A lattice Boltzmann method for solute transport

Abstract

SUMMARYA lattice Boltzmann method is developed for solute transport. Proper expressions for the local equilibrium distribution functions enable the method to be formulated on rectangular lattice with the same simple procedure as that on a square lattice. This provides an additional advantage over a lattice Boltzmann method on a square lattice for problems characterized by dominant phenomenon in one direction and relatively weak in another such as solute transport in shear flow over a narrow channel, where the problems can efficiently be approached with fine and coarse meshes, respectively, resulting in more efficient algorithm. The stability conditions are also described. The proposed method on a square lattice is naturally recovered when a square lattice is used. It is verified by solving four tests and compared with the analytical/exact solutions. They are in good agreement, demonstrating that the method is simple, accurate and robust for solute transport.

Modeling of Ti-W Solidification Microstructures Under Additive Manufacturing Conditions

“…[29] Coupled transport of fluid flow, heat, and solute would not be possible to simulate accurately in the present model for molten titanium alloys, which is limited to a single grid spacing and time step, owing to the disparate rates of heat and solute transport. The thermal Lattice Boltzmann method for coupled heat and incompressible fluid flow fields has been used previously to simulate convection problems [29,[82][83][84] and been coupled to cellular automata (CA) to simulate the solidliquid phase change for pure materials. [7,31,32,36,79] If it is assumed that the fluid is incompressible, viscous heat dissipation is negligible, and no work is done by the external pressure, an additional set of distribution functions can be introduced to the model for heat transport.…”

“…[7,85] Use of the thermal Lattice Boltzmann method with a second set of distribution functions to model the internal energy density has been shown to agree with analytical solutions for natural convection, lid-driven convection in square cavities, and other benchmark fluid dynamics problems. [78,82,85,86] Though the present work is focused on microstructure evolution and not the process-scale melt pool dynamics, we are exploring such a model for consideration of the coupled fluid flow and heat transport problem at the appropriate scale. This approach has a notable advantage over the present COMSOL model in that it enables direct tracking of the solid-liquid interface, allowing for direct coupling to the microstructure evolution model.…”

A Double-Distribution-Function Lattice Boltzmann Method for Bed-Load Sediment Transport

“…In 2002, Zhou developed a lattice Boltzmann model for shallow water equations with a source term [Zhou, 2002]. In 2009, Zhou proposed the LBM for solute transport [Zhou, 2009]. In 2010, Thang studied the lattice Boltzmann shallow equation and its coupling to build a canal network [Thang et al, 2010].…”

Rectangular lattice Boltzmann model for nonlinear convection–diffusion equations

“…Remark 2.3. In previous works on the RLB model for CDE, only linear equilibrium functions (Lfeq) are used [16,19]. Through the Chapman-Enskog expansion, we find that the macroscopic equation cannot be recovered exactly without additional assumptions on the equation.…”