Potential flow surface velocity

An asymptotic formula for Re oo is easily derived by substitution of the potential flow surface velocity. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Single gas bubbles in an inviscid liquid have hemispherical leading surfaces and somewhat flattened wakes. Their rise velocity is governed by Bernoulli s theory for potential flow of fluid around the nose of the bubble. This was first solved by G. I. Taylor to give a rise velocity Ug of ... 

A reduced scale of the model requires an increased velocity level in the experiments to obtain the correct Reynolds number if Re < Re for the prob lem considered, but the experiment can be carried out at any velocity if Re > RCj.. The influence of the turbulence level is shown in Fig. 12.40. A velocity u is measured at a location in front of the opening and divided by the exhaust flow rate in order to obtain a normalized velocity. The figure show s that the normalized velocity is constant for Reynolds numbers larger than 10 000, which means that the flow around the measuring point has a fully developed turbulent structure at that velocity level. The flow may be described as a potential flow with a normalized velocity independent of the exhaust flow rate at large distances from the exhaust opening— and far away from surfaces. 

Streamline curvature over a very extensive region, and there is infinite drift. On the axis of symmetry, the fluid velocity falls to half the sphere velocity almost two radii from the surface. The corresponding distance for potential flow is 0.7 radii. 

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

Fig. 6.6 Comparison of two alternative stagnation-flow configurations. The upper illustration shows the streamlines that result from a semi-infinite potential flow and the lower illustration shows streamlines that result from a uniform inlet velocity issuing through a manifold that is parallel to the stagnation plane. Both cases are for isothermal air flow at atmospheric pressure and T = 300 K. In both cases the axial inlet velocity is u = —5 cm/s. The separation between the manifold and the substrate is 3 cm. For the outer-potential-flow case, the streamlines are plotted over the same domain, but the flow itself varies in the entire half plane above the stagnation surface. The stagnation plane is illustrated as a 10 cm radius, but the solutions are for an infinite radius.

The pressure-gradient term Ar requires either a further boundary condition or a determination of the domain size. In the semi-infinite cases, Ar is a constant that is specified in terms of the outer potential-flow characteristics. However, the extent of the domain end must be determined in such a way that the viscous boundary layer is entirely contained within the domain. In the finite-gap cases, the inlet velocity n(zend) is specified at a specified inlet position. Since the continuity equation is first order, another degree of freedom must be introduced to accommodate the two boundary conditions on u, namely u — 0 at the stagnation surface and u specified at the inlet manifold. The value of the constant Ar is taken as a variable (an eigenvalue) that must be determined in such a way that the two velocity boundary conditions for u are satisfied. 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

Because St is assumed to be small, the velocity u can. as discussed before, be assun ed to be independent of y and equal to the value of u at the surface at the particular value of x being considered that is given by the potential flow solution, this value here being designated by i(x). Eq. (10.106) can therefore be written as ... 

Note that the slip velocity at the outer surface of the double layer results in a potential flow around the particle. This flow field, equivalent to that caused by a force quadupole at the origin [10], decays as r-3, which is faster than r ] for sedimentation of a sphere. 

As we noted above, the kinetic energy is positive definite. Furthermore, it is quadratic in the momenta. As a consequence, we can reduce the search for points of stationary flow in phase space to one of finding the stationary points of the potential energy surface. To see how this comes about, consider the Hamilton s equations for the three velocities... 

We have seen how heat transfer and thus dry deposition of SO2 is reduced on large surfaces, due to the buildup of boundary layer thickness (which reduces the local gradients). However, there are economically important structural objects composed of many elements of small dimension which show the opposite effect. These include fence wire and fittings, towers made of structural shapes (pipe, angle iron, etc.), flagpoles, columns and the like. Haynie (11) considered different damage functions for different structural elements such as these, but only from the standpoint of their effect on the potential flow in the atmospheric boundary layer. The influence of shape and size act in addition to these effects, and could also change the velocity coefficients developed by Haynie (11), which were for turbulent flow. Fence wire, for example, as shown below, is more likely to have a laminar boundary layer. 

The conclusion to be drawn from the preceding discussion is that the potential-flow theory (10-9) [or, equivalently, (10 12) and (10 13)] does not provide a uniformly valid first approximation to the solution of the Navier Stokes and continuity equations (10-1) and (10 2) for Re 1. Furthermore, our experience in Chap. 9 with the thermal boundary-layer structure for large Peclet number would lead us to believe that this is because the velocity field near the body surface is characterized by a length scale 0(aRe n), instead of the body dimension a that was used to nondimensionalize (10-2). As a consequence, the terms V2co and u V >, in (10 6), which are nondimensionalized by use of a, are not 0(1) and independent of Re everywhere in the domain, as was assumed in deriving (10-7), but instead are increasing fimctions of Re in the region very close to the body surface. Thus in... 

At this second level, the outer solution is modified to account for the weak outward displacement of the potential flow that is caused by the presence of the boundary layer at the body surface. It can be seen in (10-57) that this displacement manifests itself in the outer flow as a weak blowing velocity normal to the body surface. 

Hartree18 also obtained a family of solutions for f3 between 0 and —0.1988 that were physically acceptable in the sense that 1 from below as i] —> oo. Several such profiles are sketched in Fig. 10-7. These correspond to the boundary layer downstream of the corner in Fig. 10-6(b) (assuming that the upstream surface is either a slip surface or is short enough that one can neglect any boundary layer that forms on this surface). It should be noted that solutions of the Falkner-Skan equation exist for (l < -0.1988, but these are unacceptable on the physical ground that f —> 1 from above as r] —> oo, and this would correspond to velocities within the boundary layer that exceed the outer potential-flow value at the same streamwise position, x. It may be noted from Fig. 10-7 that the shear stress at the surface (r] = 0) decreases monotonically as (l is decreased from 0. Finally, at /3 = -0.1988, the shear stress is exactly equal to zero, i.e., /"(0) = 0. It will be noted from (10-113) that the pressure gradient... 

Thus, in a manner entirely equivalent to the two-dimensional analysis, we seek rescaled equations in the inner (boundary-layer) region very near to the sphere surface within which the tangential velocity goes from the potential-flow value (3/2)sin0 to 0 at the body surface. The only difference from the previous analysis is in the detailed form of the Navier-Stokes and continuity equations for axisymmetric geometries. When expressed in terms of spherical coordinates, these equations are... 

Suppose that a is sufficiently small, i.e., We is sufficiently large, that surface tension plays no role in determining the bubble shape, except possibly locally in the vicinity of the rim where the spherical upper surface and the flat lower surface meet. Further, suppose that the Reynolds number is sufficiently large that the motion of the liquid can be approximated to a first approximation, by means of the potential-flow theory. Denote the radius of curvature at the nose of the bubble as R(dX 6 = 0). Show that a self-consistency condition for existence of a spherical shape with radius R in the vicinity of the stagnation point, 0 = 0, is that the velocity of rise of the bubble is... 

Figure 11-5. A schematic representation showing the two-layer structure of the thermal-momentum boundary layer for large Pe, with Re 1 and Pr < C 1. In this limit, the distance required for 9 to approach its ffee-stream value is much larger than the distance required for the velocity to approach the potential-flow form. Thus convection within the thermal boundary layer is dominated by the outer limit of the momentum boundary-layer velocity distribution (or equivalently, the inner limit as we approach the body surface of the potential-flow distribution).

Reasonable investigations under these conditions are restricted by the state of the DAL theory which has been developed so far only for conditions of very strong and weak surface retardation (cf Section 8.6). Collision efficiency has been derived only for potential flow conditions (Sutherland, 1948). With increasing surfactant concentration up to c[ (Eqs 8.135 -8.136), a beginning decrease of bubble velocity may be expected. A respective rear stagnant cap results in a decrease of collision efficiency only when attachment of the particle is accomplished not due to the of instability of the water interlayer at some thickness h but under the effect of attraction forces (Appendix lOB). 

In the case of a retarded bubble surface, the velocity v exceeds that for a free surface -s/Re times since the boundary layer thickness differs slightly and the velocity gradient along the boundary layer is times greater. Thus, at a retarded surface a more substantial F can be expected. Mileva (1990) compares F and the pressing force (Dukhin Rylov 1971) for the case of a potential flow and concludes that Fl is negligible. But just from such a comparison it becomes evident that for retarded bubbles the pressing force is substantially less than F. ... 

---