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Unformatted text preview: 4.2 The Falkner-Skan equation For the case of a flat plate, we have just shown that the boundary layer solution is self-similar in the sense that two profiles u ( x,y ) at different values of x differ only by a scale factor in y . This simplified the analysis considerably, because the momentum equation reduced to an ordinary differential equation in the single similarity variable η . What about more complicated geometries? In this section, we derive the general conditions under which self-similar solutions exist. As usual we work in terms of the stream function Ψ so that the continuity equation is automatically satisfied. Consider again the boundary layer transformations 65, which tell us that the basic scales of the y coordinate and the stream function are y ′ = Re 1 / 2 y L and Ψ( x,y ) = LU Re 1 / 2 Ψ ′ ( x ′ ,y ′ ) . (106) For a flat plate, we argued that we must actually replace the overall length of the object L by the present distance travelled along it, x . This gave (in an unusual format) η = Re 1 / 2 radicalbig 2 x/L y L and Ψ = LU radicalbig 2 x/L Re 1 / 2 f ( η ) , (107) in which the scaled stream function depends on space only through the single similarity variable η . To extend this analysis to more general geometries, we recall that the only way the geometry enters the BL equations is through its influence on u e ( x ). For a flat plate just considered, u e ( x ) = U = constant. More generally, u e = u e ( x ). In what follows, we aim to find the most general form of u e ( x ) that still admits self-similar solutions....
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This note was uploaded on 01/05/2011 for the course CSU 3 taught by Professor Handsome during the Spring '10 term at CSU Pueblo.
- Spring '10