By Holt Ashley
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Additional info for Aerodynamics of Wings and Bodies
5-4. horseshoe vortex. ments of unit strength but opposite signs located infinitesimally. close together along the positive x-a~ds and joined together by an infinitesimal piece along the y-axis, also of unit strength (see Fig. 5-4). The solution for this can be obtained by integrating the solution for a doublet in the x-direction. ::::::> • Since the equations of motion and the boundary conditions are linear, solutions may be superimposed linearly. It is therefore convenient to write the solution as a sum of two terms, one giving the flow due to thickness and the other the flow due to camber and angle of attack (see Fig.
Epi + .. (4-25) (4-26) Upon substitution into the Navier-Stokes equations we obtain ('ltg. + E'lti. + ... )('ltgu + E'ltizz + ... ) - ('ltgz + E'ltiz + ... )('ltg u + E'ltizz + ... ) ° - EPlz ° + E v "'0. + . , = - Poz 2,y,0 (4-27) (4-28) (4-15) 0, The next task is to identify an appropriate small parameter do by considering an inner expansion of the form 'itO (4-14) (4-16) . (x, ,00) = 1, '2 + (E2 R e) -l,y,"'UU. i This shows that p~ is a function of x only; in 'other words, to the lowest order, the pressure is constant across the boundary layer.
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. To see how it accomplishes this, consider a transformation with a singular point at t = to, corresponding to a point Z = Zoo In the vicinity of this particular singular point, it is easy to show that the transformation can be approximated by > n 1, (2-135) = d(Z - Zo) d(t - to) = A[" _ " n) )0 1"-1 (2-136) vanishes at the point for n > 1, as it is expected to do.