Influence of dynamic boundary conditions on the computed flow patterns inside a coaxial rotating disk-cylinder system

ComputersFluidsVol.20, No. 4, pp. 387-398, 1991 Printedin GreatBritain.All rightsreserved

0045-7930/91 $3.00+ 0.00 Copyright© 1991PergamonPressplc

INFLUENCE OF DYNAMIC BOUNDARY CONDITIONS ON THE COMPUTED FLOW PATTERNS INSIDE A COAXIAL

ROTATING

DISK-CYLINDER

SYSTEM

X. Rum, M. AGUILO, J. MASSONS and F. DIAZ Departament de Quimica, Lab. Fisica Aplicada, Universitat de Barcelona, 43005 Tarragona, Spain (Received 8 March 1990; in revisedform 6 June 1991)

Abstract--In this paper, numericalsolutionsfor the steadyaxisymmetricflowof an isothermalviscousfluid driven by moderate disk and/or cylindercoaxial rotation are presented. The influenceof the two possible types of dynamicboundary conditionsfor the solid wails on the meridionalflow featuresis discussedin detail usingthe computedStokesstreamfunction,angular momentumand vorticitydistributions. Results showa cleardependencebetweenthe computed-generatedflowpatterns and the boundaryconditionsused. 1. I N T R O D U C T I O N Much interest has arisen recently in the flow of an isothermal viscous fluid inside a coaxial rotating disk-cylinder system. The main reason for this surge in interest is the need for deeper knowledge of the hydrodynamic mechanisms acting in a wide variety of technological applications, e.g. rotating machinery, chemical mixers, rotational viscometers, heat and mass exchangers with a rotating fluid, preliminary isothermal analyses of vertical crystal growth and so on. From a theoretical viewpoint, this kind of enclosed rotating systems also offers the interesting possibility of solving the Navier-Stokes equations of motion without reducing them to a set of ordinary differential equations by using the similarity transformation that results from the von K~irm~in postulates [1]. The finite spatial condition of this rotary motion seems to solve the problems concerning the degeneracy found in similarity solutions [2, 3], but, at the same time, incorporates other subsidiary questions related to the election of the appropriate dynamic boundary conditions on the solid surfaces that shroud the computational domain. As is well-known, a solid surface may be modeled numerically using either no-slip or free-slip conditions. The most usual and realistic choice is the use of the no-slip conditions but, many interesting viscous flow problems have been solved successfully by using the free-slip boundary approximation. In a coaxial rotating disk-cylinder system, steady axisymmetric forced convective patterns have been numerically simulated in several studies using no-slip conditions with fine [4-8] or spacially stretched meshes [9, 10], while meridional free-slip approximations have been attempted with coarser meshes because the size of the computational mesh used near the wails was greater than the thickness of the boundary layer that would be expected to develop in the real fluid [11]. The validity of the above justification is not rigorously proved and although it does seem to provide a meaningful approximation in some cases [11-13], no comparison between these two types of results has been attempted to date in this kind of confined geometry. Thus, the principal purpose of this work has been focused on analyzing comparatively the influences of the two possible dynamic conditions on the meridional flow details for a series of four relevant preselected situations. These correlations have been carried out in terms of the Stokes streamfunction, angular momentum per unit mass and azimuthal vorticity distributions. 2. C O M P U T A T I O N A L P R O C E D U R E S In steady-state and if axisymmetry is assumed, the dimensionless streamfunction-vorticity formulation for the laminar incompressible motion of a Newtonian fluid referred to a nonrotating cylindrical frame is given by [14]: r

dr

r

~zz=-AR2c°'

r[ruL]r

387

[

{r3[~21r}r +

388

X. RuIz et al.

It should be noted that, from a physical viewpoint, we have considered it more appropriate to work with the angular m o m e n t u m function per unit mass, L, instead of the usual azimuthal velocity component [15]. As the characteristic length scales used for the radial and axial directions are Rc and Hw, respectively in the outlined formulation, the dimensionless groups and kinematic variables used are expressed by

AR=R0//4w, n=ln u = ~z/r,

w = - ~,/r,

-ndl, L = rv,

l-[l=R2cn/v, o9 = w r / A R 2 - u z.

(2)

Because of the axial symmetry, the region of calculations sketched in Fig. 1 is bounded by three rigid walls, an axis of symmetry and a free surface. The free surface was assumed to be flat and stationary, imposing an isochoric constraint on the motion of the enclosed fluid. On all the boundaries the Stokes streamfunction was considered zero since there was no flow across the periphery of the computational domain. Angular m o m e n t u m values along the contours were simply derived starting from their own physical definition and, in addition, the axial gradient was considered zero on the free surface. No-slip and free-slip conditions on the meridional velocity components were used on all impermeable rigid walls, while the symmetry axis was assimilated to an impermeable free-slip one. Boundary conditions on the two meridional velocity components were translated to streamfunction-vorticity conditions using a first-order procedure based on a Taylor expansion of the Stokes streamfunction [16]. Because this procedure couples the vorticity at the boundaries with the streamfunction into the dimensionless squared domain of computation, a previous quantitative evaluation of ~ was obligatorily needed. Vorticity values on the free surface were taken to be zero because the tangential stresses vanish on it [17]. In summary, the dimensionless boundary conditions in analytical form finally became: Top disk:

~ = 0;

L = ~ar2/~);

o9 = - 2 7 J ( r , 1 - A z ) / r A z 2

no-slip

= 0

free-slip;

Free surface:

~ = 0;

L~ = 0;

o9 = 0;

Cylinder wall:

~v = 0;

L = ~/f~;

o9 = -271(1 - Ar, z ) / A R 2 r A r 2

no-slip

o9 = 0

free-slip

Cylinder bottom:

~ = 0;

L

=

co = - - 2 ~ ( r , A z ) / r

~er2/~;

no-slip

Az 2

co = 0 Symmetry axis:

~ = 0;

L = 0;

free-slip;

o9 = 0.

The outlined system of partial differential equations together with the above discussed boundary conditions were finite differenced on a square Ar = Az = a, which completely covered the domain of computation using central differences for the first-order derivatives appearing in the diffusive terms, and central or upwind differences to replace the ones existing in the nonlinear convective terms. In compact form, the central and upwind difference operators used were defined by:

Aox(G, _

- x(I),,_, 2a

aox(o,,j '

2a

A2 X ( i ) i , j = X ( I ) i . i + , - 2 X ( I ) , . j + X ( l ) , 4 _ , a2 ' A~ (tpz X ( I ) ) i , j = AOr(I]1z X ( I ) ) i , j

x ( o , + , , j - x ( G _ ,, '

A ~ , X ( I ) , . j = X ( I ) , + 14 - 2 X ( I ) i . j + X ( I ) i _ ,4 a2 ,

K a sgn(A0 ~O,4)A2r(~OzX(I))i4,

-T

A~(~rX(I))i,i = A° ( ~ r X ( I ) ) , . j - ~

sgn(A°$i.j)AL(¢,X(I))j.j,

(3)

where X ( I ) is a generic variable such that X ( 1 ) = ~P, X ( 2 ) = L and X ( 3 ) = to, and K is an adimensional constant with discrete values 0 or 1. Note that i f K = 0, upwind operators are reduced to the central ones.

Flow patterns in a coaxial rotating disk-cylinder

389

The substitution of the above discrete approaches leads to a system of three coupled nonlinear algebraic equations. To solve it a standard double-sweep iterative algorithm, roughly speaking "Picard type", was implemented [18]. The organization of this procedure could be synthesized as follows: (I) Initialization of the flow field variables. (II) Resolution of the Stokes streamfunction equation (the initial vorticity distribution was numerically frozen). (III) Underrelaxation of the obtained streamfunction solution. (IV) Resolution of the angular momentum equation (the obtained streamfunction and initial vorticity distributions were numerically frozen). (V) Underrelaxation of the obtained momentum distribution. (VI) Resolution of the vorticity equation (the obtained streamfunction and angular momentum distributions were numerically frozen). (VII) Underrelaxation of the obtained vorticity solution. (VIII) Test for global convergence. If not satisfied, interchange initial and final numerical results returning to step (II). (IX) Storing the converged solutions for their subsequent processing. Any direct or iterative method [18, 19] could be initially selected to solve the linearized equations in steps (II), (IV) and (VI), but, as can be seen later and for reasons of convergence, an iterative method (Gauss-Seidel relaxation) was chosen. Convergence criteria adopted in all these inner sweeps were based, as usual, on the boundedness of the norm of the maximum deviation between two successive iterations by means of a prefixed cutoff error, sufficiently small with respect to the mean value of the corresponding variable. The test on global convergence towards the steady-state was also based on the boundedness of the norm of the maximum deviation between two successive global iterations by means of a prefixed global cutoff error. The values of the relaxation parameter in steps (III), (V) and (VII) depends on the dynamical boundary conditions and the discretization mode used. The underrelaxation factor in central differencing was in all cases < 0.5, while for the upwind differencing better convergence behavior, using values closer to unity, was obtained. A residues analysis has been used in each case to evaluate the degree of accuracy of the final converged results. For each of the three variables, ~, L and o~, the maximum residue, in absolute value, has been selected to compare the magnitude of the latter with respect to the maximum variation of the corresponding distribution. At the same time, and to better describe the global residual behavior, the root-mean-square (RMS) of the residues at each mesh point has also been taken into account. To obtain a more complete evaluation of the degree of accuracy, another check using a refined mesh with double the number of points in each direction has been carried out. Based on the obtained numerical results an analysis of the differences between the results obtained using these two meshes has also been performed. Note, finally, that because the rate of convergence of the global sweeps, (I)-(VIII), is seriously affected by the number of inner sweeps requested, previous solutions for dynamic conditions with close nondimensional numbers were used in step (I) as initial values in some situations. 3. RESULTS AND DISCUSSION The digital calculations have been carried out on a 3090/200 IBM computer over a grid which completely covers a region of fixed aspect ratio Rc/Hw= 0.765. The inner cutoff errors used in all steady analyzed situations were 5 × 10 -7, 10 -4 and 5 x 10-4 for the streamfunction, angular momentum and vorticity, respectively. The global cutoff error was fixed at 5 x 10-4. Moderate disk and cylinder Reynolds numbers (Re d = R~f~,j/v, Rec = R2D~/v) were chosen to avoid any problem deriving from the existence and uniqueness of the steady solutions, since in the case of incompressible viscous flows only a unique steady solution exists below an undefined Re, several solutions exist for a range of Re above this limit, and no solutions exist above a second undefined Re [20, 21]. Although this solution severely restricts the applicability of the steady schemes, it may be noted that as the Re increases, hydrodynamic instabilities begin to appear in the flow, breaking the axial symmetry hypothesis [22], and with a further increase in Re turbulence occurs [1, 23]; which barely affect applicability of the steady-state Navier-Stokes equations themselves.

390

X. Rtaz

et al.

The first analyzed situation was a moderate disk rotation (Red = 78; Rec = 0), Figure 2 depicts the meridional distributions of the Stokes streamfunction, angular momentum and vorticity obtained numerically using a 75 x 75 mesh in the no-slip case, whereas for the free-slip one, a more coarse mesh of 38 x 38 was selected. Left/right side shows the resulting computed patterns using [

o.i/ ~-2 •5xlO "6 .

/ T~l.n (Tmin' ' z ' 4 x l O ' 3 )

o.n~

,T

T / ~ m i n ( ?rain ='3"6xI0"3) ,0.12

d

L

7

i

-

Rc

7r.~

z*

L/Lm~x (Lmax=0.12)

L/Lma x (Lmax =O.12)

0.5 '

,}ILL

'

~=~c; ~z=~ ~a=l aa-~cl

"-

' '+

1

.001

0.001

~ir~zlJ h,-a c

Fig. I. Sketch of the computational domain.

toItomm¢ (tamax=5" 51

~I~ ,.in ( ¢~min=-2.3~

Fig. 2. Comparison between the no-slip and free-slip numerical patterns for Re d = 78, Rec -- 0.

Flow patterns in a coaxial rotating disk-cylinder

391

no-slip/meridional free-slip conditions. In both cases, the negative sign of the Stokes streamfunction indicates that the sense of the meridional circulation is upwards near the centerline and downwards along the side wall of the cylinder. The geometric disposition of the streamlines indicates that the most intense meridional velocities are located in the surroundings of the top rotating disk. However, assuming no-slip conditions, the position of the absolute value of the minimum of the streamfunction, maximum normalized value, is slightly shifted towards the top disk. In the same way, under free-slip conditions the most external normalized streamline remains roughly parallel to the cylinder wall contouring a broader zone of moderate mixing. Note, also, the presence of a residual weak zone of recirculation separated from the main flow by a stagnation surface only in the no-slip case [6]. This residual flow appears as a consequence of the interaction of the vorticities generated by the cylinder wall and bottom. The angular momentum distribution shows that, in both cases, the azimuthal flow has only a moderately high intensity under the disk decaying rapidly in the upper part of the meridional plane--about 75% of the fluid in the cylinder rotates at ~ 0.Sfld and only the 25% of fluid near the top disk rotates at a speed > 0.5f~d. Isovorticity lines show, in both cases, a main negative structure. In the no-slip case, however, the inhibition of flow movement near the rigid walls--disk, cylinder wall and cylinder bottom--generates positive vorticity regions confining its spatial expansion to the upper zone of the meridional plane. The position of both vorticity centers does not coincide with the corresponding location of the normalized maxima of the Stokes streamfunction and the quantitative values are lower using free-slip conditions. The essential features of the above flow fields agree well with other published numerical works [5-8, I0, 11]. 575

& x=~ (¢~-SxtO-71 ~7 X-L (cL -io"~) O

1.2

X=~

(g =5x10-~'l

~.0.~ W

1

.

,~

ng 0.0 50

0

100

ng

575

Zl" x- $ ( ~; -5xlO"7) V x=~,(% -to'4~ r7 x=w ( ¢ -5xlO"~)

l 3.. 2

1

ng o.o 0

[~ ................... 50

1 100

b )

ng Fig. 3. G]oba! and partial convergence behaviors for Red = 78 and Re~ = 0. (a) No-slip; (b) frc¢-slip. CAF 20/4--C

392

X. Ru[z et

al.

Setting initially all variables to zero inside the computational domain, the discussed results have been satisfactorily achieved using central differencing and an underrelaxation factor of 0.35. A typical convergence behavior of the numerical algorithm is plotted in Figs 3(a, b). In both cases, the global error as a function of the global iteration number presents a sharp maximum in the initial stages of the computation and a slowly decreasing final tendence. The boundary conditions strongly

I

i i i ,I T/Tma x ( V/max=2.4x10"3)

T/~ma x

(Tmax=l.TxlO "3)

2

O.

J L/Lmax (Lmax=1.) -3.3 _.

L/Lrm~ (Lm~x=l)

-1.5

/

",

mjt~wax ( t o ~ = 0 . 8 1 )

talOJmax (minx=t-02)

Fig. 4. Comparison between the no-slip and free-slip numerical patterns for Rea = 0 and Rec = 136.

Flow patterns in a coaxial rotating disk-cylinder

393

influence these characteristics in the sense that the quantitative values for the no-slip case are roughly twice that corresponding to the meridional free-slip case. Likewise, the speed of convergence is greater for the meridional free-slip case. These facts clearly derived from the disturbance effects produced by the vorticity generation near the wails in the no-slip case due to a strong internal reorganization, as can be observed. Equivalently, a great number of inner iterations are requested for this variable in the initial stages to be converged. Finally, it can also be, noted that, independently of the boundary condition used, as the global iterative procedure converges, the number of inner iterations needed to solve each of the three linearized algebraic equations progressively decreases, thus accelerating the rate of convergence of this initially slow iterative algorithm. Both the maximum and the RMS of the six sets of residues obtained under no-slip meridional free-slip boundary conditions have been typically lower than the 0.01% of the maximum variation of their corresponding ~, L and o9 variable. In the no-slip case, the maximum difference between the values obtained for ~/', L and o9 when a mesh with double the number of points in each direction is considered are <8%, <6% and <3%, respectively. However, the magnitude of these values is not very representative of the global behavior shown by the difference surfaces, since the values are derived from the unavoidable incongruity that exists between the disk radius and the mesh width considered in both cases. Thus, in light of the former results, it can be concluded that the present algorithm is adequate to generate sufficiently accurate approximations of the meridional flow features. The second situation chosen for analysis was a moderate cylinder rotation (Red = 0, Rec = 136). Figure 4 presents the numerical results obtained for the Stokes streamfunction, angular momentum and vorticity in the meridional plane using a 95 × 95 mesh in the no-slip case and a 65 x 65 mesh in the meridional free-slip case. As in Fig. 2, the results plotted on the left-hand side of the cylinder correspond to no-slip boundary conditions while those presented on the right-hand side are obtained in the meridional free-slip case. In both cases, the positive sign of the Stokes streamfunction indicates that the sense of fluid circulation is downwards near the centerline and upwards along the side wall of the cylinder. The strong gradients of the flow in the area around the disk indicate that the most intense meridional velocities are located in this zone. A residual flow separated from the main circulating flow by a stagnation surface appears when no vorticity is generated at solid boundaries. This subsidiary flow slightly deflects the more external streamlines generating an increase in radial velocity in the bottom of the cylinder [I 1]. Finally, it can be noted that assuming free-slip conditions, the position of the maximum streamfunction is slightly shifted towards the top disk and its value is lower than that corresponding to the no-slip case. The rotating side wall increases the angular velocity of the fluid throughout the whole cylinder and two main meridional

0.8 mesh: 65x65 ~ q = 5x10 -4

%) 0.4 0.8

mesh: 95x95 ~g

= 5xi0-4 0.0

6

50 ng

0

50

i00

ng Fig. 5. Global convergence behavior of Re d = 0 and R e c = 136.

I00

394

X. Rulz et

al.

regions can be observed: the first one, under the disk, is the Taylor-Proudman cell [24], inside which the angular momentum is considerably slower than that of the cylinder and radially dependent; the second one, under the free surface, is the outer cell, in which the fluid is approximately in solid body rotation. Meridional vorticity distributions show in both cases a main vortical structure beneath the top disk with positive values roughly independent of the boundary conditions used. In the no-slip case this main structure is bounded by negative vorticity regions generated by the inhibition of flow movement near all solid boundaries, while in the free-slip case a lateral vortical nucleus with negative values and of slow intensity prevents its meridional development. As in the preceding situation, the position of both vorticity centers does not coincide with the corresponding location of the normalized maxima of the Stokes streamfunction. The essential features of the above flow fields agree well with other published numerical works [4, 6, 7, 9-11]. Computational results have been satisfactorily converged using central differencing for the no-slip case and upwind differencing for the meridional free-slip case. The underrelaxation factors used were 0.35 and 0.65, respectively. Although the behavior of the convergence is analogous to the disk rotation, a sharper decrease in the global error in the free-slip case towards the steady solution can be observed (Fig. 5). This tendency is a natural consequence of the stabilizer effects that upwind differencing generates over the numerical algorithm to ensure the diagonal dominance of each of the block-tridiagonal matrices associated to the three linearized equations to be solved simultaneously [15, 17]. In this case, the normalized residues obtained are < 0.01% and, in the no-slip case, the maximum differences obtained between the two meshes are < 5%, 2.5% and < 3% for ~/', L and co, respectively. After analyzing the flow patterns generated by the rotation of the disk and cylinder separately, two different interesting situations of differential rotation have been selected. In the first one----corotation--the disk and the cylinder rotate simultaneously around the same axis and in the same sense, whereas in the second one---counterrotation--the rotation senses are opposite. In corotation, several previous runs have shown that if the disk rotation rate is slower than the cylinder one, the dominant convective mechanism is the forced convection driven by the rotation of the cylinder. The resulting flow patterns are, consequently, quite similar to the former case. However, as the disk rotation increases the flow behavior changes drastically. Disk rotation is now the dominant mechanism of the flow, while the cylinder rotation only modulates it. Figure 6 presents the computed patterns for Red = 78 and Re c = 136. Discretization modes and underrelaxation factors were the same as in the preceding case. A detailed discussion of the convergence behavior is not possible here because the final patterns were derived by means of the three-step sequence: Re d = 78, Rec = 0; Red = 78, Re c = 68; Red = 78, Rec = 136. In both cases, the negative sign of the Stokes streamfunction indicates that the sense of fluid circulation is upwards near the centerline and downwards along the side wall of the cylinder. Also, the compression of the isolines in the neighborhood of the disk indicates that the most intense meridional velocities are located in this zone. A weak residual flow separated from the main circulating flow by a stagnation surface appears near the cylinder wall using meridional free-slip conditions. This flow pushes the main flow to the central region near the symmetry axis, intensifying the meridional velocity field in the neighborhood of the top disk and the cylinder bottom [11]. It can be noted finally, that, assuming meridional free-slip conditions, the position of the maximum streamfunction is shifted slightly towards the top disk and its value is lower than that corresponding to the no-slip case. The azimuthal flow preserves the development of the Taylor-Proudman cell, implying that most of the fluid is rotating in the same sense as the cylinder and that the azimuthal velocity, although it has a radial dependence, is approximately independent of height. Meridional vorticity distributions show in both cases a main vortical structure beneath the top disk with negative values. In the no-slip case this main structure is bounded by positive vorticity regions generated by the inhibition of flow movement near all solid boundaries, while in the meridional free-slip case these regions disappear and only a lateral vortical nucleus with positive values bounds its meridional development. As in the preceding situation, the positions of both vorticity centers do not coincide with the corresponding locations of the normalized maxima of the Stokes streamfunction. The essential features of the above flow fields agree well with other published numerical works [6, 7, 10, 1 I].

395

Flow patterns in a coaxial rotating disk-cylinder

In counterrotation, the steady flow patterns could be qualitatively described as the result of the simultaneous interaction of two centrifugal actions generated by the disk and the cylinder rotating coaxially in opposite directions. Depending on the relative magnitude of this interaction in each point of the fluid, meridional patterns could be more different. Thus, several previous runs have shown that a fast disk rotation and a small cylinder rotation generate flow patterns quite similar l

k~

/Tmi n (Tmiax=-3- ].xl0-3 ) ~IV/min (~mixt='3.2xl0 "3)

. ~ i , J T 0./.5

0./.5

II,

OO6

i/

l

L/Lmx (r~mx'o.33~

L/Lmax (Lmax=0.33 )

6.64

.~o

o

q/L

! i

i i i°/ '

-1.7

~

~±'~

a~ / tOmin (t~mlax =-1.16)

~o/a~nln(Ojmin=.2.8}

Fig. 6. Comparison between the no-slipand free-slipnumerical patterns for Re d = 78 and Rec = 136.

396

X. Rulz

et al.

to those obtained in the first analyzed situation while if the cylinder rotation rate is longer than the disk one the steady flow features are quite similar to those already discussed when the disk stays at rest and the cylinder rotates alone. Only when the disk rotation rate is greater than that of the cylinder and neither of them are too low, is a complex bicellular pattern obtained in the meridional plane. Figure 7 depicts the steady numerical results obtained for Red = - - 7 8 and Rec = 136.

T (xto -3)

T (xzO-3) -0.1

-0.1

1 52

0.(

i I

0.05

i L

L 3

W

i

]

t~

Fig. 7. Comparison between the no-slip and flee-slip numerical patterns for Red -- -78 and Rec = 136.

Flow patterns in a coaxial rotating disk-cylinder

397

Discretization modes and underrelaxation factors were the same as in the preceding case, but in the free-slip case a 75 x 75 mesh was needed to avoid convergence failures. In this situation a detailed discussion of the convergence behavior is not possible because final patterns were derived by means of the three-step sequence: Rea = - 7 8 , Rec = 0; Red = --78, Rec = 68; Red = - - 7 8 , Red = 136. In both cases, steady computational results show two counterrotating structures, one located under the disk and the other occupying almost all of the flow field. Disk rotation controls the convection in the upper oval structure above the new stagnation surface, while cylinder rotation drives the rest [22, 25]. Assuming meridional free-slip conditions, the position of the m a x i m u m streamfunction is shifted slightly towards the top disk and its value is higher than that corresponding to the no-slip case. At the same time, the compression of the isolines indicates the presence of intense meridional velocity areas near the stagnation surface---dividing the streamline = if--and the cylinder bottom [11]. On the other hand, the size of the region occupied by the upper structure is larger when no vorticity is generated in the rigid walls. Meridional vorticity distributions show in both cases two main vortical nuclei with quantitative values roughly independent of the dynamic boundary conditions applied. In the no-slip case, the negative vorticity values generated near the rigid walls slightly constrain the lower structure; while in the meridional free-slip case, a little zone of negative vorticity in the upper right-hand corner of the flow may be observed. It could be noted finally that the essential features of the above flow fields agree well with other published numerical works [6, 7, 9-11]. In view of the above results, it is absolutely necessary to correlate the quantitative information obtained using numerical models with that obtained through experimental techniques based, for instance, on flow visualizations and image processing techniques. These correlations, extended to the whole flow field, will allow the choice of the set of appropriate dynamic boundary conditions for a closer modelization of the real hydrodynamic behavior of the bulk flow. 4. C O N C L U S I O N S In the present work, the influence of dynamic boundary conditions on the axisymmetric forced convective patterns inside a coaxial rotating disk-cylinder system have been examined exhaustively. In all analyzed situations, and even in the case where the computational mesh size near the wall is greater than the thickness of the hydrodynamic boundary layer that might be expected, the substitution of a no-slip wall by a meridional free-slip wall generates computational results which clearly indicate that the meridional flow features, in terms of the Stokes streamfunction and vorticity distributions, are dependent on the boundary conditions used. Acknowledgement--The authors are grateful for the financial support provided by CICYT.

REFERENCES I. H. Schlichting, Boundary Layer Theory. McGraw-Hill, New York 0958). 2. C. Y. Lai, K. R. Rajagopal and A. Z. Szeri, Asymmetricflow between parallel rotating disks. J. Fluid Mech. 146, 203 (1984). 3. C. Y. Lai, K. R. Rajagopal and A. Z. Szeri, Asymmetric flow above a rotating disk. J. Fluid Mech. 157, 471 (1985). 4. H. P. Pao, A numerical computation of a confined rotating flow. Trans. ASME E: J. Appl. Mech. 37, 480 (1970). 5. H. P. Pao, Numerical solutions of the Navier-Stokes equations for flows in the disk-cylinder system. Phys. Fluids 15, 4 (1972). 6. N. Kobayashi and T. Arizumi, Computational analysis of the flow in a crucible. J. Crystal Growth 30, 177 (1975). 7. M. L. Adams and A. Z. Szefi, Incompressibleflow between finite disks. Trans. ASME E: J. Appl. Mech. 49, I (1982). 8. J. M. Hyun, Flow near a slowly rotating disk in a finite cylinder. J. Phys. Soc. Japan 11, 3808 0984). 9. D. Dijkstra and G. J. F. Van Heijst, The flow between finite rotating disks enclosed by a cylinder. J. Fluid Mech. 128, 123 (1983). 10. M. J. Crochet, P. J. Wouters, F. T. Geyling and A. S. Jordan, Finite-element simulation of Czochralski bulk flow. J. Crystal Growth 65, 153 (1983). 11. M. Mihel~i6, C. Schroeck-Pauli, K. Wingerath, H. Wenzl, W. Uelhoff and A. Van tier Haart, Numerical simulation of forced convection in the classical Czochralski method, in ACRT and CACRT. J. Crystal Growth 53, 337 0981). 12. P. J. Roache, Computational Fluid Dynamics. Hermosa, Albuquerque, N.M. (1972). 13. S. S. Chen, C. Y. Chow and M. S. Uberoi, Effect of slip boundary condition on flow computation in the presence of rotational body forces. Computers Fluids 9, 389 (1981). 14. H. P. Grcenspan, The Theory of Rotating Fluids. Cambridge Univ. Press, London (1968). 15. W. E. Langlois, Conservativedifferencingprocedures for rotationally symmetricflowwith swirl. Computer Mech. Appl. Mech. Engng 25, 315 (1981). 16. R. Peyret and T. D. Taylor, Computational Methods for Fluid Flow. Springer, Berlin (1985).

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17. W. E. Langlois, Vorticity-streamfunction computation of incompressible fluid flow with an almost-flat free surface. Appl. Mech. Modelling 1, 196 (1977). 18. G. D. Smith, Numerical Solution of Partial Differential Equations. Clarendon Press, Oxford (1978). 19. G. H. Golub and W. E. Langlois, Direct solution of the equation for the Stokes streamfunction. Computer Mech. Appl. Mech. Engng 19, 391 (1979). 20. W. F. Ames, Nonlinear Partial Differential Equations in Engineering. Academic Press, New York (1963). 21. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach, New York (1963). 22. X. Ruiz, J. Massons, M. Aguil6 and F. Diaz, Experimental study of the hydrodynamics in a model crystal growth crucible. Mater. Res. Bull 24, 493 (1989). 23. J. O. Hinze, Turbulence. McGraw-Hill, New York (1975). 24. A. D. W. Jones, Hydrodynamics of Czochralski growth. A review of the effects of rotation and buoyancy force. Prog. Crystal Growth Charact. 9, 139 (1984). 25. J. R. Carruthers and K. Nassau, Nonmixing cells due to crucible rotation during Czochralski crystal growth. J. Appl. Phys. 39, 5205 (1968).