Falkner-Skan flows

So far, we have been talking about the stability of zero pressure gradient flows. It is possible to extend the studies to include flows with pressure gradient using quasi-parallel flow assumption. To study the effects in a systematic manner, one can also use the equilibrium solution provided by the self-similar velocity profiles of the Falkner-Skan family. These similarity profiles are for wedge flows, whose external velocity distribution is of the form, 11 = k x . This family of similarity flow is characterized by the Hartree parameter jSh = 2 1 the shape factor, H =. Some typical non-dimensional flow profiles of this family are plotted against non-dimensional wall-normal co-ordinate in Fig. 2.7. The wall-normal distance is normalized by the boundary layer thickness of the shear layer. 

D. Streaming Flow Past a Semi-Infinite Wedge - The Falkner-Skan Solution... 

D. STREAMING FLOW PAST A SEMI-INFINITE WEDGE -THE FALKNER-SKAN SOLUTION... 

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... 

Figure 10-8. A sketch showing streaming flow past a circular cylinder. On the right is the flow as seen with a course level of resolution. On the left is the flow in the immediate vicinity of the front-stagnation point, as seen from a much finer resolution. It is evident that the local flow examined in a region close enough to the stagnation point reduces to a classic stagnation point flow as described by the Falkner-Skan equation for, 8 = 1.

These ODEs are to be solved subject to (10-135), (10-136), and (10-138). The first is just the Falkner-Skan equation for a 2D stagnation flow. This is not at all surprising because the geometry of a cylinder near x = 0 is locally just the stagnation point geometry, as sketched in Fig. 10-8. Subsequent equations are linear in / ), /-j, and so forth. There is therefore... 

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