Chapter 1 | Background and Theories
1.2 Theories
The subject of a circle or an infinitely long circular cylinder in relative motion to one or a pair of lateral plane walls (2D treatment) has been studied [26]-[30]. Either a Poiseuille flow was applied over the fixed object or the object itself was set in constant motion and the hindering force on the object was then calculated. An illustration of a two-dimensional cylinder subjected to Poiseuille flow is shown in Fig. 1.1 (a). When the cylinder translates midway at U in Fig. 1.1 (b), a wall correction factor on the drag force is calculated as
, 2b R D Fx b R/ U
(1.5)
In Eqn. (1.5), F is the actual drag and (b/R) is a dimensionless factor indicating the ratio between x
the cylinder diameter and the wall gaps indicated in Fig. 1.1. Hu (1994) [27] utilized simulation to obtain as a function of (b/R) using different particle Reynolds number and the results are represented in Fig. 1.2. It is noted that increases monotonically as the cylinder moves in a narrower gap. Further, a greater is developed at higher Re but the influence of liquid inertia is degraded as gap decreases.
Fig 1.1: 2-D physical models. (a) A circular cylinder placed midway in the channel and
subjected to Poiseuille flow. Reprinted from the Chemical Engineering Science, volume 59, A.
Ben Richou, A. Ambari, and J.K. Naciri, “Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille”, pp. 3216, Copyright (2003). (b) A cylinder translating midway between the plane walls. Reprinted from the Chemical Engineering Science, volume 60, A. Ben Richou, A. Ambari, M. Lebey, and J.K. Naciri, “Drag force on a circular cylinder midway between two parallel plates at Re= 1 Part 2: moving uniformly (numerical and experimental)”, pp. 2537, Copyright (2004).
Fig. 1.2: Numerical results of Hu (1994) for a circular cylinder descending midway between the plane walls at a constant velocity showing the wall correction factor as a function of scaled wall spacing. The trend of the curve is also shown to flatten out as Re increases. Reprinted from the Theoretical and Computational Fluid Dynamics, volume 7, Howard H. Hu, “Motion of a Circular Cylinder in a Viscous Liquid Between Parallel Plates”, pp. 447, Copyright (1994).
As for the 3D treatment ([31]-[35]), the wall correction factor is calculated as
, 3b R D Fx b R/ 6 R U
(1.6)
Goldman, Cox, and Brenner (1967) [31] derived asymptotic solutions of the Stokes equations for a sphere in constant translation parallel to a plane wall bounding a semi-infinite flow field. The
analytical approximation from the lubrication theory along with the Faxén’s approximation obtained by the method of reflections and the “exact” numerical data by O’Neill are presented in Fig. 1.4.
The Fxt 6RU on the y-axis is dynamically equivalent to and its negative sign indicates that the drag is against the translation. Note that the result from the lubrication theory in Fig. 1.4 deviates from the exact numerical solution by O’Neill because the former neglects all the higher-order terms for small gap width ( R 0) in their asymptotic solution. After they
estimated the magnitudes of the higher-order terms, the wall correction factor was improved as
0
According to Eqn. (1.7), the line of lubrication theory in Fig. 1.4 has to be adjusted by being shifted up by 0.9588 units. Thus, it would then properly conform to the numerical outcome predicted by O’Neill, and the corresponding asymptotic solution is confirmed.
Referring to Fig. 1.4, one can see that, similar to the 2D case, the wall correction factor on the Stokes drag increases as the interstitial gap diminishes.
Fig. 1.3: 3D physical model of a sphere in constant translation parallel to a plane wall bounding a semi-infinite flow field. Reprinted from the Chemical Engineering Science, volume 22, A. J. Goldman, R. G. Cox, and H.
Brenner, “Slow viscous motion of a sphere parallel to a plane wall-Ⅰ: Motion through a quiescent fluid”, pp. 638, Copyright (1966).
Fig. 1.4: Result of Brenner et al. (1966) compared with O’Neill’s and Faxén’s showing the wall correction factor as a function of scaled interstitial gap. Reprinted from the Chemical
Engineering Science, volume 22, A. J. Goldman, R. G. Cox, and H. Brenner, “Slow viscous motion of a sphere parallel to a plane wall-Ⅰ: Motion through a quiescent fluid”, pp. 638, Copyright (1966).
Later, Ganatos, Pfeffer, and Weinbaum (1980) [32] derived a semi-analytic solution for the case where a sphere of arbitrary size moves constantly at a position, not necessarily midway, between the walls as illustrated in Fig. 1.5. The sphere center position is quantified by a dimensionless parameter, s = b/(b+c), in which definition b denotes the distance between the sphere center and the nearest wall while c is the distance to the further wall. Thus, one limiting case occurs when the sphere moves midway between the walls giving s = 0.5; the other limiting case the authors considered is s = 0.0 which condition can be created only when c becomes infinitely large. Another parameter that characterizes the distance between the sphere center and the nearer wall is the ratio, b/R, that takes a limiting value b/R → 1 when the sphere is in touch with the nearer wall.
Fig. 1.5: 3D physical model of a sphere in constant translation between the plane walls.
Reprinted from the Journal of Fluid Mechanics, volume 99, part 4, Peter Ganatos, Robert
Pfeffer, and Sheldon Weinbaum, “A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion”, pp. 757, Copyright (1980).
Their result is displayed in Fig. 1.6 (a) in which the wall correction factor is denoted as Fxt rather than . We focus on the results at s = 0.5 that corresponds to a sphere translating midway between the walls. As shown by the various lines prepared for various wall gaps, increases Fxt again monotonically when the wall gap diminishes by decreasing the ratios of b/R from 5.0 to 1.1.
Apart from these theoretical treatments, Lomholt and Maxey (2002) [34] employed force-coupling scheme to simulate the wall effect using the same setup. Their results are presented by those symbols in Fig. 1.6 (b) which compare well to those lines obtained by Ganatos et al. in terms of the trends. However, this more recent numerical simulation gives slightly lower wall correction factor than those semi-analytic results.
Fig. 1.6: Results for a sphere translating at a constant distance between the walls across various scaled center-to-wall distance showing the wall correction factor as a function of the
dimensionless wall spacing. (a) —, Ganatos et al.; ---, Ho & Leal (1974), weak interaction method of reflections; ○, Faxén (1923); ●, Goldman, Cox & Brenner (1967a), exact.
Reprinted from the Journal of Fluid Mechanics, volume 99, part 4, Peter Ganatos, Robert
Pfeffer, and Sheldon Weinbaum, “A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 2. Parallel motion”, pp. 771, Copyright (1980). (b) (▲) and (─), b/R = 5.0; (■) and (---), b/R = 2.0; (▼) and ( ), b/R = 1.5; (●) and ( ), b/R = 1.25; () and (), b/R = 1.1. Reprinted from the Journal of Computational Physics, volume 184, Sune Lomholt and Martin R. Maxey, “Force-coupling method for particulate two-phase flow: Stokes flow”, pp. 397, Copyright (2003).
While some thorough investigations have been conducted via theoretical and numerical approaches, experimental validation is very sparse. One related creeping flow experiment was done by Miyamura et al. (1980) [33] who considered the drag force when one solid sphere settles at a terminal velocity in a rectangular tube, with the particle Reynolds number ranging from 0.000241 to 0.699. His results are shown in Fig. 1.7. The parameter r is the ratio of the sphere diameter, * Dp,
*f r , is different from since it is defined as
* F Wf r u u (1.8)
where u is the terminal velocity in the infinite fluid domain while F u is that developed under W
the influence of lateral walls. The fact of f being greater than unity clearly indicates the wall hindering effect on the sphere descent. In order to ensure total liquid mass conservation, the downward sphere motion would result in uprising liquid flow and it is this counter flow that leads to the greater drag and hence the slower descent.
Fig. 1.7: Results of Miyamura et al. for a sphere settling midway between the walls of a
rectangular tube showing the wall correction factor as a function of the scaled wall spacing.
Reprinted from the International Journal of Multiphase Flow, volume 7, A. Miyamura, S.
Iwasaki and T. Ishii, “Experimental Wall Correction Factors of Single Solid Spheres in Triangular and Square Cylinders, and Parallel Plates”, pp.44, Copyright (1981).
Instead of using a tube, Brooke (2005) [36] conducted experiments with the sphere descending
100% glycerol weight percentage and the other was of 80%. Two unbounded terminal particle Reynolds numbers were achieved: Re = 0.6 for the 100% mixture and Re = 72 for the 80% mixture.
In his context, the wall correction factor, f h d
/
, is defined using the force ratio as
1 2 2 f p Re QD
f h d U A F (1.9)
where the numerator is the Stokes drag with defined in Eqn. (1.4) and the denominator is the quasi-steady drag under the influence of later walls. Thus, a greater deviation of f h d
/
fromunity indicates the larger wall augmentation on the drag. In Eqn. (1.9), h/d indicates the ratio of wall spacing, h, to the sphere diameter, d. This geometry parameter is equivalent to b/R. Further, it is obvious that f (h/d) is the inverse of the wall correction factor, , defined in Eqn. (1.6).
Another point about Eqn. (1.9) worth noting is that the total quasi-steady drag, FQD, is sought
under the assumption that the liquid inertia effect at higher Re and the wall correction effect at small h/d can be accounted for by the multiplication of and . While the validity of such
multiplication is not yet verified, such treatment seems sensible and has been adopted by Joseph (2003) [37]. We will follow this custom in modifying the quasi-steady drag force. With the definition in Eqn. (1.9), Brooke calculated the wall correction factor based on his experimental data and the results are provided here in Table 1.1. The wall influence on the quasi-steady drag is observed as f deviates from 1.0 with the reduction in h/d. Such wall enhancement is more pronounced in more viscous liquid as the data taken in pure glycerol are compared to those in 80%
mixture. Take the data at h/d = 1.2 for explicit comparison, the wall influence measured in the 80%
glycerol mixture is only 45% of that developed in pure glycerol. Thus, it is sensible to expect that the wall augmenting effect would suffer from a sever reduction along the decrease in viscosity and increase in Re.
Lastly, Brooke followed Miyamura to estimate the terminal velocity ratio defined in Eqn. (1.8) and the results are re-presented in Table 1.2. The enhancement in total drag results in reduction in the wall-present terminal velocity which gives a higher f. However, such effect is slightly promoted in Miyamura’s case. This phenomenon might be attributed to the complete tube wall confinement in Miyamura’s setup in which the liquid has to rise in the opposite direction to the sphere descent and hence generates a higher relative velocity between the two phases. In Brooke’s experiment, the liquid may escape from the opening sides at the edges of the walls and the counter-flow should be weaker.
Table 1.1
Wall correction factors for quasi-steady drag. Reprinted from Warren T. Brooke, “The Transient Motion of a Solid Sphere Between Parallel Walls”, pp. 60, Copyright (2005).
Table 1.2
Wall correction factors compared with that measured by Miyamura. Reprinted from Warren T.
Brooke, “The Transient Motion of a Solid Sphere Between Parallel Walls”, Copyright (2005).
After taking the inverse of f h d
/
in Table 1.1, we observe that the experimentally measured wall correction factors are smaller than those predicted by theory and simulation (Fig.1.6). Take the low Reynolds number data measured in pure glycerol (with its unbounded terminal Re = 0.6) for example, Brooke obtained f -1 = 2.278 when h/d = 1.2. However, Ganatos predicted a factor between 2.33 and 3.9, which should fall closer to 3.9, as shown by the data at s = 0.5 in Fig.
1.6 (a).
Also, one can instantly spot that the wall correction factors computed in the 3D problem is significantly lower than those developed in the 2D case. As shown in Fig. 1.2, when the parameter of the x-axis approaches 1, the wall correction factor soars to about 100. Dividing the value of 100 with 4 , the quotient is still as high as 7.9, which is a huge difference from the limiting value of 4 (or 5) in Fig. 1.6 and 1.7. This might be due to the fact that the displaced fluid is completely confined in the 2D plane and thus can only circulate through the small gaps between the sphere and the walls. The small gap thus induces large velocity transverse gradient, and hence a greater shear
stress and wall correction factor result. In the 3D case, the displaced fluid has one more degree of freedom to escape, especially when the walls with opening ends, so the wall influence on the total drag is somewhat degraded.
It is worth noting that the above results are all confined to the steady motion of a sphere and the corresponding Re are mostly confined to the creeping flow regime. The topics of how the lateral walls may affect the hydrodynamic force when the sphere exhibits transient motion or moves at higher Re is rarely found in the context. Therefore, this thesis sets out to address this issue via systematic experiments. The unsteady motion is developed by setting the sphere motion as fully immersed pendulum midway between two lateral walls of various spacings. By changing the pendulum release angle and the liquids, various values of Re can be achieved. We will employ the instantaneous pendulum velocity v to estimate an instantaneous wall correction factor using Eqn.
(1.6) with included instead of the terminal velocity U which does not develop in our setup.
In the following, Chapter 2 introduces the experimental facility, procedures, and calibration methods. The image processing technique and its error estimation are described in Chapter 3.
Chapter 4 presents the results and discussions in three categories separated by impact pendulum Re.
Conclusions and comments are provided in Chapter 5.
Chapter 2
Apparatus and Instrumentation
2.1: Introduction
We have developed a water tank and a pendulum release mechanism in which a solid sphere of diameter D and density p was released as a pendulum in a viscous fluid of viscosity . This
impact sphere was allowed to swing not only midway between two lateral parallel plane walls with variable wall spacing W but perpendicularly towards an identical downstream target sphere, which could also conduct pendulum motion. Both spheres were made to swing in the same plane. The process was recorded by a high-speed digital camera with back illumination and the acquired images were processed by the circular Hough transform routine to obtain the position information for computing further kinematic and dynamic quantities.
The overall experiment setup is shown in Fig. 2.1. All the apparatus and their related parameters will be introduced in this chapter, while an introduction to the image processing procedure is deferred to the next chapter.
Fig. 2.1: Overall experiment setup. (a) The side view of the main structure of the tank fixtures along with a rough layout of the illumination and the high-speed cameras. (b) The front view of the tank fixtures.
2.2: Water Tank
The dimensions of the water tank (Fig. 2.2) are 500mm in length, 500mm in width, and 750mm in height on the exterior. The wall thickness is about 10mm. It is made of glass. The transparency of glass facilitates monitoring the pendulum motion from the side. Before each set of experiments, the tank wall facing the camera was carefully wiped to avoid any stain in the captured image.
Fig. 2.2: Water tank and its dimensions.
2.3: Test Fluid
The test fluid is a Newtonian fluid of glycerol-water mixture. Glycerol is a colorless, odorless, and Newtonian viscous liquid and it is soluble in water. By changing the volume proportion of water, fluids of different viscosities were prepared, resulting in various maximum velocities for the pendulum motion. Although there are tabulated relations for the mixture viscosity with varying specific weight and temperature, we directly measured its viscosity, temperature, and specific gravity with the viscometer, the thermometer, and the hydrometer (to be introduced later).
Throughout the course of experiments, the viscosity ranged from 35 to 385cP, the temperature ranged from 18 to 26.7oC, and the specific gravity fell between 1.1968 and 1.2540 with the density of water taken to be 1000kg/m3.
At low viscosity (34~36cP), it was observed that lots of tiny air bubbles were generated easily
spheres, which deteriorated the image quality. This phenomenon diminished at higher viscosity (above 70cP). On all accounts, any air bubble was carefully removed before each set of experiments.
2.4: Test Sphere
The test spheres are steel spheres of 19mm ±0.1mm in diameter. A cluster of four spheres was
measured, by an electronic scale, to be 0.114kg, which, according to the following formula
43 3
p R p
m (2.1)
was used to estimate the sphere density to be 8000kg/m3.
To enhance the accuracy of center detection, we should enhance the definition of the rim of sphere in an image. Accordingly, one should have the sphere in strong contrast with the illuminated background and avoid any light reflection on the surface of sphere. Each test sphere was colored into black by a black permanent marker pen. This layer of dye had negligible thickness, was not soluble in water, and could be easily removed with ethanol (or alcohol).
2.5: Pendulum String
SunniBond®, SB-3000 and PR-200 by Sunnico Inc. to fix the thread to the sphere. This special glue contains two parts (Fig. 2.3). SB-3000 is the cyanoacrylate adhesive (or the glue) and PR-200 is the catalyst that accelerates the hardening of adhesive.
Fig. 2.3: Instant glue used to fix the thread to the sphere.
A tiny drop of glue was first placed on the tip of a toothpick and then transferred onto the surface of sphere. One tip of the thread was laterally poked into the base of the drop (Fig. 2.4 (a)), and the catalyst was quickly sprayed onto it. The drop hardened in seconds but was put aside for10~15 minutes to ensure consolidation. Based on our experience, a single drop was usually not strong enough. Thus, the end-fixed thread was pulled to lie flat on the surface of sphere and a separate drop was placed in the vicinity to the previous drop (Fig. 2.4 (b)). After the attachment was done, the pendulum was hung in air and made to swing slightly to test its reliability. It should be noted that even if the attachment was firm, it could still be ripped apart by a sudden pull. So we should be cautious when moving the pendulum.
Fig. 2.4: Attachment of the thread to the sphere surface. (a) The tip of thread was fixed by the first glue drop. (b) The following section of thread was fixed by a consecutive drop.
The thread was kept as thin as possible to reduce its influence on the pendulum motion. A thin nylon fishing line was once used. However, during the pendulum motion, apparent oscillation was observed which persisted through the pendulum motion. This was evaluated to result from the sudden downward motion due to gravity upon release that led to alternative elongation and shortening in the radial direction. Hence, we replaced the nylon thread with a Fujino Inc.
PROGUARD 26, No.0.07 composite thread which has negligible elasticity (Fig. 2.5). The thread has an inner inelastic core covered by some elastic wrap to ensure its circular cross-section. The composite thread is 0.044mm in diameter and can sustain a pull up to 820g.
Fig. 2.5: Composite thread used as the pendulum string.
2.6: Lateral Walls – the Near-Field Solid Boundary
The walls are made of transparent acrylic allowing camera recording. Each wall has an area of 400mm-by-300mm and a thickness of 10mm. The size of it was chosen so that it covered the whole pendulum swing beneath the liquid free surface. Further, the walls were specifically located to ensure that the pendulum moved at a vertical distance greater than 5D from both the upper and bottom edges of the wall, as indicated in Fig. 2.6.
The pair of plane walls was mounted on a transverse screw rod, as shown in Fig. 2.1. When we turned the transverse screw rod, the pair of walls simultaneously moved apart at the same pace.
The pair of plane walls was mounted on a transverse screw rod, as shown in Fig. 2.1. When we turned the transverse screw rod, the pair of walls simultaneously moved apart at the same pace.