Scaling Parameters In Rarefied Flow And the Breakdown Of The Navier-Stokes Equations

In high altitude flight the average spacing between the molecules of the gas is not negligible compared to a typical dimension of the flowfield. The continuum assumption `breaks down' and the Navier-Stokes equations can, in theory, no longer be shown to be valid, particularly for high speed flight. Various scaling parameters have been suggested to identify the different regimes of continuum, near-continuum, rarefied and freemolecular flow: these are the Knudsen number, Tsein's parameter [H. S. Tsein "Superaerodynamics, mechanics of rarefied gases" J Aerospace Sci, 13, 342 (1946)], Cheng's rarefaction parameter [H. K. Cheng "Hypersonic shock-layer theory of the stagnation region at low Reynolds number" Proc 1961 heat Transfer and Fluid Mechanics Institute, Cornell Aero Lab Report IRN 13539152 (1961)], Bird's breakdown parameter [G. A. Bird "Breakdown of translational and rotational equilibrium in gaseous expansions", AIAA J, 8, 1998 (1971)], and a form of the viscous interaction parameter derived from shock-boundary layer theory. We show how all these parameters may be derived from the Boltzmann equation and interpreted in terms of the ratio of mean time between molecular collision and a characteristic flow time, or as the ratio of typical shear stress to pressure in the flow. Cheng's parameter appears to be the best correlation parameter and the best indicator of the validity of the Navier-Stokes equations for high-speed blunt body flow. For slender body flow the viscous interaction parameter based on a reference boundary layer temperature, a "modified Tsein parameter" similar to Chang's parameter, appears to be best