Layers, boundary sphere, flow around

Elzinga and Banchero (El) use Meksyn s boundary layer equation (M2) for flow around a rigid sphere, with the boundary condition that the interfacial velocity is not zero, to calculate a shift in the boundary-separation ring from an equivalent rigid-sphere location. Their calculated positions are slightly less than their observed shifts but confirm the thesis that these shifts are due to internal circulation. Similar quantitative results are reported by Garner and Tayeban (G7). 

Analytic solutions for flow around and transfer from rigid and fluid spheres are effectively limited to Re < 1 as discussed in Chapter 3. Phenomena occurring at Reynolds numbers beyond this range are discussed in the present chapter. In the absence of analytic results, sources of information include experimental observations, numerical solutions, and boundary-layer approximations. At intermediate Reynolds numbers when flow is steady and axisym-metric, numerical solutions give more information than can be obtained experimentally. Once flow becomes unsteady, complete calculation of the flow field and of the resistance to heat and mass transfer is no longer feasible. Description is then based primarily on experimental results, with additional information from boundary layer theory. 

The second boundary condition is related to the transformation of the flow profile in the surface layer into the flow profile for the Stokes problem on the flow around a sphere outside the thin surface layer near the particle surface [4] and can be written as... 

As the previous illustrations showed, the heat and mass transfer coefficients for simple flows over a body, such as those over flat or slightly curved plates, can be calculated exactly using the boundary layer equations. In flows where detachment occurs, for example around cylinders, spheres or other bodies, the heat and mass transfer coefficients are very difficult if not impossible to calculate and so can only be determined by experiments. In terms of practical applications the calculated or measured results have been described by empirical correlations of the type Nu = f(Re,Pr), some of which have already been discussed. These are summarised in the following along with some of the more frequently used correlations. All the correlations are also valid for mass transfer. This merely requires the Nusselt to be replaced by the Sherwood number and the Prandtl by the Schmidt number. 

Krahn [76] explained how the rotation of the sphere would cause the transition from laminar to turbulent boundary layers at different rotational velocities at the two sides of a sphere. The direction of the asymmetrical wake was explained based on the separation points for laminar and turbulent boundary layers. Krahn studied the flow around a cylinder. For a non-rotating cylinder the laminar boundary layer separates at 82° from the forward stagnation point, while the turbulent boundary layer separates at about 130°. Due to the rotation the laminar separation point will move further back, while the turbulent separation point will move forward. For some value of v qaa/v between 0 and 1 the laminar and turbulent separation points will be at equal distance from the stagnation point. The pressure on the turbulent side will be smaller than on the laminar side causing a negative Magnus force. 

An aerosol with particles in the micron size range flows around a smooth solid sphere a few millimeters in diameter. At sufficiently high Reynolds numbers, a laminar boundary layer develop.s over the sphere from the stagnation point up to an angle of about 110 at which. separation takes place. The removal of particles by direct interception can be calculated from the velocity distribution over the forward. surface of the sphere, up to 90 from the forward stagnation point (Fig. 4.P4). 

The curve of Cj, versus for an infinitely long cylinder normal to the flow is much like that for a sphere, but at low Reynolds numbers, does not vary inversely with because of the two-dimensional character of the flow around the cylinder. For short cylinders, such as catalyst pellets, the drag coefficient falls between the values for spheres and long cylinders and varies inversely with the Reynolds number at very low Reynolds numbers. Disks do not show the drop in drag coefficient at a critical Reynolds number, because once the separation occurs at the edge of the disk, the separated stream does not return to the back of the disk and the wake does not shrink when the boundary layer becomes turbulent. Bodies that show this type of behavior are called bluff bodies. For a disk the drag coefficient Cj, is approximately unity at Reynolds numbers above 2000. 

Now, the denominator of the final term in (11-24) can be evaluated explicitly via exact fluid dynamics solutions for creeping flow around a solid sphere, and for creeping and potential flow around a gas bubble. In the creeping or laminar flow regimes, the momentum boundary layer is not thin. Hence, the following claim ... 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

LOCALLY FLAT MOMENTUM BOUNDARY LAYER PROBLEM EOR LAMINAR FLOW AROUND SOLID SPHERES... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

A well developed theory (12) is available to deal with simple situations flow along a flat plate, around a cylinder or sphere, over an airfoil, etc. Blunt objects such as buildings are generally handled empirically. Figure 2 depicts the perturbations created by boundary layer flows whose surface characteristics differ from those of the unperturbed atmosphere nearby or upstream ... 

Fig. 10. Simulation of an electron-capture supernova following the collapse of an O-Ne core. The time evolution of the radius of various mass shells is displayed with the inner boundaries of the O+Ne, C+O and He shells marked by thick lines. The inner core of about 0.8 M is mainly made of Ne at the onset of collapse ([21], and references therein). The explosion is driven by the baryonic wind caused by neutrino heating around the PNS. The thick solid, dashed, and dash-dotted lines mark the neutrino spheres of ve, ve, and heavy-lepton neutrinos, respectively. The thin dashed line indicates the gain radius which separates the layers cooled from those heated by the neutrino flow. The thick line starting at t = 0 is the outward moving supernova shock (from [22])...

To begin our di,scussion on the diffusion of reactants from the bulk fluid to the external surface of a catalyst, we shall focus attention on the flow pa,st a single catalyst pellet. Reaction takes place only on the external catalyst surface and not in the fluid surrounding it. The fluid velocity in the vicinity of the spherical pel lei will vary with position around the sphere. The hydrodynamic boundary layer is usually defined as the distance from a solid object to where the fluid velocity is 99% of the bulk velocity Similarly, the mass transfer boundarj layer thickness, 6. is defined as the distance from a solid object to where the concentration of the diffusing species reaches 99% of the bulk concentration. 

For flow over a flat plate or around a cylinder or sphere, the velocity profile is linear near the surface but the gradient decreases as the velocity approaches that of the main stream at the outer edge of the boundary layer. Exact calculations show that the mass-transfer coefficient still varies with if is low or the Schmidt number pLjpD is 10 or larger. For Schmidt numbers of about 1, typical for gases, the predicted coefficient varies with a slightly lower power of Z>p. For boundary-layer flows, no matter what the shape of the velocity... 

So far we have considered only flows which were in one direction, as in a pipe or down a straight riverbed. In the few cases in which the fluid flow was not one-dimensional, as around a sphere or in a pipe elbow or venturi meter, we have introduced experimental data to allow us to treat the problem as if it were one-dimensional. Although this one-dimensional approach adequately covers many of the practical problems in fluid mechanics, it is not satisfactory for the complicated ones, particularly for the aerodynamics problems. To solve these more complicated problems, two additional ideas are needed potential flow and the boundary layer. 

Creeping flow of an incompressible Newtonian fluid around a solid sphere corresponds to g (0) = I sin0. For any flow regime that does not include turbulent transport mechanisms, the dimensionless boundary layer thickness is... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

Comparison of Sc(6) for creeping flow of an incompressible Newtonian fluid around stationary gas bubbles and solid spheres, where the boundary layer adjacent to a no-slip high-shear interface is... 

Compare mass transfer boundary layer thicknesses for creeping flow of identical fluids around (a) a sohd sphere, and (b) a gas bubble at the same value of the Reynolds number. In which case is the boundary layer thickness greater ... 

The interphase mass transfer coefficient of reactant A (i.e., a,mtc), in the gas-phase boundary layer external to porous solid pellets, scales as Sc for flow adjacent to high-shear no-slip interfaces, where the Schmidt number (i.e., Sc) is based on ordinary molecular diffusion. In the creeping flow regime, / a,mtc is calculated from the following Sherwood number correlation for interphase mass transfer around solid spheres (see equation 11-121 and Table 12-1) ... 

Figure 3. Possible conditions of the momentum boundary layer around a submerged solid sphere with increasing relative velocity. Key a, envelope of pseudo-stagnant fluid b, streamline flow c, flow separation and vortex formation d, vortex shedding e, localized turbulent eddy formation. Reproduced, with permission, from Ref. 38. Copyright 1981, Springer-Verlag.

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