Direct Numerical Solution Of The Boltzmann Equation

Progress in computer hardware and improvement of numerical methods made solution of the Boltzmann equation for rather complex gas dynamic problems real. The method developed by the author is based on a projection technique for evaluation of the collision operator. The computed collision integral is conservative by density, impulse, and energy, and became equal to zero when the solution has a form of the Maxwellian distribution. The later feature sharply increases its efficiency, especially for the near equilibrium flows. The method is extended on a mixture of gases and the gases with internal degrees of freedom, where it can incorporate real physical parameters of molecular potential and of internal energy spectrum. Examples of computations for a range of Mach and Knudsen numbers are presented. INTRODUCTION Solution of the Boltzmann equation was initiated by Nordsieck's group in Illinois, USA nearly 40 years ago when first big computers have appeared [1]. In this method velocity space was divided by equal cells, and collision integrals were evaluated by simulating of molecular collisions with randomly chosen velocity vectors. Then computed mean values per a cell were attributed to the middle points of the cells, and the obtained system of equations was solved by an iterative method. A few years later another method in which the collision integrals were evaluated directly at the nodes of the grid in the velocity space by Monte Carlo technique was developed by the author of this paper [2], Very soon the main deficiency of both approaches non-fulfillment of conservation laws in the computed integrals became evident. In [3], a splitting finite-difference scheme for the kinetic equation has been proposed, and then a special correction was developed to satisfy the conservation laws at the relaxation stage [4]. The new method showed itself much more efficient then that of [2], but the correction introduced some additional numerical viscosity, and required artificial assumptions in the case of gas mixtures [5, 6]. The next step in the development of conservative methods has been made after construction of a Discrete Velocity Model [7]. In this model molecular velocities are given at a uniform Cartesian lattice and impact parameters of a molecular collision are such that post collision velocities belong to the same grid. Therefore, it imposes a restriction on impact parameters, which are independent in the true Boltzmann equation. Based on a similar idea, conservative methods for the Boltzmann collision integrals have been proposed [8, 9]. The conservative method without any selection of impact parameters has been developed by the author [10, 11]. It was then improved and adopted for computing near equilibrium flows [12, 13], extended to gas mixtures [14], and gases with internal degrees of freedom [15]. The main features of the method are described bellow. NUMERICAL METHOD Consider the Boltzmann equation It is solved at a grid of 7V0 equidistant nodes £ with a step h confined in a domain O of a volume V . On the basis of Dirac's 8-functions, the distribution function and the collision integral can be presented in a form (1)