The mathematical theory of thermal choking in nozzle flows

On the stability of stationary shock waves in nozzle flows with homogeneous condensation

“…1, with reservoir temperature T 0 Ј , reservoir specific humidity 0 ͑or reservoir mixture pressure p 0 Ј), and reservoir relative humidity 0 . The differential system of equations governing the motion can then be written in normalized form as ͑for normalization, notation and other details see Delale et al 10,12 …”

“…It is also worthwhile to mention that the flow Mach number, given by Eq. ͑6͒, is different from the Mach number used by Delale et al,12 and that the normalization is carried in a similar manner, but using reservoir conditions instead of the conditions used at the saturation point in Delale et al 10 The differential system of equations ͑1͒-͑7͒ together with the nonequilibrium condensation rate equation for g constitute the basic equations of nozzle flows with nonequilibrium condensation for given cross section area of the nozzle. It is well known that the flow passes through the throat almost isentropically so that the classical saddle point singularity of the system ͑1͒-͑7͒ occurs at the throat with M ϭ1.…”

“…As long as the amount of heat released is below a critical value, the flow field remains continuous throughout the nozzle ͑subcritical flows͒. When the amount of heat exceeds this critical value ͑say, by increasing the initial relative humidity͒, the flow becomes thermally choked ͑see Delale et al 12 ͒. In this case flows with embedded normal shock waves occur ͑supercritical flows͒.…”

“…The singularities of the differential system ͑1͒-͑4͒ for given heat addition distributions were considered by Möhring 19 ͑also discussed in detail in Zierep 20 ͒ and its application to nonequilibrium condensation was given by Delale et al 12 It is well known that the classification of the singularities of the differential system of Eqs. ͑1͒-͑4͒ requires both the numerators and denominators to vanish.…”

“…In this case the flow in the condensation zones remains steady and supersonic as long as the latent heat release to the flow is below a critical value for a given nozzle geometry and reservoir conditions ͑subcritical flows͒. If this critical value is exceeded, the flow becomes thermally choked ͑e.g., see Delale et al 12 ͒ and stationary shock waves followed by local subsonic heat addition zones, where the flow is accelerated back to supersonic speeds, occur ͑supercritical flows͒. Detailed structure of the condensation zones for both subcritical and supercritical flows can be found in Delale et al 10,11 using an asymptotic predictive method given by Blythe and Shih 7 and Clarke and Delale.…”