Analytical exact solutions for unsteady collisionless plume flows in a vacuum

• Provided new formulas for detailed, unsteady, round jet starting up and shutting down.
• Provided DSMC simulations to validate the analytical solutions.
• Evaluations on the exact solutions are faster, more accurate, and lower cost.
• These new formulas complement the previous steady flow into a vacuum, so the whole process has a complete analytical solutions.
• Results have practical applications, e.g. pulsed propulsion thruster in space engineering.

This paper presents a set of gaskinetic solutions to the problem of unsteady collisionless round plume development: startup and development to a steady flow; steady flow; and shutting down from a steady flow. This can find applications in studying similar transient plume flows from space propulsion devices. Different from many past studies, here we consider the general situation that the average exit gas speed can be larger than zero, and detailed geometry of the nozzle exit radius is included. A fundamental space–velocity–time relation is adopted in this study and it plays a crucial role to obtain the complete flowfield properties of density, velocity, pressure and temperature. This study reveals that there are some internal complementary relations on density and momentum among these three processes. The results involve complex integrations involving factors of time, geometry, and specific speed ratio. Several numerical simulations with the direct simulation Monte Carlo method validate these analytical exact results.