On Reconstructing Conservative and Primitive Variables: An Eigenvector Analysis on Curvilinear Grids

TL;DR

This paper provides an algebraic foundation for the empirical success of conservative-variable eigenvector-based reconstructions in hypersonic boundary-layer simulations, emphasizing metric-invariance and contact discontinuity representation.

Contribution

It analytically explains why conservative eigenvectors yield better results than primitive ones, highlighting metric-invariance and the natural contact direction in curvilinear grids.

Findings

Conservative eigenvectors have metric-free shear and contact components.

Primitive eigenvectors contain metric-dependent shear and velocity terms.

The algebraic analysis supports the effectiveness of conservative formulations for entropy correction.

Abstract

In wall-modelled large-eddy simulations of hypersonic boundary-layer transition, Hoffmann, Chamarthi and Frankel reported that characteristic reconstruction based on conservative-variable eigenvectors produced markedly better results than the corresponding primitive-variable implementation. The observation was empirical. A subsequent wave-appropriate conservative reconstruction (WA-CR) algorithm used a rank-one entropy correction based on the premise that contact-discontinuity error lies in a single conservative entropy/contact direction. This note gives the algebraic foundation for both observations. For the standard conservative curvilinear eigenvectors, the density row of the right-eigenvector matrix contains exact, metric-free zeros in the shear columns, so shear waves carry no density perturbation and a contact discontinuity is represented by the conservative entropy eigenvector alone. The conservative left eigenvectors provide the dual projection property: the entropy amplitude is obtained with a metric-independent left eigenvector and has unit contact scaling, while total-energy perturbations have zero projection onto the shear amplitudes. In the standard primitive curvilinear eigenvectors, by contrast, shear right eigenvectors contain metric-dependent density components and the primitive entropy left eigenvector contains metric-weighted tangential-velocity terms. Thus the conservative formulation supplies the two algebraic requirements for an exact, sufficient, metric-invariant, rank-one entropy correction: metric-independent entropy projection and a metric-independent entropy update direction. Curvilinear metrics make the distinction explicit, but the conservative state-space contact direction is already the natural direction underlying WA-CR even on Cartesian grids.