A numerical study of heat diffusion using the Lagrangian particle SPH method and the Eulerian Finite-Volume method: analysis of convergence, consistency and computational cost

Abstract

In this paper, the Lagrangian Smoothed Particle Hydrodynamics (SPH) method (with different combinations of smoothing functions/numbers of particles), the Eulerian Finite-Volume method using different refinements of the meshes and the Analytical method were applied for the study of heat diffusion. The numerical simulations by the SPH method have been performed using cubic spline, quartics and quintic spline kernels. The discretization of the domain has been affected by the use of 50 x 50, 60 x 60, 70 x 70, 80 x 80 or 90 x 90 particles. It has been noticed that the phenomenon of particle inconsistency and the consequent emergence of the largest temperature differences has been noticed, when compared with the analytical solution, near the bottom corners. The lowest differences have been obtained when the interpolation smoothing function degree and the number of particles used were the highest. For the Finite-Volume method, the largest differences between temperatures have been observed near the bottom corners, however they were lower than those found with the use of the SPH method. To obtain better results it is necessary to make boundary corrections. The computational cost of the SPH method was higher than the Finite-Volume method and increased as we increased the number of particles or the interpolation kernel degree.