Lattice-Boltzmann coupled models for advection–diffusion flow on a wide range of Péclet numbers

Traditional Lattice-Boltzmann modelling of advection–diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice-Boltzmann Navier–Stokes solver is coupled to a Lattice-Boltzmann advection–diffusion model. In a novel model, the Lattice-Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection–diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme.

The models are validated using two non-trivial benchmarks, which includes discontinuous initial conditions and the case Pe$_{g}$->$\infty$ for the first time, where Pe$_{g}$ is the grid Péclet number. The evaluation of Pe$_{g}$ alongside Pe is discussed. Accuracy, stability and the order of convergence are assessed for a wide range of Péclet numbers. Recommendations are then given as to which model to select depending on the value Pe$_{g}$ - in particular, it is shown that the coupled finite-difference/Lattice-Boltzmann provide stable solutions in the case Pe->$\infty$, Pe$_{g}$->$\infty$