# Browsing by Subject "Internal energy"

Now showing 1 - 2 of 2

###### Results Per Page

###### Sort Options

Item A discrete velocity method for the Boltzmann equation with internal energy and stochastic variance reduction(2015-12) Clarke, Peter Barry; Varghese, Philip L.; Goldstein, David Benjamin; Raja, Laxminarayan; Gamba, Irene; Magin, ThierryShow more The goal of this work is to develop an accurate and efficient flow solver based upon a discrete velocity description of the Boltzmann equation. Standard particle based methods such as Direct Simulation Monte Carlo (DSMC) have a number of difficulties with complex and transient flows, stochastic noise, trace species, and high level internal energy states. To address these issues, a discrete velocity method (DVM) was developed which models the evolution of a flow through the collisions and motion of variable mass quasi-particles defined as delta functions on a truncated, discrete velocity domain. The work is an extension of a previous method developed for a single, monatomic species solved on a uniformly spaced velocity grid. The collision integral was computed using a variance reduced stochastic model where the deviation from equilibrium was calculated and operated upon. This method produces fast, smooth solutions of near-equilibrium flows. Improvements to the method include additional cross-section models, diffuse boundary conditions, simple realignment of velocity grid lines into non-uniform grids, the capability to handle multiple species (specifically trace species or species with large molecular mass ratios), and both a single valued rotational energy model and a quantized rotational and vibrational model. A variance reduced form is presented for multi-species gases and gases with internal energy in order to maintain the computational benefits of the method. Every advance in the method allows for more complex flow simulations either by extending the available physics or by increasing computational efficiency. Each addition is tested and verified for an accurate implementation through homogeneous simulations where analytic solutions exist, and the efficiency and stochastic noise are inspected for many of the cases. Further simulations are run using a variety of classical one-dimensional flow problems such as normal shock waves and channel flows.Show more Item A new method to incorporate internal energy into a discrete velocity Monte Carlo Boltzmann Equation solver(2011-08) Hegermiller, David Benjamin; Varghese, Philip L.; Goldstein, David B.Show more A new method has been developed to incorporate particles with internal structure into the framework of the Variance Reduction method [17] for solving the discrete velocity Boltzmann Equation. Internal structure in the present context refers to physical phenomena like rotation and vibration of molecules consisting of two or more atoms. A gas in equilibrium has all modes of internal energy at the same temperature as the translational temperature. If the gas is in a non-equilibrium state, translational temperature and internal temperatures tend to proceed towards an equilibrium state during equilibration, but they all do so at different relaxation rates. In this thesis, rotational energy of a distribution of molecules is modeled as a single value at a point in a discrete velocity space; this represents the average rotational energy of molecules at that specific velocity. Inelastic collisions are the sole mechanism of translational and rotational energy exchange, and are governed by a modified Landau-Teller equation. The method is tested for heat bath simulations, or homogeneous relaxations, and one dimensional shock problems. Homogeneous relaxations demonstrate that the rotational and translational temperatures equilibrate to the correct final temperature, which can be predicted by conservation of energy. Moreover, the rates of relaxation agree with the direct simulation Monte Carlo (DSMC) method with internal energy for the same input parameters. Using a fourth order method for convecting mass along with its corresponding internal energy, a one dimensional Mach 1.71 normal shock is simulated. Once the translational and rotational temperatures equilibrate downstream, the temperature, density and velocity, predicted by the Rankine-Hugoniot conditions, are obtained to within an error of 0.5%. The result is compared to a normal shock with the same upstream flow properties generated by the DSMC method. Internal vibrational energy and a method to use Larsen Borgnakke statistical sampling for inelastic collisions is formulated in this text and prepared in the code, but remains to be tested.Show more