On the structure of a Taylor column driven by a buoyant parcel in an unbounded rotating fluid

2001 ◽  
Vol 427 ◽  
pp. 131-165 ◽  
Author(s):  
DAVID E. LOPER
Keyword(s):  
Rotation Axis ◽  
Viscous Force ◽  
Momentum Balance ◽  
Return Flow ◽  
Rigid Sphere ◽  
Rotating Fluid ◽  
Ekman Number ◽  
Recirculating Flow ◽  
Structure Of Flow ◽  
Taylor Column

The velocity and pressure fields produced in a homogeneous rapidly rotating fluid driven by an isolated buoyant parcel are investigated. Gravity and rotation are allowed to have arbitrary orientations and the parcel shape is assumed Gaussian. Inertial forces and time-dependent effects are ignored. The linear problem is easily solved by three-dimensional Fourier transform, and the inversion is facilitated by assuming the Ekman number, E, to be very small. In this limit the fields form a Taylor column extended in the direction of the rotation axis. In the absence of rigid boundaries no boundary layers occur. The velocity and pressure in the vicinity of the parcel are found in closed form while elsewhere (within the Taylor column) they are expressed in terms of relatively simple scalar integrals which are easily evaluated.Within the buoyant parcel, the momentum balance is baroclinic, involving Coriolis, pressure and buoyancy forces. Outside the parcel, the balance is geostrophic at unit order. The viscous force is important at order E and determines the axial structure of the Taylor column. In contrast to the case of flow driven by a rigid body, no ‘Taylor slug’ of recirculating flow occurs. The velocity and pressure decay algebraically with distance from the parcel, with the scale of variation being a/E in the axial direction and a in the radial direction, where a is the parcel radius. In the vicinity of the parcel, the return flow occurs in a broad region surrounding the parcel. The structure of flow in the vicinity of the parcel is independent of the Ekman number. This return flow sweeps the fringes of the parcel backward, making the net rise speed significantly slower than that of a rigid sphere of identical buoyancy. The return flow also acts to deform the parcel; this deformation is quantified.

Librational forcing of a rapidly rotating fluid-filled cube

2018 ◽  
Vol 842 ◽  
pp. 469-494 ◽  
Author(s):  
Ke Wu ◽  
Bruno D. Welfert ◽  
Juan M. Lopez
Keyword(s):  
Stokes Equations ◽  
Rotation Axis ◽  
Rotation Period ◽  
Rotation Frequency ◽  
Shear Layers ◽  
Navier Stokes ◽  
Rotating Fluid ◽  
Ekman Number ◽  
Slip Boundary ◽  
The Mean

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

The motion of a rising disk in a rotating axially bounded fluid for large Taylor number

1995 ◽  
Vol 291 ◽  
pp. 1-32 ◽  
Author(s):  
Marius Ungarish ◽  
Dmitry Vedensky
Keyword(s):  
Exact Solution ◽  
Boundary Value Problem ◽  
Equations Of Motion ◽  
Taylor Number ◽  
Rossby Number ◽  
Rotating Fluid ◽  
Asymptotic Results ◽  
Dual Integral Equations ◽  
Wall Distance ◽  
Taylor Column

The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

Macroscopic Streaming Associated With Vertical, Cyclic Motion of Interface Confined in a Cylindrical Enclosure

2002 ◽  
Author(s):  
Jun Shimizu ◽  
Takahiro Ito ◽  
Yoshiyuki Tsuji ◽  
Yutaka Kukita
Keyword(s):  
Contact Line ◽  
Large Scale ◽  
Fluid Motion ◽  
Viscous Force ◽  
Wall Material ◽  
Vertical Acceleration ◽  
Stick Slip ◽  
Stable Regime ◽  
Interface Waves ◽  
Recirculating Flow

The interface between overlaid fluids can become unstable when the fluids are excited vertically. The instability caused by the variation in the vertical acceleration is known by the name of the Faraday waves. Ito et al. (1999) studied a combined excitation problem where the fluids were excited vertically in a stationary cylinder while the interface motion was restricted by the mobility of the fluid-fluid-wall contact line. They found that, under such circumstances, the symmetric fundamental mode grows on the interface, even for excitation amplitude and frequency falling in the stable regime of the Faraday wave instability. Furthermore, they found that the contact line exhibits stick-slip-like motion for the combination of fluids and wall material used in their experiments (water and kerosene oil in a cylinder made of acrylic resin). In this paper, we describe and discuss the fluid motions associated with the excitation of fluids and interface wave. It is shown that a unidirectional flow (macroscopic streaming) is induced below the center of the interface when it is excited vertically to produce axisymmetric wave of large amplitudes. This unidirectional, jet-like flow induces a large-scale recirculating flow which extends several cylinder diameters away from the interface, a spatial scale considerably greater than the wavelength or amplitude of the interface waves, and has a time scale much greater than the excitation interval. It is shown that the phase angle between the wave-induced fluid motion and the fluid motion associated with the viscous force along the interface plays an important role in establishing the large scale stream motion of the fluids.

The theory of trailing Taylor columns

1970 ◽  
Vol 68 (2) ◽  
pp. 485-491 ◽  
Author(s):  
M. J. Lighthill
Keyword(s):  
Flow Field ◽  
Experimental Evidence ◽  
Small Angle ◽  
Large Body ◽  
Physical Significance ◽  
The Body ◽  
Rossby Number ◽  
Ekman Number ◽  
Axis Of Rotation ◽  
Taylor Column

AbstractWhen Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.

Inertial oscillations in a rotating fluid cylinder

1970 ◽  
Vol 40 (3) ◽  
pp. 603-640 ◽  
Author(s):  
A. D. McEwan
Keyword(s):  
Rotation Axis ◽  
Rotation Frequency ◽  
Column Height ◽  
Rotating Fluid ◽  
Momentum Exchange ◽  
Inertial Oscillations ◽  
Large Amplitude Oscillation ◽  
Theoretical Predictions ◽  
Integral Energy ◽  
Secondary Oscillation

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.Some investigation is made of the phenomenon ofresonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.

Dynamics of a vortex ring in a rotating fluid

1996 ◽  
Vol 317 ◽  
pp. 215-239 ◽  
Author(s):  
R. Verzicco ◽  
P. Orlandi ◽  
A. H. M. Eisenga ◽  
G. J. F. Van Heijst ◽  
G. F. Carnevale
Keyword(s):  
Vortex Ring ◽  
Stokes Equations ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Swirl Flow ◽  
Degree Of Polarization ◽  
Navier Stokes ◽  
Vortex Rings ◽  
Rotating Fluid ◽  
Translation Velocity

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier–Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a ‘degree of polarization’.If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown.Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

A bathtub vortex under the influence of a protruding cylinder in a rotating tank

2013 ◽  
Vol 733 ◽  
pp. 134-157 ◽  
Author(s):  
Yin-Chung Chen ◽  
Shih-Lin Huang ◽  
Zi-Ya Li ◽  
Chien-Cheng Chang ◽  
Chin-Chou Chu
Keyword(s):  
Vortex Structure ◽  
Stokes Equations ◽  
Finite Thickness ◽  
Ekman Pumping ◽  
Height Ratio ◽  
Coriolis Parameter ◽  
Ekman Number ◽  
Drain Hole ◽  
Bathtub Vortex ◽  
Taylor Column

AbstractNumerical simulations and laboratory experiments were jointly conducted to investigate a bathtub vortex under the influence of a protruding cylinder in a rotating tank. In the set-up, a central drain hole is placed at the bottom of the tank and a top-down cylinder is suspended from the rigid top lid, with fluid supplied from the sidewall for mass conservation. The cylinder is protruded to produce the Taylor column effect. The flow pattern depends on the Rossby number ($\mathit{Ro}= U/ fR$), the Ekman number ($\mathit{Ek}= \nu / f{R}^{2} )$ and the height ratio, $h/ H$, where $R$ is the radius of the cylinder, $f$ is the Coriolis parameter, $\nu $ is the kinematic viscosity of the fluid, $h$ is the vertical length of the cylinder and $H$ is the height of the tank. It is found appropriate to choose $U$ to be the average inflow velocity of fluid entering the column beneath the cylinder. Steady-state solutions obtained by numerically solving the Navier–Stokes equations in the rotating frame are shown to have a good agreement with flow visualizations and particle tracking velocimetry (PTV) measurements. It is known that at $\mathit{Ro}\sim 1{0}^{- 2} $, the central downward flow surrounded by the neighbouring Ekman pumping forms a classic one-celled bathtub vortex structure when there is no protruding cylinder ($h/ H= 0$). The influence of a suspended cylinder ($h/ H\not = 0$) leads to several findings. The bathtub vortex exhibits an interesting two-celled structure with an inner Ekman pumping (EP) and an outer up-drafting motion, termed Taylor upwelling (TU). The two regions of up-drafting motion are separated by a notable finite-thickness structure, identified as a (thin-walled) Taylor column. The thickness ${ \delta }_{T}^{\ast } $ of the Taylor column is found to be well correlated to the height ratio and the Ekman number by ${\delta }_{T} = { \delta }_{T}^{\ast } / R= {(1- h/ H)}^{- 0. 32} {\mathit{Ek}}^{0. 095} $. The Taylor column presents a barrier to the fluid flow such that the fluid from the inlet may only flow into the inner region through the narrow gaps, one above the Taylor column and one beneath it (conveniently called Ekman gaps). As a result, five types of routes along which the fluid may flow to and exit at the drain hole could be identified for the multi-celled vortex structure. Moreover, the flow rates associated with the five routes were calculated and compared to help understand the relative importance of the component flow structures. The weaker influence of the Taylor column effect on the bathtub vortex at $\mathit{Ro}\sim 1$ or even higher $\mathit{Ro}\sim 1{0}^{2} $ is also discussed.

Experiments on turbulence in a rotating fluid

1975 ◽  
Vol 68 (4) ◽  
pp. 639-672 ◽  
Author(s):  
A. Ibbetson ◽  
D. J. Tritton
Keyword(s):  
Rotation Rate ◽  
Rotation Axis ◽  
Turbulence Energy ◽  
Length Scale ◽  
Grid Turbulence ◽  
Rotating Fluid ◽  
Mean Flow ◽  
Length Scales ◽  
Wide Range ◽  
Energy Sink

Experiments have been carried out to investigate the effect of rotation of the whole system on decaying turbulence, generally similar to grid turbulence, generated in air in an annular container on a rotating table. Measurements to determine the structure of the turbulence were made during its decay, mean quantities being determined by a mixture of time and ensemble averaging. Quantities measured (as functions of time after the turbulence generation) were turbulence intensities perpendicular to and parallel to the rotation axis, spectra of these two components with respect to a wavenumber perpendicular to the rotation axis, and some correlation coefficients, selected to detect differences in length scales perpendicular and parallel to the rotation axis. The intensity measurements were made for a wide range of rotation rates; the other measurements were made at a single rotation rate (selected to give a Rossby number varying during the decay from about 1 to small values) and, for comparison, at zero rotation. Subsidiary experiments were carried out to measure the spin-up time of the system, and to determine whether the turbulence produced any mean flow relative to the container.A principal result is that increasing the rotation rate produces faster decay of the turbulence; the nature of the additional energy sink is an important part of the interpretation. Other features of the results are as follows: the measurements with-outrotation can be satisfactorily related to wind-tunnel measurements; even with rotation, the ratio of the intensities in the two directions remains substantially constant; the normalized spectra for the rotating and the non-rotating cases show surprising similarity but do contain slight systematic differences, consistent with the length scales indicated by the correlations; rotation produces a large increase in the length scale parallel to the rotation axis and a smaller increase in that perpendicular to it; the turbulence produces no measurable mean flow.A model for the interpretation of the results is developed in terms of the action of inertial waves in carrying energy to the boundaries of the enclosure, where it is dissipated in viscous boundary layers. The model provides satisfactory explanations of the overall decay of the turbulence and of the decay of individual spectral components. Transfer of energy between wavenumbers plays a much less significant role in the dynamics of decay than in a non-rotating fluid. The relationship of the model to the interpretation of the length-scale difference in terms of the Taylor-Proudman theorem is discussed.The model implies that the overall dimensions of the system enter in an important way into the dynamics. This imposes a serious limitation on the application of the results to the geophysical situations at which experiments of this type are aimed.The paper includes some discussion of the possibility of energy transfer from the turbulence to a mean motion (the ‘vorticity expulsion’ hypothesis). It is possible, on the basis of the observations, to exclude this process as the additional turbulence energy sink. But this does not provide any evidence either for or against the hypothesis in the conditions for which it has been postulated.

Waves generated in rotating fluids by travelling forcing effects

1973 ◽  
Vol 61 (1) ◽  
pp. 129-158 ◽  
Author(s):  
G. V. Prabhakara Rao
Keyword(s):  
Wave Pattern ◽  
Far Field ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Fixed Frequency ◽  
Arbitrary Angle ◽  
Upstream Side ◽  
Axis Of Rotation ◽  
The Right ◽  
Taylor Column

The two-dimensional wave pattern produced in a homogeneous rotating fluid by a forcing effect oscillating with a frequency σ′0 and travelling with a uniform speed U along a line inclined to the axis of rotation at an arbitrary angle α is studied following Lighthill's technique. It is shown how the far field changes with α and σ′0.For all σ′0 < 2Ω, except for σ′0 = 2Ω sin α (Ω being the angular velocity of the fluid), the forcing effect excites two systems of waves. When σ′0 → 2Ω sin α one of these systems spreads out, influencing the upstream side while the other shrinks in the downstream direction. This upstream influence is to the left or to the right of the line of motion of the forcing effect (the forcing line) according as σ′0 − 2Ω sin α[lg ] 0 and increases as σ′0 − 2Ω sin α decreases. For σ′0 > 2Ω there is only a single system propagating downstream. As α varies these systems undergo a kind of rotation retaining the main features. α ≠ 0 or ½π makes the pattern asymmetric about the forcing line while a non-zero σ′0 splits the steady-case identical wave systems into two, which are otherwise coincident.When σ′0 = 2Ω sin α the forcing effect excites straight unattenuated waves of fixed frequency travelling both ahead and behind in a ‘column’ parallel to the forcing line and enclosing it. Also there are two other systems, which propagate without penetrating into an upstream wedge. It is shown that this ‘column’ is the counterpart of the ‘Taylor column’.

Unsteady flow about a sphere at low to moderate Reynolds number. Part 2. Accelerated motion

1995 ◽  
Vol 303 ◽  
pp. 133-153 ◽  
Author(s):  
Eugene J. Chang ◽  
Martin R. Maxey
Keyword(s):  
Flow Separation ◽  
Mass Term ◽  
Reynolds Numbers ◽  
Viscous Force ◽  
Recirculation Region ◽  
Rigid Sphere ◽  
Stream Velocity ◽  
Accelerated Motion ◽  
Moderate Reynolds Number ◽  
Complete Detachment

A full numerical simulation based on spectral methods is used to investigate linearly accelerating and decelerating flows past a rigid sphere. Although flow separation does not occur at Reynolds numbers below 20 for a steady flow, in the linearly decelerating flow separation is observed at much lower Reynolds numbers with complete detachment of vorticity possible in certain cases. The existence of a large recirculation region contributes to the result that a negative viscous force on the sphere is possible. The contribution of the pressure to the force includes a component that is well described by the inviscid added-mass term in both the accelerating and decelerating cases. The force on the sphere is found in general to initially decay in a power law manner after acceleration or deceleration ends followed by rapid convergence at later times to the steady state. For the cases examined this convergence is found to be exponential except for those in which the sphere is brought to rest in which case the convergence remains algebraic. This includes the special case of an infinite acceleration or deceleration where the free stream velocity is impulsively changed.

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