A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

Abstract

This paper presents a particle-based Lagrangian-Eulerian algorithm for the solution of the unsteady advectiondiffusion-reaction heat transfer equation with phase change. The algorithm combines a Lagrangian formulation for the advection + reaction problem with the Eulerian-based heat source method for the diffusion + phase change problem. The coupling between the Lagrangian and Eulerian subproblems is achieved with a phase change detector scheme based on a local latent heat balance and a consistent/conservative interpolation technique between Lagrangian particles and the Eulerian grid. This technique makes use of an auxiliary (finer) Eulerian grid that provides a simple and efficient method of tracking internal heterogeneities (e.g. phase boundaries), allows the use of higher order integration quadratures, and facilitates the implementation of multiscale techniques. The performance of the proposed algorithm is compared against one-and two-dimensional benchmark problems, i.e. pure rigid-body advection, isothermal and non-isothermal phase change, two-phase advective heat transfer and chemical reactions coupled with diffusion and advection. The numerical results confirm that the proposed solution method is accurate, oscillation-free and useful for and applicable to a wide range of fully coupled problems in science and engineering.

Numerical modelling of multiphase multicomponent reactive transport in the Earth’s interior

“…Given the presence of sharp gradients and the need to advect them over many time steps, an accurate and stable advection-diffusion algorithm is required to avoid numerical oscillations or excessive numerical diffusion. Here we use the particle-based Lagrangian-Eulerian formulation of (Oliveira et al, 2016), which combines a Lagrangian formulation for the advective part and an Eulerian-based heat source method for the diffusion and heat sources.…”

“…We have also designed an effective interpolation scheme to transfer temperature information from nodes to particles and back at every time step. (Oliveira et al, 2016) showed that the method is high-order, accurate, oscillation-free and applicable to a wide range of fully-coupled advection-diffusion-reaction problems (see Appendix C).…”

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

“…Oñate presents the solution by a stabilized FEM via the finite calculus method . Oliveira presents a particle‐based Lagrangian‐Eulerian FEM algorithm for the unsteady ADR problems considering the phase change . Kaya proposes a finite difference scheme for multidimensional steady and unsteady convection‐diffusion‐reaction problems .…”

The dual information preserving method for stiff reacting flows

“…Recent years, the Lagrangian-Eulerian (LE) approaches with the combination of Lagrangian particles and the Eulerian background grids have attracted great attention in solving the convectiondiffusion problems [20][21][22][23][24] . The LE method takes advantage of appropriate operator splitting techniques to solve different aspects of the physical model with most suitable Lagrangian or Eulerian formalism [24] .…”

“…The LE method takes advantage of appropriate operator splitting techniques to solve different aspects of the physical model with most suitable Lagrangian or Eulerian formalism [24] . Shipilova et al [25] applied a LE method (the particle transform method) to solve the convection-diffusion-reaction problems, numerical results showed that the PTM can avoid the numerical oscillation even for a very sparse grid.…”