Two-dimensional boundary surfaces for axi-symmetric external transonic flows
Al-hashel, Waleed Isa
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Investigation of two-dimensional transonic flows is extended to axi-symmetric problems. This is of considerable practical interest, for example, with regard to missiles or aircraft engines which approximate much more closely bodies of resolution and two-dimensional bodies. The main concern with axi-symmetric flows lie not only with the complexity of the governing nonlinear partial differential equation which is mixed of elliptic-hyperbolic type but also with the lack of a general method for accurately solving this type of equation. We solve the nonlinear transonic equation using separation of variables technique, which yields two nonlinear ordinary differential equations. The x-dependence can be integrated numerically, and the solution for the r-dependence can be obtained using the expansion method originated by Van Dyke. This works well with only three terms in the expansion. The sonic solution of these equations is obtained analytically since both equations are simple enough to be integrated for this case (M infinity = 1.0). The small parameter (1-M infinity(2)) plays an important role in specifying the shape of the boundary surfaces for external axi-symmetric steady flow of interest for design. A Navier-Stokes solver was used to compute the inviscid flow to confirm our results in the region over the surface where the small perturbations apply.
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