An Approximate Analytical Solution for Flow of an Under-Expanded Rarefied Gas Jet

Here we develop an analytical solution for the continuum flow and resulting plume of a gas jet into a low-density (rarefied) environment. The analytical model developed here involves posing a self-similar functional form for the radial behavior where the density is placed in a self-similar form using the hypothesis of Boynton [1] [2] and Simons [3]. The unknown angular variation of the density field is determined as a part of the solution procedure rather than specified using semi-empirical arguments. A relationship is determined that maps the self-similar mass constraint equation to a linear, variable coefficient eigenvalue problem that will satisfy the boundary conditions associated with the flow problem. Comparison of the density variation to the data of Calia and Brook [4] for flow close to the Prandtl-Meyer limit line suggests good agreement. The associated velocity solution is developed and shown to support a constant radial flow velocity field. Comparison to an isothermal fluid computational result supports this result. A simplified, quasi-ld model for under-expanded jet behavior demonstrates similar behavior. These models provide a useful, theory-based description for jet flow into a vacuum supporting exo-atmospheric propulsion problems.