By examining the modal interaction between two non-resonant shape oscillation modes of a charged liquid drop, we have identified a new route to instability via nonlinear coupling. We present numerical simulation results to show that when shape perturbation of a high-mode number Legendre mode is applied to the drop, the prolate–oblate mode of the drop may grow unbounded. Using multiple-scale analysis, we derive amplitude equations for the high-mode-number shape mode and the prolate–oblate mode to show the nonlinear coupling between the two modes.

Studies of convection in the horizontal Bridgman configuration were performed to investigate the flow structures and the nature of the convective regimes in a rectangular cavity filled with an electrically conducting liquid metal when it is subjected to a constant vertical magnetic field. Under some assumptions analytical solutions were obtained for the central region and for the turning flow region. The validity of the solutions was checked by comparison with the solutions obtained by direct numerical simulations. The main effects of the magnetic field are first to decrease the strength of the convective flow and then to cause a progressive modification of the flow structure followed by the appearance of Hartmann layers in the vicinity of the rigid walls. When the Hartmann number is large enough, Ha > 10, the decrease in the velocity asymptotically approaches a power-law dependence on Hartmann number. All these features are dependent on the dynamic boundary conditions, e.g. confined cavity or cavity with a free upper surface, and on the type of driving force, e.g. buoyancy and/or thermocapillary forces. From this study we generate scaling laws that govern the influence of applied magnetic fields on convection. Thus, the influence of various flow parameters are isolated, and succinct relationships for the influence of magnetic field on convection are obtained. A linear stability analysis was carried out in the case of an infinite horizontal layer with upper free surface. The results show essentially that the vertical magnetic field stabilizes the flow by increasing the values of the critical Grashof number at which the system becomes unstable and modifies the nature of the instability. In fact, the range of Prandtl number over which transverse oscillatory modes prevail shrinks progressively as the Hartmann number is increased from zero to 5. Therefore, longitudinal oscillatory modes become the preferred modes over a large range of Prandtl number.

The effects of a constant magnetic field on electrically conducting liquid-metal flows in a parallelepiped cavity are investigated using a spectral numerical method involving direct numerical solution of the Navier–Stokes and Ohm equations for three-dimensional flows. Three horizontal Bridgman configurations are studied: buoyancy-driven convection in a confined cavity and in a cavity where the top boundary is a stress-free surface and thirdly, thermocapillary-driven flow in a cavity where the upper boundary is subjected to effects of surface tension. The results of varying the Hartmann number (Ha) are described for a cavity with Ax = L/H = 4 and Ay = W/H = 1, where L is the length, W is the width and H is the height of the cavity. In general, an increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of thermal convection. The convective circulation progressively loses its intensity and when Ha reaches a certain critical value, which is found to depend on the direction (longitudinal or vertical) of the applied magnetic field, decrease of the flow intensity takes on an asymptotic form with important changes in the structure of the flow circulation. The flow structure may be separated into three regions: the core flow, Hartmann layers which develop in the immediate vicinity of the rigid horizontal boundaries or at the endwalls, and parallel layers appearing in the vicinity of the sidewalls. The behaviour of the maxima of velocity and of the overall flow circulation is found to depend on both the boundary conditions used and the direction of the applied magnetic field. Furthermore, the interaction of the electric current density with the applied magnetic field which leads to the structural reorganization described above can also create more subtle flow modifications, such as flow inversions which are observed mainly in the central region of the cavity.

The horizontal scale of rotating convection with rigid boundary conditions is studied. The range of Rayleigh number concerned is moderate, i.e. large enough to induce a finite-amplitude convection but small enough so that the geostrophic processes are significant.

On considering an experimental law of the Nusselt number and some constraints of elemental geostrophic processes, the horizontal scale of the convection can be estimated. This estimation strongly depends on the ratio between the thicknesses of the Ekman layer and the thermal boundary layer, and does not depend monotonically on the Rayleigh number. This dependency is compatible with the experimental results of Rossby (1969).

The estimated horizontal scale was checked by laboratory experiments. The horizontal temperature distribution was visualized by thermal liquid-crystal capsules dispersed in the working fluid. The horizontal scale was measured by counting vortices. The experimental results agree fairly well with the estimated scale.

The optimal non-modal perturbations for the neutrally stratified boundary layer in a rotating frame of reference (Ekman layer) are found for a Reynolds number characteristic of the planetary boundary layer (PBL). Two classes of non-modal instabilities are found: evanescent perturbations, with lifetimes up to about one hour, and growing instabilities. The important difference between these types of perturbations is whether or not the optimal non-modal perturbation projects onto an unstable normal mode. The evanescent instabilities are of smaller scale and are oriented at larger angles to the surface isobars compared to either the growing perturbations or normal-mode instabilities. The optimal perturbations take the form of vortices at an acute angle to the geostrophic flow that rapidly transform into streaks with associated overturning motion. The energetics of the optimal perturbations are investigated in detail to clarify the instability mechanism throughout its evolution.

Nonlinear stability analyses of the neutrally stratified Ekman layer have shown that the normal-mode instability will equilibrate with the mean flow to form boundary-layer-scale equilibrium roll eddies aligned closely with the geostrophic flow. However, numerical simulations do not generate these rolls in neutral stratification although they often realize small-scale near-surface streaks oriented at large angles to the geostrophic wind. The evanescent optimal perturbations bear a close resemblance to the simulated streaks. It is proposed that the non-model perturbation mechanism is associated with the streaks.

The stability of the incompressible attachment-line boundary layer has been studied by Hall, Malik & Poll (1984) and more recently by Lin & Malik (1996). These studies, however, ignored the effect of leading-edge curvature. In this paper, we investigate this effect. The second-order boundary-layer theory is used to account for the curvature effects on the mean flow and then a two-dimensional eigenvalue approach is applied to solve the linear stability equations which fully account for the effects of non-parallelism and leading-edge curvature. The results show that the leading-edge curvature has a stabilizing influence on the attachment-line boundary layer and that the inclusion of curvature in both the mean-flow and stability equations contributes to this stabilizing effect. The effect of curvature can be characterized by the Reynolds number Ra (based on the leading-edge radius). For Ra = 104, the critical Reynolds number R (based on the attachment-line boundary-layer length scale, see §2.2) for the onset of instability is about 637; however, when Ra increases to about 106 the critical Reynolds number approaches the value obtained earlier without curvature effect.

The upper-branch neutral modes of inviscid instability in a boundary-layer flow with significant longitudinal vortices present are shown to possess typically a logarithmically singular, non-inflectional, critical layer. This contrasts with previous linear and nonlinear suggestions implemented in vortex–wave interaction and secondary instability theories, which are re-examined. The analysis here is based first on perturbation techniques applied to a Rayleigh unstable planar motion supplemented by a vortex centred around the inflection level, followed by the extension to more general cases. Flows with order one and larger spanwise scales are considered. Multiple solutions, their limit properties and parametric continuations are illustrated with concrete examples.

A pattern-recognition technique, applied to multi-point simultaneous velocity measurements obtained with 45° X-wire anemometer probes, is used to extract and characterize the underlying organized motions, i.e. coherent structures, within the near-wake region of a turbulent round jet discharged perpendicularly from a pipe into a crossflow. This flow has been found to be quite complex owing to its three-dimensional nature and the interactions between several flow regions. Analyses of the underlying coherent structures, which play an important role in the physics of the flow, are still rare and are mostly based on flow-visualization techniques. Using a pattern-recognition technique in conjunction with hot-wire measurements, we recently examined the wake regions of the pipe and jet at levels near the tip of the pipe, and found that Kármán-like vortex structures in the wake of the pipe are locked to similar structures in the jet-wake. In this paper we expand upon our previous work and characterize these structures throughout the wake of the jet up into the region of the bent-over jet – a region where they have not been identified previously. The complex geometry of these structures in the wake of the jet as well as their interaction with the bent-over jet are discussed. The results show that these structures split before they link to similar structures on the opposite side of the symmetry plane in the jet region. The results further suggest that the vorticity due to the structures in the wake of the jet contributes to the motion of the well-known counter-rotating vortex pair.

Matched asymptotic expansions are used to obtain a reduced description of the nonlinear and viscous evolution of small, localized vorticity defects embedded in a Couette flow. This vorticity defect approximation is similar to the Vlasov equation, and to other reduced descriptions used to study forced Rossby wave critical layers and their secondary instabilities. The linear stability theory of the vorticity defect approximation is developed in a concise and complete form. The dispersion relations for the normal modes of both inviscid and viscous defects are obtained explicitly. The Nyquist method is used to obtain necessary and sufficient conditions for instability, and to understand qualitatively how changes in the basic state alter the stability properties. The linear initial value problem is solved explicitly with Laplace transforms; the resulting solutions exhibit the transient growth and eventual decay of the streamfunction associated with the continuous spectrum. The expansion scheme can be generalized to handle vorticity defects in non-Couette, but monotonic, velocity profiles.

The two-dimensional free-surface problem of an ideal jet impinging on an uneven wall is studied using complex-variable and transform techniques. A relation between the flow angle on the free surface and the wall angle is first obtained. Then, by using a Hilbert transform and the generalized Schwarz–Christoffel transformation technique, a system of nonlinear integro-differential equations for the flow angle and the wall angle is formulated. For the case of symmetric flow, a compatibility condition for the system is automatically satisfied. In some special cases, for instance when the wall is a wedge, the problem reduces to the evaluation of several integrals. Moreover, in the case of a jet impinging normally on a flat wall, the classical result is recovered. For the asymmetric case, a relation is obtained between the point in the reference ζ-plane which corresponds to the position of the stagnation point in the physical plane, the flow speed and the shape of the wall. The solution to a linearized problem is given, for comparison. Some numerical solutions are presented, showing the shape of the free surface corresponding to a number of different wall shapes.

Free-surface flow over a bottom topography with an asymptotic depth change (a ‘step’) is considered for different ranges of Froude numbers varying from subcritical, transcritical, to supercritical. For the subcritical case, a linear model indicates that a train of transient waves propagates upstream and eventually alters the conditions there. This leading-order upstream influence is shown to have profound effects on higher-order perturbation models as well as on the Froude number which has been conventionally defined in terms of the steady-state upstream depth. For the transcritical case, a forced Korteweg–de Vries (fKdV) equation is derived, and the numerical solution of this equation reveals a surprisingly conspicuous distinction between positive and negative forcings. It is shown that for a negative forcing, there exists a physically realistic nonlinear steady state and our preliminary results indicate that this steady state is very likely to be stable. Clearly in contrast to previous findings associated with other types of forcings, such a steady state in the transcritical regime has never been reported before. For transcritical flows with Froude number less than one, the upstream influence discovered for the subcritical case reappears.

A semi-analytical theory for the scattering of plane sound waves by a compressible, non-homentropic, circular-cylindrical, single vortex is presented in this paper. As a special case, the scattering of sound by a cylindrical inhomogeneity (hot spot) is investigated. Contrary to the otherwise analogous quantum-mechanical scattering problem, there are singularities in the modified acoustic wave equation for radii xs ∈ (0, ∞) when the scattering by a vortex is considered. It will be shown how these singularities can be treated.

This sound-scattering theory is applied to the problem of the interaction of weak plane shock waves with a strong cylindrical vortex. The calculated scattered sound signal has a rather complicated structure in which a cylindrical wave with an essentially quadrupolar directivity pattern is discernible. In the case of shock–hot-spot interaction a scattered sound signal with dipole-like amplitude is obtained. Both results qualitatively agree with experimental findings.

This paper and Part 2 report various new insights into the classic Kelvin–Helmholtz problem which models the instability of a plane vortex sheet and the complicated motions arising therefrom. The full nonlinear version of the hydrodynamic problem is treated, with allowance for gravity and surface tension, and the account deals in precise fashion with several inherently peculiar properties of the mathematical model. The main achievement of the paper, presented in §3, is to demonstrate that the problem admits a canonical Hamiltonian formulation, which represents a novel variational definition of a functional representing perturbations in kinetic energy. The Hamiltonian structure thus revealed is then used to account systematically for relations between symmetries and conservation laws, and none of those examined appears to have been noticed before. In §4, a generalized, non-canonical Hamiltonian structure is shown to apply when the vortex sheet becomes folded, so requiring a parametric representation, as is well known to occur in the later stages of evolution from Kelvin–Helmholtz instability. Further invariant properties are demonstrated in this context. Finally, §5, the linearized version of the problem – reviewed briefly in §2.1 – is reappraised in the light of Hamiltonian structure, and it is shown how Kelvin–Helmholtz instability can be interpreted as the coincidence of wave modes characterized respectively by positive and negative values of the Hamiltonian functional representing perturbations in total energy.

Several new results on the bifurcation and instability of nonlinear periodic travelling waves, at the interface between two fluids in relative motion, in a parametric neighbourhood of a Kelvin–Helmholtz unstable equilibrium are presented. The organizing centre for the analysis is a canonical Hamiltonian formulation of the Kelvin–Helmholtz problem presented in Part 1. When the density ratio of the upper and lower fluid layers exceeds a critical value, and surface tension is present, a pervasive superharmonic instability is found, and as u → u0, where u is the velocity difference between the two layers and u0 is the Kelvin–Helmholtz threshold, the amplitude at which the superharmonic instability occurs scales like (u0−u)1/2 with u < u0. Other results presented herein include (a) new results on the structure of the superharmonic instability, (b) the discovery of isolated branches and intersecting branches of travelling waves near a critical density ratio, (c) the appearance of Benjamin–Feir instability along branches of waves near the Kelvin–Helmholtz instability threshold and (d) the interaction between the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instability at low amplitude.

Notwithstanding the fact that the instability of the separated shear layer in the cylinder wake has been extensively studied, there remains some uncertainty regarding not only the critical Reynolds number at which the instability manifests itself, but also the variation of its characteristic frequency with Reynolds number (Re). A large disparity exists in the literature in the precise value of the critical Reynolds number, with quoted values ranging from Re = 350 to Re = 3000. In the present paper, we demonstrate that the spanwise end conditions which control the primary mode of vortex shedding significantly affect the shear-layer instability. For parallel shedding conditions, shear-layer instability manifests itself at Re ≈ 1200, whereas for oblique shedding conditions it is inhibited until a significantly higher Re ≈ 2600, implying that even in the absence of a variation in free-stream turbulence level, the oblique angle of primary vortex shedding influences the onset of shear-layer instability, and contributes to the large disparity in quoted values of the critical Reynolds number. We confirm the existence of intermittency in shear-layer fluctuations and show that it is not related to the transverse motion of the shear layers past a fixed probe, as suggested previously. Such fluctuations are due to an intermittent streamwise movement of the transition point, or the location at which fluctuations develop rapidly in the shear layer.

Following the original suggestion of Bloor (1964), it has generally been assumed in previous studies that the shear-layer frequency (normalized by the primary vortex shedding frequency) scales with Re1/2, although a careful examination of the actual data points from these studies does not support such a variation. We have reanalysed all of the actual data points from previous investigations and include our own measurements, to find that none of these studies yields a relationship which is close to Re1/2. A least-squares analysis which includes all of the previously available data produces a variation of the form Re0·67. Based on simple physical arguments that account for the variation of the characteristic velocity and length scales of the shear layer, we predict a variation for the normalized shear-layer frequency of the form Re0·7, which is in good agreement with the experimental measurements.

An inviscid analytic model is proposed for the steady separated flow around an inclined flat plate. With the plate normal to the stream, the model reduces to the wake-source model of Parkinson & Jandali originally developed for flow external to a symmetrical two-dimensional bluff body and its wake. At any other inclination, the Kutta condition is satisfied at both leading and trailing edges of the plate, and, in the limit that the angle of attack approaches zero, classical airfoil theory is recovered. A boundary condition is formulated based on some experimental results of Abernathy, but no additional empirical information is required. The predicted pressure distributions on the wetted surface for a wide range of angle attack are found to be in good agreement with experimental data, especially at smaller angles of attack. An extension to include a leading-edge separation bubble is explored and results are satisfactory.