Falkner-Skan solution

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

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

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. 

Figure 10-7. Solutions of the Falkner-Skan equation, plotted as f versus rj for several different values of ft. The values of ft that correspond to the infinite wedge configuration of Fig. 10-6 are 0 < /3 < 1. The solutions for ft > 1 and /3 < 0 are discussed in the text.

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... 

It will be noted that the pressure gradient over the front half of the cylinder is favorable -that is, the pressure decreases in the direction of motion. Beyond the halfway point (6 = x = 7i/2), on the other hand, the pressure begins to increase in this region the pressure gradient is adverse. Our experience with the solutions of the Falkner-Skan equation suggests that this latter region is a candidate for boundary-layer separation. Indeed, experimental observation... 

D. R. Hartree, On an equation occurring in Falkner and Skan s approximate treatment of the equations of the boundary-layer, Proc. Cambridge Phils. Soc. 33, 223-39 (1937) K. Stewartson, Further solutions of the Falkner-Skan equation, Proc. Cambridge Phils. Soc. 50,454-465 (1954). 

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