2003 [jong-yeon hwang] numerical study of taylor-couette flow with an axial flow

7/26/2019 2003 [Jong-Yeon Hwang] Numerical Study of Taylor-Couette Flow With an Axial Flow

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Numerical study of TaylorCouette ow with an axial ow

Jong-Yeon Hwang, Kyung-Soo Yang *

Department of Mechanical Engineering, Inha University, Incheon 402-020, Republic of Korea

Received 20 April 2002; received in revised form 30 October 2002; accepted 23 January 2003

Abstract

The ow between two concentric cylinders with the inner one rotating and with an imposed pressure-driven axial ow is studied using numerical simulation. This study considers the identical ow geometryand ow parameters as in the experiments of Wereley and Lueptow [Phys. Fluids 11 (12) (1999) 3637],where particle image velocimetry measurements were carried out to obtain detailed velocity elds in ameridional plane of the annulus. The objectives of this investigation are to numerically verify the experi-mental results of Wereley and Lueptow and to further study detailed ow elds and bifurcations related toTaylorCouette ow with an imposed axial ow. The vortices in various ow regimes such as non-wavylaminar vortex, wavy vortex, non-wavy helical vortex, helical wavy vortex and random wavy vortex are all

consistently reproduced with their experiments. It is demonstrated that

shift-and-reect symmetry holds inTaylorCouette ow without an imposed axial ow. In case of TaylorCouette ow with an imposed axialow, one can nd that the shift-and-reect symmetry is roughly valid for the remaining velocity eld aftersubtracting the annular Poiseuille ow. The axial ow stabilizes the ow eld and decreases the torquerequired by rotating the inner cylinder at a given speed. Growth rate of the ow instability is dened andused in predicting the type of the vortices. The velocity vector elds obtained also reveal the same vortexcharacteristics as found in the experiments of Wereley and Lueptow.

2003 Elsevier Ltd. All rights reserved.

Keywords: TaylorCouette ow; Axial ow; Instability; Simulation; Vortex; Shift-and-reect symmetry

1. Introduction

The ow between two concentric cylinders with the inner one rotating and the outer one sta-tionary, called TaylorCouette ow, has been studied by many researchers for decades. With alow rotating speed of the inner cylinder, the exact solution of the laminar velocity eld consists of

*

Corresponding author. Tel.: +82-32-860-7322; fax: +82-32-863-3997.E-mail address: [email protected] (K.-S. Yang).

0045-7930/04/$ - see front matter

2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0045-7930(03)00033-1

Computers & Fluids 33 (2004) 97118www.elsevier.com/locate/compuid
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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vr v z 0; vh Xir ir o=r r =r or o=r i r i=r o

; 1

where r , h and z represent the radial, azimuthal, and axial directions of the cylindrical coordinatesystem, respectively. The angular velocity of the inner cylinder is denoted by Xi; the inner andouter radii are represented by r i and r o , respectively. It should be noted that Eq. (1) is the solutionunder the idealization of innitely long cylinders where endwall effects are ignored; this solution isnot realized exactly in an physical experiment or simulation that include endwall effects. If theTaylor number ( Ta ) based on X i goes over a critical one ( Ta c1), the ow instability caused by thecurved streamlines of the main ow produces axisymmetric Taylor vortices. This fact was rstnoticed by Taylor (1923) in an analytical study of the related ow instability [1]. Since then, manyresearchers have studied the instability causing Taylor vortices [24]. In the early days of studyingTaylor vortices, researchers attention was mainly focused on determining Ta c1 by experimental oranalytical methods. As Ta further increases over a higher threshold value ( Ta c2), the Taylorvortices become unsteady and non-axisymmetric, called wavy vortices [5]. Davey et al. [5] ana-lytically determined the value of Ta c2; that was subsequently conrmed by Eagles experiment [6].Many measurements using laser doppler velocimetry (LDV) were also carried out [7,8]. However,they could not obtain detailed ow information but only local velocity elds. Even though someresearchers [9,10] have performed numerical investigation of Taylor and wavy vortices, their at-tention was paid mainly to numerical methods. However, Marcus [11] investigated wavy vorticeswithout an axial ow to get velocity elds and characteristics of traveling waves in detail.

In the case of Taylor vortices with a fully developed axial ow, Chung and Astill [12] developeda linear stability theory for the spiral ow. Gravas and Martin [13] studied Taylor vortices in thepresence of an axial ow with various ratios of the inner radius to the outer one. They reported

that the axial ow stabilizes the ow eld and Tac1 is increased. Takeuchi and Jankowski [14]made both experimental and numerical investigations into toroidal vortex structures of the spiralow; especially ows between two independently rotating concentric cylinders with wide gap wereconsidered. Meseguer and Marques [15] studied the shear instability associated with the axial owand the centrifugal instability due to rotation using linear stability theory. Recently, measure-ments of velocity elds for TaylorCouette ows with and without an imposed axial ow wereperformed by Wereley and Lueptow [16,17] using particle image velocimetry (PIV).

TaylorCouette ow is often observed in various types of engineering application, for example, journal bearing lubrication and cooling of rotating machinery among others. Since the axial ownot only stabilizes the ow eld but also decreases the torque required by rotating the innercylinder at a given speed as shown in the later part of this paper, understanding TaylorCouetteow with an imposed axial ow allows us to control the related ow elds.

In this study, we investigate the characteristics of the vortices in TaylorCouette ow havingthe identical cross-sectional geometry as in the experiments of Wereley and Lueptow [16,17].Technical limitations of experimental methods keep researchers from getting detailed ow in-formation. They have been limited to obtain only temporally and spatially averaged torque on theinner cylinder or unsteady ow characteristics on a meridional plane in the ow eld. Numericalsimulations are suitable for efficiently studying effects of ow geometry and ow parameters, andcan make up for the shortcomings of experimental techniques by providing detailed informationon full three-dimensional (3D) time-dependent ow elds. The objectives of this investigation are

98 J.-Y. Hwang, K.-S. Yang / Computers & Fluids 33 (2004) 97118


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to numerically verify the experimental results of Wereley and Lueptow [16,17] and to furtherstudy detailed ow elds and bifurcations related to TaylorCouette ow with an imposed axialow.

2. Formulation

The governing incompressible continuity and momentum equations are

r u 0; 2

o uo t

u ru 1q

r p mr 2u; 3

where u, q, p and m denote velocity, density, pressure, and kinematic viscosity of the uid, re-spectively. The governing equations are discretized using a nite-volume method in a generalizedcoordinate system. Spatial discretization is second-order accurate. A hybrid scheme is used fortime advancement; non-linear terms and cross diffusion terms are explicitly advanced by a third-order RungeKutta scheme, and the other terms are implicitly advanced by the CrankNicolsonscheme. A fractional step method [18] is employed to decouple the continuity and momentumequations. The resulting Poisson s equation is solved by a multigrid method. For the details of thenumerical algorithm used in the code, see [18].

3. Choice of parameters and boundary conditions

There are two non-dimensional parameters of importance, namely, Ta and Re. Although Ta canbe dened in several different ways, we choose Ta r iXid =m, where d is the difference between r oand r i. The axial Reynolds number is dened as Re wd =m, where w is the averaged axial velocity.Radius ratio, g, is selected 0.83 as in the experiments of Wereley and Lueptow [16,17].

A special care is required in selecting the size of the axial domain ( H ). A considerably largeaxial domain, H 27d 32d depending on Ta ; Re, is employed for natural selection of thedominant axial wavelength of the vortex pairs ( k). In each case of Ta ; Re, we rst estimate kusing Fig. 11 in Ref. [17], and take 16 multiples of the estimated k as the size of H , anticipatingappearance of 16 pairs of vortices. It turns out that exactly 16 pairs of vortices show up in all casesreported here. Since the dominant axial wavelength is not imposed a priori but naturally selected,it is estimated that the axial computational domain allows at worst 3.13% error in k. In fact, theerror should be much smaller than our conservative estimate because the axial domain istuned according to the experimental measurements of k.

Temporal instability is considered in association with the axial ow; a periodic boundarycondition is employed in the axial direction. For convective instability related to the spiral ow,see [14,15]. Selecting the size of the azimuthal domain needs a special care, too. Once wavyvortices are formed, traveling waves show up. In order to capture the longest wavelength spanningthe whole azimuthal domain, if any, we have to consider the full azimuthal domain. Fig. 1 showsthe computational grid system employed in this study. Fig. 1(a) and (b) exhibit the computational

J.-Y. Hwang, K.-S. Yang / Computers & Fluids 33 (2004) 97118 99


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domain at one cylinder cross-section and at one meridional section, respectively. The grid is abody-tted O-grid system which is the most adequate for this ow; it has more resolution near thecylinder walls where gradients are steep. The number of grid points determined by grid-renementstudy was 128 32 256 in the azimuthal, radial, and axial directions, respectively. No signicantdifference in the ow elds is noticed in comparison with those obtained by a 256 64 512 gridsystem; qualitative difference is hardly visible and the difference in growth rate of the primaryinstability is less than 1.42% between the two.

No-slip boundary condition is imposed on the solid boundaries. An initial ow eld for the caseof a particular Ta ; Re is constructed using the exact steady solution corresponding to the par-ticular Ta ; Re which would be present at the same Ta ; Re if there were no instability present. Inthis study, 12 cases with various values of Ta ; Re are computed. For each case, about 15,000 timesteps are computed before the ow eld is taken for ow analysis; the time span is correspondingto dimensionless time of 35006500 based on d and r iXi or 114212 revolutions of the inner

cylinder depending on Ta ; Re. The vortices appear rather quickly between r iXit =d 150 and 600depending on Ta ; Re.

4. Results and discussion

4.1. TaylorCouette ow without an axial ow

Wereley and Lueptow [16] reported that the value of Ta c1 over which Taylor vortices are formedis 102 and that the value of Ta c2 is in the range of 124 < Ta c2 < 131. Wereley and Lueptow s result

Fig. 1. Computational domain and grid system; (a) cylinder cross-section, (b) meridional section.



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in Fig. 2(a) shows the velocity vector eld on an rz plane at Ta 124; Fig. 2(b) shows the resultof present numerical study at Ta 123. The solid curves denote the magnitude contours of azi-muthal velocity, while the lower and upper lines represent the inner and the outer cylinder sur-faces, respectively. One can notice in both gures that on the boundaries of Taylor vortices themagnitude of the outer directional velocity is larger than that of the inner directional velocity, and

that the two adjacent vortices which have outer directional velocity at their boundary (hereaftercalled Vortex Pair) get closer to each other. Distributions of the radial and axial velocitycomponents along the axial and radial directions on an azimuthal plane are compared with thecorresponding results of Wereley and Lueptow s experiment in Fig. 3(a) and (b), respectively.Each velocity is non-dimensionalized by the surface velocity of the inner cylinder. Radial locationis denoted by n r r i=d , and axial location by f z =d . One can notice that the shape of eachpeak of the experimental result in Fig. 3(a) is different from one another; this could be explained inpart by the end effects of the cylinder in the experimental setup. Since the axial direction isassumed homogeneous in our simulations, the shape of each peak is identical in the computa-tional result. The peak corresponding to the outow is narrower and stronger than the onecorresponding to the inow; that is the case not only in the simulation but also in the experiment.This is due to the fact that the distance between the two vortex centers of a Vortex Pair is shorterthan that of two adjacent Vortex Pairs, and the outow between the vortex centers of a VortexPair is stronger than the inow between two adjacent Vortex Pairs as explained in Fig. 2. In Fig.3(b), computation slightly overpredicts v z along the radial line; this situation may be due to thefact that experimentally measured value of v z at a given point is actually smaller than the cor-responding real value of v z . This comes about because the measurement actually reects the ve-locity at a region around the measurement point called interrogation region, not just themeasurement point. As a result, at a maximum, the value is slightly reduced due to spatial ave-raging because the nearby values are slightly less than the maximum [19]. One can also notice that

Fig. 2. Velocity vectors and magnitude contours of azimuthal velocity on an rz plane without an axial ow, contourincrement: 0 :1Xr i; (a) Wereley and Lueptow s experiment [16], Ta 124, (b) current study, Ta 123. The lower andupper lines represent the inner and the outer cylinder surfaces, respectively.



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the numerical results can accurately predict the peak locations and the tendency of higher mag-nitudes near the inner cylinder.

Wavy vortices exhibit oscillations in the azimuthal direction as well as in the axial direction.This fact can be explained by a traveling wave. On a given rz plane, oscillation is noticed as atraveling wave passes through that plane. Therefore, time-dependent characteristics on that planemust be consistent with those along the azimuthal direction. Fig. 4(a) shows time-dependentresults of Wereley and Leuptow [16] on a typical rz plane at Ta 131, where each frame is takenat a xed azimuthal location during passage of one wavelength of the traveling wave. Fig. 4(b)shows our results at Ta 129, where each frame is taken along the azimuthal direction at a giventime and at the same phase as the corresponding frame of Fig. 4(a). The symbol ( ) denotes thecenters of the vortices. In order to show the computed ow eld more clearly, velocity vectorswere slightly magnied in Fig. 4(b). The two gures are consistent with each other as shown,conrming the traveling wave found in Wereley and Lueptow s experiment. The averaged oscil-lation distance of the centers of the vortices (1 :1d ) in Fig. 4(b) is slightly longer than that (0 :9d ) inFig. 4(a). In addition, the vortex centers oscillate slightly in the radial direction consistent withexperimental results [16].

If we compare each frame of Fig. 4(b) with that of half period later, a symmetry rule can beidentied. For instance, comparing the second frame with the sixth one in Fig. 4(b), one cannotice that vortex 1 and vortex 2 in the second frame are shifted to the counterparts in the sixth

1 2 3

-0.05

0

0.05

vr

(a)

presentWereley andLueptow [16]

/ri i

0.25 0.5 0.75

-0.05

0

0.05

vz

(b)

presentWereley andLueptow [16] /ri i

Fig. 3. (a) Distribution of the radial velocity component along an axial line through the center of the annular gap, (b)distribution of the axial velocity component along a radial line through a Taylor vortex center; n r r i=d , f z =d ,without an axial ow, Ta 123.



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frame and that vortex 2 in the sixth frame is a reection of vortex 1 in the second frame withrespect to the boundary between vortex 1 and vortex 2. This is called

shift-and-reect symmetrywhich can be expressed as follows;

vr r ; h; z ; t vr r ; h D h; z ; t ; 4

vhr ; h; z ; t vhr ; h D h; z ; t ; 5

v z r ; h; z ; t v z r ; h D h; z ; t ; 6

where z 0 corresponds to a vortex boundary, and Dh represents the angle corresponding to onehalf of azimuthal wavelength. Marcus [11] claimed that shift-and-reect symmetry is adequate to

Fig. 4. Velocity vectors on rz planes; (a) Wereley and Lueptow s experiment [16]: Each frame is taken on a xedazimuthal plane during passage of one wavelength of the traveling wave (from top to bottom), Ta 131. (b) Currentstudy: at a given instant, each frame is taken along the azimuthal direction at the same phase as the correspondingframe of Fig. 4(a), Ta 129. The lower and upper lines of each frame represent the inner and the outer cylindersurfaces, respectively.



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be implemented in numerical simulation. Fig. 5(a) shows vortex 1 and vortex 2 in the second

frame of Fig. 4(b) after reection with respect to their boundary and Fig. 5(b) shows the corre-sponding vortices in the sixth frame of Fig. 4(b). Fig. 5 conrms that shift-and-reect symmetry iscorrect, based on our calculation in which such a symmetry was not imposed. In order to quantifythe degree of shift-and-reect symmetry, one can dene a root-mean-squared ratio ( c) as fol-lows;

c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijua ubj2q ffiffiffiffiffiffiffiffiffijubj2q

: 7

Here, ua and ub represent two velocity elds on rz planes which are one half of azimuthalwavelength apart, and ua is the reected eld. For example, ua and ub are corresponding to thevelocity elds in Fig. 5(a) and (b), respectively. The overline denotes averaging over all the gridpoints on an rz plane containing a pair of vortices. In Fig. 5, it turns out that c 0:00262.

4.2. TaylorCouette ow with an axial ow

Since the axial ow stabilizes the ow eld, both Tac1 and Tac2 increase in the presence of theaxial ow [17]. Fig. 6(a) shows the velocity eld at an rz plane at Ta 123 and Re 4:9, cor-responding to non-wavy Taylor vortices. One can nd that the axial ow makes the uid ow upand down alternately around the vortices. Wereley and Lueptow [17] called this a winding ow.

1.5 2 2.5 30

0.2

0.40.6

0.8

1

(a)

1.5 2 2.50

0.2

0.4

0.6

0.8

1

(b)

Fig. 5. Velocity vectors on an rz plane, n r r i=d , f z =d ; (a) Reected velocity vectors of vortex 1 and vortex 2in the second frame of Fig. 4(b) with respect to their boundary, (b) velocity vectors corresponding to vortex 1 andvortex 2 in the sixth frame of Fig. 4(b). Solid lines represent magnitude contours of azimuthal velocity, contour in-crement: 0 :14Xr i.



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The vortices shrink due to the axial ow and the vortices located toward the inner cylinder isslightly larger than those toward the outer cylinder. Fig. 6(b) reveals the remaining velocity eldafter subtracting the exact laminar axial velocity prole of annular Poiseuille ow at Re 4:9,

wr K r 2o r 2i lnr i=r o

lnr o=r K r 2 r 2o; 8

where K 1=4l d p =d z and l is the viscosity coefficient of the uid. Fig. 6(b) is very similar toFig. 2. Thus one can speculate that the TaylorCouette ow with an imposed axial ow (Fig. 6(a))is nearly a linear superposition of non-wavy Taylor vortex ow at Ta 123 and annular Poiseuilleow at Re 4:9, as reported in [17]. In order to make an estimate of legitimacy of the spec-ulation, one can substract wr from the axial velocity component at Ta 123 and Re 4:9, andcompare the remainder with that of Ta 123 and Re 0. In Fig. 7, the radial distributions of thetwo are presented in the same format as Fig. 3(b). Even though there exists a slight quantitativedifference in the peak magnitudes, it is not difficult to expect a qualitative resemblance betweenthe two ow elds. The reduction in peak magnitudes implies reduction of circulation around aTaylor vortex; this is consistent with the notion that axial ow stabilizes TaylorCouette ow.

Fig. 8 shows the velocity eld on an rz plane at Ta 129 and Re 13:1, corresponding tonon-wavy helical vortices. Vortices are hidden by the relatively strong axial ow; no vortices areobserved in Fig. 8(a).

Helical vortices do not occur without an axial ow. To visualize helical-type structure of Taylor vortices, Fig. 9 presents instantaneous contours of azimuthal velocity component on aportion of the center surface of the annulus for various types of Taylor vortices for comparison.The regions of higher (lower) azimuthal velocity than the median is marked by black (white). Fig.9(a) represents the characteristics of laminar vortex (LV) at Ta 123 and Re 0. Travelingwaves are not seen and the vortices are stationary. Traveling waves are observed in Fig. 9(b) for

1 2 30

0.5

1

(a)

1 2 30

0.5

1

(b)

Fig. 6. Velocity vectors and magnitude contours of azimuthal velocity on an rz plane at Ta 123 and Re 4:9,n r r i=d , f z =d , contour increment: 0 :11Xr i; (a) including the axial velocity prole, (b) with the axial velocityprole removed.



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the case of Ta 139 and Re 4:9; this type of Taylor vortex is called wavy vortex (WV). Twowavelengths are seen in Fig. 9(b). Fig. 9(c) shows the features of helical vortex (HV) appearingat Ta 129 and Re 24. Traveling waves do not occur in this case, and the helix angle is ap-proximately 2.9 . Traveling waves do appear at a higher Ta than HV; two wavelengths are con-tained in Fig. 9(d) for the case of Ta 167 and Re 13:1. This type of Taylor vortex is calledhelical wavy vortex (HWV). Fig. 9(e) reveals the features of random wavy vortex (RWV) atTa 215 and Re 24. Approximately four wavelengths of dominant traveling waves are iden-tied in the gure. It turned out, however, that unlike the other types of Taylor vortices, circu-lation around an RWV signicantly varies in h within one wavelength as explained later.Nevertheless, RWV maintains quasi-periodicity both in h and in z .

0 0.25 0.5 0.75 1

-0.05

0

0.05vz

/ri i

Ta=123, Re=0Ta=123, Re=4.9

Fig. 7. Distribution of the axial velocity component along a radial line through a Taylor vortex center; n r r i=d ,without an axial ow ( Ta 123 and Re 0) and with the exact laminar axial velocity prole substracted ( Ta 123 and Re 4:9).

Fig. 8. Velocity vectors and magnitude contours of azimuthal velocity on an rz plane at Ta 129 and Re 13:1,n r r i=d , f z =d , contour increment: 0.11 Xr i; (a) including the axial velocity prole, (b) with the axial velocityprole removed.



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More detailed description on the helical structure is given in Figs. 10 and 11 for the case of Ta 129 and Re 24 (HV). Fig. 10 shows instantaneous velocity vectors of helical vortices on aportion of an rz plane in the ow domain along with arrows pointing the corresponding vortexon the other side. Solid and dotted lines represent foreground and background pointers,respectively. The helical vortices form a negative angle with respect to the axis of rotation, andthey are shifted by a vortex pair after one revolution. Three-dimensional vortical structures of HVare presented in Fig. 11 for the case of Ta 129 and Re 24. Fig. 11(a) shows contours of az-imuthal component of vorticity ( x h) in a portion of the ow domain; regions of positive (negative)x h are denoted by dark (bright) color. Nevertheless, the helical structure is clearly seen. See thearrows which point corresponding vortices. To highlight vortex cores only, positive values of J 0:25D 2 are plotted in Fig. 11(b) where J represents Jacobian of shear-stress vectors and Ddenotes divergence of shear-stresses. This method is based on the topological consideration that avortex core constitutes a focus which satises J > 0:25D 2 [20]. Vortex cores are clearly visualizedby this method (Fig. 11(b)), even though their rotational direction cannot be distinguished.

Fig. 9. Instantaneous contours of azimuthal velocity component on a portion of the center surface of the annulus; (a)laminar vortex (LV) at Ta 123 and Re 0, (b) wavy vortex (WV) at Ta 139 and Re 4:9, (c) helical vortex (HV) atTa 129 and Re 24, (d) helical wavy vortex (HWV) at Ta 167 and Re 13:2, (e) random wavy vortex (RWV) atTa 215 and Re 24. The regions of higher (lower) azimuthal velocity than the median is marked by black (white).



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Fig. 11(a) and (b) are complementary to each other; one can grasp 3D structures of helical vorticesusing those two gures.

A Ta Re map is presented in Fig. 12, where solid lines, taken from [17] denote the approximateboundaries of various types of vortices in TaylorCouette ow. As Re increases, so does Ta c2 . Forexample, the four data points corresponding to Ta 139 show that Hopf bifurcation to wavyvortices occurs between Re 13:1 and Re 24, while the four data points at Ta 129 imply alower Re for the bifurcation. Random wavy vortices do not occur without an imposed axial ow.It should be also noted that all the numerical data points are in accordance with the Ta Re map.

Fig. 13 shows velocity vectors and magnitude contours of azimuthal velocity on rz planes atTa 139 and Re 4:9 for one azimuthal wavelength of the traveling wave. Each frame was takenat a given instant at equal circumferential distance in the azimuthal direction. In Fig. 13(a), theaxial velocity prole is included; it is removed in Fig. 13(b). The axial ow is winding in Fig.13(a). Oscillation of the vortices is clearly seen as in Fig. 4(b) after the axial velocity prole issubtracted from the vectors. This spatial observation reveals the oscillation pattern of the wavyvortices better than the temporal approach reported in [17]. They measured time-dependent ve-locity elds on a given meridional plane; their measurements include not only the oscillationpattern but also the translation movement along the axis due to the imposed axial ow (see Fig. 7

O.C.

O.C.

I.C.

I.C.

axis ofrotation

Fig. 10. Instantaneous velocity vectors of helical vortices on a portion of an rz plane in the ow domain along witharrows pointing the corresponding vortex on the other side; Ta 129 and Re 24 (HV). Solid and dotted lines rep-resent foreground and background pointers, respectively. OC and IC denote outer cylinder and inner cylinder , re-spectively.



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in [17]). For direct comparison of the numerical result with the experimental one, a time-sequenceof the same case is shown on a typical rz plane in Fig. 14. Both the winding axial ow (Fig.14(a)) and the non-uniform translation of the vortex centers (Fig. 14(b)) are clearly seen and wellmatch Fig. 7 in [17]. Especially, the retrograde motion of the vortex centers is also identied asnoticed in Fig. 7(b) in [17]. See the retrograde axial motion of the center of vortex 1 in frames 45of Fig. 14(b).

In the case with an imposed axial ow, shift-and-reect symmetry is expressed as follows;

vr r ; h; z ; t vr r ; h D h; z ; t ; 9

Fig. 11. Three-dimensional structures of helical vortices; (a) contours of azimuthal component of vorticity ( x h) in aportion of the ow domain, dark color: positive x h , bright color: negative x h , (b) contours of positive J 0:25D 2 where J represents Jacobian of shear-stress vectors and D denotes divergence of shear-stresses.



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vhr ; h; z ; t vhr ; h D h; z ; t ; 10

v z r ; h; z ; t wr v z r ; h D h; z ; t wr : 11

As shown in Fig. 15, shift-and-reect symmetry is roughly valid in the case with an imposed axialow; the symmetry is less perfect than that in the case without an axial ow ( c 0:0294 in Fig.15). It should be noted that unlike TaylorCouette ow without an imposed axial ow, this

symmetry is applicable only to the remaining ow eld after subtracting the annular Poiseuilleow from the computed ow eld.

In the following two gures, are shown velocity vectors for HWV ow at Ta 167, Re 13:1(Fig. 16), and for RWV ow at Ta 215, Re 24:0 (Fig. 17), respectively. Each frame is taken ona typical azimuthal plane during passage of one wavelength of the traveling wave for directcomparison with Wereley and Lueptow s result [17]. In Fig. 16, the vortex centers persistentlymove downstream without the retrograde motion (Fig. 16(b)). Fig. 17 shows that the behaviors of the vortices are somewhat random, and some of the vortices sometimes almost disappear (the2nd and 6th frames of Fig. 17(b)). This kind of signicant variation in vortex strength does notoccur in the other types of Taylor vortex. All these features are also experimentally observed. SeeFig. 8 for HWV ow and Fig. 9 for RWV ow in [17], respectively. It should be noted that thebehaviors of RWV are random in one wavelength of the traveling wave, but quasi-periodic in theazimuthal and the axial directions from the viewpoint of the entire ow domain (Fig. 9(e)).Therefore, velocity vectors for a given rz plane reveal a random behavior quasi-periodicallyin time.

Fig. 18 exhibits the speed of the center of a vortex in TaylorCouette ow in the axial direction(wvortex ) at various Ta= Re, where wvortex is non-dimensionalized with the mean axial velocity. Thetheoretical value [21] was obtained for the wavy vortices in the range of 1 :6 < Re < 20. Thetheoretical, experimental, and computational results are all consistent with one another. Fur-thermore, the tendency persists even beyond Ta= Re 20.

100 150 2000

5

10

15

20

25

Ta

Re CP

HV

LV

HWV

WV

RWV

Fig. 12. Flow types indicated on the Ta Re plane, non-vortical CouettePoiseuille ow (CP, M), non-wavy laminarvortex (LV, ), wavy vortex (WV, j ), non-wavy helical vortex (HV, ), helical wavy vortex (HWV, ), random wavyvortex (RWV, } ). Notations and solid lines are taken from [17].



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Phase speeds of the traveling waves of various wavy vortices are presented in Fig. 19; they arenormalized by corresponding angular velocity of the inner cylinder. One can notice that thenormalized phase speed is almost constant at c=X i 0:42 regardless of Ta or Re.

In order to quantify the ow instability, we dene a non-dimensional variable ( V cl) on the h z surface at the center of the annulus ( r r i 0:5d );

V cl 1

Xir i2p r H Z H

0Z

2p

0jvr h; r ; z jr dh d z ; 12

(a) (b)

Fig. 13. Velocity vectors and magnitude contours of azimuthal velocity on rz planes at Ta 139 and Re 4:9 for oneazimuthal wavelength of the traveling wave (from top to bottom), contour increment: 0 :14Xr i. Each frame was takenat a given instant at equal circumferential distance in the azimuthal direction; (a) including the axial velocity prole,(b) with the axial velocity prole removed. The lower and upper lines of each frame represent the inner and theouter cylinder surfaces, respectively.



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which is a quantitative measure of absolute ow rate through the center surface. As ow insta-bility induces vr , the onset of bifurcation can be detected by exponential temporal growth of V cl.Fig. 20 shows time history of V cl at various Ta ; Re in a linear-log scale, where time ( t ) is non-dimensionalized with d =Xir i. In the case of higher Ta for a xed Re, V cl grows more rapidly.However, the slopes of the curves for Ta 123, Re 4:9 and for Ta 129, Re 13:1 are almostidentical, which means that a stronger axial ow stabilizes the ow eld. Growth rate( r ) of V cl isdened as V cl exp r t . Fig. 21 reveals growth rates of V cl in various cases of Ta and Re, computedfrom the sloped portion of the curves in Fig. 20; the solid line denotes an approximate boundarybetween non-wavy and wavy vortices. The smaller the value of Re is, the larger the value of r is ata given Ta . The simulation results corresponding to Ta 139 tell us that wavy vortices do notshow up in the case of stronger axial ow ( Re 24:0). The type of vortices at particular values of Ta and Re can be predicted using Fig. 21 and the corresponding growth rate of V cl without a fullsimulation. In other words, a big saving in computing time can be achieved because the growthrate of V cl can be computed in an early stage of full simulation.

Fig. 14. Velocity vectors for wavy vortex ow on an rz plane at Ta 139, Re 4:9. Each frame is taken on a typicalazimuthal plane during passage of one wavelength of the traveling wave (from top to bottom); (a) including the axialvelocity prole, (b) with the axial velocity prole removed. The lower and upper lines of each frame represent the innerand the outer cylinder surfaces, respectively.



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The torque coefficient ( C M ) is dened as

C M T i=0:5p qXir i2r 2i H ; 13

where T i is the torque required by rotating the inner cylinder at a given speed. Fig. 22 shows C M atvarious Ta and Re, where the solid line denotes the exact values [22] for non-vortical Couette Poiseuille ow. One can see that the axial ow signicantly reduces C M . In order to elucidatethis point more clearly, instantaneous wall shear-stresses ( s) along a typical axial line on theinner cylinder surface are presented in Fig. 23, where s is normalized by the wall shear stress ( so )obtained analytically by assuming that no vortices are present. It can be noticed that s=so islarger than unity in most of the range, and this tendency is more evident as Ta increases for a xed Re. This tells us that the vortices in TaylorCouette ow actually increase the wall shear stress.This is also consistent with a well-known fact that vortices increase the shear stress. Further-more, the peaks and valleys in each curve are corresponding to the inner and outer directionalboundary ows of neighboring vortices, respectively; for example, compare the case of Ta 139, Re 4:9 in Fig. 23 with Fig. 15(b). In Fig. 22, it is not surprising that the torque co-efficient decreases in the range of Ta > 139 as Ta increases. This is due to the particular way of normalization of T i; the denominator of C M contains a square of Xi term. Axial ow, however, hasthe opposite effect on s=so . Compare the case of Ta 123, Re 0 with that of Ta 123, Re 4:9in Fig. 23.

1 1.25 1.5 1.75 2 2.25 2.5 2.75 30

0.2

0.4

0.6

0.8

1

(b)

5 5.25 5.5 5.75 6 6.25 6.5 6.750

0.2

0.4

0.6

0.8

1

(a)

Fig. 15. Velocity vectors and magnitude contours of azimuthal velocity on an rz plane, n r r i=d , f z =d ,contour increment: 0 :13Xr i; (a) reected velocity vectors of vortex 1 and vortex 2 in the rst frame of Fig. 13(b),(b) velocity vectors of vortex 1 and vortex 2 in the fth frame of Fig. 13(b).



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5. Conclusion

In this study, numerical simulations of TaylorCouette ow with an imposed axial ow werecarried out for the identical ow geometry and ow parameters as in the recent PIV measurementsof Wereley and Lueptow [17]. The numerical results obtained are in excellent agreement with theirexperimental results both qualitatively and quantitatively. The vortices in various ow regimessuch as non-wavy laminar vortex, wavy vortex, non-wavy helical vortex, helical wavy vortex andrandom wavy vortex are all consistently reproduced with their experiments. It is conrmed thatthe shift-and-reect symmetry is valid for TaylorCouette ow without an imposed axial ow.The symmetry is roughly valid for TaylorCouette ow with an imposed axial ow, if the annularPoiseuille ow at the corresponding Reynolds number is subtracted from the computed ow eld.

Fig. 16. Velocity vectors for helical wavy vortex ow on an rz plane at Ta 167, Re 13:1. Each frame is taken on atypical azimuthal plane during passage of one wavelength of the traveling wave (from top to bottom); (a) including theaxial velocity prole, (b) with the axial velocity prole removed. The lower and upper lines of each frame represent theinner and the outer cylinder surfaces, respectively.



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It is revealed that an axial ow stabilizes the ow eld as expected. A traveling wave along theazimuthal direction can be detected from instantaneous velocity elds; it is veried that thetemporal measurements of Wereley and Lueptow on a given meridional section are valid. Invarious cases of Taylor and Reynolds numbers, detailed ow information is obtained and the typeof the vortices is identied. The computed speed of vortex translation is in consistent with theexperimental and theoretical values, and the normalized phase speed of traveling waves is almostconstant in the range of Taylor and Reynolds numbers considered in this study. Growth rate of the ow instability is dened and used in predicting the type of the vortices. It is also shown thatthe torque coefficient on the inner cylinder surface is signicantly reduced by the axial ow. Theresults obtained in this study demonstrate that the numerical simulation employed for the currentinvestigation is suitable and economical.

Fig. 17. Velocity vectors for random wavy vortex ow on an rz plane at Ta 215, Re 24:0. Each frame is taken on atypical azimuthal plane during passage of one wavelength of the traveling wave (from top to bottom); (a) including theaxial velocity prole, (b) with the axial velocity prole removed. The lower and upper lines of each frame represent theinner and the outer cylinder surfaces, respectively.



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w v o r t e x

/ w

5 10 15 20 25 30

0.8

1

1.2

1.4

1.6

1.8

2

Ta/Re

Wereley and Lueptow [17]presenttheory [21]

Fig. 18. Vortex translation speed as a function of Ta = Re. Here, w is the mean axial velocity.

100 120 140 160 180 200 220 2400

0.2

0.4

0.6

0.8

1

1.2

Ta

c/ Re=4.9 Re=13.2 Re=24.0 Re=0

i

Fig. 19. Normalized phase speeds of traveling waves.

500 1000 150010 -910 -810 -710 -610 -510 -4

10 -310 -210 -110 0

Ta=123, Re=4.9Ta=129, Re=13.1Ta=139, Re=4.9Ta=167, Re=13.2Ta=215, Re=24.0

V cl

tr i i /d

Fig. 20. Time history of V cl.



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120 140 160 180

0.05

0.1

Re=0 Re=4.9 Re=13.1 Re=24.0

Ta

wavy vortex

non-wavy vortex

Fig. 21. Growth rates of V cl

in various cases of Ta and Re .

Fig. 22. Torque coefficient ( C M ) for the inner cylinder vs. Ta at various Re, Re 0:0 (j ), Re 4:9 (M), Re 13:1 ( ), Re 24:0 (} ). The linear theory is taken from [22].

Fig. 23. Instantaneous wall shear-stresses along an axial line on the inner cylinder surface.



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Acknowledgement

This work was supported by grant no. R01-2002-000-00060-0 from the Basic Research Program

of the Korea Science & Engineering Foundation.

References

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[12] Chung KC, Astill KN. Hydrodynamic instability of viscous ow between rotating coaxial cylinders with fullydeveloped axial ow. J Fluid Mech 1977;81:64155.

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Stokes equations in generalized coordinate systems. J Comput Phys 1994;94:10237.[19] Lueptow RM. Personal communication, 2001.[20] Hunt JCR, Abell CJ, Peterka JA, Woo H. Kinematical studies of the ows around free or surface-mounted

obstacles; applying topology to ow visualization. J Fluid Mech 1978;86:179200.[21] Recktenwald A, Lucke M, Muller HW. Taylor vortex formation in axial through-ow: Linear and weakly

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2003 [jong-yeon hwang] numerical study of taylor-couette flow with an axial flow

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