tail. The wave packet carries with it sound energy proportional to the kinetic energy of the falling drop and to the cube of the impact Mach number. Applications of these analytic results to the study of noise from natural rain are discussed, and an illustrative example is given where the noise level due to rain showers is linearly related to the rainfall rate, which is shown to be consistent with observations.

Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

The ignition of lean H2/air mixtures under microgravity (μg) conditions can lead to the formation of spherical premixed flames (flame balls) with small Péclet number (Pe). A central question concerning these structures is the existence of appropriate stationary stable solutions of the combustion equations. In this paper we examine an individual flame ball that is suspended in a fluid whose velocity far from the flame is steady and varies linearly in space. Detailed results are obtained for simple shear flows and simple straining flows, both axisymmetric and plane.

Convection enhances the flux of heat from the flame and the flux of mixture to the flame, but because the Lewis number (Le) is less than unity the relative impact on the former is greater than on the latter. Consequently, there is a net loss of energy from the flame to the far field, and if large enough this will quench the flame. For values of shear or strain less than the quenching value there are two possible stationary solutions, but one of these is unstable to spherically symmetric disturbances of the flame ball. The radius of the other solution is unbounded as Pe goes to zero. Examination of a class of three-dimensional disturbances reveals no additional instability when the energy losses are due only to convection, but sufficiently large flame balls are unstable when volumetric heat losses from radiation are accounted for. This last result is in agreement with previous results that have been obtained for zero Pe, albeit with inadequate accounting for the flow field generated by the perturbations.

A three-dimensional asymptotic analysis of the oscillations of electrically charged drops in an external electric field is carried out by means of the multiple-parameter perturbation method. The mathematical framework allows separate treatments of the quiescent deformation due to the electric field and the oscillatory motions caused by other physical factors. Without oscillations, the solution for the quiescent drop shape exhibits a prolate deformation with a slight asymmetry about the drop's equatorial plane. This axisymmetric quiescent deformation of the equilibrium drop shape is shown to modify the oscillation characteristics of axisymmetric as well as asymmetric modes. The expression of the characteristic frequency modification is derived for the oscillation modes, manifesting fine structure in the frequency spectrum so the degeneracy of Rayleigh's normal modes for charged drops is removed in the presence of an electric field. Physical reasoning indicates that the degeneracy of the oscillation modes is associated with the spherical symmetry of the system, so the removal of the degeneracy may be regarded as a consequence of the symmetry breaking caused by the electric field. In addition, the small-amplitude oscillation mode shapes are also modified as a result of the coupling between the oscillatory motions and the electric field as well as the quiescent deformation.

The upstream influence in an inviscid two-dimensional shear flow around a semicircular ‘cape’ (radius A) is computed using a piecewise uniform vorticity model of a boundary-layer current. The area of this layer upstream from the cape increases as the square root of time t when A is small, and increases as t for larger A. Complete blocking occurs when A is approximately three times the boundary-layer thickness, in which case all oncoming particles accumulate in a large upstream vortex. The numerical results obtained from the contour dynamical method also show the generation of large eddies downstream from the obstacle.

It is shown that the dilatational terms that need to be modelled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of the compressible velocity field, which generates these dilatational terms, is explored by asymptotic analysis of the compressible Navier-Stokes equations in the case of homogeneous turbulence. The lowest-order equations for the compressible field are solved and explicit expressions for some of the associated one-point moments are obtained. For low Mach numbers, the compressible mode has a fast timescale relative to the incompressible mode. Therefore, it is proposed that, in moderate Mach number homogeneous turbulence, the compressible component of the turbulence is in quasi-equilibrium with respect to the incompressible turbulence. A non-dimensional parameter which characterizes this equilibrium structure of the compressible mode is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.