Strongly conservative discretization of governing equations in cylindrical coordinates

Abstract

Cylindrical coordinates are often used in computational fluid dynamics (CFD), particularly when one considers axisymmetric flow. Although cylindrical coordinates have several advantages in accurately describing the rotational flow as well as the complex boundary, there are limits to how far they can be taken. These are (1) the governing equations are mostly in a non-conservative form in current standard models, which cannot preserve the conservation of numerical simulations, (2) the singularity along the axis at the coordinate origin (z-axis), leading to difficulty reproducing the flow across the z-axis, and (3) the time step is extremely shortened to satisfy the CFL condition near the z-axis because the numerical cell thereof is narrow in the azimuthal direction for a given angular resolution. A strongly conservative model is proposed in this paper to overcome these difficulties. First, the strongly conservative form of the governing equations is derived by considering changes in the direction of the base vector within a cell, and a concept of direction error is defined to reveal the inherent error of the standard model. Second, inspired by the derivation of the first, a strongly conservative method for evaluating the flux across each cell surface is proposed, preserving the conservation of discretization and enlarging the time step. Finally, the proposed model is tested by three classical problems, i.e., uniform flow, Sod shock tube problem, and expanding outflows. The results imply that the proposed model is strongly conservative for the cylindrical coordinate system, and it can overcome the limits of the standard model and eliminate the inherent error in the standard model. Furthermore, the strongly conservative model and method are extended to arbitrary coordinates for the numerical conservation in general curvilinear coordinates of the problem with complex boundaries.