Relaxation and reversibility of extended Taylor dispersion from a Markovian-Lagrangian point of view

摘要

Taylor dispersion in a two-dimensional (2D) stratified velocity field describes a transition, called relaxation, from convective behaviour for short times, towards Fickian behaviour for large times and is partially reversible upon reversal of the flow direction. In 2D the physics are assumed to be governed by the unidirectional convection diffusion equation (2D uCDE). The approximate height-averaged 1D Generalised Telegraph Equation (GTE) catches an essential part of the longitudinal spreading. Contrary to the 1D Fickian approach, it explicitly accounts for the transient reversible nature [Camacho J. Purely global model for Taylor dispersion. Phys Rev E 1993/2;48(1); Berentsen CWJ, Verlaan ML, van Kruijsdijk CPJW. Upscaling and reversibility of Taylor dispersion in heterogeneous porous media. Phys Rev E 2005;71:046308]. Here we approach Taylor dispersion from a Lagrangian-Markovian point of view. In the redistribution model (RM) that we present, the Euler forward method accounts for convection and a probability redistribution matrix generates the transverse movement by diffusion over time. We consider two redistribution matrices. The first results from the direct discretisation of the Gaussian distribution function associated with the transverse mixing of the 2D uCDE. The resulting Gaussian redistribution model (GRM) is able to capture the multi-scale relaxation and reversibility behaviour of the full 2D uCDE. The correlated redistribution model (CRM) is the RM model with a redistribution matrix based on auto-correlation. The CRM is a generalisation of the correlated random walk model of [Scheideg-ger AE. The random walk model with auto-correlation of flow through porous media, Can J Phys 1958;36]. For uniform auto-correlation, the CRM model approximates the multi-scale relaxation nature of the 2D uCDE as a single scale relaxation process similar to the GTE. Moreover, it has the same variance as the GTE in the limit of the time step over relaxation time ratio to zero. For specific conditions the equality of the CRM model and the GTE is proven up to △t~2 order accuracy.