A perturbation scheme is constructed to describe the evolution of stable, localized Rankine-type hydrodynamic vortices under the action of disturbances such as density stratification. It is based on the elimination of singularities in perturbations by using the necessary orthogonality conditions which determine the vortex motion. Along with the discrete-spectrum modes of the linearized problem which can be kept finite by imposing the orthogonality conditions, the continuous-spectrum perturbations play a crucial role. It is shown that in a stratified fluid, a single (monopole) vortex can be destroyed due to the latter modes before it drifts very far, whereas a vortex pair preserves its stability for a longer time. The motion of the latter is studied in two cases: smooth stratification and a density jump. For the motion of a pair under a small angle to the interface, a complete description is given in the framework of our theory, including the effect of reflection of the pair from a region with slightly larger density.

The possibility of acoustic control of the two-dimensional instabilities of a lossless plane shear layer of vanishing thickness is studied. The shear layer is formed from a body of incompressible fluid sliding over another fluid at rest. It is unstable through the generation of Kelvin–Helmholtz waves. We consider the possibility of adding to this linearly unstable flow a simple source, driven in such a way that its field interferes destructively with the instability to render the flow stable. The required strength of the unsteady control source is determined in terms of the fluctuating velocity at some fixed position in the moving fluid. We show that no unstable Kelvin–Helmholtz wave could survive the action of such a source. Next, we examine the scope for constructing the control signal from a measurement of the flow velocity at some fixed position. The source is a linear functional of the monitored velocity and we give the transfer function that would be required for the instabilities to be controlled. We prove that such control action would completely stabilize the otherwise unstable vortex sheet, and that other alternative sensor/actuator arrangements could also be effective. We go on to show that our particular very simple arrangement could not in fact be realized because, if required to work at all frequencies, it would not be causal. If we insisted on causality the vortex sheet would then only be stabilized over most frequencies. That would of course make the controlled flow completely different from the vortex sheet whose instabilities are so well known – and troublesome. We conjecture that there will exist some variations of the basic control arrangement described here that are both physically realizable and effective over the required frequency range. From our study of the initial value problem we have concluded that short perturbations would be attenuated very rapidly.

Turbulent flow over idealized water waves with varying wave slope ak and wave age c/u∗ is investigated using direct numerical simulations at a bulk Reynolds number Re = 8000. In the present idealization, the shape of the water wave and the associated orbital velocities are prescribed and do not evolve dynamically under the action of the wind. The results show that the imposed waves significantly influence the mean flow, vertical momentum fluxes, velocity variances, pressure, and form stress (drag). Compared to a stationary wave, slow (fast) moving waves increase (decrease) the form stress. At small c/u∗, waves act similarly to increasing surface roughness zo resulting in mean vertical velocity profiles with shorter buffer and longer logarithmic regions. With increasing wave age, zo decreases so that the wavy lower surface is nearly as smooth as a flat lower boundary. Vertical profiles of turbulence statistics show that the wave effects depend on wave age and wave slope but are confined to a region kz < 1 (where k is the wavenumber of the surface undulation and z is the vertical coordinate). The turbulent momentum flux can be altered by as much as 40% by the waves. A region of closed streamlines (or cat's-eye pattern) centred about the critical layer height was found to be dynamically important at low to moderate values of c/u∗. The wave-correlated velocity and flux fields are strongly dependent on the variation of the critical layer height and to a lesser extent the surface orbital velocities. Above the critical layer zcr the positions of the maximum and minimum wave-correlated vertical velocity ww occur upwind and downwind of the peak in zcr, like a stationary surface. The wave-correlated flux uwww is positive (negative) above (below) the critical layer height.

The generation of Reynolds stress, turbulent kinetic energy and dissipation in the turbulent boundary layer simulation of Spalart (1988) is studied using the invariants of the velocity gradient tensor. This technique enables the study of the whole range of scales in the flow using a single unified approach. In addition, it also provides a rational basis for relating the flow structure in physical space to an appropriate statistical measure in the space of invariants. The general characteristics of the turbulent motion are analysed using a combination of computer-based visualization of flow variables together with joint probability distributions of the invariants. The quantities studied are of direct interest in the development of turbulence models. The cubic discriminant of the velocity gradient tensor provides a useful marker for distinguishing regions of active and passive turbulence. It is found that the strongest Reynolds-stress and turbulent-kinetic-energy generating events occur where the discriminant has a rapid change of sign. Finally, the time evolution of the invariants is studied by computing along particle paths in a Lagrangian frame of reference. It is found that the invariants tend to evolve toward two distinct asymptotes in the plane of invariants. Several simplified models for the evolution of the velocity gradient tensor are described. These models compare well with several of the important features observed in the Lagrangian computation. The picture of the turbulent boundary layer which emerges is consistent with the ideas of Townsend (1956) and with the physical picture of turbulent structure set forth by Theodorsen (1955).

In order to gain insight into the hydraulics of rotating-channel flow, a set of initial-value problems analogous to Long's towing experiments is considered. Specifically, we calculate the adjustment caused by the introduction of a stationary obstacle into a steady, single-layer flow in a rotating channel of infinite length. Using the semigeostrophic approximation and the assumption of uniform potential vorticity, we predict the critical obstacle height above which upstream influence occurs. This height is a function of the initial Froude number, the ratio of the channel width to an appropriately defined Rossby radius of deformation, and a third parameter governing how the initial volume flux in sidewall boundary layers is partitioned. (In all cases, the latter is held to a fixed value specifying zero flow in the right-hand (facing downstream) boundary layer.) The temporal development of the flow according to the full, two-dimensional shallow water equations is calculated numerically, revealing numerous interesting features such as upstream-propagating shocks and separated rarefying intrusions, downstream hydraulic jumps in both depth and stream width, flow separation, and two types of recirculations. The semigeostrophic prediction of the critical obstacle height proves accurate for relatively narrow channels and moderately accurate for wide channels. Significantly, we find that contact with the left-hand wall (facing downstream) is crucial to most of the interesting and important features. For example, no instances are found of hydraulic control of flow that is separated from the left-hand wall at the sill, despite the fact that such states have been predicted by previous semigeostrophic theories. The calculations result in a series of regime diagrams that should be very helpful for investigators who wish to gain insight into rotating, hydraulically driven flow.

The process by which a liquid jet falling into a liquid pool entrains air is studied experimentally and theoretically. It is shown that, provided the nozzle from which the jet issues is properly contoured, an undisturbed jet does not entrap air even at relatively high Reynolds numbers. When surface disturbances are generated on the jet by a rapid increase of the liquid flow rate, on the other hand, large air cavities are formed. Their collapse under the action of gravity causes the entrapment of bubbles in the liquid. This sequence of events is recorded with a CCD and a high-speed camera. A boundary-integral method is used to simulate the process numerically with results in good agreement with the observations. An unexpected finding is that the role of the jet is not simply that of conveying the disturbance to the pool surface. Rather, both the observed energy budget and the simulations imply the presence of a mechanism by which part of the jet energy is used in creating the cavity. A hypothesis on the nature of this mechanism is presented.

A linear stability analysis of the mixing layer in the presence of fibre additives is presented. Using a formulation based on moments of the probability distribution function to determine the particle orientation, we extend the classical linear stability theory and derive a modified Orr–Sommerfeld equation. It is found that, for large Reynolds numbers, the flow instability is governed by two parameters: a dimensionless group H, analogous to a reciprocal Reynolds number representing the importance of inertial forces versus viscous forces associated with the anisotropic elongational viscosity of the suspension; and a coefficient CI that accounts for inter-particle hydrodynamic interactions. A parametric study reveals that both parameters can induce an important attenuation of the flow instability. Furthermore, we show that the stabilizing effects arise from the orientation diffusion due to hydrodynamic interactions, and not from the anisotropy induced by the presence of fibres in the flow, as speculated before. The examination of profile contours of different perturbation terms and the analysis of the rate of production of enstrophy show clearly that the main factor behind the reduction of the flow instability is associated with the fibre shear stress disturbance. This disturbance acts as a dissipative term as the fibres, due to the orientational diffusivity arising from hydrodynamic interactions, deviate from the fully aligned anisotropic orientation. On the other hand, fibre normal stresses act as a destabilizing factor and are important only in the absence of hydrodynamic interactions.

We study the linear stability of thermocapillary-driven convection in a planar unbounded layer of an electrically conducting low-Prandtl-number liquid heated from the side and subjected to a transverse magnetic field. The thresholds of convective instability for both longitudinal and oblique disturbances are calculated numerically and also asymptotically by considering the Hartmann and Prandtl numbers as large and small parameters, respectively. The magnetic field has a stabilizing effect on the flow with the critical temperature gradient for the transition from steady to oscillatory convection increasing as square of the field strength, as also does the critical frequency, while the critical wavelength reduces inversely with field strength. These asymptotics develop in a strong enough magnetic field when the instability is entirely due to the jet of the base flow confined in the Hartmann layer at the free surface. In contrast to the base flow, the critical disturbances, having a long wavelength at small Prandtl numbers, extend from the free surface into the bulk of the liquid layer over a distance exceeding the thickness of the Hartmann layer by a factor O(Pr−1/2). For Ha [lsim ] Pr−1/2 the instability is influenced by the actual depth of the layer. For such moderate magnetic fields the instability threshold is sensitive to the thermal properties of the bottom of the layer and the dependences of the critical parameters on the field strength are more complicated. In the latter case, various instability modes are possible depending on the thermal boundary conditions and the relative magnitudes of Prandtl and Hartmann numbers.

In most past theories on Bragg reflection of waves by a finite patch of rigid bars, only outgoing waves are allowed on the transmission side, simulating the effect of an idealized shoreline where all the incident wave energy is consumed by breaking. In these theories the amplitudes of both the incident and reflected waves are found to decrease monotonically over the bar patch in the shoreward direction. This result has motivated the idea of artificially constructing bars to protect a beach from incident waves. However, some numerical calculations have suggested that this tendency does not always hold when there is some reflection from the shore. We show here that with finite reflection by the shoreline the spatial distribution of wave energy over the patch can indeed be reversed, indicating that the mechanism can increase the hazards to the beach. The phase relation between the bars and the shoreline reflection is found to be the key to this qualitative change of wave response.

We consider the linear stability of exact, temporally periodic solutions of the Euler equations of incompressible, inviscid flow in an ellipsoidal domain. The problem of linear stability is reduced, without approximation, to a hierarchy of finite-dimensional Floquet problems governing fluid-dynamical perturbations of differing spatial scales and symmetries. We study two of these Floquet problems in detail, emphasizing parameter regimes of special physical significance. One of these regimes includes periodic flows differing only slightly from steady flows. Another includes long-period flows representing the nonlinear outcome of an instability of steady flows. In both cases much of the parameter space corresponds to instability, excepting a region adjacent to the spherical configuration. In the second case, even if the ellipsoid departs only moderately from a sphere, there are filamentary regions of instability in the parameter space. We relate this and other features of our results to properties of reversible and Hamiltonian systems, and compare our results with related studies of periodic flows.

The dependence on initial conditions of the three-dimensional algebraic spatial instability of the Blasius boundary layer is examined by a recently developed method of receptivity analysis based on the upstream integration of adjoint equations. This method allows us to determine optimal perturbations, i.e. initial perturbations that maximize the energy growth, even in the wavenumber range where the problem is not amenable to a mode analysis, and thus to complement a previous paper in which the small-wavenumber regime was described.

The plane layer Childress–Soward dynamo model, consisting of a rotating layer of electrically conducting fluid between horizontal planes heated from below, is studied. Solutions periodic in the horizontal directions are sought, with electrically insulating boundary conditions applied. The large Prandtl number limit is used.

Fully three-dimensional convection-driven dynamos have been studied numerically for this problem. Both the kinematic and the magnetically saturated regimes are studied, and a simple model of the dynamo mechanism is proposed. The dependence of the dynamo on the Rayleigh number, Ekman number and diffusivity ratio is studied, and the role of Taylor's constraint in low Ekman number convection-driven dynamos is considered.

The effect of base suction on a plane wake was found to produce significant changes in wake dynamics. The wake is produced by merging two boundary layers from the trailing edge of a splitter plate in a two-stream water tunnel. A threshold suction speed exists which is approximately equal to half of the free-stream velocity. If the suction speed is below the threshold, the wake flow is unstable. If the suction speed is above the threshold, the wake becomes stable and no vortex shedding is observed. In the present experiment, the suction technique can stabilize a wake at a maximum tested Reynolds number of 2000.

The suction significantly reduces the length of the absolutely unstable region in the immediate vicinity of the trailing edge of the splitter plate and produces a non-parallel flow pattern, resulting in the breakdown of global instability. The global growth rate changes from positive (unstable flow) to negative (stable flow) at the suction speed equalling 0.46 of the free-stream velocity. The threshold suction speed can be accurately predicted by the global linear theory of Monkewitz et al. (1993) with a non-parallel flow correction.