The stability limit of stationary normal shock waves in supercritical nozzle flows with homogeneous condensation is investigated by the singularity theory of the quasi-one-dimensional steady-state differential equations of motion. In this case it is shown that a catastrophic change in the phase portraits of the flow variables, where the spiral point turns into a nodal point, occurs when the initial relative humidity exceeds a critical value, resulting in the alteration of the quasi-one-dimensional steady-state condensation structures. In particular, a criterion for the limit of stability, based on the break-up of structural stability of the steady equations of motion in the quasi-one-dimensional approximation, is established by variational analysis and a correlation for the critical initial relative humidity is derived for fixed nozzle geometry keeping appropriate reservoir conditions fixed in the same approximation. A comparison of the values of the critical initial relative humidity, calculated by this correlation, shows excellent agreement with those of experiments and/or numerical simulations for moist air expansions in various slender nozzles under different reservoir conditions.
Delale, C. F., Lamanna, G., & Dongen, van, M. E. H. (2001). On the stability of stationary shock waves in nozzle flows with homogeneous condensation. Physics of Fluids, 13(9), 2706-2719. https://doi.org/10.1063/1.1388206