Multiple-relaxation-time lattice Boltzmann model for the axisymmetric convection diffusion equation

A multiple-relaxation-time (MRT) lattice Boltzmann (LB) model with five discrete velocities in two dimensions (D2Q5) for simulating the axisymmetric convection-diffusion equation (CDE) is proposed. The axisymmetric CDE in the cylindrical coordinate system is rearranged as an ordinary CDE in the Cartesian coordinate system with the extra terms considered as additional sources. The gradient of the macroscopic scalar in the source terms is directly obtained from the non-equilibrium components of the distribution functions; thus no finite-difference calculation is required. The proposed model is consistent with the LB models for the axisymmetric hydrodynamic equations and they can be directly coupled. A series of numerical tests are conducted to validate the applicability and accuracy of the proposed model, including: (i) steady convection-diffusion in an annulus with mass injection on the boundaries, (ii) steady and transient convection-diffusion in a circular pipe, (iii) steady heat conduction inside a sphere, (iv) steady natural convection in an annulus between two coaxial vertical cylinders, and (v) steady swirling flows in a vertical cylindrical container. Analytical solutions are available for tests (i)-(iii), and the numerical results show that the proposed model is second-order accurate in space and first-order accurate in time. The surface-averaged Nusselt numbers for test (iv), the flow pattern and the velocity and temperature distributions for test (v) agree well with published results. (C) 2013 Elsevier Ltd. All rights reserved.