Wakes fluid spheres

As the Reynolds number increases, a wake is formed behind the fluid sphere or ellipsoid [22, 45]. The formation of a wake behind a fluid particle is delayed compared to a solid sphere due to the internal circulation of the gas. The recirculating wake may be completely detached from the fluid sphere. A secondary internal vortex will then not be formed. For smaller particle Reynolds numbers the wake is symmetrical, but as the Reynolds number increases further the vortex sheet breaks down to vortex rings. Further increase of the Re molds number cause the vortex rings to shed asymmetrically, and the drop or bubble will show a rocking motion. This is one of the two types of secondary motion defined. The other is oscillations (shape dilations), and is also thought to be due to the vortex shedding. 

As the fluid flows over the forward part of the sphere, the velocity increases because the available flow area decreases, and the pressure decreases as a result of the conservation of energy. Conversely, as the fluid flows around the back side of the body, the velocity decreases and the pressure increases. This is not unlike the flow in a diffuser or a converging-diverging duct. The flow behind the sphere into an adverse pressure gradient is inherently unstable, so as the velocity (and lVRe) increase it becomes more difficult for the streamlines to follow the contour of the body, and they eventually break away from the surface. This condition is called separation, although it is the smooth streamline that is separating from the surface, not the fluid itself. When separation occurs eddies or vortices form behind the body as illustrated in Fig. 11-1 and form a wake behind the sphere. 

If Re is of the order of 105, the drag on the sphere may be reduced if the fluid stream is turbulent. The flow in the boundary layer changes from streamline to turbulent and the size of the eddies in the wake of the particle is reduced. The higher the turbulence of the fluid, the lower is the value of Re at which the transition from region (c) to region (d) occurs. The value of Re at which R /pu2 is 0.15 is known as the turbulence number and is taken as an indication of the degree of turbulence in the fluid. 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

Many authors (B4, G3, H2, L2, Sll) have considered the flow pattern and wakes involved in the flow of a fluid past a rigid sphere. Nearly every book on fluid mechanics contains a chapter on flow around submerged shapes. Flow around fluid shapes is only touched upon by a few advanced treatises such as that by Lamb (L2). 

Predicted and observed wake lengths and wake volumes agree closely for Re = 100 (Figs. 5.7 and 5.8). For Re > 100, the excess pressure over the leading surface of the sphere approaches that for an ideal fluid, but there is little recovery in the wake. As Re increases, the importance of skin friction decreases relative to form drag. 

At Re = 130, a weak long-period oscillation appears in the tip of the wake (T2). Its amplitude increases with Re, but the flow behind the attached wake remains laminar to Re above 200. The amplitude of oscillation at the tip reaches 10% of the sphere diameter at Re = 270 (GIO). At about this Re, large vortices, associated with pulsations of the fluid circulating in the wake, periodically form and move downstream (S6). Vortex shedding appears to result from flow instability, originating in the free surface layer and moving downstream to affect the position of the wake tip (Rll, R12, S6). 

Closed wakes have been modeled as completing the sphere or spheroid of which the particle forms the cap [e.g. (C5, P2)]. However, the wake is smaller than that required to complete a spheroid for Re < 5 and greater for larger Re (B3). The wake becomes more nearly spherical as Re 100, but is still somewhat egg-shaped (B3, H5). Wake volumes, normalized with respect to the volume of the fluid particle, are shown in Fig. 8.6 for Re up to 110. Note the close agreement with results (Kl) for solid spherical caps of the same aspect ratio. This is not surprising since separation necessarily occurs at the rim of the... 

Numerical solutions have been reported for fluid motion around spheres falling freely from rest (H4, L5, L7). The value of Re at which wake separation first occurs may be much higher than in steady motion (Re = 20 see Chapter 5) and increases with Re (H4). Figure 11.12 shows typical results for the case where y = 1.72, Re = 145 (Re s = 770). Lateral wake development occurs quickly so that the separation circle rapidly approaches its steady position. Downstream growth is considerably slower. Similar trends are predicted for a sphere started impulsively (Rl, R4). 

Free-fall experiments with Re >10 show that a sphere released from rest initially accelerates vertically, and then moves horizontally while its vertical velocity falls sharply (R3, S2, S3, V2). As for steady motion discussed in Chapter 5, secondary motion results from asymmetric shedding of fluid from the wake (S3, V2). Wake-shedding limits applicability of the equations given above. Data on the point at which wake-shedding occurs are scant, but lateral motion has been detected for in the range 4-5 (C7). Deceleration occurs for Re > 0.9 Re. The first asymmetric shedding occurs at much higher Re than in steady motion (Re = 200 see Chapter 5), due to the relatively slow downstream development, as shown in Fig. 11.12. 

Fig. 11.12 Fluid streamlines relative to sphere falling from rest showing development of wake, after (L5), y = 1.72, Re, = 145. Conditions as above.

In case 3 the relative size of the particles (with respect to the computational cells) is large enough that they contain many hundreds or even thousands of computational cells. It should be noted that the geometry of the particles is not exactly represented by the computational mesh and special, approximate techniques (i.e., body force methods) have to be used to satisfy the appropriate boundary conditions for the continuous phase at the particle surface (see Pan and Banerjee, 1996b). Despite this approximate method, the empirically known dependence of the drag coefficient versus Reynolds number for an isolated sphere could be correctly reproduced using the body force method. Although these computations are at present limited to a relatively low number of particles they clearly have their utility because they can provide detailed information on fluid-particle interaction phenomena (i.e., wake interactions) in turbulent flows. 

Kolev [46] discussed the validity of these relations for fluid particle collisions considering the obvious discrepancies resulting from the different nature of the fluid particle collisions compared with the random molecular collisions. The basic assumptions in kinetic theory that the molecules are hard spheres and that the collisions are perfectly elastic and obey the classical conservation laws do not hold for real fluid particles because these particles are deformable, elastic and may agglomerate or even coalescence after random collisions. The collision density is thus not really an independent function of the coalescence probability. For bubbly flow Colella et al [15] also found the basic kinetic theory assumption that the particles are interacting only during collision violated, as the bubbles influence each other by means of their wakes. 

Beginning right at the sphere surface, heat is transferred radially outward by conduction. However, because Pe is large, very small convection velocities can overwhelm thermal conduction, and the heat transfer process becomes convection dominated at a very short distance from the sphere surface. Thus, before the heat released from the sphere can propagate very far outward in the radial direction, it is swept around the sphere and downstream into a wake. Very near the sphere surface where convection comes into play, the dominant velocity component is in the tangential direction. As a result, the region of heated fluid... 

Flow separation in the case of a drop is delayed compared with the case of a solid particle, and the vorticity region (wake) is considerably narrower. While in the case of a solid sphere, the flow separates and the rear wake occurs at Re 10 (the number Re is determined by the sphere radius), in the case of a drop there may be no separation until Re = 50. For 1 < Re < 50, numerical methods are widely used. The results of numerical calculations are discussed in [94], For these Reynolds numbers, the internal circulation is more intensive than is predicted by the Hadamard-Rybczynski solution. The velocity at the drop boundary increases rapidly with the Reynolds number even for highly viscous drops, In the limit case of small viscosity of the disperse phase, /3 —> 0 (this corresponds to the case of a gas bubble), one can use the approximation of ideal fluid for the outer flow at Re > 1. 

It is of interest that the onset of circulation of liquid drops is associated with deformation of the spherical drop into an ellipsoid (R8, C6, G2). Deformation of the spherical drops was found to be associated with oscillations and with higher transfer rates (C6) even in highly viscous fluids where stagnant drop characteristics are generally assumed. Garner (GI2), who studied the wake behind the deformed drops, concluded that the higher transfer rates were due to the onset of instability and oscillation of the wake at this relatively low Reynolds number (200) compared with 400 to 450 for solid spheres. However, no quantitative relationship is available. 

Oliver (02) observed that a sphere settling slowly near the wall of a tube in an otherwise stagnant fluid moved inwardly, towards the tube axis. Karnis (K4a, K5b), in a series of more detailed measurements, made simitar observations and followed the motion of the sphere all the way to the tube axis. It is on the basis of these observations that F, iP) is concluded to be negative. This conclusion accords with Oseen s (04) theoretical finding [also summarized in Berker (B4, p. 328)1 that, when small inertial effects are considered, a sphere moving parallel to a plane wall in a semi-infinite fluid experiences a repulsive hydrodynamic force, due to the sourcelike behavior of the flow at points distant from the sphere not lying within the wake [cf. Lamb (L5, p. 613)]. 

Based on detailed analysis of experimental results and numerical simulations, the creeping flow occurs in shear-thinning fluids up to about Re 1, even though a visible wake appears only when Re for the sphere reaches the value of 20 [Tripathi et al, 1994]. 

The type of drag occurring when fluid flows by a bluff or blunt shape such as a sphere or cylinder, which is mostly caused by a pressure difference, is termed form drag. This drag predominates in flow past such objects at all except low values of the Reynolds numbers, and often a wake is present. Skin friction and form drag both occur in flow past a bluff shape, and the total drag is the sum of the skin friction and the form drag. (See also Section 3.1 A). 

The velocity of the sphere St with respect to the fluid, i.e. the velocity v,(fr), modified by the wake-effect can be calculated with the help of the simple following model the relative velocity of a sphere is supposed to be the difference between the velocity expressed in the coordinate system in which the fluid is at rest at infinity and the velocity that the fluid would have if the sphere were not existent. With regard to the sphere St follows... 

Figure 10 shows representative streamline patterns for oblates, prolates, and spheres in Newtonian and shear-thinning fluids similar results (not shown here) are obtained for dilatant fluids. The streamline patterns for sphere match with the literature predictions for example see Clift et al. (1978) for Newtonian fluids and Adachi et al. (1973) for power-law fluids (n< 1). The effect of the flow behavior index on streamline patterns for a sphere is found to be negligible, except the fact that the wake formation is somewhat delayed. For prolate spheroids (E = 5), no wake formation occurs even at Re = 100, whereas for oblates, a visible wake is formed even at Re= 10 for = 0.2. To recap, the flow patterns appear to be much more sensitive to the... 

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