Exact solutions for transonic flows of dense gases in two-dimensional and axisymmetric nozzles

Flows of dense gases in two-dimensional and axisymmetric nozzles are considered. If during an isentropic expansion the fundamental derivative changes the sign within the flow region, nozzles with two throats and one antithroat are necessary for a shock-free acceleration to supersonic conditions. Exact solutions to the transonic inviscid small-disturbance equation for flows of dense gases in plane and axisymmetric nozzles are obtained. The exact nozzle shapes have only one throat, one antithroat, and an inflection point. In the latter, the minimum or maximum of the Mach number is attained. The flow in a throat-antithroat-throat configuration may be constructed by solving the small-disturbance equation locally for a sequence of points along the isentrope.