Variational Method in the Theory of Compressible Fluid

The variational principle in hydrodynamics was first studied by Hargreaves, who has shown that, when the hydrodynamic equations are satisfied, the integrand in the variational principle is a linear function of the pressure. For steady, irrotational flows, Bateman ,3 has shown that the integrand is the pressure only, which can be written as a function of the velocity potential 4>. A direct method to obtain approximate solution in the case of subsonic flow of a compressible fluid can therefore be formulated following the Rayleigh-Ritz method instead of solving the original nonlinear differential equations. In applying this method to the case of a compressible fluid passing an arbitrary airfoil, however, there are two difficulties: first, since the fluid region is infinite, Bateman's integral is not directly applicable; second, it is not easy to assume the potential function satisfying the boundary conditions. The nonlinear problem of compressible flow passing an arbitrary airfoil has been formulated in this paper. I t is proved that, by conformal transformation, the velocity potential can be set up to satisfy the boundary conditions in the general case, and, hence, the Rayleigh-Ritz method can be applied. I t is also shown that, by using the Theory of Residues in the integration and a modified Crout method in solving the resulting simultaneous equations, the computational labor can be mate' daily reduced. Since the variational principle is valid for subsonic, sonic, and supersonic flows, the possibility of applying it to the study of mixed flow is indicated. A numerical example of the nonlinear problem of the compressible flow around a circular cylinder is carried out, and the results compare favorably with those found by the existing methods.