A Novel Highly Efficient Scheme for the Boltzmann Equation

The work completed during the course of the NSTRF grant period was focused on the development of an accurate, low cost flow solver through a discrete velocity analysis of the Boltzmann equation. According to kinetic theory, at the microscopic level the state of a system is defined by the motion of molecules and their interactions through collisions. The nature of these interactions is determined by the intermolecular potentials, and collisions result in exchanges of momentum and sometimes internal energy between molecules. The Boltzmann equation accounts for these molecular interactions and is necessary for the description of flows where the continuum (NavierStokes) equations fail. More specifically, the continuum equations break down when the ratio between the mean free path and the length scale describing flow gradients, known as the Knudsen number, approaches and passes 0.01. Rarefied flows can occur in micro- and nano-scale devices, shocks, satellite attitude control thruster plumes, around satellites in low-earth orbit, and during hypersonic re-entry. These flows can be very complex and might include changes to internal energy states, unsteadiness, and intricate geometries. The basic premise behind the discrete velocity method (DVM) is characterized by defining the velocity distribution function only at discrete internals in velocity space. This is advantageous by decreasing the number of possible collision events and the number of quasi-particles stored in memory. The discretized velocity distribution function is used as the framework for solving the full Boltzmann equation. The equation is split into convection (left hand side) and collision (right hand side) parts, each is solved in sequence, and time is advanced. Solving the collision integral is done through selecting a simulated collisions weighted by the number density at each velocity location, depleting the appropriate number density adjusted by the total collision cross-section and relative velocity, determining a post-collision replenishing location, returning the number density in a manner which conserves mass, momentum, and energy, and finally repeating the process until a total number of simulated collisions has been reached. The number of simulated collisions is chosen to obtain a desired level of computational efficiency and stochastic noise, and the density depleted per collision is determined from the total depletion indicated by the collision term in the Boltzmann equation divided by the number of simulated collisions. Convection is performed using a finite difference method appropriate to the flow parameters and desired accuracy. The discrete velocity method developed in the PhD work is an extension of a previous method completed by Aaron Morris for a single, monatomic species solved on a uniform (evenly space) velocity grid with either a hard sphere or pseudo-Maxwell collision cross-section model. The additions to the method included: • Variable hard sphere and variable soft sphere collision models were implemented to model the collision cross-section of molecules. • A variety of new boundary conditions were added including diffuse and accommodated wall reflection. • Adjustments were made to account for a range of Knudsen numbers. • Using simple realignment of grid lines, non-uniform velocity grids were implemented with the intention of focusing computational effort in a more efficient manner. • The method was extended to include gas mixtures. A major focus of this work was on mixtures with large number density or mass ratios. • The final improvement to the discrete velocity method was the implementation of a quantized internal energy model where full rotational and vibrational distributions are stored and updated during elastic and inelastic collisions. • Variance reduced formulations for each of the preceding bullet points. Variance reduction methods reduce stochastic noise and the number of simulated collisions, especially in nearequilibrium flows. The research completed during the fellowship period extended a basic Boltzmann solver to include new physics to allow for a variety of flow conditions, and made use of computational methods to improve the efficiency of flow simulations. The completed work is only some of the possible improvements to the discrete velocity code that could have been made, and continuing work by future graduate students will introduce new physics and further improve computational efficiency.