Critical Point Analysis of Transonic Flow Profiles with Heat Conduction

Critical points arising in transonic flow profiles of the steady compressible Euler equations with heat conduction are investigated. Two examples of transonic flow profiles are studied, namely, radial outflow in a gravitational field, and flow in a quasi-one-dimensional nozzle. The stationary flow equations are reformulated in terms of a dynamical system that parametrizes the flow profiles. Adding the heat conduction term introduces a critical point that is of a type different from the well-known sonic critical point that occurs at the transition from subsonic to supersonic flow when there is no heat conduction. This thermal critical point takes over the saddle-point role of the sonic critical point in the flow profile. Both the sonic and the thermal critical points are present in radial outflow profiles, and the type of the sonic critical point is changed from a saddle point to a simple node by the addition of the heat conduction term. In the nozzle case, the sonic point is no longer a critical point of the dynamical system when heat conduction is added. It is illustrated how the results of this analysis can be used for efficient and accurate numerical calculation of transonic flow trajectories and boundary value problems with heat conduction that are of interest in applications like supersonic planetary escape and solar wind models, and in aerospace applications. The analysis also elucidates how many boundary conditions are required for a well-posed transonic boundary value problem with heat conduction and clarifies the mathematical structure of transonic flow profiles with heat conduction that have been calculated numerically in the literature for various applications.