Wall Effects in Two-Dimensional Axisymmetric Flow Over a Circular Disk Oriented Normal to Flow in a Cylindrical Tube

Abstract

T he presence of fi nite boundaries is known to exert an extra retardation force on freely settling objects. Conversely, the terminal falling velocity of a solid particle is known to be lower in the presence of fi nite boundaries than that in an unconfi ned medium but otherwise under identical conditions. Qualitatively speaking, the extra retardation effect arises from the upward fl ow of the liquid, and obviously closer are the walls to the falling object, stronger is the upward fl ux and more severe is the wall effect. The extent of this extra retardation force must be known in order to evaluate the net hydrodynamic drag on the particle. Evidently, the extent of such wall effects is strongly dependent on the shape and/or orientation of the falling particle and the type of confi nement, e.g., circular or square tube, slit or a plane wall just on one side, etc. axially in cylindrical tubes fi lled with Newtonian and nonNewtonian liquids. The voluminous literature available on the extent of wall effects on a sphere has been critically evaluated and reviewed recently in many papers (Chhabra, 1993(Chhabra, , 2002Chhabra et al., 1996bChhabra et al., , 2003Uhlherr and Chhabra, 1995;Wham et al., 1996;Kehlenbeck and DiFelice, 1999). Based on a combination of analytical and/or numerical results and of experimental results, it is now possible to estimate the wall effects over the complete range of sphere-to-tube diameter ratio and the Reynolds number. In contrast, much less is known about the extent of wall effects on non-spherical particles, especially circular discs falling in viscous liquids in cylindrical tubesThe effect of cylindrical walls on the drag of a neutrally buoyant disk (with zero inertia) oriented normal to the direction of fl ow has been investigated numerically. Using FLUENT, the fi eld equations have been solved to obtain the values of the total, pressure and friction components of drag as a function of the Reynolds number and of the blockage ratio. The results presented herein encompass the following ranges of the conditions: Reynolds number: 1 to 100; and the blockage ratio (disk diameter/cylinder diameter): 0.02 to 0.5. As expected, the drag is found to be higher under the confi ned fl ow conditions than that in the unconfi ned fl ow conditions. However, the confi ning boundaries appear to exert only a weak infl uence on the formation and size of the wake region. The drag values are also weakly infl uenced by the thickness of the disk. However, this effect progressively disappears as the Reynolds number is gradually increased. The present numerical values corresponding to the unconfi ned fl ow conditions are in excellent agreement with the previous numerical and experimental results available in the literature.On a étudié numériquement l'effet des parois cylindriques sur la traînée d'un disque non soumis à la force d'Archimède (avec une inertie nulle) orienté en position normale dans le sens du courant. À l'aide de FLUENT, on a résolu les équations d'échange afi n d'obtenir les valeurs des composante...

Sedimentation of a circular disk in power law fluids

“…This qualitative inference is also in line with the subsequent experimental findings of Chhabra et al [23] and Rami et al [24]. Both Chhabra et al [23] and Rami et al [24] concluded that the drag coefficient of disks in power law fluids was in line with the corresponding expression for drag in Newtonian liquids [4,16,25]. Combined, these results embrace the following ranges of conditions: Re 100 and power law index from 0.48 to 1.…”

“…3) for the Newtonian fluids [4] in the range 1 < Re < 100 and 0.48 n 0.79. The following equation [4] appears to consolidate the experimental and numerical results [16,25] for circular disks settling in Newtonian fluids, in the absence of wall effects, i.e., e = 0: The good correspondence seen in Fig. 3 is primarily due to the fact that the total drag is dominated by the pressure component and this is little influenced by the shear thinning behavior of the liquid.…”

“…are available elsewhere [16,28], it is not repeated here. It suffices to add here that the field equations have been solved numerically using FLUENT (version 6) to obtain the values of the individual and total drag coefficients as functions of the power law index (0.4 n 1): 1 Re 100 and 0.02 e 0.5 for one value of disk thickness t/d = 0.025.…”

“…A domain extending a distance of 40R upstream of the disk and of 60R downstream from the disk has been used. These values were selected after extensive exploration as outlined in our previous paper for a disk settling in Newtonian fluids [16]. The results reported herein have been checked for grid independence and for domain independence, and are believed to be reliable within 1-1.5%.…”

“…Hence, this work presents the results of a numerical study on the drag and wall effects for a thin circular disk sedimenting broad-face-wise in a long cylindrical tube filled with a power law fluid. However, prior to presenting the new results obtained in this study, it is instructive to review briefly the scant literature available in this field, especially relating to the sedimentation of thin disks and plates in power law liquids, whereas the corresponding body of knowledge for disks falling in Newtonian fluids has been recently summarized elsewhere [16].…”