An improved formulation of equations of flow of a general gas mixture includes consistent boundary conditions that are applicable to real gases. An analysis of prior formulations, with focus on boundary conditions, led to the conclusion that boundary conditions based on ideal mixtures and/or perfect gases can lead to errors in computed flows of real gases. The improved formulation makes it possible to achieve greater accuracy in computation of flows of real (including chemically reactive) gas mixtures, and is expected to be especially beneficial in computing flows of supercritical fluids like those in diesel engines, gas turbine engines, rocket engines, supercritical-fluid extraction processes, and crude oil under high pressure.

Pressure as a Function of Position Across the Flow was computed after an acoustic wave reached the nonreflecting side boundaries. The simplistic formulation yielded spurious reflected waves. The small,high-spatial-frequency oscillations are artifacts of a numerical-integration procedure used to solve the equations of flow.
The improved formulation is derived from equations of conservation of mass, chemical species, energy, and momentum of a real gas mixture. These equations have the typical form of the Navier- Stokes equations augmented by the species- and energy-conservation equations and by an equation of state, except that the species and energy equations contain additional terms: The traditional Fick mass-diffusion and Fourier heat-diffusion terms are now respectively complemented by the Soret and Dufour terms that represent, respectively, the thermal contribution to diffusion of species and the transport of heat due to gradients in concentrations of species.

The conservation equations are for general viscous fluids; however, the boundary conditions are calculated from equations of inviscid flow (Euler equations) augmented by species and energy equations. Characteristic boundary conditions are derived from a wave decomposition of the Euler equations, and wave-amplitude variations are determined from the prescribed boundary conditions on the flow variables in conjunction with a general equation of state for a real gas.

The improved formulation was tested in computations of the one-dimensional propagation of acoustic waves in a flowing supercritical mixture of nitrogen and heptane in a two-dimensional domain with nonreflecting boundaries. The results obtained with this formulation were compared with those from another formulation in which real-gas thermodynamic properties were simplistically substituted into characteristic equations derived previously for a perfect gas. The waves computed in the improved formulation were found to leave the computational domain with minimal reflection at a subsonic outflow boundary, whereas the waves computed in the simplistic formulation exhibited significant reflections at the boundaries (see figure).

Although the superiority of the improved formulation has been shown by this test, caution is in order because the characteristic-wave analysis inherently incorporates the assumption that elliptic terms act only as corrections to an essentially hyperbolic operator. Thus, diffusional terms are not parts of the consistent- boundary-condition analysis, but are used in the governing equations once a solution is sought. The condition of weak ellipticity may not always be satisfied when thermal-diffusion effects are large enough to augment the effective thermal conductivity to the point of making heat diffusion processes comparable in magnitude to convective processes.

This work was done by Josette Bellan, Nora Okong'o, and Kenneth Harstad of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at under the Physical Sciences category.


This Brief includes a Technical Support Package (TSP).
Boundary Conditions for Computing Flows of Real Gas Mixtures

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This article first appeared in the January, 2002 issue of NASA Tech Briefs Magazine.

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