Viscous Shock Profiles and Primitive Formulations

Weak solutions of hyperbolic systems in primitive (nonconservative) form for which a consistent conservation form exists are considered. It is shown that for primitive formulations, shock relations are not uniquely defined by the states to either side of the shock, but also depend on the viscous path connecting the two. Consistent viscous shock profiles are enforced by adding scheme-dependent small viscous perturbations that account for leading order conservation errors. The resulting primitive algorithm is conservative to the order of the approximation. One-dimensional Euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms. While not eliminating conservation errors entirely, it is found that for a wide range of problems, both conservative and primitive flow calculations are of comparable quality.