Local, Point-Wise Rotational Transformations of the Conservation Equations into Stream-Wise Coordinates

TLDR: In this article, it was shown that a subset of the multidimensional Euler equations can be diagonalized, but not the entire set, and the authors extended the formulation of the conservation equations into the local stream-wise coordinate system to the time-dependent, 2D and 3D conservation equations.

Abstract: In dealing with multidimensional simulations, many authors have shown that a major cause of numerical dispersion errors is due to the flow being skewed to the coordinate axes. Crane and Blunt [1] have shown that the stream-wise transformations can reduce the numerical errors associated with the multidimensional transport equations. However, it has been proven that no transformation can completely diagonalize the multidimension conservation equations. It shall be demonstrated that a subset of the multidimensional Euler equations can be diagonalized, but not the entire set. The formulation of the conservation equations into the local stream-wise coordinate system is extended to the time-dependent, two- and three-dimensional (2D and 3D) conservation equations. At any point in space, there exists a set of local rotations that aligns the fluid velocity vector coincident with the stream-wise coordinate; hence, the fluid velocity components orthogonal to the stream-wise coordinate are identically zero. Such transformations result in a subset of PDEs that are diagonalized, namely, the mass, total energy, and principal momentum density PDEs. However, the orthogonal momentum component conservation PDEs are not diagonalized and are multidimensional; these PDEs are responsible for streamline bending.