KSME International Journal, VoL 18 No.2, pp. 313-324, 2004

Numerical Study on the Motion of AzimuthalVortices in Axisymmetric Rotating Flows

Yong Kweon Suh*Division of Mechanical and Industrial System Engineering, Dong-A University,

840 Hadan-dong, Saha-gu; Busan 604-714, Korea

A rich phenomenon in the dynamics of azimuthal vortices in a circular cylinder caused by theinertial oscillation is investigated numerically at high Reynolds numbers and moderate Rossbynumbers. In the actual spin-up flow where both the Ekman circulation and the bottom frictioneffects are included, the first appearance of a seed vortex is generated by the Ekman boun­dary-layer on the bottom wall and the subsequent roll-up near the corner bounded by the sidewall. The existence of the small vortex then rapidly propagates toward the inviscid region andinduces a complicated pattern in the distribution of azimuthal vorticity, i.e. inertial oscillation.The inertial oscillation however does not deteriorate the classical Ekman-pumping model in thetime scale larger than that of the oscillatory motion. Motions of single vortex and a pair ofvortices are further investigated under a slip boundary-condition on the solid walls. For the caseof single vortex, repeated change of the vorticity sign is observed together with typical pro­pagation of inertial waves. For the case of a pair of vortices with a two-step profile in the initialazimuthal velocity, the vortices' movement toward the outer region is resisted by the crescent­shape vortices surrounding the pair. After touching the border between the core and outerregions, the pair vortices weaken very fast.

Key Words: Rotating Flows, Inertial Oscillation, Spin-up, Circular Cylinder, AzimuthalVortex

313

1. Introduction

Rotating flows are ubiquitous in natural andtechnological environment. In view of its conse­quence in engineering and scholarly interest, ithas drawn much attentions of the fluid dynami­cists from long time ago. Its application can befound in broad areas such as turbomachinery,oceanic flows, atmospheric flows, the motion ofliquid fuel within space crafts, mantle movementinside the Earth, and spin coating, etc. In morespecific terms, the rotating flows are characterized

• E-mail: yksuh@daunet.donga.ac.krTEL: +82-5!-200-7648; FAX: +82-51-200-7656Division of Mechanical and Industrial System Engi­neering, Dong-A University, 840 Hadan-dong, Saha­gu, Busan 604-714, Korea. (Manuscript Received July9,2003; Revised November 17,2003)

by relative flows under a background rotation;the ratio of the representative magnitude of rela­tive-flow velocity and that of the backgroundrotational velocity is defined as the Rossby num­ber. The relative flows are intrigued by variousmechanisms. In the spin-up flows, the rotationalspeed is just increased abruptly, and the subse­quent flow development, i.e. decay, in time be­comes the primary interest. The relative flows canbe generated by other forces such as the buoyantforce, the magnetic force, and the surface forcedriven by winds. Among these, the body forcegiven by the increase (or decrease in the 'spin­down' case) of the rotational speed presents themost fundamental subject known as 'spin-up'flow.

Early investigation on the spin-up flows hasfocussed on axisymmetric containers and aimedto understand the basic underlying mechanism

314 Yang K wean Suh

(2)

associated with the decaying of the primaryflow. Greenspan and Howard (1963) studied onthe spin-up of fluid in between two infiniteparallel plates. They showed that at small Rossbynumbers the linearized version of the governingequations can well predict the spin-up time.Wedemeyer (1964) proposed an approximationmethod for treating the non-linear equationsfor the spin-up from rest (Rossby number = 1).He applied his approximation method to thecircular-cylinder case and showed that duringthe spin-up process a shear discontinuity frontappears near the side wall between the regions ofpositive and negative vorticity and it moves to­ward the central axis.

On the other hand, recently there have beenincreased interests in improving the Ekmanpumping models (e.g. Hart 1995, 2000; ZavalaSanson and van Heijst 2000; Suh and Choi2002). The proposed models compare well withexperimental results even in the non-axisymme­tric container such as a rectangle. However it stilldemands further improvement especially in thevery beginning stage and at moderate Rossbynumbers. As a fundamental motivation of thisstudy, it is assumed that the inertial oscillations(see e.g. Greenspan 1968) may be one of the keyreasons for the discrepancy. For instance Cederlof(1988) suggested that the inertial oscillation inthe spin-up inside a circular cylinder may giverise to Lagrangian drift motion. Dolchanskii,Krymov and Manin (1992) showed experiment­ally that considerable amount of inertial oscilla­tions takes place in the circular cylinder of smallratio of height to a diameter. The inertial oscil­lations also have been observed numerically byHyun, Fowlis and Wam-Varnas (1982). How­ever there have been no detailed reports on thestudy of inertial oscillations and its effect on thedecay of the primary azimuthal flows.

In the literature however most studies concern­ing the inertial oscillation are oriented to thestructure of the wave field, and no studies havebeen given to its connection with the bizarrebehavior of the horizontal vortices (here the term'horizontal' indicates 'normal' to the rotationalaxis which is vertical) within the rotating fluids.

The present study is thus purposed to investi­gate the motion of the horizontal vortices andthe influence of the inertial oscillation to thevortices' motions. For simplicity, we focuss ourattention to the axisymmetric-flow configuration,and use numerical methods to obtain the solu­tions of the basically two-dimensional Navier­Stokes equations. In Section 2, the problem isformulated and the numerical methods are des­cribed. To verify the two-dimensional numericalcode, we also employed one-dimensional equa­tions by using the Ekman pumping models deve­loped by Suh and Choi (2002). The Wedeme­yer's solution is also presented in this sectionfor further verification of the two-dimensionalcode. A simple linear analysis for the inertialoscillation is given in Section 3. The numericalresults and discussions are then presented inSection 4, and final conclusions are summarizedin Section 5.

2. Formulation, NumericalMethods and Analysis

2.1 Two-dimensional equationsConsider a circular cylinder of radius R con­

taining a viscous fluid of density p and viscosityII with a free surface and rotating in a constantangular velocity Qs. The flow is maintained in asolid body rotation. Then at an instant of time therotational speed is suddenly increased to 1.1=.Qs+LJ.Q. Assuming that the flow is axisymmetricand taking 1/LJ.Q, R, RLJ.Q, pRz.QLJ.Q as thereference quantities for the time, length, velocity,and pressure, we can write the non-dimensionalgoverning equations as follows.

3v QV av uv 2 Idf+uar+waz+y+S-u= Re £::"zv (I)

K+uK+wK- ul; +.l- avz+1.-~

at ar 3z r r az e QZ

I= Re £::"21;

(3)

where t is time, (r, z) the coordinates alongthe radial and axial directions, respectively, u,v and w the velocity components in the radial,

Numerical Study on the Motion of Azimuthal Vortices in Axisymmetric Rotating Flows 315

azimuthal, and axial directions, respectively, pthe pressure, and Sthe azimuthal vorticity; s=owlor-ouloz. Here the coordinates are coro­tating with the circular cylinder. The stream func­tion ¢ is related to the velocity components as

the time integration for the S-equation is per­formed by the forth-order Runge-Kutta method,while the stream function equation is solved by aconjugate gradient method.

v=vo(r), s=so(r, a) at t=O (6)

(8)

(10)

I rrU= hr Jo rwEdr

Analysis for the bottom boundary-layer flowprovides the Ekman pumping velocity WE;

2.2 One-Dimensional equations with animproved ekman pumping model

When the Rossby number e is small enough,the Taylor-Proudman theorem applies and thequasi-geostrophic flow is governed by

Note that the first term on the right-hand sidecorresponds to the classical linear pumping mo­del. The higher-order correction on the right­hand side is in fact consistent with that derivedby Hart (2000) except that his paper has typogra­phical errors.

Although U at the side wall is in principlezero, a small non-zero of U at this wall canlead to the overall numerical instability. Thus, atevery time step we subtract from WE the averagedquantity iiJ defined as

where U and V denote the depth averages of uand v, respectively, and f::,.1=()2!ar2+alrar­

1/r 2• The radial velocity component U is given

by the integration of the continuity equation interms of the Ekman pumping velocity WE fromthe edge of the boundary layer near the bottomwall;

WE=J.. IS [(oV +JC)2VRe or r(9)

(9 V sv+ 7 V!fV+ 7 sv oV)]

r s SOrar 80 or2 80arar

(4)u=~ w=--!.2J:1!...oz' r or

Impermeable conditions are employed on allthe boundaries surrounding the fluid. The freesurface at z=h is set with vanishing shear stress.On the solid walls at z=O and r= I, the no­slip conditions are applied in Section 4.1, wherethe symptom of the inertial oscillation causedby the corner vortex in the actual spin-up flowsis presented, and the fluid's slip is permitted inSection 4.2, where the numerical solutions forthe behavior of isolated vortices without beingbothered by the presence of the solid boundariesare presented.

The initial conditions for the equations (1) and(2) are

Re= R2~Q , e= ~, h=~ (5)

and the operator .6.2 denotes .6.2=()2lor2+o1

rar -II r2+()2I oz2. Three dimensionless para­meters are Re, the Reynolds number, e, theRossby number and h, the dimensionless liquiddepth, defined as

In Section 4.1 we specify Vo= - r and ~o=O cor­responding to the spin-up from rest. In Section4.2 we set essentially an arbitrary profile for Voand set so=O all over the domain except withinone or two small circles in which uniform vor­ticity is specified.

To enhance the grid resolution near the bottomwall, an exponential function is used with twoparameters band Zo;

where 1J is a new variable to be used In thenumerics with a unform grid size.

The governing equations (I) through (3) arediscretized by the centered difference method and

2.3 Analysis for one-dimensional inviscidflows with linear pumping law

If the linear pumping law is considered, Ubecomes from (8) and (9)

316 Yong K weon Suh

( IS)

(16)

( 17)2kS

In terms of the stream function this is written as

EJk2+t2

Typically the frequency S is of 0 (IIE). Also

note that the frequency takes a maximum value

The separable form of the solution can be ex­

pressed as

ISmax=-;

4. Results of the NumericalComputations

where k is the wave number along the z-direc­tion, l corresponds to the wave number along the

r-direction, and S is the frequency given by

when the flows in the axial plane are purely

radial, i.e. l =0, while it becomes zero, or the

flows are steady, when the streamlines are verti­

cal, i.e. k=O. Otherwise, the wave propagates

vertically with the phase velocity sl k, or 2/

(EJk2+/2) . Thus the mode propagates faster at

lower Rossby numbers.

Equation (14) also implies that vertical gra­

dient in the radial velocity u causes an increase

of the time rate of the azimuthal vorticity ~.

Therefore, even when the flow field is initially

irrotational, if there is a region where the radial

velocity has a vertical gradient, then a vorticity

field is induced in the region as will be shown in

the following section.

4.1 Spin-up flows within a circular cylinderThe two-dimensional numerical computations

for the spin-up process of a viscous fluid in a

circular cylinder are performed with grids 201 X

101 at Re=IOOOO, E=h=0.5 and with grids

301 X 151 at Re=30000, E=h=O.5. For the

strain of the z-coordinate, b=3 and Zo=O.1 are

chosen. In most cases .1t=O.OI gives the con­

verged solutions. These computational experi­

ments are purposed firstly to verify the numerical(14)

( (2)

(lla)

(II b)

(13b)

-r

r-Ilrv

v

for r < rs, and

for r ~ rs; where re is determined by

re= [I - E+ Ee-JEtll.£]1/2

This equation admits a separation-of-variables

form as a solution. It can be shown that the

solution can be written as

3. Inertial Oscillations

where E=e/Re indicates the Ekman number

based on the radius of the cylinder (not on the

liquid depth). Substituting this into the inviscid

version of (7) gives

The most important phenomenon in asso­

ciation with the motion of the azimuthal vor­

tices under the background rotation is the 'iner­

tial oscillation' or the 'inertial wave' (see e.g.

Greenspan, (968). The time scale of the inertial

motion is much smaller than the one of the advec­

tion in the axial plane at low Rossby numbers.

Thus in the inviscid and low-Rossby-number

limit, the only terms which make balance with

the Coriolis terms in equations (I) and (2) are

avIat and a~lOt. Then (I) and (2) reduce to

av 2 ( )af=--;u 13a

K=_~.E!:!-at E az

Eliminating v from these two gives

The above solutions are firstly derived by

Wedemeyer (1964), and later by Maas (1993)

and van de Konijnenberg and van Heijst (1995).

Numerical Study on the Motion of Azimuthal Vortices in Axisymmetric Rotating Flows 317

code by comparing the solutions with those givenby the one-dimensional model, and secondlyto visualize the complex evolution of the cornervortex caused by the background rotation, i.e.inertial oscillation.

Figure I shows the time history of distribu­tions of the vertical vorticity at the free surfaceto (z, h. t) = [a(rv) / riirl~=h obtained by thetwo-dimensional calculation. Also shown in thisfigure are distributions of the vertical vorticitycomputed by the one-dimensional equation aswell as the Wedemeyer's analytic solutions. It isseen that the one-dimensional numerical resultsare in reasonable agreement with the two-di­mensional ones. On the other hand due to the lackof the viscous effects the Wedemeyer's solutionsare only useful in predicting the location of thepropagating front and the level of vorticity withinthe core, i.e. for r< re. We also note that the two

numerical results show some discrepancy in theouter region. However these results suffice toprove the validity of the two-dimensional code.

Figure 2 reveals, in the axial plane, the velo­city vectors composed of u and w at two ins­tants of time obtained by two-dimensional com­putation for h=€=O.5 and Re=3OOO0. Indeedthe velocity magnitude in this plane is shown tobe much smaller than that of the azimuthal ve­locity (refer to the magnitude of the referencevector on the top of each figure). However, it isthis flow that is responsible for the much fasterspin-up than would be attained if only themolecular diffusion were considered (e.g. ZavalaSanson and van Heijst 2000; Suh and Choi2002). Thus understanding the evolution of flowpatterns in the axial plane seems to be importantfor e.g. improvement of the Ekman pumpingmodel. The vortical flows shown in this plane

154

3

2

s

o ---..-.

o 0.2

(b)

O~-----_··········__·-01-------

•.•.•.•.•.•.•.•.•..•.•.•.•.•._.c: '

0.75

I

I "'I ,.

I.i

.(I,.

.. ---I' ------j ,

,. I,. I

I

0.25 0.5r

(d) t=105

a

0.5

-0.5

s

0.750.25 0.5

r(c) t=75

·1 0

-0.5

S 0.5

Fig. 1 Distribution of the vertical vorticity (J) at z= h given by two-dimensional numerics (solid line), one­

dimensional numerics (dash-dot) and Wedemeyer's solution (dashed) for Re=3000 and h=e=O.5 at

four instant of times

318 Yong Kweon Suh

the two-dimensional numerics for e=0.2 and0.5. It clearly exhibits the oscillatory motionsthe period being much shorter than the spin-up

time scale 0 (!If/ he). It also demonstrates thehigher frequency at lower Rossby numbers inline with the analysis presented in the previoussection. On the other hand, the fundamental as­sumption behind the Ekman pumping modelsproposed hitherto is that the radial component ofvelocity remains z-Independent, except of coursein the Ekman boundary-layer on the bottomwall. In this sense the considerable variation ofU shown in the axial plane (Figure 2) may castdoubt on the validity of the model. Some authors(e.g. Zavala, Sanson and van Heijst, 2000) sug­gest to use depth-averaged quantities instead ofrequiring the z-independency; however the re­sultant equations for the fluid motions in theaxial plane are the same as far as the effect of theoscillatory motions is not considered.

Figure 4 shows the time average of the velo­city vectors presented in Figure 2. We can seethat in the region r >0.4 the radial component uis almost uniform along the vertical coordinate.This means that the z-independent nature of uis guaranteed only in a time-average sense orin the spin-up time scale but not in the timescale of the inertial oscillations. Then the nextquestion is ; what will be the effect of the inertialoscillation on the momentum transfer betweenthe Ekman layer and the bulk fluid? This issuemay play one of the key roles in improvement ofthe Ekman-pumping model.

Fig.4 Velocity vectors in the axial plane averagedfor 30-2i'l'~t~30 given by two-dimension­al numerics for the same parameter set asfigure I

0.010.5 -- ~ - ,

~ - .- ..- - ...0.' ~ - .- ..- - ...

~, - ", ",. - ........, / ",. - ;::.I'~

,,- ....0.3 .I'

~....

~~ ~,,-.. % ~ ~2 ~ ~ :::: ~0.2 ~ ~ z ~ ~

10 15 20Z Z Z

t 0.1

0.5 0.01_.-

I f f, - --- - ,, ".,..",. --I ",-- - ...

0.4~ <1'."'--,~2~::::=

0.3 ~ ~ .: ~ ~ ~ -" - -..... Z ~ ,- -.. -~ Z ~ ~ -0.2 -

~ ~ :

0.1

0

-0.5

-0.65

·0.45

however evolve very fast and the evolution is notcharacterized by advection but by inertial wavesor oscillations.

Figure 3 shows the temporal behavior of v (0.7,h, t) and v(0.7, h/2, t) up to t=20 given by

(b) t=45

(a) t= 15

0.1 r~;:::;~0.2

.. 0.3

0.4

'U -0.55;>

t:;> -0.6

Fig.2 Velocity vectors in the axial plane computedby two-dimensional numerics for the sameparameter set as figure 1

Fig. 3 History of Vn (the azimuthal velocity com­ponent evaluated at r=O.7 and at z=h, i.e.at the free surface; solid lines) and urr (theone at r=O.7 and z= h/2; dashed lines),given by two-dimensional numerics for thesame parameter set as figure 1

Numerical Study on the Motion of Azimuthal Vortices in Axisymmetric Rotating Flows 319

Figure 5 shows a sequence of initial develop­ments of the streamlines and the azimuthal vor­ticity contours in the axial plane. This sequencereveals the generation mechanism of the inertialwaves. At first (/=2), a corner vortex is gen­erated by the roll-up of the vortex sheet of posi­tive vorticity on the bottom wall. Due to theimage vortex behind the side wall as well as thestrong blowing from the Ekman boundary-layer,this vortex moves up (t=3). As a result, the fluidsituated in the left-hand side of this vortex movesdown toward the right-hand side and then turnto the left to be finally sucked into the Ekmanboundary-layer on the bottom wall (t=3). Onthe other hand the region very close to the vor­tex core at the left-down side (i.e. near r=0.9and z=O.1 at t=2) is characterized by a negativevalue in au!az. As shown in the previous section(via the equation (14)) this gradient induces anegative vorticity in the same place (t=3, 4). Thevortex is propagated upward and toward the wallwhile becoming slender. Simultaneously in theregion between the mentioned vortex and the bot­tom wall a positive vorticity is generated (/=5).After this, the mentioned positive vortex patterntravels toward the wall while inducing anothernegative vortex behind it, and so on. Althoughthe vortices' propagation is toward the side wall,the movement of the fluid element is toward thecentral region in a time-average sense as shownin Figure 4. Vortical flows become more pro­nounced in the core at later time (see Figure 2) ;we can see from Figure 4 that a slightly wavypattern still exists in the core even if the velocitycomponents are time-averaged.

Hitherto we have seen the axisymmetric flowevolutions in the actual spin-up flows. In sum­mary, the complex flow pattern shown in the axialplane is caused by the mechanism of inertial os­cillation the perturbation being the corner vortexgenerated at the beginning of the spin -up process.The corner vortex is of course generated by boththe Ekman circulation and the bottom friction. Inthe next section, to focuss our attention on themotion of the azimuthal vortex we will consider asingle vortex or a pair of vortices, in a form ofvortex patch, without the Ekman circulation or

frictions of the bottom and side walls.

4.2 Motion of azimuthal vortices under abackground rotation

Figure 6 shows the time evolution of strea­mlines and vorticity contours obtained by thenumerical computation of (I) through (3) nowwith homogeneous boundary conditions

v==IjJ=t,=O at r=O, 1 (I8a)

avaz = 1jJ= t,=0 at z=O, h (I8b)

The flow is induced only by a circular vortex­patch of radius 0.05 having a uniform vorticitywith magnitude ~= I situated initially at theposition, r=0.9 and z=0.2, in the similitude ofthe corner vortex shown in Figure 5 ; otherwise,the entire domain is initially quiescent. All theother parameter values used in the computationis the same as Figure 1. Since the boundary con­ditions on the lower and side walls are set as(18a) and (18b), slip of the fluid element ispermitted on the walls. Thus, we have neitherEkman pumping effect from the lower wall norfriction effect on the side wall so that we canstudy the vortical motions in a completely solitarystatus.

The only no-slip condition at the side wall,i.e. the condition at the wall r= I, shown bythe equation (18a) in fact exerts no fundamentaleffect on the vortices' dynamics because the boun­dary layer adjacent to the side wall caused by theno-slip condition for v remains very thin, beingattached to the wall during the course of thevortical development.

At t=l most of the domain is characterized bya weak stream except near the side wall where thevortex patch resides; since the vortex patch isinitially positive, the vortical flow in this region iscounterclockwise. At t=2 however the flow di­rection is reversed because the central region ofthe vortex patch is switched from positive tonegative vorticity. It can be shown that for a smallcircular patch the vorticity keeps its uniform dis­tribution but the magnitude is changed repeatedlywith the angular frequency 2/e. On the otherhand each of the vortices with negative vorticity

320 Yong Kweon Suh

generated on the right-and left-hand sides res­

pectively of the initial patch (shown at t= 1)

0.5,.-------------------,

0.5 [

04

10.3 10.2

o I

tends to take a 'X' shape ([=3) and is pushed

toward the side wall. Near r=O.8 and z=O.2

os

0.4

0.3

0.2

01

O,t---n---7!'r--....,n----lI"Ir------!

Fig. 5 Streamlines (left) and vorticity countours (right) obtained by two-dimensional numerics for the same

parameter set as figure I. Increment of stream functions is 0.0002, and the vorticity is colored white for

the value greater than 0.2 and black for lower than -0.2

Numerical Study on the Motion of Azimuthal Vortices in Axisymmetric Rotating Flows 321

another vortex with positive vorticity is generated (t=3) and rapidly grows to be split into two

0.5

'0.4

0.3

02

0.1

0 0 0.4 0.6

0.5

0.4

0.3

"0.2

0.1

0

0.5

0.4

0.3..0.2

0.1

0.5~-------------r------....,

0.4

0.3

0.2

0.1

0.5~------r---------.----....,

0.4

0.3

0.2

0.1

0.5 r---::--:===:::::::-r-:::=:::::::,-i0.4

0.3

0.2

0.1

Fig. 6 Streamlines (left) and vorticity countours (right) obtained by two-dimensional numerics for the case

with single vortex patch of initial so= I under the slip boundary conditions on the solid walls; Re=30000, h=e=0.5. Increment of stream functions is 0.0002, and the vorticity is colored white for the value

greater than 0.2 and black for the one lower than -0.2

322 Yang Kweon Suh

(t=4). At t=5 near r=0.5 and z=0.3 another vortex with negative vorticity is also generated.

..

o.a0.'

tF2

0.1

0.3

0.2

0.5.----...,------------......,-----,0.'

..

..

..

0.•0.2

t=4

0.'

0.3t=6

0.2

0.1

0.2

G.,

0.•

0.5.----------------,-......,--...,

o.s..---,.------------_----,

..

..

0.6 080.'02

t=8

05r-------,----------~-_,_...,

0.3

a.'

0.'

0.2

..

0.5..-----------------,....,--,--,

0.'

0.3

0.2

0.1

Fig.7 Streamlines (left) and vorticity countours (right) obtained by two-dimensional numerics at Re=30000,

c= I and h= c=0.5 with the initial azimuthal velocity distribution and the intial vorticity as specified

in the text for the slip boundary conditions on the solid walls. Increment of stream functions is 0.0002,

and the vorticity is colored white for the value greater than 1.0 and black for lower than -1.0

Numerical Study on the Motion of Azimuthal Vortices in Axisymmetric Rotating Flows 323

Although the vorticity generation point moveswith time toward the central axis, all the vorticespropagate toward the side wall. In fact, Figure 6implies that the continuous generation of theazimuthal vortices and subsequent oscillations inthe flow field shown in Figure 5 is basically dueto the corner vortex (shown at t=2 and 3 inFigure 5), the result of roll-up of the positive­vorticity sheet on the Ekman boundary-layer.

Figure 7 shows the numerical results for thecase when a pair of vortices, with opposite signsof vorticity with magnitudes ~=± 10, was situ­ated initially at r=O.6, z=h/2±O.03. The initialazimuthal velocity Vo was set as

{-r for r~O.75

Vo=B(r-r-131ll

) for r>O.75

where B is chosen in such a way that the pro­file is continuous at r=O.75. The profile for theouter region is similar to (11b), but here it wasobtained through the linear analysis to the in­viscid version of (7) with the first-order correc­tion for the Ekman pumping law included. Sincethe Rossby number in this case is I, this profilein fact mimics the azimuthal velocity distributionof the spin-up from rest at the time when the frontreaches r=O.75, which was arbitrarily chosen.The core region bounded by the front thereforehas a zero azimuthal velocity in the inertial frameof reference, and thus the vortices (these vorticeswill be referred to as 'primary' hereafter) simplymove radially outward until they reach the frontr=O.75. After the contact, however, the evolutionis quite different from the ordinarily observedpattern. As soon as the vortices contact the bor­der, they are surrounded by a pair of crescent­shape vortices on the outer-region side, whichare of course generated by the primary vortices;see the frames for t=2 in Fig. 7. As a result, theprimary vortices become almost stationary after­ward. During and after this event, the outer regionis disturbed by the presence of the primaryvortices and shows the inertial waves as seen inFig. 7, especially after t=8. This means that theouter region is characteristic of rotating flows,whereas the core region is free from the rotation­al effect. We can also observe that the primary

vortices weaken very rapidly after they reside atthe border, as can be distinctively captured fromthe streamline patterns.

Penetration of vortices from the core to theouter region may be realized by, for instance,ejecting a vortex ring into a rotating fluid duringa spin-up process from the bottom of the con­tainer at the time when the front reaches r=O.75.In this case the vortex ring at first may travel allthe way up to the free surface of the fluid andthen turn toward the outer region. Preparationfor the experimental study on such occasion iscurrently performed in the author's laboratory.

5. Conclusions

It was shown that the two-dimensional com­putation for the spin-up flow within a circularcylinder is in close agreement, in the core regionbetween the rotational axis and the front, with theone-dimensional computation with an Ekmanpumping model which includes the first-ordercorrection. In the outer region between the frontand the side wall, however, the discrepancy isconsiderable, and the profile given by the two­dimensional numerics is smoother than that givenby one-dimensional computation. At e=0.5, theaxial plane is dominated by the typical inertial­wave motion superimposed by the well knownEkman circulation in the spin-up process. Thewave motions turn out to be stronger than ex­pected. However they do not deteriorate, in atime scale larger than the period of inertial­oscillations, the underlying assumption for theclassical Ekman pumping law; that is, the radialvelocity component is uniform over and the verti­cal component is proportional to the depth fromthe free surface, in the inviscid region.

When a small vortex patch is introduced at e=0.5 without any other perturbation, the vorticitychanges its sign repeatedly and the whole flowfield in the axial plane is dominated by a complexinertial-wave pattern. Finally, when a pair ofsmall vortex patches is introduced with initialdistribution of azimuthal velocity component intwo steps, which mimics the situation of a spin­up from rest, the motion of vortices is completely

324 Yong Kweon Suh

different for the core and outer regions. At theinitial stage when the pair is at the core region, itsimply moves straight outward. As soon as theouter region is touched by the pair vortices at theborder, it blocks the vortices' penetration by sur­rounding them with the crescent-shape vortices,making the pair motionless. The outer region isthen dominated by the typical inertial waves, andthe pair vortices weaken very fast.

In the future, numerical computations for thepair vortices' motion at various initial distribu­tion of the azimuthal velocity component arerequired to investigate its effect on the motion ofthe pair vortices as well as on the inertial-wavestructure. The final goal in the series of studieson the inertial oscillations and vortical motions isto develop an improved Ekman pumping modelwhich has a broad range of applications in thefluid mechanics area.

Acknowledgment

This work was supported by the Dong-A Uni­versity Research Fund in 2002. This work hasalso been supported by grant No. RI2-2002-058­01004-02003 from the CANSMC of KOSEF.

References

Cederlof, U., 1988, "Free-Surface Effects onSpin-up," J. Fluid Mech., Vol. 187, pp. 395-407.

Dolzhanskii, F. V., Krymov, V. A. and Marrin,D. Y., 1992, "Self-Similar Spin-up and Spin­down in a Circular Cylinder of Small Ratio ofHeight to Diameter," J. Fluid Meeh., Vol. 234,pp.473-486.

Greenspan, H. P., 1968, The Theory of Rota­ting Fluids, Cambridge University Press.

Greenspan, H. P. and Howard, L. N., 1963,"On a Time Dependent Motion of a RotatingFluid," 1. Fluid Meeh., Vol. 17, pp. 385-404.

Hart, J. E., 1995, "Nonlinear Ekman Suctionand Ageostrophic Effects in Rapidly RotatingFlows," Geophys. Astrophys. Fluid Dynamics,Vol. 79, pp.201-222.

Hart, J. E., 2000, "A Note on Nonlinear Cor­rection to the Ekman Layer Pumping Velocity,"Phys. Fluids, Vol. 12, No.1, pp. 131-135.

Hyun, J. M., Fowlis, W. W. and Warn-Varnas,A., 1982, "Numerical Solutions for the Spin-upof a Strafied Fluid," J. Fluid Mech., Vol. 117,pp.71-90.

Maas, L. R. M., 1993, "Nonlinear and Free­Surface Effects on the Spin-down of BarotropicAxisymmetric Vortices," J. Fluid Mech., Vol. 246,pp.117-141.

Suh, Y. K. and Choi, Y. H., 2002, "Study onthe Spin-up of Fluid in a Rectangular ContainerUsing Ekman Pumping Models," J. Fluid Mech.,Vol. 458, pp. 103-132.

van de Konijnenberg, J. A. and van Heijst,G. J. F., 1995, "Nonlinear Spin-up in a CircularCylinder," Phys. Fluids, Vol. 7. No. 12, pp. 2989­2999.

Wedemeyer, E. H., 1964, "The Unsteady Flowwithin a Spinning Cylinder," J. Fluid Mech., Vol.20, pp. 383- 399.

Zavala Sanson, L. and van Heijst, G. J. F.,2000, "Nonlinear Ekman Effects in RotatingBarotropic Flows," J. Fluid Mech., Vol. 412,pp.75-91.