Unstructured lattice Boltzmann method in three dimensions

Abstract

SUMMARYOver the last decade, the lattice Boltzmann method (LBM) has evolved into a valuable alternative to continuum computational uid dynamics (CFD) methods for the numerical simulation of several complex uid-dynamic problems. Recent advances in lattice Boltzmann research have considerably extended the capability of LBM to handle complex geometries. Among these, a particularly remarkable option is represented by cell-vertex ÿnite-volume formulations which permit LBM to operate on fully unstructured grids. The two-dimensional implementation of unstructured LBM, based on the use of triangular elements, has shown capability of tolerating signiÿcant grid distortions without su ering any appreciable numerical viscosity e ects, to second-order in the mesh size. In this work, we present the ÿrst three-dimensional generalization of the unstructured lattice Boltzmann technique (ULBE as unstructured lattice Boltzmann equation), in which geometrical exibility is achieved by coarse-graining the lattice Boltzmann equation in di erential form, using tetrahedrical grids. This 3D extension is demonstrated for the case of 3D pipe ow and moderate Reynolds numbers ow past a sphere. The results provide evidence that the ULBE has signiÿcant potential for the accurate calculation of ows in complex 3D geometries.

Detailed analysis of the lattice Boltzmann method on unstructured grids

“…However, we emphasize that this result is valid only for the forward Euler time integration and might not hold for different time integration schemes. Our analysis of the operator splitting based time integration, introduced in Rossi et al (2005), show that the kinematic viscosity is proportional to the difference between the relaxation time and the time step ν = c 2 s (τ − δt), resembling the results for the finite difference LB methods on regular grids. This interesting result might have far reaching consequences since it overcomes the constraint on the relaxation and the time step to obtain very low viscosities.…”

“…Also, by analysing the dispersion relation, they find that the kinematic viscosity is given by ν = c 2 s τ , indicating that numerical viscosity effects are absent (with the exception of the numerical diffusion proportional to the square of the grid spacing). This fact, as pointed out in Rossi et al (2005), requires a more careful theoretical examination. We have addressed this standing problem by means of the Chapman-Enskog expansion.…”

“…This is primarily due to the fact, that in order to obtain an accurate boundary representation, a high resolution grid is required, increasing the overall size of the system (the boundary representation's accuracy is on the order of O(h), while the volumetric grid's size scales like h −2 in 2D and h −3 in 3D, where h is the grid spacing). In the recent years, various types of off-lattice Boltzmann methods have been developed in order to allow for enhanced geometric flexibility of such schemes, which might challenge the standard LB methods (see for instance Ubertini et al (2003), Rossi et al (2005), Ubertini et al (2006), Bardow et al (2008) and references therein).…”

“…Our focus here is on the finite volume schemes developed in Ubertini et al (2003) and Rossi et al (2005). A prominent feature of these schemes is the independence of the velocity and space discretizations.…”

“…Our results, as demonstrated later, corroborate those findings for the forward Euler time integration. As stated in Rossi et al (2005) lack of numerical viscosity implies no mesh limitations on the highest Reynolds number that can be simulated, nonetheless, small viscosities can only be achieved with vanishingly small relaxation times. These, together with the Courant-FriedrichsLewy (CFL) stability condition, δt < 2τ , would imply prohibitively small time step size.…”