A study has been performed of the interaction of periodic vortex rings with a central columnar vortex, both for the case of identical vortex rings and the case of rings of alternating sign. Numerical calculations, both based on an adaptation of the Lundgren–Ashurst (1989) model for the columnar vortex dynamics and by numerical solution of the axisymmetric Navier–Stokes and Euler equations in the vorticity–velocity formulation using a viscous vorticity collocation method, are used to investigate the response of the columnar vortex to the ring-induced velocity field. In all cases, waves of variable core radius are observed to build up on the columnar vortex core due to the periodic axial straining and compression exerted by the vortex rings. For sufficiently weak vortex rings, the forcing by the rings serves primarily to set an initial value for the axial velocity, after which the columnar vortex waves oscillate approximately as free standing waves. For the case of identical rings, the columnar vortex waves exhibit a slow upstream propagation due to the nonlinear forcing. The cores of the vortex rings can also become unstable due to the straining flow induced by the other vortex rings when the ring spacing is sufficiently small. This instability causes the ring vorticity to spread out into a sheath surrounding the columnar vortex. For the case of rings of alternating sign, the wave in core radius of the columnar vortex becomes progressively narrower with time as rings of opposite sign approach each other. Strong vortex rings cause the waves on the columnar vortex to grow until they form a sharp cusp at the crest, after which an abrupt ejection of vorticity from the columnar vortex is observed. For inviscid flow with identical rings, the ejected vorticity forms a thin spike, which wraps around the rings. The thickness of this spike increases in a viscous flow as the Reynolds number is decreased. Cases have also been observed, for identical rings, where a critical point forms on the columnar vortex core due to the ring-induced flow, at which the propagation velocity of upstream waves is exactly balanced by the axial flow within the vortex core when measured in a frame translating with the vortex rings. The occurrence of this critical point leads to trapping of wave energy downstream of the critical point, which results in large-amplitude wave growth in both the direct and model simulations. In the case of rings of alternating sign, the ejected vorticity from the columnar vortex is entrained and carried off by pairs of rings of opposite sign, which move toward each other and radially outward under their self- and mutually induced velocity fields, respectively.

This paper treats the steady three-dimensional thermocapillary convection in a cylindrical liquid-metal zone between the isothermal ends of two coaxial solid cylinders and surrounded by an atmosphere. There is a uniform steady magnetic field which is parallel to the common centrelines of the liquid zone and solid cylinders, and there is a non-axisymmetric heat flux into the liquid's free surface. The magnetic field is sufficiently strong that inertial effects and convective heat transfer are negligible, and that viscous effects are confined to thin boundary layers adjacent to the free surface and to the liquid–solid interfaces. With an axisymmetric heat flux, the axisymmetric thermocapillary convection is confined to the thin layer adjacent to the free surface, but with a non-axisymmetric heat flux, there is an azimuthal flow inside the free-surface layer from the hot spot to the cold spot with the circulation completed by flow across the inviscid central core region. This problem is related to the magnetic damping of thermocapillary convection for the floating-zone growth of semiconductor crystals in Space.

Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s−1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

We derive a deterministic directional shoaling model and a stochastic directional shoaling model for a gravity surface wave field, valid for a beach with parallel depth contours accounting for refraction and nonlinear quadratic (three wave) interactions. A new phenomenon of non-resonant spectral evolution arises due to the medium inhomogeneity. The kernels of the kinetic equation depend on the bathymetry through an integral operator. Preliminary tests carried out on laboratory data for a unidirectional case indicate that the stochastic model also works rather well beyond the region where the waves may be regarded as nearly Gaussian. The limit of its applicability is decided by the dispersivity of the medium (relative to the nonlinearity). Good agreement with both laboratory data and the underlying deterministic model is found up to a value of about 1.5 for the spectral peak Ursell number. Beyond that only the deterministic model matches the measurements.

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

In the present study, the effects of an isolated stationary spherical particle on the transition process in a flat-plate boundary layer are examined by a spatial direct numerical simulation. The full three-dimensional time-dependent incompressible Navier–Stokes equations are integrated by a time-splitting method and discretized spatially by a high-order finite difference/spectral method. A virtual boundary technique defining the no-slip boundary of a sphere is implemented within the Cartesian geometry of the computational grid.

Two numerical simulations which consider the effects of the sphere on the boundary layer are presented. The subcritical Reynolds number case reveals the appearance of hairpin vortices shed into the sphere wake which decay as they are convected downstream. The initial interaction of the sphere and the boundary layer produces a three-dimensional isolated disturbance comprising a wave part and a transient part. The decaying transient part is convected downstream at the local mean velocity, while the wave part induces a decaying Tollmien–Schlichting wave in the flow field.

In the second case, an increase in the Reynolds number results in a wedge of incipient turbulent flow downstream of the sphere. The development of the wake of the sphere is dominated by the appearance of an isolated disturbance which rapidly breaks down forming a structure resembling a turbulent spot. It is demonstrated that the transition induced by a sphere in the boundary layer is due to a mechanism related to bypass transition.

The strain-induced evolution of shielded monopolar vortices has been investigated in a stratified fluid. A steady strain flow was generated by an arrangement of four rotating horizontal discs, whereas the monopolar vortex was created by a small spinning sphere. Quantitative information about the flow field was obtained by tracking passive tracer particles. The vortex was observed to deform into a tripolar-like structure, followed by the shedding of the accompanying satellites. During this stage, the remaining vortex core evolved quasi-steadily, which was evident from the functional relationship between the vorticity and the stream function. Furthermore, it was shown that the removal of the surrounding ring of oppositely signed vorticity induces an accelerated horizontal growth of the vortex core. Owing to the diffusive decay of vorticity, the vortex was finally torn apart along the horizontal strain axis. The dynamics of the vortex core appeared to be very similar to that of an elliptic patch of uniform vorticity. The instantaneous vorticity contours at high vorticity levels were close to ellipses with nearly the same aspect ratios and orientations. Moreover, the observed vortex evolution was in qualitative agreement with the calculated motion of an elliptic patch of uniform vorticity. As a second approach, the full two-dimensional vorticity equation was solved numerically by a finite-difference method in order to account for both the non-uniform spatial vorticity distribution of the laboratory vortex and the diffusion of vorticity in the horizontal directions. The numerically obtained vortex evolution was in good agreement with that observed in the laboratory.

A hollow vortex in the form of a straight tube, parallel to the z-axis, and of radius a, moves in a uniform stream of fluid with velocity U in the x-direction, with U small compared with the sound speed c. This steady flow is disturbed by the presence of a thin symmetric fixed aerofoil. With a change of x-coordinate, the problem is equivalent to that of a moving aerofoil cutting through an initially fixed vortex in still fluid. The aim of this work is to determine the resulting perturbed flow, and to estimate the distant sound field. A detailed calculation is given for the perturbed velocity potential in the incompressible flow case, for the linearized equations in the limit of small aerofoil thickness. A formally exact solution involves a four-fold integral and an infinite sum over all mode numbers. For the important special case where the vortex tube has small radius a compared with the aerofoil width, the deformed vortex is characterized by a hypothetical vortex filament located at the ‘mean centre’ x¯(z, t), y¯(z, t) of the tube. Explicit results are given for x¯(z, t), y¯(z, t) for the case where the aerofoil has the elementary rectangular profile; results can then be obtained for more general and realistic cylindrical aerofoils by a single integral weighted with the derivative of the aerofoil thickness function. Finally the distant sound field is estimated, representing the aerofoil by a distribution of moving monopole sources and representing the effect of the deformed vortex in terms of compressible dipoles along the mean centre of the vortex.

Spatially modulated vortex pairs were generated below a free surface by two counter-rotating flaps whose edges approximate a sinusoid. The surface interactions of the vertically approaching vortex pairs were visualized by the shadowgraph technique. Two limiting cases were investigated in detail: the interaction with a surfactant-rich (contaminated) surface and with a surfactant-poor (‘clean’) surface. In the latter case shadowgraph images showed that the underlying vortex core formed a line of circular surface depressions. Subsequent measurements of the temporally evolving velocity fields using digital particle image velocimetry (DPIV) of the vortex pair cross-sections and the subsurface plane confirmed the connection process of the main vortex core with the surface. As a result of the connection the initially modulated vortex tube was broken into a line of U-vortices. In the presence of surfactants this connection could not be observed; rather a Reynolds ridge (or stagnation line) was formed and a very weak connection of the secondary separation vortex could be seen in the shadowgraphs as well as measured with the time-resolved DPIV technique.

A prerequisite for connection of the vortex with the surface is that the flow's kinematics force the vortex core, that is, regions of concentrated vorticity, toward the surface. The ensuing locally concentrated viscous flux of surface-parallel vorticity through the surface is balanced by a local surface deceleration. Surface-normal vorticity appears on each side of the decelerated region whose gradually increasing circulation is directly balanced by the loss of circulation of the surface-parallel vortex. However, the shear forces caused by small amounts of surface contamination and its associated subsurface boundary layer inhibit the connection process by preventing the essential viscous flux of parallel vorticity through the surface. Instead, the subsurface boundary layer is associated with a flux of parallel vorticity into the surface which then concentrates into the observable secondary separation vortex.

A detailed analysis of the predictability of observed Monin–Obukhov (MO) similarity within the near-ground region of near-neutral to moderately convective atmospheric boundary layers (ABL) from large-eddy simulation (LES) fields is reported in this work. High-resolution LES predictions of means, variances, budgets of turbulent kinetic energy and temperature variance, and the velocity and temperature spectra from three ABL states (−zi/L=0.44, 3 and 8) are analysed under MO scaling. The resolution in the near-ground region is increased by using ‘nested meshes.’ For the close-to-neutral case (−zi/L=0.44) the relative roles of grid resolution and subgrid-scale scale (SGS) parameterization on the predictability of MO-similarity are also studied. The simulated temperature field is found to satisfy the MO hypothesis and agree well with observations. The simulated velocity field, on the other hand, shows significant departures. Except for the horizontal variance, MO scales are the appropriate normalizing scales for the near-ground-layer statistics. However, the LES suggest that the boundary layer depth zi has an ‘indirect’ influence on all near-ground-layer variables except temperature, and the LES-predicted MO-scaled variables exhibit a functional dependence on both z/L and z/zi. The simulated two-dimensional spectra of velocity and temperature fluctuations, however, suggest that while large scales deviate from MO-similarity, inertial subrange scales are MO-similar. Discrepancies with field observations raise important questions of the non-dimensional depth z/zi over which MO-similarity holds for a particular variable. Surface-layer field studies generally do not document zi. It is also not clear to what extent these discrepancies are due to approximations made in LES. Measurements are needed designed specifically for comparing with LES predictions.

The slow advective-timescale dynamics of the atmosphere and oceans is referred to as balanced dynamics. An extensive body of theory for disturbances to basic flows exists for the quasi-geostrophic (QG) model of balanced dynamics, based on wave-activity invariants and nonlinear stability theorems associated with exact symmetry-based conservation laws. In attempting to extend this theory to the semi-geostrophic (SG) model of balanced dynamics, Kushner & Shepherd discovered lateral boundary contributions to the SG wave-activity invariants which are not present in the QG theory, and which affect the stability theorems. However, because of technical difficulties associated with the SG model, the analysis of Kushner & Shepherd was not fully nonlinear.

This paper examines the issue of lateral boundary contributions to wave-activity invariants for balanced dynamics in the context of Salmon's nearly geostrophic model of rotating shallow-water flow. Salmon's model has certain similarities with the SG model, but also has important differences that allow the present analysis to be carried to finite amplitude. In the process, the way in which constraints produce boundary contributions to wave-activity invariants, and additional conditions in the associated stability theorems, is clarified. It is shown that Salmon's model possesses two kinds of stability theorems: an analogue of Ripa's small-amplitude stability theorem for shallow-water flow, and a finite-amplitude analogue of Kushner & Shepherd's SG stability theorem in which the ‘subsonic’ condition of Ripa's theorem is replaced by a condition that the flow be cyclonic along lateral boundaries. As with the SG theorem, this last condition has a simple physical interpretation involving the coastal Kelvin waves that exist in both models.

Salmon's model has recently emerged as an important prototype for constrained Hamiltonian balanced models. The extent to which the present analysis applies to this general class of models is discussed.

A set of integro-differential equations in the Lagrangian renormalized approximation (Kaneda 1981) is rederived by applying a perturbation method developed by Kraichnan (1959), which is based upon an extraction of direct interactions among Fourier modes of a velocity field and was applied to the Eulerian velocity correlation and response functions, to the Lagrangian ones for homogeneous isotropic turbulence. The resultant set of integro-differential equations for these functions has no adjustable free parameters. The shape of the energy spectrum function is determined numerically in the universal range for stationary turbulence, and in the whole wavenumber range in a similarly evolving form for the freely decaying case. The energy spectrum in the universal range takes the same shape in both cases, which also agrees excellently with many measurements of various kinds of real turbulence as well as numerical results obtained by Gotoh et al. (1988) for a decaying case as an initial value problem. The skewness factor of the longitudinal velocity derivative is calculated to be −0.66 for stationary turbulence. The wavenumber dependence of the eddy viscosity is also determined.

The recent work of Rooney & Linden (1996) is generalized to describe the motion of non-Boussinesq plumes in both uniform and stratified environments. Using an integral model in which the horizontal entrainment velocity is assumed to take the form uε=α(ρ¯/ρe) 1/2w where α is the entrainment coefficient, ρ¯ is the plume density, w the plume velocity and ρe the ambient density, it is shown that the vertical scale over which non-Boussinesq effects are significant is given by zB=5/3 (B2o/ (20α4g3))1/5 where Bo is the buoyancy flux at the source. In a uniform environment, the system admits similarity solutions such that the location of the source of a real plume lies a distance zB[mid ]ρo/Δρ[mid ] −5/3 beyond the point source of the similarity solution. The above entrainment law implies a fundamental difference between the motion of upward and downward propagating non-Boussinesq plumes, with the radius of upward propagating plumes being greater than that of the equivalent Boussinesq plume, while the radius of downward propagating plumes is smaller. In a stratified but incompressible environment the model predicts that non-Boussinesq effects are confined close to the source and that at each height, the plume velocity and the fluxes of mass, momentum and buoyancy coincide exactly with those of the equivalent Boussinesq plume. Furthermore, at the neutral buoyancy height, the plume radius equals that of the equivalent Boussinesq plume.

An investigation of the near-field flow structure of coaxial jets with large outer to inner velocity ratio ru has been conducted. Since in all cases ru>1, the outer jet dominates the near-field flow structure. Two flow regimes are identified depending on whether ru is larger or smaller than a critical value ruc. When ru<ruc, the fast annular jet periodically pinches the central, slow jet near the end of the inner potential cone. The pinching frequency corresponds to the outer-jet mode. The length of the inner potential cone is strongly dependent on ru and behaves like A/ru, where A depends weakly on the initial conditions. When ru>ruc, the inner potential cone is truncated and is followed by an unsteady recirculation bubble with low-frequency oscillation.

The transition from one regime to another is explained by a simple model whose ingredients are the turbulent entrainment rate, governed by the outer-jet mixing layers and mass conservation. This model satisfactorily predicts the dependence of the inner potential cone length on ru and the critical velocity ratio ruc. The recirculation bubble has a wake-type instability. It oscillates at a low frequency and a large amplitude compared to the Kelvin–Helmholtz mode. Angular cross-correlations in the plane parallel to the jet outlet show moreover that this oscillation displays an azimuthal precession such that the rotation time of the phase of the oscillation equals the oscillation period. These salient features are discussed in the framework of the nonlinear delayed saturation (NLDS) model.

The linear stability analysis of hypersonic flow over a sharp slender cone with an attached shock is described. Attention is focused on the viscous modes of instability which may be described by a triple-deck structure. The situation in which both the effect of the shock and the influence of curvature are important is considered in the weak-interaction region. Both neutral and non-neutral solutions are presented for both axisymmetric and non-axisymmetric disturbances. The results obtained suggest that the effect of curvature on the stability of hypersonic flow is significant when the attached shock is taken into account.