Toroidal Eddies Research Articles

The particle image velocimetry of vortical electrohydrodynamic flows of oil near a high-voltage electrode tip

A dual-domain particle image velocimetry (PIV) is employed to characterize the electrohydrodynamic flows arising in an oil bath with a grounded bottom near a high-voltage pin electrode with its tip close to the bottom. The applied voltage is varied to explore its effect on the arising flow field. Spatially, PIV observations are conducted using a dual-domain approach by synchronizing two cameras on the sampling plane. One camera is focused on the electrode tip and the second one on the fluid bulk. By capturing these synchronized domains, the velocity field is explored in its entirety, demonstrating a broad scope of morphology spanning the domain with the high velocity within the inter-electrode gap to the slow eddies in the fluid bulk. At high applied voltages, a transitional regime is observed in which the flow field bifurcates into toroidal Moffatt-like eddies. The experimental technique employed has its limitations. For example, at high applied voltages, the peak velocity domain could not be adequately resolved in every trial due to the deposition of fatty acids on the electrodes. Accordingly, the peak velocity has been measured only in the trials where fatty acid build-up did not interfere. Additionally, due to the axisymmetric geometry, reflections interfere with the observations of the near-surface velocity field.

Axisymmetric slow viscous liquid flow around a spherical bubble translating in a circular tube

We investigate the axisymmetric slow viscous liquid flow around a spherical bubble located on the axis of a long circular tube analytically based on the Stokes approximation. The bubble translates along the axis of the tube with a constant velocity within Hagen–Poiseuille flow flowing far from the bubble. The translating velocity of the bubble and mean velocity of the Hagen–Poiseuille flow are given arbitrarily. To analyze Stokes equation, we use the method of complex eigenfunction expansions and the method of least squared error. As results, the streamline patterns and the pressure contour lines in the flow field are drawn for some radii of the bubble. The drag exerted on the bubble and the pressure change induced by the bubble are determined according to the radius of the bubble. For a small bubble radius in the tube, we compared results with previous asymptotic results. The drift velocity of the bubble due to the Hagen–Poiseuille flow in the circular tube is calculated according to the bubble radius. We show that extra pressure drop due to the drift bubble in the Hagen–Poiseuille flow becomes positive or negative depending on the radius of the bubble. When the bubble moves along a circular tube in the absence of the Hagen–Poiseuille flow, a series of viscous toroidal eddies appears in the anterior and posterior directions of the bubble. Moreover, we discuss the pressure and normal stress distributions on the bubble surface, which depend on the radius of the bubble.

Axisymmetric Stokes flow due to a point-force singularity acting between two coaxially positioned rigid no-slip disks

Abstract

Axisymmetric Stokes flow around a sphere translating along the axis of a circular cylinder

Abstract We investigate the axisymmetric slow viscous flow around a sphere located on the axis of a long circular cylinder analytically based on the Stokes approximation. The sphere translates along the centerline of the cylinder with a constant velocity within Hagen–Poiseuille flow flowing far from the sphere. The translating velocity of the sphere and mean velocity of the Hagen–Poiseuille flow are arbitrary constant. To analyze Stokes equation, we use the method of complex eigenfunction expansions and the method of least squared error. As results, the streamline patterns and the pressure contour lines in the flow field are drawn for some radii of the sphere. The drag exerted on the sphere and the pressure change by the sphere are determined as the radius of the sphere. For a small sphere radius and for a sphere fitted closely in the cylinder, we compared results with previous asymptotic results and lubrication theory results, respectively. The velocity of the drifting sphere by the Hagen–Poiseuille flow in the circular cylinder is determined as the sphere radius. The pressure change induced by the drifting sphere in the Hagen–Poiseuille flow is also obtained. When the sphere translates along the plugged circular cylinder, a series of viscous toroidal eddies appears each side of the cylinder expectably. Moreover, we discussed the pressure and shear stress on the sphere surface and showed some critical range of these stresses.

Axisymmetric Stokes flow due to the motion of a thin disk along the axis of a circular tube

The axisymmetric Stokes flow around a disk in a circular tube is investigated based on a Stokes approximation, where the radius of the disk, located coaxially with the circular tube, is arbitrary. The disk translates perpendicularly to its own plane along the centerline of the tube, and a Hagen–Poiseuille flow exists far upstream and downstream from the disk. The velocities of the translating disk and the Hagen–Poiseuille flow are given arbitrarily. The problem is investigated through an analysis of the Stokes equation using a complex eigenfunction expansion and the least squares method. The streamline patterns from the stream function, and the pressure distributions of the flow fields, are shown for some typical radii of the disk. The force exerted on the disk, and the pressure drop created by the disk itself, are both calculated as functions of the disk radius. For a small disk radius, the results of the force are coincident with previous asymptotic expressions. For a given velocity of the Hagen–Poiseuille flow in the tube, the translational drift velocity of the disk is determined as a function of the disk radius. The results show that this drift velocity is slightly lower than the mean velocity of the Hagen–Poiseuille flow component projected by the disk. The induced pressure drop from the disk drifting in the Hagen–Poiseuille flow is quite small. When the disk translates in a stagnant circular tube, a series of viscous toroidal eddies appears in the tube apart from the disk, as expected.

Stratlets: Low Reynolds Number Point-Force Solutions in a Stratified Fluid

We present fundamental solutions of low Reynolds number flows in a stratified fluid, including the case of a point force (Stokeslet) and a doublet. Stratification dramatically alters the flow by creating toroidal eddies, and velocity decays much faster than in a homogeneous fluid. The fundamental length scale is set by the competition of buoyancy, diffusion and viscosity, and is O(100 μm-1 mm) in aquatic environments. Stratification can therefore affect the swimming of small organisms and the sinking of marine snow particles, and diminish the effectiveness of mechanosensing in the ocean.

Effects of large scale eddies and stagnation surfaces on microcrystallization

A Lagrangian marker particle (LMP) method is applied to measure the toroidal large-scale eddies (LSEs) and their enveloping stagnation surfaces in a 280 l bottom-sweeping model crystallizer. The trajectories of a 0.4 cm diameter LMP show that these stagnation surfaces inhibit transport. Analysis shows that the velocity component normal to stagnation surfaces vanish. Therefore, stagnation surfaces act as a semi-permeable barriers to particle transport. Microconductivity measurements show that the stagnation surfaces are leaky at the molecular scale. Thus particle transport through stagnation surfaces is size-dependent. The LMP measurements reveal the structure of the LSEs. This consists of (1) an upward-swirling flow adjacent to the tank perimeter extending from the bottom to the top of the tank, (2) a central, quiescent zone, and (3) a downward return flow between (1) and (2) through a system of nested, smaller diameter, secondary toroidal flows concentric with the impeller axis. A cylindrical stagnation surface surrounds the central quiescent zone. These results are corroborated by measurements of inhomogeneous concentration profiles in an industrial scale 2000 l batch crystallizer. This leads to an understanding of the effects of LSEs on silver halide microcrystal particle size distribution in the industrial scale crystallizer.

Exact solutions for Stokes flow in and around a sphere and between concentric spheres

A general method is suggested for deriving exact solutions to the Stokes equations in spherical geometries. The method is applied to derive exact solutions for a class of flows in and around a sphere or between concentric spheres, which are generated by meridional driving on the spherical boundaries. The resulting flow fields consist of toroidal eddies or pairs of counter-rotating toroidal eddies. For the concentric sphere case the exact solution when the inner sphere is in instantaneous translation is also derived. Although these solutions are axisymmetric, they can be combined with swirl about a different axis to generate fully three-dimensional fields described exactly by simple formulae. Examples of such complex fields are given. The solutions given here should be useful for, among other things, studying the mixing properties of three-dimensional flows.

Evolution of toroidal magnetic eddies in an ideal fluid

The magnetohydrodynamic evolution of axisymmetric magnetic eddies within which the magnetic field is purely toroidal with . When the fluid is initially at rest, the magnetic eddy first contracts towards the axis of symmetry under the action of its Lorentz force distribution; then two spherical fronts form, which propagate in the two opposite directions along the axis of symmetry, in a manner captured well by the exact solution. The magnetic energy remains bounded away from zero despite the fact that there is no topological barrier to its further decrease. Magnetic eddy evolution and the possible existence of steady states under a uniform compressive strain field is also numerically investigated.

Analytical solutions for a spherical particle near a wall in axisymmetrical polynomial creeping flows

Analytical solutions are presented for axisymmetric creeping flows around a spherical particle near a wall, when the unperturbed flow velocity varies as polynomial with the coordinates. The perturbed fluid velocity and pressure are calculated directly with the method of bipolar coordinates. They are obtained with a 10−16 precision, even for small gaps of the order of 10−6 sphere radius. For a constant unperturbed flow, the problem is equivalent to that of a sphere moving perpendicularly to a wall in a fluid at rest. An alternative indirect solution for the fluid velocity and pressure is obtained in this case from the solution of Brenner [Chem. Eng. Sci. 16, 242 (1961)] and Maude [Brit. J. Appl. Phys. 12, 293 (1961)] for the stream function. For a small gap between the sphere and the wall, the values of the pressure are compatible with the ones from the lubrication approximation but are systematically larger; this may be important for applications. Calculations are also performed when the unperturbed flow velocity is a polynomial of degree 2 and 3 in the coordinates, viz., for different types of stagnation point flows. Various flow structures are obtained, depending on the particle to wall distance. When the sphere approaches the wall, there is an increasing number of nested toroidal eddies, providing a link between the case of a single toroidal eddy when the sphere is far from the wall and the infinite set of Moffatt eddies in the gap between bodies in contact. The flow structure is analogous to that for two equal spheres in a uniform flow field, cf. Davis, O’Neill, Dorrepaal, and Ranger [J. Fluid Mech. 77, 625 (1976)]. Precise results for the force on the sphere are provided.

Turbulent flow dynamics, heat transfer and mass exchange in the melt of induction furnaces

Experimental investigations of the turbulent flow velocities measured in the melt of experimental induction furnaces show, that beside the intensive local turbulence pulsations, macroscopic low‐frequency oscillations of the recirculated toroidal main flow eddies play an important role in the heat and mass exchange processes. Traditional numerical calculations of the flow and transfer processes, based on wide spread commercial codes using various modifications of the k‐ε turbulence model show that these models do not take into account the low‐frequency oscillations of the melt flow and the calculated temperature and concentration distributions in the melt essentially differs from experimental results. Therefore, the melt flow dynamics in an induction crucible furnace was numerically simulated with help of transient three‐dimensional calculations using the large eddy simulation turbulence model. This leads to a good agreement between calculated and measured periods of low‐frequency oscillations and heat and mass transfer between the toroidal flow eddies.

Enhanced efficiency of feeding and mixing due to chaotic flow patterns around choanoflagellates

The motion of particles and feeding currents created by micro-organisms due to a flagellum are considered. The calculations are pertinent to a range of sessile organisms, but we concentrate on a particular organism, namely Salpingoeca amphoridium (SA) (a choanoflagellate), due to the availability of experimental data (Pettitt, 2000). These flow fields are characterized as having very small Reynolds numbers, which implies that viscous forces dominate over inertial ones consistent with using the Stokes flow equations. The flow generated by the flagellum is modelled via the consideration of a point force known as a stokeslet. The interaction between the boundary, to which the organism is attached, and its flagellum leads to toroidal eddies, which serve to transport particles towards the micro-organism, promoting filtering of nutrients by the microvilli which constitute the cell's collar (the filtering mechanism in SA). It is our conjecture that the interaction of multiple toroidal eddies will lead to chaotic advection and hence enhance the domain of feeding for these organisms. The degree of mixing in the region around SA is investigated using chaotic and statistical measures to study the influence the flagellum has on the surrounding fluid. The three-dimensional particle paths around such organisms are also considered with the aim of showing that the plane within which they are situated is an attractor.

Enhanced efficiency of feeding and mixing due to chaotic flow patterns around choanoflagellates

The motion of particles and feeding currents created by micro-organisms due to a flagellum are considered. The calculations are pertinent to a range of sessile organisms, but we concentrate on a particular organism, namely Salpingoeca amphoridium (SA) (a choanoflagellate), due to the availability of experimental data (Pettitt, 2000). These flow fields are characterized as having very small Reynolds numbers, which implies that viscous forces dominate over inertial ones consistent with using the Stokes flow equations. The flow generated by the flagellum is modelled via the consideration of a point force known as a stokeslet. The interaction between the boundary, to which the organism is attached, and its flagellum leads to toroidal eddies, which serve to transport particles towards the micro-organism, promoting filtering of nutrients by the microvilli which constitute the cell's collar (the filtering mechanism in SA). It is our conjecture that the interaction of multiple toroidal eddies will lead to chaotic advection and hence enhance the domain of feeding for these organisms. The degree of mixing in the region around SA is investigated using chaotic and statistical measures to study the influence the flagellum has on the surrounding fluid. The three-dimensional particle paths around such organisms are also considered with the aim of showing that the plane within which they are situated is an attractor.

Instantaneous Stokes Flow in a Conical Apex Aligned with Gravity and Bounded by a Stress-Free Surface

The instantaneous viscous motion near the apex of a fluid cone of included angle $2\theta_0$ with stress-free boundary conditions is studied in the limit of zero Reynolds number. Gravity acts parallel to the conical axis, and surface tension is neglected. When $2\theta_0 > 134.6^{\circ}$ the dominant term of the solution close to the apex is independent of gravity and depends only on the far-field boundary conditions; the leading behavior is thus a self-similar solution of the second kind. In this case the instantaneous flow is akin to the steady flow in a rigid cone reported by Liu and Joseph [SIAM J. Appl. Math., 34 (1978), pp. 286--296] who showed that toroidal eddies appear below a critical apex angle. For $2\theta_0 < 134.6^{\circ}$ the flow close to the apex is dominated by gravity and represents a self-similar solution of the first kind. The complete eigenvalue problem is solved and example streamline patterns are presented. We estimate the short-time behavior of flow in the neighborhood of the ape...

Simulation of the Piston Driven Flow Inside a Cylinder With an Eccentric Port

A grid-free Lagrangian approach is applied to simulate the high Reynolds number unsteady flow inside a three-dimensional domain with moving boundaries. For this purpose, the Navier-Stokes equations are expressed in terms of the vorticity transport formulation. The convection and stretch of vorticity are obtained using the Lagrangian vortex method, while diffusion is approximated by the random walk method. The boundary-element method is used to solve a potential flow problem formulated to impose the normal flux condition on the boundary of the domain. The no-slip condition is satisfied by a vortex tile generation mechanism at the solid boundary, which takes into account the time-varying boundary surfaces due to, e.g., a moving piston. The approach is entirely grid-free within the fluid domain, requiring only meshing of the surface boundary, and virtually free of numerical diffusion. The method is applied to study the evolution of the complex vortical structure forming inside the time-varying semi-confined geometry of a cylinder equipped with an eccentric inlet port and a harmonically driven piston. Results show that vortical structures resembling those observed experimentally in similar configurations dominate this unsteady flow. The roll-up of the incoming jet is responsible for the formation of eddies whose axes are nearly parallel to the cylinder axis. These eddies retain their coherence for most of the stroke length. Instabilities resembling conventional vortex ring azimuthal modes are found to be responsible for the breakup of these toroidal eddies near the end of the piston motion. The nondiffusive nature of the numerical approach allows the prediction of these essentially inviscid phenomena without resorting to a turbulence model or the need for extremely fine, adaptive volumetric meshes.

Extension of the k-ε model for the numerical simulation of the melt flow in induction crucible furnaces

Checking the calculations of turbulent melt flow in induction crucible furnaces that were carried out with various modifications of the two-dimensional (2-D)k-e model using experimental findings has shown basic differences in the distributions of the specific generation of the turbulent energy and the kinetic energy of the turbulence. The discrepancies are explained by the distinctive three-dimensional (3-D) character of the pulsations and the low-frequency fluctuations of the macroscopic toroidal eddy; these are not taken into account in the numerical methods mentioned. With the aid of this 3-D model, the additional component of turbulent kinetic energy involved is estimated, and an approximation formula for the low-frequency component of the specific generation of turbulence is given. This results in an extension of the 2-Dk-e model for a recirculated flow with several toroidal eddies, leading to good qualitative agreement of the characteristics of turbulent flow with the experimental findings. Since the numerical simulation—in agreement with industrial practice and the experiments carried out—demonstrates good effective mixing of the entire flow region, there is thus a possibility for the simulation of aterial transport in the melt of induction crucible furnaces as part of the widespread 2-D computation methods.

Axisymmetric Couette flow at large Taylor numbers

Measurements have been made in flow between concentric cylinders with only the inner one rotating, for Reynolds numbers (based on flow gap) from 17 000 to 120 000, corresponding to Taylor numbers from 8 × 107 to 4 × 109. At the lower speeds (Reynolds numbers less than 30 000), the large-scale motion consists of toroidal eddies, highly uniform in spacing and intensity and convected by a slow axial flow past fixed sensors. By synchronizing an external oscillator with the passage frequency, flow velocity and small-scale turbulent intensity may be sampled at defined stages of the passage cycle and averaged to provide maps of the velocity fields and the associated distributions of small-scale intensity and Reynolds stress.At higher speeds, the passage of toroidal eddies becomes too irregular to establish the passage cycle, but, by comparing the velocity fluctuations from four inclined hot wires placed near the outer cylinder, the current location of large-scale flow separation from the outer cylinder can be approximately determined. Statistics of the temporal variations of the location show that the large-scale motion still approximates to the toroidal form, but that there are azimuthal variations of separation position and velocity that indicate a change from toroidal to helical eddies. Conditional averages of flow velocity and small-scale turbulent intensity, based on relative distance from the position of flow separation, are very similar in form and magnitude to phase-selected averages obtained at lower speeds.The considerable changes in the large-scale motion that occur as the Reynolds number increases from 300 to 1200 times the critical value are believed to arise from increase in the ‘turbulent Taylor number’ of the central flow, based on variation of angular momentum and on the eddy diffusion coefficients for linear and angular momentum. Effects on the motion of the slow axial flow, always less than 1% of the peripheral flow velocity, are also discussed.

Turbulent Couette flow between concentric cylinders at large Taylor numbers

Measurements have been made of the Couette flow in the annular space between concentric cylinders with a radius ratio of 1.5, the outer cylinder being held stationary and the inner one rotated at speeds to give Taylor numbers in the range 1.0 × 104−2.3 × l06 times the critical value for first instability of the steady viscous flow. Mean velocities have been measured both with Pitot tubes and with linearized hot-wire anemometers, and turbulent intensities and stresses, frequency spectra and space-time correlations have been obtained using single hot-wire anemometers of X-form and linear arrays of eight single-wire anemometers. For Taylor-number ratios to the critical number less than 3 × l05, the most prominent feature of the flow is a system of toroidal eddies, encircling the inner cylinder and uniformly spaced in the axial direction with nearly the separation of the Taylor vortices of the viscous instability. They are superimposed on a background of irregular motion and, except within the thin wall layers, the toroidal eddies contribute more to the total intensity. With increase of rotation speed, the toroidal eddies lose their regularity, and they cannot be clearly distinguished at Taylor-number ratios beyond 5 × l05.The change of flow type from quasi-regular toroidal to fully irregular turbulent takes place over an extensive range of Taylor-number ratio centred near 3 × l05, and it may be linked with changes in the thin wall layers that separate the flow boundaries from the central region of nearly constant circulation. For ratios over 5 × l05, an appreciable part of the wall layers is comparatively unaffected by flow curvature and has a logarithmic distribution of mean velocity similar to that found in channel flows. It is suggested that the motion in the wall layers changes from a set of Gortler vortices characteristic of curved-wall flow to the more irregular motion found on plane walls, causing the toroidal eddies to break into sections of length ranging from a considerable fraction of the flow perimeter to nearly the separation of the cylinders. Changes in the frequency spectra of the radial and azimuthal velocit'y fluctuations are consistent with such a change.

Do eddington–sweet circulations exist?

Abstract The initial value problem for a rotating baroclinic star is considered. Arguments are derived suggesting that the steady meridional circulations first postulated by Vogt (1925) and Eddington (1925) do not exist and that instead a state of a purely zonal velocity field is approached from rather arbitrary initial conditions. By extending previous criteria for stability with respect to axisymmetric disturbances to the case of non-vanishing Prandtl number, it is shown that the purely zonal velocity field may be stable in special cases. In realistic situations, it tends to be unstable, but the preferred instability is likely to introduce nonaxisymmetric toroidal eddies rather than meridional circulations.

On the generation of viscous toroidal eddies in a cylinder

The streamlines due to a stokeslet on the axis in a finite, semi-infinite and infinite cylinder are obtained together with the case of a Stokes-doublet and source-doublet in an infinite cylinder. In the infinite and semi-infinite cylinder examples an infinite set of toroidal eddies are obtained. The eddies alternate in sign and the magnitude of the stream function decays exponentially with distance from the driving singularity. In the finite cylinder a primary interior eddy adjacent to the singularity is always obtained and, depending on location of the singularity within the cylinder and the ratio of cylinder length to radius, a finite number of secondary interior eddies. In the case of long cylinders, the eddies are generated along the axis, whereas, for squat cylinders, secondary eddies occur in the radial direction. The interior eddies emerge from the corner as the length of the cylinder is increased. Moffatt corner eddies exist but they are very much smaller than the interior eddies.