Evaluation of diffusive flux across faces of arbitrary shaped finite volume cells

Abstract

A new procedure based on principles of directional derivatives of vector calculus has been presented for the computation of normal gradient across faces of arbitrarily skewed finite volume cells. The equations developed are compact, computationally less expensive and enable easy development of set of algebraic equations for implicit methods. The uniqueness and boundedness of the derivatives have been examined and confirmed with the help of typical test functions for a second order accurate implementation of the scheme. The accuracy of the gradient evaluation on arbitrarily skewed finite volume cells has been demonstrated using the order of convergence of solution to Poisson’s equation and viscous flow in a driven square cavity. A straight forward extension of the scheme for three dimensional problems has been illustrated by computing three dimensional potential flow over a sphere in Cartesian coordinates and viscous flow in a cubic cavity. A new interpretation to Coirier diamond path integration presented here shows that the diamond path can be integrated in a compact manner identical to that of a typical implementation of the present approach.