Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level

Perspectives on hypersonic viscous and nonequilibrium flow research

An attempt is made to reflect on current focuses in certain areas of hypersonic flow research by examining recent works and their issues. Aspects of viscous interaction, flow instability, and nonequilibrium aerothermodynamics pertaining to theoretical interest are focused upon. The field is a diverse one, and many exciting works may have either escaped the writer's notice or been abandoned for the sake of space. Students of hypersonic viscous flow must face the transition problems towards the two opposite ends of the Reynolds or Knudsen number range, which represents two regimes where unresolved fluid/gas dynamic problems abound. Central to the hypersonic flow studies is high-temperature physical gas dynamics; here, a number of issues on modelling the intermolecular potentials and inelastic collisions remain the obstacles to quantitative predictions. Research in combustion and scramjet propulsion will certainly be benefitted by advances in turbulent mixing and new computational fluid dynamics (CFD) strategies on multi-scaled complex reactions. Even for the sake of theoretical development, the lack of pertinent experimental data in the right energy and density ranges is believed to be among the major obstacles to progress in aerothermodynamic research for hypersonic flight. To enable laboratory simulation of nonequilibrium effects anticipated for transatmospheric flight, facilities capable of generating high enthalpy flow at density levels higher than in existing laboratories are needed (Hornung 1988). A new free-piston shock tunnel capable of realizing a test-section stagnation temperature of 10(exp 5) at Reynolds number 50 x 10(exp 6)/cm is being completed and preliminary tests has begun (H. Hornung et al. 1992). Another laboratory study worthy of note as well as theoretical support is the nonequilibrium flow experiment of iodine vapor which has low activation energies for vibrational excitation and dissociation, and can be studied in a laboratory with modest resources (Pham-Van-Diep et al. 1992)

Constructive Methods of Invariant Manifolds for Kinetic Problems

We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability. A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems. The systematic use of thermodynamics structures and of the quasi--chemical representation allow to construct approximations which are in concordance with physical restrictions. We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing procedures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy significantly. The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; invariant grids for a two-dimensional catalytic reaction and a four-dimensional oxidation reaction (six species, two balances); universal continuous media description of dilute polymeric solution; the limits of macroscopic description for polymer molecules, etc

THE LATTICE BOLTZMANN METHOD FOR FLOWS WITH SLIP AND NO-SLIP BOUNDARIES

This thesis assesses and extends a modern method to study the physics of simple and complex flows by using the lattice Boltzmann method (LBM). With the moment-based boundary conditions, different problems with no-slip and slip boundaries are simulated. The moment method is based on the specification of the appropriate hydrodynamic moments of LBM. Throughout this thesis, distinct collision operators of D2Q9 LBM are presented and examined; the models include the Bhatnager-Gross-Krook (BGK), multiple relaxation time (MRT) and a special case of the last model which is two relaxation times (TRT-LBM). Simple numerical simulations are given and the LBM proved its accuracy when it is compared with other numerical methods. The accuracy of the LBM with the no-slip and slip moment-based boundary conditions is examined numerically by studying the dipole wall collision flow. The two relaxation times lattice Boltzmann model is used to simulate this flow and the results are compared with other numerical methods. Our implementation shows excellent agreement with other numerical results. The vorticity generation on the wall shows interesting behaviour after the dipole collides with no-slip wall. The angle of the incidence effects the behaviour of the dipole after the wall collision, the dissipation of the energy and the growth of the enstrophy. Throughout this thesis the impact of the slip length and Reynolds number on the dipole wall collision is studied. By applying the Navier-slip condition with moment boundary conditions the behaviour of the flow changes and the dissipation of the energy is affected by slip length and the peaks of the enstrophy decreases with higher slip lengths. The dissipation of the energy and its relation to the enstrophy over dipole wall collision are also investigated for different types of boundaries and angles. The theoretical and the numerical investigation shows that the presence of the wall modifies this relation. Moreover, the dissipation of the energy in the absence and the existence of the viscosity effect are studied. Finally, an analysis is done of the stress field of the LBM by using the same boundary conditions for simple flow.the Higher committee for Education Development in Iraq (HCEDiraq

Lectures in Applied Mathematics. Volume 7 - Space Mathematics, Part 3

Fluid mechanic and radiation pressure effects on satellite orbits and orbital decay, and space rendezvous maneuvers - mathematics conferenc

A Kinetic Theory Description of Rarefied Gas Flows

An approach to the kinetic theory of gas flows is developed which starts with Maxwell's original integral equations of transfer, rather than with the Maxwell-Boltzmann equation for the velocity distribution function itself. In this procedure the Maxwell-Boltzmann equation is satisfied in a certain average sense, rather than at every point. The advantage of this method is that relatively simple distribution functions are utilized which contain a small number of unknown functions to be determined by applying the conservation laws, plus several additional higher moments. For simplicity a "two-stream Maxwellian" is employed, which is a natural extension and generalization of Mott-Smith's function for a normal shock, but differs from it in certain essential respects. As an illustration, the method is applied to linearized plane Couette flow and Rayleigh's problem. Reasonable results are obtained for macroscopic quantities such as mean velocity and shear stress over the whole range of densities from free-molecule flow to the Navier-Stokes regime. This technique is now being applied to some typical non-linear rarefied gas flows

Numerical Modelling of Transient and Droplet Transport for Pulsed Pressure - Chemical Vapour Deposition (PP-CVD) Process

The objective of this thesis is to develop an easy-to-use and computationally economical numerical tool to investigate the flow field in the Pulsed Pressure Chemical Vapour Deposition (PP-CVD) reactor. The PP-CVD process is a novel thin film deposition technique with some advantages over traditional CVD methods. The numerical modelling of the PP-CVD flow field is carried out using the Quiet Direct Simulation (QDS) method, which is a flux-based kinetic-theory approach. Two approaches are considered for the flux reconstruction, which are the true directional manner and the directional splitting method. Both the true directional and the directional decoupled QDS codes are validated against various numerical methods which include EFM, direct simulation, Riemann solver and the Godunov method. Both two dimensional and axisymmetric test problems are considered. Simulations are conducted to investigate the PP-CVD reactor flow field at 1 Pa and 1 kPa reactor base pressures. A droplet flash evaporation model is presented to model the evaporation and transport of the liquid droplets injected. The solution of the droplet flash evaporation model is used as the inlet conditions for the QDS gas phase solver. The droplet model is found to be able to provide pressure rise in the reactor at the predicted rate. A series of parametric studies are conducted for the PP-CVD process. The numerical study confirms the hypothesis that the flow field uniformity is insensitive to the reactor geometry. However, a sufficient distance from the injection inlet is required to allow the injected precursor solution to diffuse uniformly before reaching the substrate. It is also recommended that placement of the substrate at the reactor’s centre axis should be avoided

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

Auto-diffusion de particules dans un ecoulement cisaille : des interactions hydrodynamiques aux effets collisionnels

Ce travail aborde, à l'aide de simulations Lagrangiennes, la description du comportement rhéophysique d'une suspension de particules solides sphériques en écoulement cisaillé. Les effets de l’inertie du fluide, de l'agitation Brownienne et de la gravité sont négligés. Les suspensions étudiées sont classées en deux grandes familles, en fonction de l'inertie de la phase dispersée caractérisée par le nombre de Stokes. A très petit nombre de Stokes, les suspensions sont de type liquide-solide où le fluide est très visqueux. Le modèle de simulation "Force Coupling Method" est utilisée pour simuler les interactions hydrodynamiques qui contrôlent la dynamique de ces suspensions. Cette méthode se base sur un développement multipolaire de la perturbation de vitesse induite par la présence des particules dans le fluide porteur. L'évolution de quantités macroscopiques en fonction de la fraction volumique du solide [φ=1-20%] est analysée dans des suspensions monodisperses. Les résultats (fluctuations de vitesse, auto-diffusion, auto-corrélation des vitesses et distribution spatiale de paires de particules…) confortent les tendances observées dans plusieurs études de la littérature. Nous montrons que l’agitation des particules induit un comportement diffusif dont l’intensité est une fonction croissante de la concentration. Le niveau d’agitation mais aussi le temps de diffusion augmentent lorsque les interactions multi-corps contrôlent la dynamique de la suspension. Les effets de lubrification associés à des particules proches du contact sont résolus précisément. Ceci permet d'utiliser la FCM pour simuler des suspensions de concentration plus élevée (allant jusqu'à 35%), et de quantifier leur viscosité effective. Le modèle de simulation est étendu aux cas de suspensions bidisperses. L'impact de la variation du rapport de taille ou de concentration sur les statistiques (des deux espèces) est examiné pour une fraction volumique constante de la phase dispersée. Pour un rapport de concentration fixe, nous avons trouvé qu'un rapport de taille croissant entraîne une augmentation (resp. diminution) du niveau de fluctuation des petites (resp. grosses) particules. Quand le rapport de taille et la concentration totale sont fixes, l'augmentation du nombre de grosses particules entraîne l'augmentation du taux de fluctuation et de la diffusion des deux espèces. Les suspensions caractérisées par un nombre de Stokes modéré ou grand sont en général de type gaz-solide. Un modèle de simulation basé sur l’intégration des trajectoires de particules assimilées à des sphères dures est utilisé pour simuler la dynamique de la suspension. Le mouvement des particules est uniquement contrôlé par les collisions et par la force de traînée sur une particule isolée. Les simulations montrent que les propriétés de la suspension dépendent fortement de l'inertie des particules et de la concentration. La variation du nombre de Stokes de 1 à 10 induit une augmentation de l'agitation des particules de trois ordres de grandeur, et une évolution de la distribution de vitesse d'une forme très piquée (proche d’un Dirac) à une forme Maxwellienne. Les résultats numériques sont confrontés aux prédictions de deux modèles issus de la théorie cinétique des milieux granulaires adaptés aux nombres de Stokes modérés: la fonction Dirac (resp. Maxwellienne déviée) est utilisée pour décrire les suspensions faiblement (resp. fortement) agitées. Une nouvelle théorie pour déterminer les coefficients du tenseur d'auto-diffusion Lagrangienne est développée et validée avec les résultats des simulations. Les coefficients de diffusion et la viscosité de la phase solide sont également confrontés aux modèles théoriques utilisés pour la prédiction d’écoulements complexes. L'effet de l'inélasticité sur les quantités statistiques est également discuté. La conclusion de ce document fait la synthèse de tous ces résultats en proposant une approche unifiée de l’évolution de la viscosité effective du mélange fluide/particules. Une modification de la méthode FCM est proposée pour modéliser simultanément l'inertie des particules et les interactions hydrodynamiques

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described