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The motion of a neutrally buoyant circular cylinder in bounded shear flows
Pergamon
ChmticA Engineering Scimce, Vol. 49. No. 22. pp. 3765-3772 1994 Copyright 0 1994 Elaevier Scieme Ltd Printed in Great Britain All rights resmwd axw-zo9/94 $7.00 + 0.00
OOW-2509(94)001&i-3
THE MOTION OF A NEUTRALLY CYLINDER IN BOUNDED
BUOYANT CIRCULAR SHEAR FLOWS
E. EKLUND Fluid Dynamics Int., 500 Davis St., Evanston, IL 60201, U.S.A. and A. JERNQVIST’ Department of Chemical Engineering I, University of Lund, Lund, Sweden (Received
23 February
1994; accepted for publication
2 June 1994)
Abstract-The complete motion of a solid, rigid, circular cylinder, floating freely in bounded Poisenille and Couette flow has been calculated at Reynolds numbers between 0 and 10. The results include velocities and forces for cylinders at different positions and sizes. The calculations have been done in an iterative way using a finite element analysis package-FIDAP. The results obtained for the force acting on a cylinder at the center line in Stokes flow agree up to four decimal places with the analytical perturbation solutions at small ratios of cylinder radius to channel width. Stokes solution of the problem shows excellent agreement with the results obtained by Sugihara and Niimi. It has been proved that the Segrt-Silberberg effect occurs in the two-dimensional case. The results also show that both the lateral velocity and the equilibrium position are strongly dependent on the cylinder size. Comparison has been made with theoretical and experimental results of the lateral behavior in three-dimensions giving both qualitative similarities and differences.
INTRODUCTKON The behavior of freely floating neutrally buoyant particles in bounded domains is of fundamental importance in rheology and in the mechanics of suspensions, especially in the study of blood and other medical particles, membrane filtration, FFF and similar analytical methods, food production and transport, pulp and paper production, etc. The research in this area increased rapidly after the discovery of the “tubular-pinch” effect (SegrB and Silberberg, 1962a, b). Their experiment showed the behavior of neutrally buoyant solid spherical particles in laminar tube flow. They found that the particles are subjected to inertia-induced forces which tend to move the particles to a certain stable equilibrium position at about Q.6R, where R is the tube radius. This discovery initiated a number of experimental and theoretical studies concerning particle behavior in different types of flows and domains. Many of these investigations were summarized and critically analyzed in two excellent review articles (Brenner, 1966; Goldsmith and Mason, 1967). The first successful attempt to theoretically describe the lateral behavior of a single particle of infinite size in a general manner was made in the late 1960s (Cox and Brenner, 1968) using the method of asymptotic expansions. The results were expressed in terms of volume integrals and were not evaluated explicitly. Some investigators (Vasseur and Cox, 1976; Cox and ‘Author to whom correspondence
should be addressed.
Hsu, 1977) used Fourier transformations of the flow field and succeeded in determining the lateral motion of a single solid sphere for a number of particle and flow situations at an infinite plane and between two parallel planes, respectively. An investigation of the latter case was also made using the method of reflections (Ho and Leal, 1974). However, close to the boundaries their results do not seem to be in agreement with the asymptotic behavior predicted by Cox and Hsu (1977). Despite the quite large amount of analytical work on this phenomenon, no two-dimensional solution has as yet been presented. In the Stokes regime, where the tubular-pinch effect does not occur, translateral and angular motions of a circular body between parallel plates have been described in a general manner by several investigators. Earlier analytical and numerical solutions of this problem have been restricted to the symmetry position of the cylinder and to small ratios of cylinder radius to channel width. The method of reflection was used to determine the force acting on a quiescent cylindrical particle in Poiseuille flow (Harrison, 1924; Fax&n, 1946; Takaisi, 1956). Fax&n’s results are the most accurate, since he used higher-order terms than did the other two investigators. A finite difference method with boundary conforming coordinates was developed to solve general twodimensional particle problems in Stokes flow (Dvinsky and Popel, 1986). This method was applied on the transverse creeping motion of a solid cylinder
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E. EKLUND~~~
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between two plane walls. The results, containing forces, torques, velocities and pressures for a large range of cylinder radius and positions, were presented in two parts. The first contains the motion of a circular cylinder in a quiescent fluid with and without sedimentation (Dvinsky and Popel, 1987a), while the second part deals with cylinder motion in Poiseuille and Couette flow (Dvinsky and Popel, 1987b). In another paper, the finite element method was used to calculate the motion of neutrally buoyant cylindrical and elliptical cylinders in bounded Poiseuille flow (Sugihara and Niimi, 1984). Their results were presented only at some selected positions and sizes. The present paper presents the complete motion of a circular cylinder between two parallel plates at low Reynolds numbers. The simultaneous calculations of the translateral, angular and lateral velocities have been done iteratively on an IBM 3090. The finite element-package FIDAP (Engelman et al., 1989) has been used to solve the Navier-Stokes equations and to calculate the forces on the cylinder surface. The mathematical formulations, geometric considerations conditions, solution procedure, and boundary accuracy estimates, and discussion of the results for a neutrally buoyant circular cylinder in plane Poiseuille and Couette flows are presented in the following sections. THEORY
Mathematical formulations The basic equations to be solved are the steady, isothermal, Newtonian Navier-Stokes equations together with the continuity equation. The most important parts of the finite element formulation for this problem are summarized by Eklund (1990), with more detailed discussions found in the FIDAP-manuals (Engelman et al., 1989). In the current problem a combination of two solution methods have been used to solve these equations. First, the fixed-point procedure known as successive substitution (SS) which may be written as K,(ui)ui+ I = F,
(1)
~JERNQVIST
where K, and F, represents the global left-hand matrix and global right-hand vector, respectively. The second solution method used was the Quasi-Newton updates (QN) which is similar to the ordinary Newton-Raphson method (NR). The SS method is robust and usually converges, though only after many iteration steps. On the other hand, the QN method converges relatively fast if the initial guess is good. Stokes solution has been used as the initial guess for the lowest non-zero Reynolds number. Two convergence criteria were set for this solution scheme. Both the difference in the latest two solution vectors ui and the residual vector R(ui) must bc less than 1.0 x 10e4. For low Reynolds numbers the solution usually converged after one SS followed by three or four QNs. Geometrical considerations and boundary conditions The geometry of the model is shown in Fig. 1. A circular cylinder with radius R and position P = - Y is considered in a Cartesian coordinate system. The position of the two plane parallel boundaries is y = f D/2. The characteristic length scale is D. The distance between the wall and the cylinder center is W, which leads to the simple relation D = 2(P + W). The total length of the channel L is L = 60 with the cylinder situated at half of this distance_ The cylinder motion contains the translateral, lateral and angular velocities denoted V,, V,, and 0, respectively. All these velocities are dimensionless according to the characteristic fluid velocity U. In the case of a plane Poiseuille flow entering the channel, the velocity at the inlet U,.; satisfies, u.x,i
0.6
V
"Y
-
(2Y/D)Z) + UW
UX.i = 2Y + WW.1 + 1)
D
u=uw
uc(l
(2)
(3)
where U,,, denotes the velocity of the lower wall. The velocity of the upper wall, U,.,, is always equal to U,_I
AY
A
=
where U, denotes the velocity of the walls and UC is the center line velocity when U, = 0. If U, is equal to zero then the mean velocity of the fluid is equal to the characteristic velocity W. In the case of a Couette flow the inlet velocity equals
-0.5
Fig. 1. Geometry of the model.
The motion of a neutrally buoyant circular cylinder + 2 and if U,., equals - 1 then the net-volume flow is zero. The Reynolds number is defined by Re = DUp/p, where p is the density and p is the viscosity of the fluid. The undisturbed flow field must satisfy the steady isothermal Newtonian Navier-Stokes equations and the continuity equation for incompressible fluid flow. To make the cylinder free-floating, the following boundary conditions should also be satisfied: tl
Solution
=
V + 51R
on the cylinder surface
(4)
u = u,
on the walls
(5)
u=
at the inlet and outlet.
03
UXi
procedure
Because of the non-linearity of the Navier-Stokes equations the velocities U, and Q have to be adjusted in an iterative way until the corresponding forces on the surface of the body, F, and Fn, are approximately zero. The lateral force FY is then obtained and the lateral velocity V,, is later determined by adjusting it in an analogous way. These forces are evaluated at the Gaussian integration points in each of the cylinder boundary elements. The above calculation plays a central role in the global iteration method, which is described in the following way: 1. The translateral ( Vxl) and angular (Q,) velocities obtained by earlier investigators are used as an initial guess. 2. FIDAP calculates the flow field, and the translateral (F,,) and angular (F,,) forces acting on the cylinder are obtained by post-processing. 3. If F,, > 50F,,, then it is used together with a factor K,, to adjust V,, in the next iteration step. K, is estimated through previous runs. 4. Step 2 is repeated to give Fx2 and a new value of FW 5. The factor K, is now adjusted by the use of F,,, F x7., V,, and V,, according to the secant method. This factor is then used the next time when F, > 50F,. The iteration of V,, is done in an analogous way. This iteration procedure continues until the forces are smaller than a certain limit E. The factor K, = 50 was used since a change in V, strongly affects the value of F, at an unchanged R. The limits for the translateral and angular forces were set as E, = .sr, = 1.0 x 1O-4 and for the lateral force as a,. = 2.0 x10-5.
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At least three calculations were done for every combination of size and position. The translateral and angular velocities were calculated for the creeping motion. These velocities are valid up to at least Re = 1, where the lateral force is obtained. The lateral velocity was also determined at this Reynolds number. Accuracy
estimates
Several meshes and element formulations have been tested to make sure that the very small lateral forces and velocities have an adequate accuracy. This has been achieved by calculating the translateral and angular velocities of an asymmetrically positioned cylinder in Stokes flow. The lateral force should always be zero for a neutrally buoyant symmetrical particle, despite the position in the domain. The test case has a cylinder radius of 0.15 and a position of 0.20. An example of an element mesh is shown in Fig. 2. The translateral and angular forces are, by adjusting the velocities, equal to zero. Three different variations have been tested; the pressure option, the element type and the mesh density. It has been shown that the combination of ninenode elements and continuous pressure approximation is preferable (Eklund, 1990). Satisfactory testing results were achieved with mesh densities and mesh grading based on the basic model. Variations of the selected model were tested against analytical solutions by calculation of the Stokes solution for the drag on a quiescent solid cylinder at the center line between two parallel walls in Poiseuille flow. These results are compared in Table 1 with both analytical (Faxin, 1946) and numerical results (Dvinsky and Pope], 1987b). The present method agrees up to four decimal figures with the analytical solution as long as the cylinder radius R is less than or equal to 0.15. At higher Rvalues, the analytical solution fails and finally it reaches a singularity at R = 0.33. The difference between the present results and those of Dvinsky and Pope1 (1987b) is approximately 1% in the radius range investigated. A comparison with the results of Sugihara and Niimi (1984) would not be appropriate in this case, since their primary data are not available. RESULTS Results are presented for the behavior of a solid circular cylinder in Poiseuille or Couette flow between two parallel plates. Translateral and angular motions are constant, in the range 0 c Re =S 1.0. The quotient
Fig. 2. Mesh for the basic model; 2726 nodes.
E.
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EKLUND
and A. JERNQVIST
Table 1. Calculation of the force acting on a solid cylinder fixed at the centerline betweentwo planewalls in Poiseuille flow Radius 0.05 0.10 0.15 0.20 0.25
Faxen
Present
13.359 24.3 11 41.847 72.935 138.48
13.362 24.312 41.845 72.725 132.35
Dvinsky-Pope1 24.084 41.393 71.784 130.76
between the lateral variables and Reynolds number was calculated and found to be constant in the range 0 < Re < 1.0. The cylinder radius ranges between 0.05 and 0.25, and the position between 0.0 (center line) and wall distance W= 0.02. Extrapolations are done (dotted lines) in the figures to the wall distance of w= 0.
0
.l
.I
P
3
The curves marked R + 0 and eq. represent the behavior of an infinitesimal cylinder and for the lateral equilibrium positions, respectively.
.I
.6
Fig. 3. Translateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at ) present work, (- -) Sugihara and Niimi Red 1. ((1984). Curve marked eq.; P, = 0.
Poiseuille flow The undisturbed Poiseuille flow, far from the cylinder, is given by eq. (2). Results have been obtained at four different Reynolds numbers; Re = 0, 0.1, 1.0 and 10.0. Translateral and angular motions have proved to be totally independent of Reynolds number as long as Re < 1.0.The divergence at Re = 10 is in the range of f O-l% and it also depends on the position and type of flow, which is commented on for each specific result. The relationship between the lateral variables and Reynolds number is found to be linear in the range 0 -Z Re < 1.0, and that the divergence from linearity is up to 5% at Re = 10, according to Fig. 6. Figure 3 shows the translateral motion of a solid cylinder between parallel plates in Poiseuille flow at Re < 1.0. As can be seen, the velocity decreases sharply towards zero as the distance to the wall decreases and it also decreases with increasing radius at all positions. The present results almost coincide with those of Sugihara and Niimi (1984) at the radius of 0.25. When Reynolds number increases, the translateral motion decreases if the direction of the lateral velocity is towards the wall; otherwise it increases. The angular velocity of a cylinder between two plates in Poiseuille flow is shown in Fig. 4. The vorticity is zero at the center line and reaches a maximum at some lateral position. It then rapidly decreases towards zero, as the cylinder touches the wall. The vorticity decreases with increasing radius at all positions. A comparison is made with the results of Dvinsky and Pope1 (1987b) for the radius of 0.15. For Re > 1, the angular velocity decreases irrespective of cylinder positions.
2.6-
2-m
1.5- R l--
.5-’
R~0.~~R=O.2OiR=O.1~R=0.10iR=0.05i I
Fig. 4. Vorticity of a neutrally buoyant circular cylinder between two parallel plates in Poiseuihe flow at 1. () present work, (- -) Dvinsky and Pope1 (1987b) for R = 0.15; (-- .-_) Sugihara and Niimi (1984) for R = 0.25.
Re 4
3769
The motion of a neutrallybuoyant circular cylinder
F; '10'
v; 10"
Rd.16
-200 0
.1
.a
.2
.4
-14 :
I .5
:
0
:
.I
:
:
.2
P Fig. 5. Lateralforce of a neutrallybuoyant circular cylinder between two parallel platesin Poiseuilleflow at 0 < Re G 1.
Figure 5 displays the lateral force acting on a circular cylinder between two plane boundaries in Poiseuille flow at 0 < Re G 1.0. The lateral force Fi, defined by FL = FJRe, is zero at the center line-an unstable equilibrium position (UEP). The force increases to a maximum, before it reaches the stable lateral equilibrium position (SLEP). The force then decreases to a finite value as the cylinder touches the wall. It is found that the outward lateral force reaches an absolute maximum of 0.013 for a radius of about O.KS and at a position of approximately 0.11. The lateral force decreases at Re > 1 irrespective of the cylinder position. The lateral velocity for the cylinder between two plane walls in Poiseuille flow, defined by V ; = VJRe, is depicted in Fig. 6. As in the case of the force, the UEP is midway between the two planes. Starting from that position, the lateral velocity increases to a maximum and then decreases, reaching the SLEP. It is likely that there also exists two symmetrical UEP by the wall. Starting from there, the inwnrd lateral velocity sharply increases to a maximum velocity, and decreases until it also reaches the SLEP. The extrema in Fig. 6 vary with the radius in a very interesting way. The maximum inward lateral velocity increases until a radius of about 0.125, where it reaches its highest value of about 0.85 x 10-j at P z 0.31. The analogous value for the outward lateral velocityis0.52x10-3atPx0.11forR~0.15.When Reynolds number is larger than unity, the lateral velocity Y; decreases irrespective of the position. The variation of the stable lateral equilibrium position (SLEP) with the radius is presented in Fig. 7. By extrapolation of the data it is shown that the SLEP for a cylinder of infinitely small radius is P* z 0.243. At the other end of the curve, the SLEP must approach the center line as the radius becomes 0.5. Some extra runs were done to confirm the shape of the curve up to the radius 0.33, which is the smallest cylinder radius where the cylinder always moves laterally towards the centerline, The SLEP seems to move towards the wall as Reynolds number increases. Figure 8 shows results for the variable A, defined as A = VJF,,, for different cylinder radii and positions.
:
:
.a
:
:
.4
I .6
P
Fig. 6. Lateral velocity of a neutrallybuoyant circular cylinder between two parallel plates in Poiseuille flow at 0 c Re < 1. (- -) Values obtained at Re = 10.
.25-’
R
.2-b .15-.
.1-.06” o-l
0
:
:
.05
:
:
.l
:
:
.lti
:
:
.2
:
‘4 .!22
P
Fig. 7. Stable lateral equilibrium positions for a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at 0 < Re < 10.
A*lO'
Fig. 8. The variable A, defined as A = VJF,, for a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at Re < 1.0.
This variable could be used to estimate the lateral velocities from the lateral forces in a number of cases, such as non-neutrally buoyant cylinders, other flow types, etc. The results show that the force that is needed to achieve a certain velocity increases with increasing cylinder radius and position.
E.
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and A. JERNQVIST
EKLUND
Couette flow
The undisturbed Couette flow, far from the cyiinder, is given by eq. (3). Results have been obtained for Re = 0 and 1.0, where translateral and angular motions are constant. Figure 9 displays the translateral motion of a cylinder between two walls in Couette flow at Re < 1.0. The velocity is zero at the center line between the walls, and it decreases to - 1.0 as the cylinder approaches the lower wall. The translateral velocity decreases with increasing radius at all positions. The angular velocity, represented by the vorticity, of a cylinder between two boundaries in Couette flow is presented in Fig. 10. At the center line, the vorticity asymptotically approaches 1.0 as the cylinder radius decreases to zero. For all other values of cylinder radius, the vorticity decreases with increasing lateral position and radius. A comparison is also shown with the translateral and angular motions obtained by Dvinsky and Pope1 (1987b) for R = 0.15. Figure 11 shows the quotient between the lateral force, acting on a neutrally buoyant circular cylinder between two parallel walls in Couette flow, and Reynolds number at Re = 1. The force (FJ, = F,/Re) is
P; lo=
Fig. 11. Lateral force of a neutrally buoyant circular cylin-
der between two parallel plates in Couette flow at Re = 1.
v; .lOS
P
-.a ’
Fig. 12. Lateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at
-.I
Re=
VX
1.
.
-.*
-.8’. -14 Cl
:
: .1
:
: .2
: P
:
.a
!
: .I
. :
--i .6
Fig. 9. Translateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at Re < 1. () Present work. (- -) Values obtained by Dvinsky and Pope1 (1987b) for R = 0.15.
.s- -
c-2 .4’’
.a-.I
.2
P
.a
.4
.s
Fig. 10, Vorticity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at Re c 1. ) Present work. (- -) Values obtained by Dvinsky (and Popcl(1987b) for R = 0.15.
zero at the center line and it then decreases both with R and with increasing values of P towards a finite value, as the cylinder touches the wall. The relation between the lateral velocity and the lateral force was previously determined in Poiseuille flow as shown in Fig. 8. From this relation, the lateral velocity for the cylinder, defined by ff; = VJRe, in Couette flow could be obtained at Re = 1 and the results are depicted in Fig. 12. Comparison with results obtained in an iterative way gives a difference which is less than O.l%, irrespective of the position and radius of the cylinder. The stable equilibrium position is always at the center line in Couette flow. It is likely, as in the case of Poiseuille flow, that there also exist two symmetrical unstable lateral equilibrium positions by the wall. Starting from there, the inward lateral velocity sharply increases up to a maximum velocity and it then slowly decreases until it reaches the center line. It was found that the maximal inward lateral velocity is about 0.01 at P x 0.18 for R z 0.20. DlSCUSSlON AND CONCLUSIONS
The complete motion of a neutrally buoyant circular cylinder between two parallel plates has been determined for both Poiseuille and Couette flow. A few major results should be further commented upon.
The motion of a neutrally buoyant circular cylinder For Reynolds numbers other than zero, the Navier-Stokes equation is solved iteratively. All velocities associated with the particle are then quasisteady, until the lateral equilibrium position is reached. The question is then one of whether this influences the accuracy of the instantaneous velocity or not. Another problem is that the lateral velocity acts on the surface of the cylinder, but that the mesh is fixed. The calculations should then be changing both in time and space to give reliable results at all Reynolds numbers. The following four statements should prove that the present calculations of the lateral variables are not influenced by the quasisteady velocities, as long as Re -c 1. 1. The translateral and angular motions do not change. 2. The lateral velocity was also determined by the motion of the walls in the negative perpendicular direction without any differences in the results or irregularities in the streamline plot. 3. All streamline figures are perfectly symmetrical. 4. The lateral velocity in Couette flow was obtained both through the relationship shown in Fig. 8 and in an iterative way. The differences in the results between these two methods are less than O.l%, irrespective of the position and radius of the cylinder. Resides calculations at the equilibrium positions, none of these statements is valid at Re = 10. Results at this Reynolds number should therefore be regarded as somewhat less accurate. Complete results in Couette flow were only calculated at Re = 0 and 1.0. However, a number of calculations were performed with the basic model, verifying that the variations of the velocities with Reynolds number follow the same rules as in Poiseuille flow. Consequently, since the lateral velocity in Couette flow is always towards the center line, the translateral velocity increases with increasing Reynolds number when Re > 1. The opposite behavior is expected with the vorticity and both these changes are about O-l% at Re = 10 depending on the radius. It was also established that Figs 11 and 12 are valid in the range 0 c Re < 1.0 and that the lateral variables, divided by the radius, are decreasing at Re = 10 with approximately O-10% in a way similar to the results in Fig. 6. The present solution is compared with the results of Sugihara and Niimi (1984) in Fig. 4, indicating that they have an estimated accuracy of about 1% in their results. However, the comparison with Dvinsky and Pope1 (1987b) in Figs 4, 9 and 10 indicates that their primary results do not achieve the same level of accuracy. No other analytical or numerical solutions of the lateral behavior of a cylinder between two plane walls have previously been presented. Consequently, it would be instructive to compare the present results with related three-dimensional problems. A few of the
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previous theoretical investigations deal with the case of a sphere of infinite size between parallel plates. The results obtained by Vasseur and Cox (1976) are considered to be the most accurate and complete of these. These authors present the lateral migration V;, defined by V,,/[(2R/D)ZRU~p/p], of a neutrally buoyant sphere in Poiseuille and Couette flow. Their results are qualitatively similar to the present results in the region close to the center line of the domain, However, close to the wall, the velocity in the two-dimensional cases reaches a minimum before it approaches the value of zero, instead of the monotonical decrease towards some finite value, as in the three-dimensionai cases. The variations of the stable lateral equilibrium positions (SLEP) with the particle radius have been observed experimentally in the flow of single particles through a rectangular channel (Yanizeski, 1968). The flow was essentially a two-dimensional Poiseuille flow, since the channel was of high aspect ratio. The SLEP was found to be between 0.22 and the center line as the radius of the particles increased, in the size R/D range of 0.25-0.5. The SLEP for a particle of infinite size is 0.3 according to Vasseur and Cox (1976). This behavior differs somewhat from the present results where the SLEP varies with the radius of the cylinder in the range o-0.33. As with the lateral velocity, the large interaction with the walls in twodimensions is probably the reason why the lateral equilibrium position only exists in a limited range of cylinder sizes. In Poiseuille flow, it was experimentally confirmed (Karnis et al., 1966) that the lateral velocity increases with increasing particle radius up to the size R/D of 0.2 in tube flow. For larger particles the lateral velocity decreases. The present results, as indicated in Fig. 6, show that such a maximal lateral velocity exists in two-dimensions at R = 0.12. This is further proof that the lateral behavior close to the center line in twodimensions is in a qualitative agreement with the behavior in three-dimensions. The variations of the lateral behavior with Reynolds number in Poiseuille flow for an infinite sphere between two plane walls have recently been studied (Schonberg, 1986, Schonberg and Hinch, 1989). These studies have theoretically proved that the quotient between the lateral velocity and Reynolds number decreases, and that the equilibrium positions are moving towards the wall, with increasing Reynolds number when Re r=-30. These results are in qualitative agreement with both the present results and the experimental results of Segre and Silberberg (1962b). In Couette flow, the dependence of the lateral velocity on the radius and position in the radius range of O-0.15 was established, both experimentally and theoretically (Halow, 1967; Halow and Wills, 1970a, b). The lateral velocity, defined as Vy/(R3U,), is represented by a single curve in this radius range, close to the results obtained by Vasseur and Cox (1976) for a sphere of infinite size. The accuracy of this
E.
3772
EKLUND
and A.
curve is about 10% compared with the experiments. The maximal lateral velocity in Couette flow that occurs in the present study (for R = 0.20) has not been experimentally confirmed in three-dimensions, since nobody has carried out any studies on R/D larger than 0.15. For the same reason the variations of the lateral velocity with Reynolds number at Re > 10 in Couette flow are also experimentally unconfirmed. Acknowledgements-The authors wish to express their gratitude to Dr G. Aly for many helpful suggestions. The financial support of the Swedish National Board for In-
dustrial and Technical Development is 8ratefully acknowledged.
NOTATION
A D F F K K L P P= R R Re U u
V W x, Y
variable, defined as Vy/Fp, f&t channel width, m force, N global right-hand system matrix, dimensionless factor global left-hand system matrix, dimensionless length of model, m cylinder position from the center line, m equilibrium position of the cylinder from the center line, m cylinder radius, m residual vector Reynolds number, dimensionless vector of unknowns, dimensionless fluid velocity, m/s cylinder velocity, m/s distance between wall and cylinder centre, m Cartesian coordinates, dimensionless
Greek letters E convergence limit, dimensionless
c1 P n Subscripts c i W W.1
W’U X
Y
dynamic viscosity, Pas density, kg/m3 vorticity, angular velocity, rad/s
value at centre line at the inlet, index for a global vector at the wall lower wall upper wall translateral lateral REFERENCES
Brenner, H., 1966, Hydrodynamic resistance of particles at small Reynolds numbers, in Advances in Chemical Engineering (Edited by T. B. Drew, J. W. Hoopes Jr and T. Vermuelen), Vol. 6, pp. 287438. Academic Press, New York.
JERNQVIST
Cox, R. G. and Brenner, H., 1968, The lateral migration of solid particles in Poiseuille flow: I. Theory. Chem. Engng Sci. 23, 147- 173. Cox, R. G. and Hsu, S. K., 1977, The lateral migration of solid particles in a laminar flow near a plane. .J. Multiphase FIQW 3, 201-222. Dvinsky, A. S. and Pop&, A. S., 1986, Numerical solution of 2-D Stokes equations for flow with particles in a channel of arbitrary shape using boundary conforming coordinates. J. Comput. Phys. 67, 73-90. Dvinskv. A. S. and Pooel. _. A. S.. 1987a. Motion of a riaid cylinder between parallel plates in Stokes flow-l. Mot&n in a quiescent fluid and sedimentation. Comput. Fluids 15, 39 l-404. Dvinsky, A. S. and Popel, A. S., 1987b, Motion of a rigid cylinder between parallel plates in Stokes flow-2. Poiseuille and Couette flow. Compur. Fluids 15, 405-419. Eklund, E., 1990, A study of particle-fluid phenomena in bounded shear flows-numerical simulation and experimental modelling. Dissertation, Department of Chemical Engineering I, University of Lund, Sweden. Engelman, M. S. et al., 1989, FfDAP Theoretical Manual, ch. 4.5, Fluid Dynamics International Inc., Evanston, IL. Fax&n, H., 1946, Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes. Proc. Swed. Inst. Engng Res. 187, 1-13. Goldsmith, H. L. and Mason, S. G., 1967, The microrheology of dispersions, in Rheology: Theory and Applications (Edited by F. R. Eirich), ch. 4, pp. 86-246. Acedemic Press, New York. Halow. J. S., 1967, Radial migration of solid spheres in Couette systems. Ph.D. dissertation, Virginia Polytechnic Institute, Blacksburg, Virginia. Halow, J. S. and Wills, G. B., 197Oa, Radial migration of solid spheres in Couette systems. A.1.Ch.E. J. 16.281-286. Halow, J. S. and Wills, G. B., 1970b, Experimental observations of sphere migration in Couette systems. Ind. Engag Chem. Fundam. 9, 603-607. Harrison, W. J., 1924, Trans. Cnmbr. Phil. Sot. 23, 71. Ho, B. P. and teal, L. G., 1974, Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65. 27-43. Karnis, A., Goldsmith, H. L. and Mason, S. G., 1966, The flow of suspensions through tubes. V. Inertial effects. Con. J. Chem. Engng 44, 181-193. Segrd, G. and Silberberg, A., 1962a, Behaviour of macroscopic rigid spheres in Poiseuille flow of suspensions-l. Determination of local concentration by statistical analysis of particle passages through crossed light beams. J. Fluid Mech. 14, 115-135. Segre, G. and Silberberg, A., 1962b, Behaviour of macroscopic rigid spheres in Poiseuille flow of suspensions-2. Experimental results and interpretation. J. Fluid Mech. 14, 136-157. Shonberg, J. A., 1986, Ph.D. dissertation, Rensselaer Polytechnic Institute, NY. Shonberg, J. A. and Hinch E. J., 1989, Inertial migration of sphere in Poiseuilleflows. J. Fluid Mech. 203, 517-524. Sugihara, ‘M. and Niimi, H., 1984, Asymmetric flow of a cylindrical particle through a narrrow channel. J. appt. Mech.
51.
879-884.
Takaisi, Y., 1956, The drag on a circular cylinder placed in a stream of viscous liquid midway between two parallel manes. Trans. J. Phvs. Sot. Jaoan 11. 1092-1095. Va$seur, P. and COY.,R. G., 1976;The lateral migration of a spherical particle in two-dimensional shear flows. J. Ffuid A&h. 72;385-413. Yanizesky, G. M., 1968, Ph.D. dissertation, Carnegie-Mellon University, Pittsburgh, PA.
ChmticA Engineering Scimce, Vol. 49. No. 22. pp. 3765-3772 1994 Copyright 0 1994 Elaevier Scieme Ltd Printed in Great Britain All rights resmwd axw-zo9/94 $7.00 + 0.00
OOW-2509(94)001&i-3
THE MOTION OF A NEUTRALLY CYLINDER IN BOUNDED
BUOYANT CIRCULAR SHEAR FLOWS
E. EKLUND Fluid Dynamics Int., 500 Davis St., Evanston, IL 60201, U.S.A. and A. JERNQVIST’ Department of Chemical Engineering I, University of Lund, Lund, Sweden (Received
23 February
1994; accepted for publication
2 June 1994)
Abstract-The complete motion of a solid, rigid, circular cylinder, floating freely in bounded Poisenille and Couette flow has been calculated at Reynolds numbers between 0 and 10. The results include velocities and forces for cylinders at different positions and sizes. The calculations have been done in an iterative way using a finite element analysis package-FIDAP. The results obtained for the force acting on a cylinder at the center line in Stokes flow agree up to four decimal places with the analytical perturbation solutions at small ratios of cylinder radius to channel width. Stokes solution of the problem shows excellent agreement with the results obtained by Sugihara and Niimi. It has been proved that the Segrt-Silberberg effect occurs in the two-dimensional case. The results also show that both the lateral velocity and the equilibrium position are strongly dependent on the cylinder size. Comparison has been made with theoretical and experimental results of the lateral behavior in three-dimensions giving both qualitative similarities and differences.
INTRODUCTKON The behavior of freely floating neutrally buoyant particles in bounded domains is of fundamental importance in rheology and in the mechanics of suspensions, especially in the study of blood and other medical particles, membrane filtration, FFF and similar analytical methods, food production and transport, pulp and paper production, etc. The research in this area increased rapidly after the discovery of the “tubular-pinch” effect (SegrB and Silberberg, 1962a, b). Their experiment showed the behavior of neutrally buoyant solid spherical particles in laminar tube flow. They found that the particles are subjected to inertia-induced forces which tend to move the particles to a certain stable equilibrium position at about Q.6R, where R is the tube radius. This discovery initiated a number of experimental and theoretical studies concerning particle behavior in different types of flows and domains. Many of these investigations were summarized and critically analyzed in two excellent review articles (Brenner, 1966; Goldsmith and Mason, 1967). The first successful attempt to theoretically describe the lateral behavior of a single particle of infinite size in a general manner was made in the late 1960s (Cox and Brenner, 1968) using the method of asymptotic expansions. The results were expressed in terms of volume integrals and were not evaluated explicitly. Some investigators (Vasseur and Cox, 1976; Cox and ‘Author to whom correspondence
should be addressed.
Hsu, 1977) used Fourier transformations of the flow field and succeeded in determining the lateral motion of a single solid sphere for a number of particle and flow situations at an infinite plane and between two parallel planes, respectively. An investigation of the latter case was also made using the method of reflections (Ho and Leal, 1974). However, close to the boundaries their results do not seem to be in agreement with the asymptotic behavior predicted by Cox and Hsu (1977). Despite the quite large amount of analytical work on this phenomenon, no two-dimensional solution has as yet been presented. In the Stokes regime, where the tubular-pinch effect does not occur, translateral and angular motions of a circular body between parallel plates have been described in a general manner by several investigators. Earlier analytical and numerical solutions of this problem have been restricted to the symmetry position of the cylinder and to small ratios of cylinder radius to channel width. The method of reflection was used to determine the force acting on a quiescent cylindrical particle in Poiseuille flow (Harrison, 1924; Fax&n, 1946; Takaisi, 1956). Fax&n’s results are the most accurate, since he used higher-order terms than did the other two investigators. A finite difference method with boundary conforming coordinates was developed to solve general twodimensional particle problems in Stokes flow (Dvinsky and Popel, 1986). This method was applied on the transverse creeping motion of a solid cylinder
3765
E. EKLUND~~~
3766
between two plane walls. The results, containing forces, torques, velocities and pressures for a large range of cylinder radius and positions, were presented in two parts. The first contains the motion of a circular cylinder in a quiescent fluid with and without sedimentation (Dvinsky and Popel, 1987a), while the second part deals with cylinder motion in Poiseuille and Couette flow (Dvinsky and Popel, 1987b). In another paper, the finite element method was used to calculate the motion of neutrally buoyant cylindrical and elliptical cylinders in bounded Poiseuille flow (Sugihara and Niimi, 1984). Their results were presented only at some selected positions and sizes. The present paper presents the complete motion of a circular cylinder between two parallel plates at low Reynolds numbers. The simultaneous calculations of the translateral, angular and lateral velocities have been done iteratively on an IBM 3090. The finite element-package FIDAP (Engelman et al., 1989) has been used to solve the Navier-Stokes equations and to calculate the forces on the cylinder surface. The mathematical formulations, geometric considerations conditions, solution procedure, and boundary accuracy estimates, and discussion of the results for a neutrally buoyant circular cylinder in plane Poiseuille and Couette flows are presented in the following sections. THEORY
Mathematical formulations The basic equations to be solved are the steady, isothermal, Newtonian Navier-Stokes equations together with the continuity equation. The most important parts of the finite element formulation for this problem are summarized by Eklund (1990), with more detailed discussions found in the FIDAP-manuals (Engelman et al., 1989). In the current problem a combination of two solution methods have been used to solve these equations. First, the fixed-point procedure known as successive substitution (SS) which may be written as K,(ui)ui+ I = F,
(1)
~JERNQVIST
where K, and F, represents the global left-hand matrix and global right-hand vector, respectively. The second solution method used was the Quasi-Newton updates (QN) which is similar to the ordinary Newton-Raphson method (NR). The SS method is robust and usually converges, though only after many iteration steps. On the other hand, the QN method converges relatively fast if the initial guess is good. Stokes solution has been used as the initial guess for the lowest non-zero Reynolds number. Two convergence criteria were set for this solution scheme. Both the difference in the latest two solution vectors ui and the residual vector R(ui) must bc less than 1.0 x 10e4. For low Reynolds numbers the solution usually converged after one SS followed by three or four QNs. Geometrical considerations and boundary conditions The geometry of the model is shown in Fig. 1. A circular cylinder with radius R and position P = - Y is considered in a Cartesian coordinate system. The position of the two plane parallel boundaries is y = f D/2. The characteristic length scale is D. The distance between the wall and the cylinder center is W, which leads to the simple relation D = 2(P + W). The total length of the channel L is L = 60 with the cylinder situated at half of this distance_ The cylinder motion contains the translateral, lateral and angular velocities denoted V,, V,, and 0, respectively. All these velocities are dimensionless according to the characteristic fluid velocity U. In the case of a plane Poiseuille flow entering the channel, the velocity at the inlet U,.; satisfies, u.x,i
0.6
V
"Y
-
(2Y/D)Z) + UW
UX.i = 2Y + WW.1 + 1)
D
u=uw
uc(l
(2)
(3)
where U,,, denotes the velocity of the lower wall. The velocity of the upper wall, U,.,, is always equal to U,_I
AY
A
=
where U, denotes the velocity of the walls and UC is the center line velocity when U, = 0. If U, is equal to zero then the mean velocity of the fluid is equal to the characteristic velocity W. In the case of a Couette flow the inlet velocity equals
-0.5
Fig. 1. Geometry of the model.
The motion of a neutrally buoyant circular cylinder + 2 and if U,., equals - 1 then the net-volume flow is zero. The Reynolds number is defined by Re = DUp/p, where p is the density and p is the viscosity of the fluid. The undisturbed flow field must satisfy the steady isothermal Newtonian Navier-Stokes equations and the continuity equation for incompressible fluid flow. To make the cylinder free-floating, the following boundary conditions should also be satisfied: tl
Solution
=
V + 51R
on the cylinder surface
(4)
u = u,
on the walls
(5)
u=
at the inlet and outlet.
03
UXi
procedure
Because of the non-linearity of the Navier-Stokes equations the velocities U, and Q have to be adjusted in an iterative way until the corresponding forces on the surface of the body, F, and Fn, are approximately zero. The lateral force FY is then obtained and the lateral velocity V,, is later determined by adjusting it in an analogous way. These forces are evaluated at the Gaussian integration points in each of the cylinder boundary elements. The above calculation plays a central role in the global iteration method, which is described in the following way: 1. The translateral ( Vxl) and angular (Q,) velocities obtained by earlier investigators are used as an initial guess. 2. FIDAP calculates the flow field, and the translateral (F,,) and angular (F,,) forces acting on the cylinder are obtained by post-processing. 3. If F,, > 50F,,, then it is used together with a factor K,, to adjust V,, in the next iteration step. K, is estimated through previous runs. 4. Step 2 is repeated to give Fx2 and a new value of FW 5. The factor K, is now adjusted by the use of F,,, F x7., V,, and V,, according to the secant method. This factor is then used the next time when F, > 50F,. The iteration of V,, is done in an analogous way. This iteration procedure continues until the forces are smaller than a certain limit E. The factor K, = 50 was used since a change in V, strongly affects the value of F, at an unchanged R. The limits for the translateral and angular forces were set as E, = .sr, = 1.0 x 1O-4 and for the lateral force as a,. = 2.0 x10-5.
3767
At least three calculations were done for every combination of size and position. The translateral and angular velocities were calculated for the creeping motion. These velocities are valid up to at least Re = 1, where the lateral force is obtained. The lateral velocity was also determined at this Reynolds number. Accuracy
estimates
Several meshes and element formulations have been tested to make sure that the very small lateral forces and velocities have an adequate accuracy. This has been achieved by calculating the translateral and angular velocities of an asymmetrically positioned cylinder in Stokes flow. The lateral force should always be zero for a neutrally buoyant symmetrical particle, despite the position in the domain. The test case has a cylinder radius of 0.15 and a position of 0.20. An example of an element mesh is shown in Fig. 2. The translateral and angular forces are, by adjusting the velocities, equal to zero. Three different variations have been tested; the pressure option, the element type and the mesh density. It has been shown that the combination of ninenode elements and continuous pressure approximation is preferable (Eklund, 1990). Satisfactory testing results were achieved with mesh densities and mesh grading based on the basic model. Variations of the selected model were tested against analytical solutions by calculation of the Stokes solution for the drag on a quiescent solid cylinder at the center line between two parallel walls in Poiseuille flow. These results are compared in Table 1 with both analytical (Faxin, 1946) and numerical results (Dvinsky and Pope], 1987b). The present method agrees up to four decimal figures with the analytical solution as long as the cylinder radius R is less than or equal to 0.15. At higher Rvalues, the analytical solution fails and finally it reaches a singularity at R = 0.33. The difference between the present results and those of Dvinsky and Pope1 (1987b) is approximately 1% in the radius range investigated. A comparison with the results of Sugihara and Niimi (1984) would not be appropriate in this case, since their primary data are not available. RESULTS Results are presented for the behavior of a solid circular cylinder in Poiseuille or Couette flow between two parallel plates. Translateral and angular motions are constant, in the range 0 c Re =S 1.0. The quotient
Fig. 2. Mesh for the basic model; 2726 nodes.
E.
3768
EKLUND
and A. JERNQVIST
Table 1. Calculation of the force acting on a solid cylinder fixed at the centerline betweentwo planewalls in Poiseuille flow Radius 0.05 0.10 0.15 0.20 0.25
Faxen
Present
13.359 24.3 11 41.847 72.935 138.48
13.362 24.312 41.845 72.725 132.35
Dvinsky-Pope1 24.084 41.393 71.784 130.76
between the lateral variables and Reynolds number was calculated and found to be constant in the range 0 < Re < 1.0. The cylinder radius ranges between 0.05 and 0.25, and the position between 0.0 (center line) and wall distance W= 0.02. Extrapolations are done (dotted lines) in the figures to the wall distance of w= 0.
0
.l
.I
P
3
The curves marked R + 0 and eq. represent the behavior of an infinitesimal cylinder and for the lateral equilibrium positions, respectively.
.I
.6
Fig. 3. Translateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at ) present work, (- -) Sugihara and Niimi Red 1. ((1984). Curve marked eq.; P, = 0.
Poiseuille flow The undisturbed Poiseuille flow, far from the cylinder, is given by eq. (2). Results have been obtained at four different Reynolds numbers; Re = 0, 0.1, 1.0 and 10.0. Translateral and angular motions have proved to be totally independent of Reynolds number as long as Re < 1.0.The divergence at Re = 10 is in the range of f O-l% and it also depends on the position and type of flow, which is commented on for each specific result. The relationship between the lateral variables and Reynolds number is found to be linear in the range 0 -Z Re < 1.0, and that the divergence from linearity is up to 5% at Re = 10, according to Fig. 6. Figure 3 shows the translateral motion of a solid cylinder between parallel plates in Poiseuille flow at Re < 1.0. As can be seen, the velocity decreases sharply towards zero as the distance to the wall decreases and it also decreases with increasing radius at all positions. The present results almost coincide with those of Sugihara and Niimi (1984) at the radius of 0.25. When Reynolds number increases, the translateral motion decreases if the direction of the lateral velocity is towards the wall; otherwise it increases. The angular velocity of a cylinder between two plates in Poiseuille flow is shown in Fig. 4. The vorticity is zero at the center line and reaches a maximum at some lateral position. It then rapidly decreases towards zero, as the cylinder touches the wall. The vorticity decreases with increasing radius at all positions. A comparison is made with the results of Dvinsky and Pope1 (1987b) for the radius of 0.15. For Re > 1, the angular velocity decreases irrespective of cylinder positions.
2.6-
2-m
1.5- R l--
.5-’
R~0.~~R=O.2OiR=O.1~R=0.10iR=0.05i I
Fig. 4. Vorticity of a neutrally buoyant circular cylinder between two parallel plates in Poiseuihe flow at 1. () present work, (- -) Dvinsky and Pope1 (1987b) for R = 0.15; (-- .-_) Sugihara and Niimi (1984) for R = 0.25.
Re 4
3769
The motion of a neutrallybuoyant circular cylinder
F; '10'
v; 10"
Rd.16
-200 0
.1
.a
.2
.4
-14 :
I .5
:
0
:
.I
:
:
.2
P Fig. 5. Lateralforce of a neutrallybuoyant circular cylinder between two parallel platesin Poiseuilleflow at 0 < Re G 1.
Figure 5 displays the lateral force acting on a circular cylinder between two plane boundaries in Poiseuille flow at 0 < Re G 1.0. The lateral force Fi, defined by FL = FJRe, is zero at the center line-an unstable equilibrium position (UEP). The force increases to a maximum, before it reaches the stable lateral equilibrium position (SLEP). The force then decreases to a finite value as the cylinder touches the wall. It is found that the outward lateral force reaches an absolute maximum of 0.013 for a radius of about O.KS and at a position of approximately 0.11. The lateral force decreases at Re > 1 irrespective of the cylinder position. The lateral velocity for the cylinder between two plane walls in Poiseuille flow, defined by V ; = VJRe, is depicted in Fig. 6. As in the case of the force, the UEP is midway between the two planes. Starting from that position, the lateral velocity increases to a maximum and then decreases, reaching the SLEP. It is likely that there also exists two symmetrical UEP by the wall. Starting from there, the inwnrd lateral velocity sharply increases to a maximum velocity, and decreases until it also reaches the SLEP. The extrema in Fig. 6 vary with the radius in a very interesting way. The maximum inward lateral velocity increases until a radius of about 0.125, where it reaches its highest value of about 0.85 x 10-j at P z 0.31. The analogous value for the outward lateral velocityis0.52x10-3atPx0.11forR~0.15.When Reynolds number is larger than unity, the lateral velocity Y; decreases irrespective of the position. The variation of the stable lateral equilibrium position (SLEP) with the radius is presented in Fig. 7. By extrapolation of the data it is shown that the SLEP for a cylinder of infinitely small radius is P* z 0.243. At the other end of the curve, the SLEP must approach the center line as the radius becomes 0.5. Some extra runs were done to confirm the shape of the curve up to the radius 0.33, which is the smallest cylinder radius where the cylinder always moves laterally towards the centerline, The SLEP seems to move towards the wall as Reynolds number increases. Figure 8 shows results for the variable A, defined as A = VJF,,, for different cylinder radii and positions.
:
:
.a
:
:
.4
I .6
P
Fig. 6. Lateral velocity of a neutrallybuoyant circular cylinder between two parallel plates in Poiseuille flow at 0 c Re < 1. (- -) Values obtained at Re = 10.
.25-’
R
.2-b .15-.
.1-.06” o-l
0
:
:
.05
:
:
.l
:
:
.lti
:
:
.2
:
‘4 .!22
P
Fig. 7. Stable lateral equilibrium positions for a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at 0 < Re < 10.
A*lO'
Fig. 8. The variable A, defined as A = VJF,, for a neutrally buoyant circular cylinder between two parallel plates in Poiseuille flow at Re < 1.0.
This variable could be used to estimate the lateral velocities from the lateral forces in a number of cases, such as non-neutrally buoyant cylinders, other flow types, etc. The results show that the force that is needed to achieve a certain velocity increases with increasing cylinder radius and position.
E.
3770
and A. JERNQVIST
EKLUND
Couette flow
The undisturbed Couette flow, far from the cyiinder, is given by eq. (3). Results have been obtained for Re = 0 and 1.0, where translateral and angular motions are constant. Figure 9 displays the translateral motion of a cylinder between two walls in Couette flow at Re < 1.0. The velocity is zero at the center line between the walls, and it decreases to - 1.0 as the cylinder approaches the lower wall. The translateral velocity decreases with increasing radius at all positions. The angular velocity, represented by the vorticity, of a cylinder between two boundaries in Couette flow is presented in Fig. 10. At the center line, the vorticity asymptotically approaches 1.0 as the cylinder radius decreases to zero. For all other values of cylinder radius, the vorticity decreases with increasing lateral position and radius. A comparison is also shown with the translateral and angular motions obtained by Dvinsky and Pope1 (1987b) for R = 0.15. Figure 11 shows the quotient between the lateral force, acting on a neutrally buoyant circular cylinder between two parallel walls in Couette flow, and Reynolds number at Re = 1. The force (FJ, = F,/Re) is
P; lo=
Fig. 11. Lateral force of a neutrally buoyant circular cylin-
der between two parallel plates in Couette flow at Re = 1.
v; .lOS
P
-.a ’
Fig. 12. Lateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at
-.I
Re=
VX
1.
.
-.*
-.8’. -14 Cl
:
: .1
:
: .2
: P
:
.a
!
: .I
. :
--i .6
Fig. 9. Translateral velocity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at Re < 1. () Present work. (- -) Values obtained by Dvinsky and Pope1 (1987b) for R = 0.15.
.s- -
c-2 .4’’
.a-.I
.2
P
.a
.4
.s
Fig. 10, Vorticity of a neutrally buoyant circular cylinder between two parallel plates in Couette flow at Re c 1. ) Present work. (- -) Values obtained by Dvinsky (and Popcl(1987b) for R = 0.15.
zero at the center line and it then decreases both with R and with increasing values of P towards a finite value, as the cylinder touches the wall. The relation between the lateral velocity and the lateral force was previously determined in Poiseuille flow as shown in Fig. 8. From this relation, the lateral velocity for the cylinder, defined by ff; = VJRe, in Couette flow could be obtained at Re = 1 and the results are depicted in Fig. 12. Comparison with results obtained in an iterative way gives a difference which is less than O.l%, irrespective of the position and radius of the cylinder. The stable equilibrium position is always at the center line in Couette flow. It is likely, as in the case of Poiseuille flow, that there also exist two symmetrical unstable lateral equilibrium positions by the wall. Starting from there, the inward lateral velocity sharply increases up to a maximum velocity and it then slowly decreases until it reaches the center line. It was found that the maximal inward lateral velocity is about 0.01 at P x 0.18 for R z 0.20. DlSCUSSlON AND CONCLUSIONS
The complete motion of a neutrally buoyant circular cylinder between two parallel plates has been determined for both Poiseuille and Couette flow. A few major results should be further commented upon.
The motion of a neutrally buoyant circular cylinder For Reynolds numbers other than zero, the Navier-Stokes equation is solved iteratively. All velocities associated with the particle are then quasisteady, until the lateral equilibrium position is reached. The question is then one of whether this influences the accuracy of the instantaneous velocity or not. Another problem is that the lateral velocity acts on the surface of the cylinder, but that the mesh is fixed. The calculations should then be changing both in time and space to give reliable results at all Reynolds numbers. The following four statements should prove that the present calculations of the lateral variables are not influenced by the quasisteady velocities, as long as Re -c 1. 1. The translateral and angular motions do not change. 2. The lateral velocity was also determined by the motion of the walls in the negative perpendicular direction without any differences in the results or irregularities in the streamline plot. 3. All streamline figures are perfectly symmetrical. 4. The lateral velocity in Couette flow was obtained both through the relationship shown in Fig. 8 and in an iterative way. The differences in the results between these two methods are less than O.l%, irrespective of the position and radius of the cylinder. Resides calculations at the equilibrium positions, none of these statements is valid at Re = 10. Results at this Reynolds number should therefore be regarded as somewhat less accurate. Complete results in Couette flow were only calculated at Re = 0 and 1.0. However, a number of calculations were performed with the basic model, verifying that the variations of the velocities with Reynolds number follow the same rules as in Poiseuille flow. Consequently, since the lateral velocity in Couette flow is always towards the center line, the translateral velocity increases with increasing Reynolds number when Re > 1. The opposite behavior is expected with the vorticity and both these changes are about O-l% at Re = 10 depending on the radius. It was also established that Figs 11 and 12 are valid in the range 0 c Re < 1.0 and that the lateral variables, divided by the radius, are decreasing at Re = 10 with approximately O-10% in a way similar to the results in Fig. 6. The present solution is compared with the results of Sugihara and Niimi (1984) in Fig. 4, indicating that they have an estimated accuracy of about 1% in their results. However, the comparison with Dvinsky and Pope1 (1987b) in Figs 4, 9 and 10 indicates that their primary results do not achieve the same level of accuracy. No other analytical or numerical solutions of the lateral behavior of a cylinder between two plane walls have previously been presented. Consequently, it would be instructive to compare the present results with related three-dimensional problems. A few of the
3771
previous theoretical investigations deal with the case of a sphere of infinite size between parallel plates. The results obtained by Vasseur and Cox (1976) are considered to be the most accurate and complete of these. These authors present the lateral migration V;, defined by V,,/[(2R/D)ZRU~p/p], of a neutrally buoyant sphere in Poiseuille and Couette flow. Their results are qualitatively similar to the present results in the region close to the center line of the domain, However, close to the wall, the velocity in the two-dimensional cases reaches a minimum before it approaches the value of zero, instead of the monotonical decrease towards some finite value, as in the three-dimensionai cases. The variations of the stable lateral equilibrium positions (SLEP) with the particle radius have been observed experimentally in the flow of single particles through a rectangular channel (Yanizeski, 1968). The flow was essentially a two-dimensional Poiseuille flow, since the channel was of high aspect ratio. The SLEP was found to be between 0.22 and the center line as the radius of the particles increased, in the size R/D range of 0.25-0.5. The SLEP for a particle of infinite size is 0.3 according to Vasseur and Cox (1976). This behavior differs somewhat from the present results where the SLEP varies with the radius of the cylinder in the range o-0.33. As with the lateral velocity, the large interaction with the walls in twodimensions is probably the reason why the lateral equilibrium position only exists in a limited range of cylinder sizes. In Poiseuille flow, it was experimentally confirmed (Karnis et al., 1966) that the lateral velocity increases with increasing particle radius up to the size R/D of 0.2 in tube flow. For larger particles the lateral velocity decreases. The present results, as indicated in Fig. 6, show that such a maximal lateral velocity exists in two-dimensions at R = 0.12. This is further proof that the lateral behavior close to the center line in twodimensions is in a qualitative agreement with the behavior in three-dimensions. The variations of the lateral behavior with Reynolds number in Poiseuille flow for an infinite sphere between two plane walls have recently been studied (Schonberg, 1986, Schonberg and Hinch, 1989). These studies have theoretically proved that the quotient between the lateral velocity and Reynolds number decreases, and that the equilibrium positions are moving towards the wall, with increasing Reynolds number when Re r=-30. These results are in qualitative agreement with both the present results and the experimental results of Segre and Silberberg (1962b). In Couette flow, the dependence of the lateral velocity on the radius and position in the radius range of O-0.15 was established, both experimentally and theoretically (Halow, 1967; Halow and Wills, 1970a, b). The lateral velocity, defined as Vy/(R3U,), is represented by a single curve in this radius range, close to the results obtained by Vasseur and Cox (1976) for a sphere of infinite size. The accuracy of this
E.
3772
EKLUND
and A.
curve is about 10% compared with the experiments. The maximal lateral velocity in Couette flow that occurs in the present study (for R = 0.20) has not been experimentally confirmed in three-dimensions, since nobody has carried out any studies on R/D larger than 0.15. For the same reason the variations of the lateral velocity with Reynolds number at Re > 10 in Couette flow are also experimentally unconfirmed. Acknowledgements-The authors wish to express their gratitude to Dr G. Aly for many helpful suggestions. The financial support of the Swedish National Board for In-
dustrial and Technical Development is 8ratefully acknowledged.
NOTATION
A D F F K K L P P= R R Re U u
V W x, Y
variable, defined as Vy/Fp, f&t channel width, m force, N global right-hand system matrix, dimensionless factor global left-hand system matrix, dimensionless length of model, m cylinder position from the center line, m equilibrium position of the cylinder from the center line, m cylinder radius, m residual vector Reynolds number, dimensionless vector of unknowns, dimensionless fluid velocity, m/s cylinder velocity, m/s distance between wall and cylinder centre, m Cartesian coordinates, dimensionless
Greek letters E convergence limit, dimensionless
c1 P n Subscripts c i W W.1
W’U X
Y
dynamic viscosity, Pas density, kg/m3 vorticity, angular velocity, rad/s
value at centre line at the inlet, index for a global vector at the wall lower wall upper wall translateral lateral REFERENCES
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JERNQVIST
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