Performance of 1-D and 2-D Lattice Boltzmann (LB) in Solution of the Shock Tube Problem

In this paper we presented a lattice Boltzmann with square grid for compressible flow problems. Triple level velocity is considered for each cell. Migration step use discrete velocity but continuous parameters are utilized to calculate density, velocity, and energy. So, we called this semi-discrete method. To evaluate the performance of the method the well-known shock tube problem is solved, using 1-D and 2-D version of the lattice Boltzmann method. The results of these versions are compared with each other and with the results of the analytical solution. Keywords___ distribution function, Lattice Boltzmann, phase velocity, shock tube, sound velocity. I. INRTODUCTION T is about 20 years since Frisch first time (1987) succeeded to use the Lattice Boltzmann Equations to calculate the viscosity of Lattice Gas Cellular Automata (LGCA) [1]. Then, this method has been considered as a new solver of the fluid flow in the various regimes. Ranging from low speed [2-4] to high speed flows [5-8]. In 1987 Alexander introduced a modified LB method to model the compressible flows [9]. The Sung model modifies the LB in such a manner that the fluid velocity is added to particles’ velocity [10-17]. Its method regains the results of the Euler and Navierstockes solutions with first-order and second-order precision, respectively. In the present work, we intend to develop the Sung method to model 1-D and 2-D compressible flow fields and to evaluate the performance of the model for solution of 1-D problems. II. LATTICE BOLTZMANN EQUATIONS The first step in Lattice Boltzmann models is determination of a distribution function. It defines the probability of finding a particle on the specific position and the specific time with a definite velocity. While, the distribution function is identified throughout the domain. We will be able to calculate the Microscopic quantities (i.e. density, velocity, pressure, temperature, and energy). The relations between the microscopic quantities and the distribution function are