A Unified Asymptotic Theory of Supersonic, Transonic, and Hypersonic Far Fields

The problems of steady, inviscid, isentropic, irrotational supersonic plane flow passing a body with a small thickness ratio was solved by the linearized theory, which is a first approximation at and near the surface but fails at far fields from the body. Such a problem with far fields was solved by W.D. Hayes' "pseudo-transonic" nonlinear theory in 1954. This far field small disturbance theory is reexamined in this study first by using asymptotic expansion theory. A systematic approach is adopted to obtain the nonlinear Burgers' equation for supersonic far fields. We also use the similarity method to solve this boundary value problem (BVP) of the inviscid Burgers' equation and obtain the nonlinear flow patterns, including the jump condition for the shock wave. Secondly, the transonic and hypersonic far field equations were obtained from the supersonic Burgers' equation by stretching the coordinate in the y direction and considering an expansion of the freestream Mach number in terms of the transonic and hypersonic similarity parameters. The mathematical structures of the far fields of the supersonic, transonic, and hypersonic flows are unified to be the same. The similar far field flow patterns including the shock positions of a parabolic airfoil for the supersonic, transonic, and hypersonic flow regimes are exemplified and discussed.