Browsing by Subject "Rarefied gas dynamics"
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Item A conservative deterministic spectral method for rarefied gas flows(2008-08) Tharkabhushanam, Sri Harsha, 1979-; Martínez Gamba, Irene, 1957-The mathematical analysis of the Boltzmann equation for a wide range of important models is well developed. It describes physical phenomena which are often of great engineering importance (in aerospace industry, semiconductor design, etc.). For that reason, analytical and computational methods of solving the Boltzmann equation are studied extensively. The idea of describing processes on a scale of the order of the relaxation scales of time and space has been realized. The nonlinear Boltzmann equation possesses the important essence of a physically realistic equation, so it is possible not only to consider the flows of simple media but to formulate new problems due to the ability of this equation to describe nonequilibrium states. In this dissertation, a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation for variable hard potential (VHP) collision kernels with conservative or non-conservative binary interactions is proposed. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in the collision integral computation is reduced to a separate integral over the unit sphere S2. In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or to elastic model of slowing down processes. We prove the accuracy, consistency and conservation properties of the proposed conservative spectral method. Existing spectral methods have consistency proofs which are only for elastic collisions, and also such methods do not conserve all the necessary moments of the collision integral. In this dissertation, error estimates for the conservation routine are provided. Such conservation correction is implemented as an extended isoperimetric problem with the moment conservation properties as the constraints. We use and extend an existing bound estimate of Gamba, Panferov and Villani for the inelastic/elastic space homogeneous Boltzmann collision operator. The result is an original extension to the work of Gustaffson. Using these estimates along with projection error estimates and conservation correction estimates, we prove that the conservation correction is bounded by the spectral accuracy. Simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the self-similar simulations involves the selection of a time rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods. Recognizing the importance of the Boltzmann equation in the analysis of shock structures and nonequilibrium states, such a study is done for 1D(x) × 3D(v). The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Sone and Aoki, where the wall temperature is suddenly changed, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.Item Deterministic and random particle methods applied to Vlasov-Poisson-Fokker-Planck kinetic equations(Texas Tech University, 1996-05) Havlak, KarlWe devise and study two different particle methods for approximating Vlasov-Poisson-Fokker-Planck systems. We first consider a random particle method. Such a proposed scheme takes into account the fact that the trajectories of a particle, undergoing Brownian motion due to collisions with the medium or background particles, can be obtained as the solutions of stochastic differential equations, i.e., the Langevin equations. These equations are the precise analogs of the (deterministic) Hamiltonian system in the collisionless model. The particle approximation, in particular, simulates the action of viscosity by the use of independent Wiener processes (Brownian motions). The analysis relies heavily on the machinery developed by K. Ganguly and H.D. Victory, Jr. [SIAM J. Numer. Anal., 26 (1989), pp. 249-288] to treat the sampling errors due to random motions of the particles. For example, the key idea in the consistency error analysis is to separate the moment and discretization errors - accounting for the deterministic portions of the error - from the sampling errors introduced by the random motion of the particles. The latter errors constitute the dominant component of the overall consistency error in terms of order and are gauged by applying Bennett's Inequality utilized to estimate tail probabilities for standardized sums of independent random variables with zero means. Moreover, the stability estimates for the particle approximations to the collisionless model are extended to the Vlasov-Poisson-Fokker-Planck setting by means of this inequality. We then consider a deterministic method. Such a proposed scheme is a splitting method, whereby particle methods are used to treat the convective part and the diffusion is simulated by convolving the particle approximation with the fieldfree Fokker-Planck kernel. The states of the particles are not affected by the diffusion per se, but the charge or mass on the particles in their previous states is redistributed via the diffusion. Because of this redistribution of mass or charge, it is necessary to monitor the growth in time of the velocity moments of the approximate distribution. Convergence of the errors in both the density and the fields is shown to be first order in time with respect to both the uniform and L^- senses. This treatment is the first application of the velocity moment analysis by P.L. Lions and B. Perthame [Invent. Math., 105 (1991), 415- 30] in a numerical analysis of Vlasov-type kinetic equations. Our study is made feasible by some formulas by F. Bouchut [J. Fund Anal, 111 (1993), 239-258] concerning the field-free fundamental solution and recent extensions to the periodic setting. The splitting procedure we employ is related to the viscous splitting or fractional step procedure of G.H. Cottet and S. Mas-Gallic [Numer. Math., 57 (1990), 805-827] for treating Navier-Stokes equations modeling viscous, incompressible flow.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, ThierryThe 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.Item Fluid-structure interactions in microstructures(2013-05) Das, Shankhadeep; Murthy, Jayathi; Mathur, SanjayRadio-frequency microelectromechanical systems (RF MEMS) are widely used for contact actuators and capacitive switches. These devices typically consist of a metallic membrane which is activated by a time-periodic electrostatic force and makes periodic contact with a contact pad. The increase in switch capacitance at contact causes the RF signal to be deflected and the switch thus closes. Membrane motion is damped by the surrounding gas, typically air or nitrogen. As the switch opens and closes, the flow transitions between the continuum and rarefied regimes. Furthermore, creep is a critical physical mechanism responsible for the failure in these devices, especially those operating at high RF power. Simultaneous and accurate modeling of all these different physics is required to understand the dynamical membrane response in these devices and to estimate device lifetime and to improve MEMS reliability. It is advantageous to model fluid and structural mechanics and electrostatics within a single comprehensive numerical framework to facilitate coupling between them. In this work, we develop a single unified finite volume method based numerical framework to study this multi-physics problem in RF MEMS. Our objective required us to develop structural solvers, fluid flow solvers, and electrostatic solvers using the finite volume method, and efficient mechanisms to couple these different solvers. A particular focus is the development of flow solvers which work efficiently across continuum and rarefied regimes. A number of novel contributions have been made in this process. Structural solvers based on a fully implicit finite volume method have been developed for the first time. Furthermore, strongly implicit fluid flow solvers have also been developed that are valid for both continuum and rarefied flow regimes and which show an order of magnitude speed-up over conventional algorithms on serial platforms. On parallel platforms, the solution techniques developed in this thesis are shown to be significantly more scalable than existing algorithms. The numerical methods developed are used to compute the static and dynamic response of MEMS. Our results indicate that our numerical framework can become a computationally efficient tool to model the dynamics of RF MEMS switches under electrostatic actuation and gas damping.