Generalised finite volume strategies for simulating transport in strongly orthotropic porous media

Abstract

In this work two different finite volume computational strategies for solving a representative two-dimensional diffusion equation in an orthotropic medium are considered. When the diffusivity tensor is treated as linear, this problem admits an analytic solution used for analysing the accuracy of the proposed numerical methods. In the first method, the gradient approximation techniques discussed by Jayantha and Turner [Numerical Heat Transfer, Part B: Fundamentals, 40, pp.367--390, 2001] are applied directly to the diffusion equation. In the second method, the diffusion equation is transformed via scaling parameters to an isotropic model and then the control volume techniques discussed by Jayantha and Turner are used to obtain the numerical results on the transformed domain. Both methods are shown to produce reasonable results in comparison with the exact solution for a range of anisotropy ratios typical of wood. However, only the first method is appropriate for use in non-linear coupled transport systems. This work highlights the necessity of determining a higher order gradient approximation to improve the numerical results on the untransformed domain.