Nonlinear Hypersonic Aerodynamics in Terms of Invariant Integrals: Some Exact Self-Similar Solutions for Jets and Projectiles

An analysis of the gas flow near a jet or projectile moving at the Mach number higher than about 6 called the hypersonic flow will be of major interest for the science, industry and military of the 21th century. In this paper, the basic differential equation of gas dynamics for any Mach numbers is derived for the steady, irrotational, adiabatic and homoentropic flows by using the general approach of invariant integrals of gas dynamics. For hypersonic flows past thin jets and projectiles, this equation is a nonlinear differential equation in partial derivatives of second order. Similar to the transonic gas dynamics, the nonlinearity of the hypersonic gas dynamics is caused by a very small value of the coefficient before the senior derivative which depends on a junior derivative of the sought solution. This nonlinearity of hypersonic flows was usually ignored. Several exact self-similar solutions of this nonlinear problem of the compressible fluid dynamics are given for the hypersonic flows around some profiles, including thin wedge, thin cone, and some new finds for other more sophisticated profiles. Manned hypersonic aviation as well as military hypersonic missiles for use in the Earth’s atmosphere along known trajectories are shown to be almost ineffective.