An Exact Solution for Incompressible Flow Through a Two-Dimensional Laval Nozzle

A careful examination of the variation of the velocity along the centerline and the contour of a Laval nozzle in the physical plane shows that either the upper or the lower half of the Laval nozzle assumes the same form of a slitted thick airfoil with tandem trailing edges. These two airfoils lie on different Ricmann sheets in the hodograph plane. The interior of the airfoil is then mapped onto an infinite strip in the complex potential plane. Making use of these results, we obtained an exact solution for the incompressible potential flow through a two-dimensional Laval nozzle. The solution is applicable for nozzles with any given contraction ratio n₁ expansion ratio n₂ and throat wall radius R^*. As examples of the method, various nozzle contours, the velocity distribution of the flow, and the locations of the fluid particles at different time intervals are presented.