Lattice Boltzmann Methods for Navier-Stokes Equations in General Orthogonal Coordinates for Efficient Flow Simulations Using Nonuniform Clustered Grids

Resolving fluid flow structures with multiscale features or boundary layers that arise in many real-world applications effectively requires the use of spatially nonuniform meshes with localized clustering of grids. The lattice Boltzmann (LB) method, which is based on models from kinetic theory and involves the use of collide-and-stream steps of the particle distribution functions, is a modern computational technique for fluid dynamics. However, its standard formulation restricts it to the use of uniform Cartesian grids that involve high computational resource requirements in simulating such flows. In this paper, we present new and improved formulations of LB methods that accommodate the use of continuously varying spatial grids via coordinate transformations to simulate the Navier-Stokes equations (NSE) in the general orthogonal curvilinear coordinates (GOC). They are constructed using a Chapman-Enskog analysis to specify the equilibrium moments of the distribution functions and the geometric force terms used in the collision step to be dependent on the local metric factors and their spatial derivatives, which parameterize the local variations in grids as a tensor, along with the density, pressure, momentum and their fluxes, and some correction terms related to the normal velocity gradients so as to represent the conservative form of the NSE in the GOC in a consistent manner. The latter enables recovering the normal viscous stress tensor accurately while the shear stress components emerge naturally from the formulations through appropriate choices for the third order moment equilibria. The resulting LB algorithm using the GOC, and referred to as the GOC-LBM, importantly maintains the simplicity of the collide-and-stream approach in the computational domain based on the D2Q9 lattice while allowing grid clustering/stretching in the physical domain, and is Galilean invariant that is free of the cubic velocity artifacts. Our approach for the GOC-LBM is general and modular in that it can be used with any collision model with appropriate modifications to equilibria and forcing terms. We present the implementation details of the GOC-LBM using a variety of collision models while the central moments-based model using multiple relaxation times was found to be the most robust in practical implementations. We validate the GOC-LBM through numerical simulations for various canonical benchmark flow problems, which demonstrate its good accuracy and Galilean invariance, and also its benefits over the Cartesian grids-based standard LBM in naturally representing flows in curvilinear geometries using body-fitted grids. Moreover, we show significant computational advantages of our approach over the standard uniform grids-based LBM for a case study on simulating boundary layer flows efficiently that involves coupling the GOC-LBM for the NSE with another newly constructed GOC-LB scheme for solving the magnetic induction equation for magnetohydrodynamics (MHD) that involves thin Hartmann layers near walls. Such a configuration is important in fusion engineering applications and the results show that the GOC-LBM can perform simulations of high Hartmann number flows using fewer grid points by about a third of that of the standard LBM with equivalent accuracy.