Arithmetic Averages of Viscosity Coefficient are Sufficient for Second-Order Finite-Volume Viscous Discretization on Unstructured Grids

Abstract

In this short note, we discuss the use of arithmetic averages for the evaluation of viscous coefficients such as temperature and velocity components at a face as required in a cell-centered finite-volume viscous discretization on unstructured grids, and show that second-order accuracy can be achieved even when the arithmetic average is not linearly-exact second-order reconstruction at a face center (e.g., the face center is not located exactly halfway between two adjacent cell centroids) as typical in unstructured grids. Unlike inviscid discretizations, where the solution has to be reconstructed in a linearly exact manner to the face center for second-order accuracy, the viscous discretization does not require the linear exactness for computing viscous coefficients at a face. There are two requirements for second-order accuracy, and the arithmetic average satisfies both of them. Second-order accuracy is numerically demonstrated for a simple one-dimensional nonlinear diffusion problem and for a three-dimensional viscous problem based on methods of manufactured solutions.