Jan-Bert Flór

Internal waves generated by a moving sphere and its wake in a stratified fluid

The internal gravity waves and the turbulent wake of a sphere moving through stratified fluid were studied by the fluorescent dye technique. The Reynolds number Re=U·2a/v was kept nearly constant at about 3 · 103 and the Froude number F;U/a N ranged from 0.5 to 12.5. It is observed that waves generated by the body are dominant only when F<4 and are replaced by waves generated by the large scale coherent structures of the wake when F>4.

Dynamics of monopolar vortices on a topographic beta-plane

The dynamics of a cyclonic monopolar vortex on a topographic beta-plane are studied experimentally and theoretically. Detailed measurements of the vortex structure are conducted using high-resolution quantitative velocity measurements. The initial velocity profiles were described in terms of a radius R vm , maximum azimuthal velocity v θ m , and a dimensionless parameter α which characterizes the steepness of the velocity profile. The initial direction of motion of the monopolar vortex is critically dependent on α and weakly dependent of the initial strength and size of the vortex: isolated vortices (α ∼ 3) move north, whereas non-isolated vortices characterized by α ∼ 1 move northwest. When the azimuthal velocity decays slowly with radial distance (α < 1.4), Rossby wave generation dominates the vortex dynamics and the translational speed of the vortex correlates with the Rossby wave speed. When the azimuthal velocity decays rapidly with radial distance (α > 1.4) the vortex is isolated and the translational speed is much slower than the Rossby wave speed. To interpret the effect of the vortex structure on the direction of motion, a mechanistic model is developed which includes the Rossby force and a lift force arising from circulation around the vortex, but does not include the effect of Rossby waves. The Rossby force results from the integrated effect of the Coriolis force on the vortex and drives the vortex north; the lift force is determined from the circulation around the vortex and drives the vortex west. Comparison with the experimental data reveals two regimes: α < 1.4, where the vortex dynamics are dominated by Rossby waves whereas for α > 1.4 Rossby waves are weak and favourable agreement is found with the mechanistic model.

Formation of a Tripolar Vortex in a Stratified Fluid

This paper reports on an experimental study of vortices in a stratified fluid. The vortices were generated by two different stirring devices, viz. a rotatting sphere and a rotating bent rod. It was found that the vortices created with the rotating sphere are mostly axisymmetric and stable, whereas the vortices produced with the bent rod generally show instabilities, under certain conditions leading to the formation of a tripolar vortex. This report concentrates on this tripolar structure and presents quantitative information about the flow obtained through streak photography of tracer particles.

Internal wave generation by oscillation of a sphere, with application to internal tides

A joint theoretical and experimental study is performed on the generation of internal gravity waves by an oscillating sphere, as a paradigm of the generation of internal tides by barotropic tidal flow over three-dimensional supercritical topography. The theory is linear and three-dimensional, applies both near and far from the sphere, and takes into account viscosity and the unsteadiness arising from the interference with transients generated at the start-up. The waves propagate in conical beams, evolving with distance from a bimodal to unimodal wave profile. In the near field, the profile is asymmetric with its major peak towards the axis of the cones. The experiments involve horizontal oscillations and develop a cross-correlation technique for the measurement of the deformation of fluorescent dye planes to sub-pixel accuracy. At an oscillation amplitude of one fifth of the radius of the sphere, the waves are linear and the agreement between experiment and theory is excellent. As the amplitude increases to half the radius, nonlinear effects cause the wave amplitude to saturate at a value that is 20% lower than its linear estimate. Application of the theory to the conversion rate of barotropic tidal energy into internal tides confirms the expected scaling for flat topography, and shows its transformation for hemispherical topography. In the ocean, viscous and unsteady effects have an essentially local role, in keeping the wave amplitude finite at the edges of the beams, and otherwise dissipate energy on such large distances that they hardly induce any decay.

Fluid transport by dipolar vortices

Abstract The transport properties of dipolar vortices propagating on an f -plane are studied experimentally by examining the distortion of a series of material surfaces. The observations are compared with a model based on characterising the flow around the dipole as irrotational flow past a rigid cylinder of volume V . Measurements made of the volume of fluid permanently displaced forward by the vortices, agree to within 20% of that predicted by the proposition of Darwin [Darwin, C., 1953. A note on hydrodynamics. Proc. Cambridge Philos. Soc., 49, 342–354], namely that the vortex will displace a volume C M V forward, where C M =1 for a Lambs dipole. The results are applied to examine fluid transport by dipolar vortices propagating on the β -plane, where the ambient potential vorticity field causes easterly propagating dipolar vortices to meander sinusoidally between the North and South. We demonstrate that as the vortex moves between the North and South, it exchanges a volume C M V sin α by the drift effect (where α is the angle between the velocity of the dipole and the material surface), which is generally larger than that attributed to other mechanisms such as lobe shedding. The results are applied to give new insight to the effect of vortices in enhancing diffusion, and the secondary flow generated by the transport of ambient potential vorticity.

The evolution of an isolated turbulent region in a two‐layer fluid

A turbulent region is generated by horizontal pulsed injection at the interface of a two‐layer fluid. Flow visualization studies reveal the existence of three stages in the evolution of the vertical size of this region: growth, maximum height, and collapse. A scaling analysis for the height of the turbulent region is presented, which appears to be in good agreement with the measurements. Comparable results were obtained by Fernando, van Heijst, and Fonseka (submitted to J. Fluid Mech.) for similar experiments in a linearly stratified fluid. Thorpe‐scale measurements of the turbulent region reveal that the ratio of the rms displacement Lt and the maximum displacement Ltmax remain constant with time. The eventual formation process of the dipolar vortices after the collapse and the influence of interfacial wave motions on these dipolar vortices are discussed.

Turbulent mixing at a stable density interface : the variation of the buoyancy flux–gradient relation

Experiments conducted on mixing across a stable density interface in a turbulent Taylor–Couette flow show, for the first time, experimental evidence of an increase in mixing efficiency at large Richardson numbers. With increasing buoyancy gradient the buoyancy flux first passes a maximum, then decreases and at large values of the buoyancy gradient the flux increases again. Thus, the curve of buoyancy flux versus buoyancy gradient tends to be N-shaped (rather than simply bell shaped), a behaviour suggested by the model of Balmforth et al. (J. Fluid Mech. vol. 428, 1998, p. 349). The increase in mixing efficiency at large Richardson numbers is attributed to a scale separation of the eddies active in mixing at the interface; when the buoyancy gradient is large mean kinetic energy is injected at scales much smaller than the eddy size fixed by the gap width, thus decreasing the eddy turnover time. Observations show that there is no noticeable change in interface thickness when the mixing efficiency increases; it is the mixing mechanism that changes. The curves of buoyancy flux versus buoyancy gradient also show a large variability for identical experimental conditions. These variations occur at time scales one to two orders of magnitude larger than the eddy turnover time scale.

An experimental investigation of spin-up from rest of a stratified fluid

We examine the spin-up from rest of a stratified fluid with initial Brunt–Väisälä frequency N bound within a cylindrical container of height 2H and radius R which is set to rotate impulsively with angular speed f/2. Particular attention is given to characterizing the dependence of the form of the resulting flow on the governing parameters. Our experimental study reveals a wealth of flow behaviours and instabilities. In all experiments, the initial phase of motion is marked by the establishment of mixed axisymmetric corner regions fed by radial Ekman transport, a process detailed in Flór et al. (Flór, J.B., Ungarish, M. and Bush, J.W.M., “Spin-up from rest of a stratified fluid: boundary flows”, J. Fluid Mech., 472, 51–82 (2002).). The subsequent evolution of the central vortex depends critically on the Burger number , where is the buoyancy frequency of the central core following the establishment of the corner regions. For B > 1.0, the axisymmetry of the system is retained throughout the spin-up process: the central vortex attains a state of near solid body rotation by the diffusion of vorticity from the sidewalls. For B < 1.0, the central core becomes baroclinically unstable, and its streamlines strained from circles into ellipses. Subsequently, for Nc/f < 1 (and B < 1.0), the symmetry of the central core is broken in a manner reminiscent of the elliptical instability. For short tanks (2H/R < 1), the instability is marked by a simple tip-over of the central core in the laboratory frame that is resisted by the core stratification. For 2H/R > 1, the centreline of the stratified core is deflected into a helical form before the core breaks into a series of stacked vortices. A Burger number criterion, , for the baroclinic instability of the central core is derived and found to be consistent with the experimental observations.

Spatial structure of first and higher harmonic internal waves from a horizontally oscillating sphere

An experimental study is presented on the spatial structure of the internal wave field emitted by a horizontally oscillating sphere in a uniformly stratified fluid. The limits of linear theory and the nonlinear features of the waves are considered as functions of oscillation amplitude. Fourier decomposition is applied to separate first harmonic waves at the fundamental frequency and higher harmonic waves at multiples of this frequency. For low oscillation amplitude, of 10 % of the sphere radius, only the first harmonic is significant and the agreement between linear theory and experiment is excellent. As the oscillation amplitude increases up to 30 % of the radius, the first harmonic becomes slightly smaller than its linear theoretical prediction and the second and third harmonics become detectable. Two distinct cases emerge depending on the ratio Ω between the oscillation frequency and the buoyancy frequency. When Ω > 0.5, the second harmonic is evanescent and localized near the sphere in the plane through its centre perpendicular to the direction of oscillation, while the third harmonic is negligible. When Ω < 0.5, the second harmonic is propagative and appears to have an amplitude that exceeds the amplitude of the first harmonic, while the third harmonic is evanescent and localized near the sphere on either side of the plane through its centre perpendicular to the direction of oscillation. Moreover, the propagative first and second harmonics have radically different horizontal radiation patterns and are of dipole and quadrupole types, respectively.

Inviscid coupling between point symmetric bodies and singular distributions of vorticity

We study the inviscid coupled motion of a rigid body (of density ρ b , in a fluid of density ρ) and singular distributions of vorticity in the absence of gravity, using for illustration a cylinder moving near a point vortex or dipolar vortex, and the axisymmetric interaction between a vortex ring and sphere. The coupled motion of a cylinder (radius a) and a point vortex, initially separated by a distance R and with zero total momentum, is governed by the parameter R 4 /(ρ b /ρ+ 1)a 4 . When R 4 /(ρ b /ρ+1)a 4 «1, a (positive) point vortex moves anticlockwise around the cylinder which executes an oscillatory clockwise motion, with a mixture of two frequencies, centred around its initial position. When R 4 /(ρ b /ρ + 1)a 4 »1, the initial velocity of the cylinder is sufficiently large that the dynamics become uncoupled, with the cylinder moving off to infinity. The final velocity of the cylinder is related to the permanent displacement of the point vortex. The interaction between a cylinder (initially at rest) and a dipolar vortex starting at infinity depends on the distance of the vortex from the centreline (h), the initial separation of the vortical elements (2d), and ρ b /ρ. For a symmetric encounter (h =0) with a dense cylinder, the vortical elements pass around the cylinder and move off to infinity, with the cylinder being displaced a finite distance forward. However, when ρ b /ρ 0) leads to the cylinder moving off in the opposite direction to the dipolar vortex. To illustrate the difference between two- and three-dimensional flows, we consider the axisymmetric interaction between a vortex ring and a rigid sphere. The velocity perturbation decays so rapidly with distance that the interaction between the sphere and vortex ring is localized, but the underlying processes are similar to two-dimensional flows. We briefly discuss the general implications of these results for turbulent multiphase flows.