A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann--BGK equation

In this work, we present a positivity-preserving high-order flux reconstruction method for the polyatomic Boltzmann--BGK equation augmented with a discrete velocity model that ensures the scheme is discretely conservative. Through modeling the internal degrees of freedom, the approach is further extended to polyatomic molecules and can encompass arbitrary constitutive laws. The approach is validated on a series of large-scale complex numerical experiments, ranging from shock-dominated flows computed on unstructured grids to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. The results show the ability of the scheme to directly resolve shock structures without any ad hoc numerical shock capturing method and correctly approximate turbulent flow phenomena in a consistent manner with the hydrodynamic equations.Comment: 31 pages, 20 figure

On the Fokker-Planck approximation to the Boltzmann collision operator

The Boltzmann equation (BE) is a mesoscopic model that provides a description of how gases undergoing a binary collision process evolve in time, however there is no general analytical approach for finding its solutions and direct numerical treatment using quadrature methods is prohibitively expensive due to the dimensions of the problem. For this reason, models that are able to capture the behaviour of solutions to the BE, but which are simpler to treat numerically and analytically are highly desirable. The Fokker-Planck collision operator is one such collision model, which is suited well to numerical solutions using stochastic particle methods, and is the subject of this thesis. The stochastic numerical solutions of the Fokker-Planck model suffer heavily from noise when the speed of the flow is low. We develop two methods that are able to reduced the variance of the estimators of the particle method. The first is a common random number method, which produces a correlated equilibrium solution where thermodynamic fields are known. The second is a importance sampling method, where weights are attached to the particles. This means that particles close to equilibrium do not contribute to the noise of the estimators. We also develop a randomised quasi-Monte Carlo scheme for solving the diffusion equation, which has a faster rate of convergence than simple Monte Carlo methods. The relative simplicity of the functional form of the Fokker-Planck collision operator makes it possible to find analytic solutions in simple cases. We consider a spatially homogeneous, isotropic gas with elastic collisions in the presence of forcing and dissipation and derive self-consistent non-equilibrium steady-state solutions. Previous numerical evidence exists that suggest such forcing and dissipation mechanisms, widely separated, give rise to steady-states of the BE that are close to Maxwellian, with a direct energy cascade and an inverse particle cascade. Using our analytic solutions, we are able to investigate the dependence of such solutions on the forcing and dissipation scales, and find that in the inertial range, the interaction is non-local. We then show that the “extreme driving” mechanism, responsible for a family of non-universal power-law solutions for inelastic granular gases, where the flux of energy is towards lower scales, is also able to produce inverse energy cascades for the elastic system

Lattice Boltzmann method for warm fluid simulations of plasma wakefield acceleration

A comprehensive characterization of lattice Boltzmann (LB) schemes to perform warm fluid numerical simulations of particle wakefield acceleration (PWFA) processes is discussed in this paper. The LB schemes we develop hinge on the moment matching procedure, allowing the fluid description of a warm relativistic plasma wake generated by a driver pulse propagating in a neutral plasma. We focus on fluid models equations resulting from two popular closure assumptions of the relativistic kinetic equations, i.e., the local equilibrium and the warm plasma closure assumptions. The developed LB schemes can thus be used to disclose insights on the quantitative differences between the two closure approaches in the dynamics of PWFA processes. Comparisons between the proposed schemes and available analytical results are extensively addressed.Comment: 8 figure

Modélisation des écoulements de gaz raréfiés au travers de filtres fibreux par la méthode de Boltzmann sur réseau

RÉSUMÉ: Les particules fines suspendues dans l’air (aussi nommées aérosols) sont nocives pour la santé humaine et pour l’environnement. La filtration des aérosols (ou la séparation de ces particules de l’air) est donc un procédé d’une importance cruciale. Les filtres fibreux sont généralement choisis pour leur haute performance et leur compacité. L’ajout de nanofibres (<1 μm) déposées sur une couche de microfibres ou mélangées à des microfibres a été proposé pour améliorer ces filtres. La théorie de la fibre unique est souvent utilisée pour prédire la performance des filtres à aérosols. Cependant, cette théorie prend pour acquis que les fibres d’un filtre sont toutes du même diamètre et ignore donc les impacts potentiels de la structure multicouche. La simulation numérique directe des écoulements gazeux au travers de milieux fibreux doit être utilisée pour tenir compte des interactions entre les fibres. Or, les effets de raréfaction qui apparaissent autour des nanofibres doivent être considérés pour prédire quantitativement la performance des milieux filtrants.----------ABSTRACT: Suspensions of fine particles (also called aerosols) are harmful to human health and the environment. The filtration of airborne particles (or the separation of these particles from the air) is therefore a process of crucial importance. Fibrous filters are generally chosen for their high performance and compactness. The addition of nanofibers (<1 μm) deposited on a layer of microfibers or mixed with microfibers has been proposed to improve these filters. The single fiber theory is often used to predict the performance of aerosol filters. However, this theory assumes that the fibers of a filter are all the same diameter and therefore ignores the potential impacts of the multilayer structure. Direct numerical simulation of gas flows through fibrous media must be used to account for the interactions between the fibers. However, the rarefaction effects that occur around nanofibers must be considered to quantitatively predict the performance of the filter media

A kinetic Fokker-Planck algorithm for simulating multiscale gas flows

Numerical, aerodynamic analysis of spacecraft requires the modeling of rarefied hypersonic flows. Such flow regimes are usually dominated by broad shock waves and strong expansion flows. In such areas of the flow the gas is far from its equilibrium state and therefore conventional modeling approaches such as the Euler or Navier-Stokes equations cannot be used. Instead, non-equilibrium modeling approaches must be applied. While most non-equilibrium flow solvers are computationally expensive, a recently introduced kinetic Fokker-Planck (FP) method shows the potential of describing non-equilibrium flows with satisfactory accuracy and, at the same time, significantly reducing computational costs. However, the application of kinetic FP solvers was so far still limited to simple, single species gases. The aim of this study is to extend the capabilities of the kinetic FP approach for describing complex gas flows. Particular attention is paid to the modeling of non-equilibrium aerodynamics, as it is relevant for describing spacecraft related gas flows. Methods for describing polyatomic species as well as gas mixtures within the kinetic FP framework are constructed. All models are intensively validated by comparison to already established numerical methods, as well as in comparison to experimental studies. Excited energy states are modeled by a stochastic jump process described by a master equation. This approach allows the description of both continuous and discrete energy levels. Gas mixtures are modeled based on the hard-sphere and variable hard-sphere collision potentials. For both cases, FP models are constructed for an arbitrary number of species. The efficiency of the described models is investigated and different strategies are proposed to use kinetic FP methods efficiently. The expansion of synthetic air from an axially symmetric orifice is numerically reproduced using the developed models and results are compared with experimental measurements. Although the numerical simulations capture several magnitudes of Knudsen numbers, from the continuum flow in the reservoir up to the free-molecular far field, good agreement between simulation and experiment is seen

Hydrodynamics

The phenomena related to the flow of fluids are generally complex, and difficult to quantify. New approaches - considering points of view still not explored - may introduce useful tools in the study of Hydrodynamics and the related transport phenomena. The details of the flows and the properties of the fluids must be considered on a very small scale perspective. Consequently, new concepts and tools are generated to better describe the fluids and their properties. This volume presents conclusions about advanced topics of calculated and observed flows. It contains eighteen chapters, organized in five sections: 1) Mathematical Models in Fluid Mechanics, 2) Biological Applications and Biohydrodynamics, 3) Detailed Experimental Analyses of Fluids and Flows, 4) Radiation-, Electro-, Magnetohydrodynamics, and Magnetorheology, 5) Special Topics on Simulations and Experimental Data. These chapters present new points of view about methods and tools used in Hydrodynamics

Application of wave mechanics theory to fluid dynamics problems: Fundamentals

The application of the basic formalistic elements of wave mechanics theory is discussed. The theory is used to describe the physical phenomena on the microscopic level, the fluid dynamics of gases and liquids, and the analysis of physical phenomena on the macroscopic (visually observable) level. The practical advantages of relating the two fields of wave mechanics and fluid mechanics through the use of the Schroedinger equation constitute the approach to this relationship. Some of the subjects include: (1) fundamental aspects of wave mechanics theory, (2) laminarity of flow, (3) velocity potential, (4) disturbances in fluids, (5) introductory elements of the bifurcation theory, and (6) physiological aspects in fluid dynamics