rotating flows driven by axisymmetric blades in a cylinder*rotatingflows 275 the fluid are at rest....

International Journal of Rotating MachinelT1999, Vol. 5, No. 4, pp. 273--281

Reprints available directly from the publisherPhotocopying permitted by license only

1999 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science

Publishers imprint.Printed in Malaysia.

Rotating Flows Driven by Axisymmetric Bladesin a Cylinder*

JAE WON KIM

Department of Mechanical Engineering, Sun Moon University, Asan City, Chung Nam 336-840, South Korea

(Received 24 April 1998," Revised 8 May 1998," In ji’na/ form 7 July 1998)

A study is made of steady state flow of a viscous fluid, driven by a rotating endwall diskhaving radial blades in a finite geometry. Numerical solutions to the Navier-Stokesequations are obtained for the flows in the cylindrical cavity. The bottom endwall disk of thevessel is impulsively rotating at a constant rotating speed ft with respect to the longitudinalaxis of it. Details of the three components velocity field are examined at high Reynoldsnumber for its engineering application. The main parameter for this study is the number ofthe radial blades of a rotating pulsator. The numerical results for the fluid flows and pressuredistribution, for both an odd and an even number of the blades are procured. The presentoutput offers an optimal number of the blades for rotating machinery such as agitator. Thegrid-net for the numerical computation is constructed on a body-fitted coordinate systemtransformed from physical coordinates. It is also flexible to suit any number of bladesattached on the rotating bottom disk. The algorithm for the numerical computation is basedon the SIMPLE release by Patankar, and the results are validated with prior published data.In addition, a characteristic model is prepared for the pressure measurement. The pressuremeasurements performed for the present model are consistent with this computational work.The explicit effect of the blade on the overall flow character is scrutinized. The numericaldata are processed to describe the behavior of the meridional velocities under different bladeconditions. Also, the traces of particles are plotted to assess the effects. Pronounceddifferences are noted and these results supply comprehensive data for practical application.

Keywords." Rotating flow, Rotating blade, Meridional flow, Numerical work,Body-fitted coordinate, Computational work

INTRODUCTION

Fluid flows in a circular cylinder with a rotatingexciter which is, in fact, radial blades on its surfacehas been studied in this paper. The present model

describes the characteristics of the flow system in anupright washing machine. The characteristic modelconsist of a closed circular cavity fully filled withviscous fluid of kinematic viscosity u, and itsendwall disks. The top disk of the cylinder is

This paper was originally presented at ISROMAC-7.Tel." +82 418 530 2337. Fax: +82 418 5420141. E-mail: [email protected].

273

274 J.W. KIM

stationary, whereas the rotating bottom one causesfluid motion inside the vessel of radius R. Theaspect ratio of the container is 2.5. Figureschematically shows the flow system.

Rotating flows in a circular cylinder with flatendwall disks have been extensively investigated inrecent years (Larrousse, 1987; Altgeld et al., 1983;Escudier, 1984). This internal flow "serves as a

classical model, exhibiting the dynamic ingredientsintrinsic to rotating fluid systems. Also, the flowsconfined between two parallel coaxial disks, one

rotating and the other stationary, are of greatrelevance to a host of technological applications,i.e., fluid machinery, chemical mixers, centrifuges,computer disk drive systems, and washing machinesystems, to name a few. For a definitive problemformulation, it will be envisioned that the cylinderaxis is aligned vertically and the spinning disk formsthe bottom endwall of the cylinder. One principaldynamic component is the Ekman layer that formson the endwall disks perpendicular to the rotation

FIGURE Schematic diagram of flow system andcoordinates.

axis. Owing to the fluid pumping mechanism by theEkman layer, a small axial flow is induced; thisin turn gives rise to the secondary flows in themeridional plane. In practice, the basic idea for this

type washing machine with a rotating pulsator at

the bottom of the tub is to utilize the secondaryflows in a meridional plane of a cavity as majormass transfer mechanism. The prior works are

mainly on the flows system by a fiat plate disk nota topological disk. Engineering data from theprevious results are concerned with the flows by a

rotating flat disk. In fact the difference betweenthe rotating flows by a fiat disk and bumpy disk ismore pronounced as the rotational Reynolds num-ber (Re-gtR2/u), based on radius of the cavity,increases. In engineering applications, Reynoldsnumber is sufficiently large. Consequently, thepresent consideration is focused on the dramaticflows by a projecting part, i.e., a rotating convexblades attached on the endwall disk.

This investigation tries to find out the optimalnumber of blades for the rotating pulsator of thewasher used by Asian customers. The number ofblades of a pulsator in the present running modelof washing machine in a market is equal to 6. Inthis work, the estimation of the normal systemand comparison between odd and even number ofblades are both made for engineering applica-tions. Both proper numerical scheme and grid sys-tem was prepared for this investigation. For thecurved shape of the blades, the present mesh is

non-orthogonal system and governing equationsare also transformed into the body-fitted coordi-nate system. In detail, the blade is made of bothtypical sine curves and a horizontal line connectingeach end points of the curved line. The flows in a

meridional plane and spiral motion are describedfor several system with different number of vanes.

NUMERICAL MODEL

A schematic of the flow configuration, togetherwith the physical coordinate system (x,y,z) isshown in Fig. 1. For < 0, both the cylinder and

ROTATING FLOWS 275

the fluid are at rest. At the initial instant 0, thebottom disk with blades begins impulsively rotat-ing about the longitudinal axis z with a constantspeed, . As for usual engineering applications,the situation in which the rotational Reynoldsnumber is large shall be considered. The task is todescribe the induced flow motion in a tub by thetopological rotating plate.

This numerical work is composed of two steps;first work is to run a program for numerical gridgeneration and then a solver program integrateseach derivative term of governing equations. Thegrid system is constructed of four parts. Figure 2displays the four groups, i.e., the disk with convex

vanes, the vertical grid wall through the meridi-onal plane defined by the two endwall disksand shrouded wall, the top disk, and the side wallof the cylindrical container. The present grid gen-eration of the pulsator is so flexible that arbitrarynumber of blades is represented on the flat disk.In Fig. 2, typical pulsator with three radial bladeson the flat plate bottom disk is constructed at thelower endwall disk.The governing equations are the three dimen-

sional incompressible Navier-Stokes equations(Eq. (1)) and continuity equation (Eq. (2)). Numer-ical calculation was carried out on the computa-tional domain in the transformed coordinates(G r/, ) from the physical one (x, y, z). Written forcomputational coordinates (G r/, ) with respective

<

FIGURE 2 Grid net for flow configuration.

non-dimensional velocity components (U, V, W),these are

o0. (2)

J- x<y,z + xyzr + x,lyz

G11 j(x2 q_ y2 q_ z2),j( x& + +

G23 G32 J(T]xCx q- 77yCy -q-

G 12 G2l J({xrlx + T]y q- zT]z)

G22 J(rl2x q- rl2y q- ,z2).Consider a vertically mounted right circular

cylinder of radius R and height 2.5R filled with afluid having kinematic viscosity u. The aspect ratioof the present cavity, A, is equal to 2.5 in thisnumerical work. The rotating bottom disk with

276 J.W. KIM

convex radial vanes cause internal flows of a vessel.The bottom disk rotates steadily about the centralaxis at constant angular frequency f. The char-acteristic variables for normalization are R, Rf,and 1,/9 for length, velocity, and time, respectively.The actual computational procedure is as fol-

lowing. At the absolutely silent flow, the impuls-ively rotation of the uneven bottom disk is imposedin the flow field.The main focus is on the flow field at almost

steady state. In summary, the initial and boundaryconditions can be expressed as

U- V- W-0 for < 0 (initial state), (3)

U-V-W-O at r- R for 0 < z < 2.5R,

> 0 (the side wall), (4a)

U-V-W-O at z- 2.5R for 0 < r < R,

> 0 (the top disk), (4b)

v- at z- 0 for 0 < r < R,

> 0 (the bottom disk). (4c)

The time-marching integration continued untilan approximate steady state was attained. Thespin-up time scale, "r-Rel/2f-1, gauged the lapseof time for global adjustment in unsteady rotatingflows (Hyun and Kim, 1989; Kim and Hyun, 1997).For convenience, the approximate steady state in thepresent work was arbitrarily taken to be the statewhen the flow variables at r/R =0.5, z/R 1.25,changed less than 0.1% over the time interval 0.1 r.

The finite-volume methods developed byPatankar (1980) were amended to solve the systemof equations. This method has been shown to

provide accurate and reliable numerical solutionsto a large class of rotating fluid flows in themanuscripts by Kim and Noh (1997) and Langet al. (1994). This model used the primitive variables

on a regular nonuniform mesh. The pressure wasfound from a Poisson equation, which was solvedby a pressure substitution method proposed byHobson and Lakshminarayana (1991). The specif-ics of the numerical techniques adopted, includingthe finite-volume method, were elaborated in Kimand Noh (1997). Also this numerical result obtainsits accuracy by validation both with the prior workand comparison with laboratory experiments ofpressure in a model system.

In order to validate the present numerical resultscomprehensive experiment has been performed.Pressure variation in the flow system is measuredand compared with the numerical data. Figure 3shows the axial profiles of pressure at a radiallocation, r/R =0.75. The continuous line denotesthe numerical results and the symbols for themeasurements by a pressure transducer that isconnected to a hole via a fitting tube. The holesfor the pressure gauging are prepared on a rodmounted at a wanted position. Uncertainty error

due to the rod and tube for pressure transferis rigourously excluded by removing bubbles inthe experiment system. It is observed that thediscrepancy between the two data is in a reasonablerange.

2.5

2.0

1.0

0.5

0.00.00 0.01 0.02 0.03 0.04 0.05

FIGURE 3 Comparison of pressure by computational work(solid line) and measurements (symbols) along axial locationsat r/R =0.75.

ROTATING FLOWS 277

RESULTS AND DISCUSSION

Several sets of numerical computations wereconducted to render comprehensive flow details.For all the calculations performed, Re=2 x 1.03and A 2.5. These values are comparable to thoseused in the real running model. In line with thestated objectives of the present study, the numberblades were varied to scrutinize the effects ofsuction in finite geometry. In the ensuring discus-sion, only the exemplary and physical illuminatingresults will be presented.

Figure 4 displays the vector plot of the meridi-onal velocities in a vertical cross plane (x-0plane) according to the number of blades. Figure4(a) shows the plot for the case of B 2, where Bdenotes the number of blade on the rotating disk.Also the flows for B= 3, 5 and 6 are depicted inFig. 4(b)-(d), respectively. The cross sectional

plane for this plot is the vertical meridional oneconstituted by x- and z-axis. The flows are asym-metric for the case ofodd number blades. Otherwisesymmetric flow is found for even number blades(see Fig. 4(d) for B= 6).

Similar plot in the y=0 plane for B=3 is

displayed in Fig. 5. Flows are still asymmetric andonly right half of the wing is shown in the plane.

In the plot of meridional velocities, it is observedthat there are secondary circulations due to theaction of suction by a rotating pulsator. Majoremphasis is on the enhanced meridional circulationby adjustment of the number ofwings. In this work,the magnitude of the meridional velocity is in-directly gauged by the quantity of the axial velocityat a pre-fixed depth of fluid in x=0 plane. Forconvenience, the depth of water for the measure-ment is at z/R=0.5. Figure 6 shows the radialvariation of the axial velocity in x=0 planedepending on the number of wings. The meridionalflow is axisymmetric with respect to the vertical axisin the case of even number of B.

In contrast, the flows are not axisymmetric forodd number of B. For the present rotationalcondition, in the case of Re 2000, it is found thatthe downward flows toward the bottom disk for

(a) 2.5

2.0

1.5

FIGURE 4Ca) and (b)

278 J.W. KIM

2.0

r/F

1.5

1.0

0.5

0.0

tltllh.

lllllh,

1111h..

FIGURE 4(c) and (d)

FIGURE 4 Meridional velocity vector plot in the x=0plane according to (a) B=2, (b) B=3, (c) B=5, and (d)B=6.

//’//.__-.:-..x,\\

............ ,,,, ,,,,*’""-,,,,,,-,,1.5--

lllll/,, \\P "\\\\

’,T ..,,,,,,,, .,,.) -’\\\\\,,,,,,\ \"..\’...".,’.....,\\..o O:

,,’,,’’X,,,,,:..,r ..... \\\\\ X.\\\\\\\\. \\.............. ,.!\\\!\,\\\ .... t/fit,0.,5 {{,............... ’,,,MI.,\\ ,’ ’ ,,’, ,. \\\\ \\

0.01-:---’- I,,,, I.,, ,!"q

-1.0 -0.5 0.0 0.5 1.0

rlF

z/R

FIGURE 5 Vector plot for B 3 in the y 0 plane.

both cases of B=5 and 6 are obstructed andrestricted because of excess vanes. The analysis ofFig. 6 shows the optimal number of blades forRe 2000 is less than 5.

Let us compare the suction velocity in theanother meridional plane. Figure 7 shows the com-parison of the magnitude of the vertical velocityaccording to the meridional planes for same bladenumber, B 3. Figure 7(a) displays the radial varia-tion of the vertical velocity (IV) in x 0 plane and7(b) in y 0 plane which is cross sectional plane con-structed by x-axis and the vertical axis. In the twoplots, the magnitude and shape are changed depend-ing on the view-plane. However, the characteristicsof asymmetry are still found in the both two plots.As stated earlier, the resulting flows are sym-

metric or asymmetric depending on the number ofblades. It is recommended for the enhanced radialflows to select the odd number of wings because theasymmetric behavior of the meridional flows causethe inward and outward flows along the verticalaxis.

ROTATING FLOWS

o.s S-2, z/R-O.5 B--3, z/R-O.5(b) 0.4

0o4

0.3

0.2

0.1

0.10.0

-1 1.0

0.0 -0.o .o

-0. -o2

279

0.2

0.1

r/F (xWO.O

5 0.0 0

-0.1

-0.2

0.3 I0.2

0.1

0.0,

0

FIGURE 6 Radial variation of the axial velocity (W) near the bottom disk in the x--0 plane. (a) B= 2, (b) B= 3, (c) B--5,and (d) B= 6.

REMARKS

Numerical calculation for the flow filed in amodel of a washing tub having a pulsator is carried

out for the selection of the number of blades of a

rotating agitator. The present grid system repre-sents the overall domain of the flow configurationinvolving the uneven shape of a pulsator. The result

280 J.W. KIM

B--3, z/F -O.50.4

0.3

0.2

0.0-I 1.0

-0.1

-0.2

0.4-

0,3-

’-1.0 0.5 ..00./-0

,10O c

FIGURE 7 Plots of vertical velocity in the x 0 plane (a) and y 0 plane (b).

1.0

of the present numerical calculations is comparedwith the advanced work by Lang et al. (1994) and isin good agreement. Also the solution by thisnumerical work is directly validated by laboratoryexperiments.For this rotating flows at Re 2000, the optimal

number of wings of the rotating pulsator is 3 forodd number and 4 for even number. The meridionalflows for mixing mechanism when the number ofwings is five or six becomes weaker and weakerdue to the inconsistency between the time intervalof the rotating blade and the phase of downwardvelocity by the suction of the rotating endwall. Thesuction speed by the pulsator with 3 or 4 blades isfaster than that induced by the rotating plate having5 or 6 blades. The role of the vane of a pulsator isdefined as a generator of suction motion in thisresearch.

Meridional flows are axisymmetric for evennumber of risen vanes from the bottom disk. In

the case of asymmetric flow, the blade number isodd. The difference of odd and even blade numbersis the symmetric or asymmetric behavior of theflows by the pulsator.

Acknowledgment

This work was supported by the Korea Science andEngineering Foundation through the RegionalResearch Center for Climate Control Technologyat Sun Moon University, Korea.

NOMENCLATURE

ABJ

PR

aspect ratio of the cavitynumber of bladesJacobian matrixpressure normalized by pfR2

radius of cylinder

ROTATING FLOWS 281

Re

(x, y, z)(, , )P

T

rotational Reynolds number

physical timevelocity components in (x, y, z)direction

physical coordinate

computational coordinatefluid densitykinematic viscosity of working fluidcharacteristic time scale[ Re1/2f-1

source termrotational speed of the bottom disk

References

Altgeld, H., Jones, W.P. and Wilhelmi, H., 1983, Velocitymeasurements in a confined swirl driven recirculating flow,Experiments in Fluids; 1,236--242.

Escudier, M.P., 1984, Observations of the flow produced in acylindrical container by a rotating endwall, Experiments inFluids’, 2, 189-196.

Hobson, G.V. and Lakshminarayana, B., 1991, Prediction ofcascade performance using an incompressible Navier-Stokestechnique, ASME Journal oj’ Turbomachinery, 113, 561-572.

Hyun, J.M. and Kim, J.W., 1989, Buoyant convection driven byan encapsuled spinning disk with axial suction, AIAA J.Thermophysics and Heat Tranfer, 3(2), 189-194.

Kim, J.W. and Hyun, J.M., 1997, Axisymmetric inertial oscilla-tions in transient rotating flows in a cylinder, ASME J. FluidsEngineering, 119(2), 390-396.

Kim, J.W. and Noh, S.K., 1997, Enhancement of mixing effectsin fluid flows having an inclined endwall, Proceedings ofExHFT-4, Brussels, Vol. III, pp. 1505-1510.

Lang, E., Sridhar, K. and Wilson, N.M., 1994, Computationalstudy of disk driven rotating flow in a cylindrical enclosure,ASME J. Fluids Engineering, 116(4), 621-629.

Larrousse, M.F., 1987, Transport Phenomena During Spin-up/Spin-down in the Bridgman-Stokbarger Technique, ClarksonUniversity, New York.

Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow,Hemisphere Publishing Company, New York.

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