By Holt Ashley
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Extra info for Aerodynamics of Wings and Bodies
D d d (3-7) • This problem was used by Prandtl to illustrate and explain his boundary layer theory (see Schlichting, 1960, p. 63). I I I I 1 I 64 SINGULAR PERTURBATION PROBLEMS The mass will have reached its maximum deviation when u(t) (3-7) we thus obtain, approximately, I X max = (j' [CHAP. 3 = O. From This has the solution ,(3-10) \ In order to determine the integration constant A approximately, we notice that the initial phase of the motion as described by (3-6)-(3-8) takes place almost instantaneously for vanishing m.
See Fig. ) At time t = 0 the mass is given an impulse I (for example, by shooting off a charge to the left). "· This problem will be solved in essentially three different ways. First, an exact solution may easily be obtained. The governing differential equation and the associated boundary conditions are (3-1) = u(O) (dX) = dt '=0 (3-2) 11m, (3-3) x(O) = O. ed mathematically by the property that, in the limit as the small parameter vanishes, one or more highest-orderderivative terms in the governing differential equation drop out so that the differential equation degenerfl,tes.
2-133) The derivative of the transformation is written (2-134) and subsequently integrated in closed form. By the rather laborious process of trial-and-error location of singular points, many practical airfoils were developed during the 1920's. It is possible to adjust the thickness and camber distributions in a very general way. Interesting examples of von Mises and other airfoils will be found discussed in a recent book by Riegels (1961). 3. The von Karman-Trefftz. Transformation. This method derives from a scheme for getting rid of the cusp at the trailing edge, produced by the foregoing classes of transformations, and replacing it by a corner with a finite angle T.