A lattice-Boltzmann model for the Advection-Diffusion equation in generalized coordinates (2019)
Lattice-Boltzmann models (LBM) are very powerful simulation techniques for fluid dynamics, diffusion processes, mechanical waves, magneto- and electrodynamics. However, one of the main complications when working with these LBM is the necessity of an accurate implementation of the boundary conditions. Ideally, the boundaries are rectangular and parallel to the computational mesh, but most of the times, real-life problems have complex geometries and, therefore, boundaries are not easy to implement. This work develops an alternative lattice-Boltzmann model to reproduce the Advection-Diffusion Equation (ADE) on generalized coordinates in two and three dimensions. Our model introduces the geometry as a source term, which makes it much easier and more flexible to simulate curved geometries in two and three dimensions like disks, cylinders, torii, sinusoidal curved channels and any complex shape that can be described as an orthogonal coordinate transformation. The proposed LBM, which shows second-order accuracy, allows also to perform mesh refinements without losing isotropy, to avoid staircase approximations and to take advantage of geometrical symmetries, when present. Our simulation results are in excellent agreement with the theoretical predictions in all studied cases in two and three dimensions, with or without symmetries, and even reproduce with great accuracy experimental results. In fact, we have defined our model in such a way that it facilitates to deal with real physical units (like centimeters, seconds, etc) something that is not obvious when dealing with non-uniform cell sizes, making easier the comparison with experimental data. Our model can be used on a broad range of applications, like heat diffusion in complex geometries, pollutant spreading in channels or pipes, and sediment transport in rivers. Because each geometry is defined by few parameters, and those parameters can be time-dependent, our model could be also used to simulate the ADE on time-varying geometries, like pulsing blood vessels, by computing our model with a Navier-Stokes equations solver with changing boundary conditions. This work contains a valuable contribution for the study of advection-diffusion phenomena. Because this phenomenon is relevant to many scientific, industrial and environmental applications, we expect it would be of great usefulness in future research.