Investigation by the Hodograph Method of Flow Through a Symmetrical Nozzle With Locally Supersonic Regions.

摘要

The flow of a compressible fluid through a channel having locally supersonic regions is studied by using the Tricomi equation in the hodograph variables as an approximation in the sonic region to the equation of flow of an irrotational, inviscid gas. It is shown that this is equivalent to studying the flow of a gas having a pressure-density relation matching the isentropic relation to the third derivative at the sonic point. A one-parameter family of solutions of the Tricomi equation is used which provides symmetrical accelerated-decelerated flows. The variation of this parameter alters the Mach number at the center of the throat, the velocity distribution and gradient along the center streamline, as well as the shape of the channel, that is, the curvature of the bounding streamline. As specific examples, flows are computed having Mach number equal to unity and to 0.86 at the center of the throat section. Constant-velocity lines are plotted and it is found that the velocity gradient becomes zero at three places along each streamline outside of a limiting streamline for values of the parameter greater than zero (M < 1 at center of throat section). For the parameter equal to zero (M = 1 at center of throat section), the velocity gradient along the streamlines and the curvature is discontinuous at all points of the two-characteristics which meet the center streamline. Other solutions to the Tricomi equation are discussed which may be used to formulate channel flows. The exact nature of these flows has not yet been investigated.