In this approach, the LBM flow solver is used to model the conservation of mass and momentum, while the temperature and species fields were handled using other sets of distribution functions. The advection-diffusion or passive scalar LBM is widely used in the literature, especially under the assumption of incompressible flow. Despite the latter issue, the lattice Boltzmann method has emerged as a potential alternative for simulating a range of complex flows throughout the last decade. The disadvantage of LBM is that as the number of lattice velocity sets increases, more memory storage will be required. The local nature of the time evolution of the LBM helps to simplify the numerical coding process, improve the parallel scalability of the algorithm, and make the implementation of the complex boundary conditions straightforward. Although initially developed to solve the Navier–Stokes (NS) equation, lattice Boltzmann method have been extended to a variety of applications and flows. Conventional numerical methods for solving these equations include finite difference methods, finite volume methods, and finite element methods.
The governing equations consist of three main components: (a) Navier–Stokes equations for the fluid, (b) heat transport equation and (c) a set of transport equations for the species. Species transport and heat transfer flows are widely present in real-world systems with complicated geometry (gas turbines, burners, furnaces, and so on), and accurate numerical simulation of these processes is key to initial design. In addition, the adaptive filtering method can also improve the numerical stability of the lattice Boltzmann method with limited numerical dissipation. The results show that the hybrid regularized collision operator has advantages in simulating the scalar advection-diffusion problem with small diffusion coefficient. The results are compared to the classical finite difference method and to the lattice Boltzmann method using the projection-based regularized and standard Bahtnagar-Gross-Krook collision operator. The adaptive dynamic filtering method is also tested. Then the scalar transport in doubly periodic shear layer flow is tested, which is sensitive to numerical stability. The advection–diffusion lattice Boltzmann method is first tested in uniform flow with smooth and discontinuous initial conditions. In order to improve the stability of advection–diffusion lattice Boltzmann method to simulate scalar transport in complex flow, a hybrid regularized collision operators and a dynamic filtering method which is suitable for the convection-diffusion lattice Boltzmann method are proposed in this paper. The stability conditions will become more severe, when there are high gradient regions in the computational domain. In multi-component flow and/or thermal flows, when the diffusion coefficient of the advection–diffusion equation is relatively small, the relaxation coefficient in the lattice Boltzmann method will be close to 0.5, which will lead to numerical instability.