*Article* **Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics**

**Oleg Ilyin**

Dorodnicyn Computing Center, Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Vavilova-40, 119333 Moscow, Russia; oilyin@gmail.com

**Abstract:** In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the *H*-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasiincompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

**Keywords:** discrete velocity method; lattice Boltzmann method; computational fluid dynamics
