Дисертації з теми "Boltzmann Scheme"

The Helicon Plasma Thruster is a space propulsion system composed of a Helicon plasma source, with a properly designed magnetic nozzle. It is a very attractive concept due to the expected range of specific impulse and thrust-to-weight ratio, the scalability of the design, and the simplicity of construction; moreover, it is electrode-less and has no moving parts, and hence it can be expected to have an extended lifetime. Although Helicon plasma sources have been used for decades in laboratories for producing high density plasmas, they are not fully understood yet. In fact, despite the simple geometry, a whole range of physical phenomena take place in the source: atomic physics, fluid kinetics, electrostatics and electromagnetism must be taken into account, and they are strongly interconnected to each other. As a result, a Helicon plasma source is a very complex system to model, and to the author's knowledge, there is no reliable set of predictive tools for the design and optimization of such a source. This thesis work focusses on Helicon plasma sources for space propulsion applications, and more precisely, it studies the configuration proposed in the HPH.com project (Helicon Plasma Hydrazine. COmbined Micro), in the Seventh Framework Programme of the European Union. The plasma source under study is small (approximately 15 cm in length), and the thruster is expected to provide just about 2 mN of thrust with 50 W of electric power consumption; as such, it is intended for use in the attitude control system of micro-satellites. In order to optimize the computational resources available, a hybrid model is preferred to a monolithic model. In the former approach, the physical system is decomposed into subsystems, and each of these is simulated by a dedicated submodel, which (ideally) would use the most appropriate level of detail. As no extensive theory on hybrid modeling exists, part of this thesis is dedicated to the investigation of the 'best way' to construct a hybrid model. An original approach is proposed, which is based on constructing submodels which rely on many different levels of detail, instead of just 'the best one'. This approach is natural, and it is believed to provide flexibility, robustness and physical insight. According to the ideas above, a series of increasingly complicated models have been developed. Since the detailed and self-consistent simulation of the whole plasma source falls way beyond the scope of a single PhD thesis, most of the effort has been put into understanding the coupled dynamics of electrons and neutrals, which has not been throughly investigated yet. In order to assess the ionization efficiency of the source, 0D and 1D analytic models of the neutral depletion process are presented. The comparison of the two models show the regimes where a higher level of detail is necessary, and the conditions under which the 1D model asymptotically recovers the 0D solution. Subsequently, the neutral dynamics is coupled to the electron dynamics, by means of a semi-analytic 0D model which assumes Maxwellian electrons. The solution obtained gives a first estimate of the plasma parameters in the source, so that proper ranges for the characteristic lengths and time-scales of the various physical processes are calculated. Those results are essential to the preliminary design of a bounce averaged electron kinetic model, which is still 0D in space, but which calculates the electron energy distribution function self-consistently with the various processes. After that, the 0D-1V electron kinetic model is designed in detail, including the effect of electromagnetic heating and various collisional processes. Accelerated convergence to the self-consistent steady-state is obtained by means of time-scale separation, fixed-point iteration, implicit time integration with Newton's solver and variable time-step, and a reduced auxiliary model. The neutral density in the source is obtained from the aforementioned 1D analytic model. When the necessity for a detailed kinetic model for neutrals was realized, a 3D-3V semi-Lagrangian Convected Scheme was developed, which solves the Boltzmann equation in six-dimensional phase-space, plus time. Being the first implementation of the Convected Scheme to be 3D in space, several computational problems arose, and new solutions had to be found. For this reason, a considerable part of this thesis work had to deal with a new method for implementing diffuse boundary conditions, a new injector model, a new mass-, momentum- and energy-conserving collision operator for the Bhatnagar-Gross-Krook model, and a new angular mesh. Moreover, a novel third-order positivity-preserving remapping method with low numerical diffusion was developed.
Un propulsore al plasma di tipo Helicon è un sistema di propulsione spaziale composto da una sorgente Helicon e da un ugello magnetico appositamente progettato. Tale tipo di propulsore attrae molto interesse per via dell'intervallo atteso per l'impulso specifico ed il rapporto spinta-su-peso, nonché per la scalabilità del concetto e la semplicità costruttiva. Inoltre, un propulsore Helicon è privo di elettrodi e di parti in movimento, dunque ci si aspetta una lunga durata di funzionamento. Malgrado le sorgenti Helicon siamo stata impiegate per decenni per produrre plasmi ad elevata densità, il loro funzionamento non è ancora del tutto compreso. Infatti, sebbene la geometria sia semplice, una vasta gamma di fenomeni fisici convivono all'interno della sorgente: vanno presi in considerazione la fisica atomica, la cinetica dei fluidi, l'elettrostatica e l'elettromagnetismo, e tutti questi sono strettamente interdipendenti. La sorgente Helicon è dunque un sistema molto complesso da modellare e, a conoscenza dell'autore, non è ancora stato sviluppato un sistema di strumenti per la progettazione e l'ottimizzazione di tale tipo di sorgente. Il lavoro svolto all'interno di questa tesi si concentra sullo studio di una sorgente Helicon da applicarsi nella propulsione spaziale e, più precisamente, sullo studio della configurazione proposta del progetto HPH.com (Helicon Plasma Hydrazine. COmbined Micro), nel settimo Framework Programme dell'Unione Europea. La sorgente di plasma considerata è di piccole dimensioni (circa 15 cm in lunghezza), e ci si aspetta che il propulsore fornisca circa 2 mN di spinta a fronte di 50 W di potenza elettrica fornita. Con queste caratteristiche, il propulsore è pensato per l'utilizzo nel controllo d'assetto di micro-satelliti. Con il fine di ottimizzare le risorse computazionali a disposizione, un modello ibrido risulta preferibile rispetto ad un modello monolitico. Secondo il primo approccio, il sistema fisico è decomposto in sotto-sistemi, ed ognuno di essi è simulato da un sotto-modello dedicato, che (idealmente) dovrebbe utilizzare un livello di dettaglio appropriato. Non esiste alcuna teoria esaustiva su come sviluppare modelli ibridi, e parte di questa tesi è dedicata ad investigare la 'via migliore' di costruire un modello ibrido. Viene qui proposto un approccio originale, basato sulla costruzione di sotto-modelli che si affidano a diversi livelli di dettaglio, invece che semplicemente sul miglior modello possibile. Tale approccio è naturale, e ci si aspetta che sia flessibile, robusto e che fornisca una migliore comprensione del fenomeno fisico. Seguendo tale metodologia, è stata sviluppata una serie di modelli via via più complessi. Poiché una simulazione dettagliata ed autoconsistente dell'intera sorgente non può essere completata in una singola tesi di Dottorato, la maggior parte di questo lavoro si concentra sulla comprensione della dinamica accoppiata di elettroni e neutri, che in questo sistema non è mai stata approfonditamente investigata. Per valutare l'efficienza di ionizzazione all'interno della sorgente, modelli analitici 0D e 1D del processo di deplezione dei neutri sono presentati. Il confronto dei due modelli suggerisce i regimi in cui è necessario un livello di dettaglio più elevato, e mostra le condizioni in cui il modello 1D converge asintoticamente alla soluzione 0D. Successivamente, la dinamica dei neutri è accoppiata alla dinamica degli elettroni, per mezzo di un modello semi-analitico 0D che assume che gli elettroni abbiano una distribuzione Maxwelliana. La soluzione ottenuta fornisce valori preliminari per i parametri di plasma all’interno della sorgente, dai quali è possibile valutare un intervallo di lunghezze caratteristiche e di scale temporali che caratterizzano i diversi processi fisici. Questi risultati sono essenziali per la progettazione preliminare di un modello cinetico per gli elettroni mediato su un elevato numero di oscillazioni all'interno della sorgente ('bounce averaged'); tale modello rimane 0D nello spazio, ma esso calcola la distribuzione energetica degli elettroni in modo autoconsistente con i vari processi. Successivamente, un modello 0D-1V cinetico per gli elettroni è stato progettato nel dettaglio, includendo l’effetto del riscaldamento elettromagnetico e dei diversi processi collisionali. La convergenza a regime stazionario è stata accelerata attraverso la separazione delle diverse scale temporali, iterazioni di punto fisso, integrazione implicita con un solutore di Newton a passo temporale variabile, ed un modello ausiliario ridotto. La densità dei neutri nella sorgente è ottenuta dal modello analitico 1D sopra citato. Quando si è ritenuto necessario un modello dettagliato dei neutri, è stato sviluppato un modello cinetico 3D-3V, che impiega un solutore semi-Lagrangiano chiamato Convected Scheme. Questo modello risolve l'equazione di Boltzmann nello spazio nelle fasi a sei dimensioni, più il tempo. Trattandosi della prima implementazione del Convective Scheme in tre dimensioni spaziali, si sono incontrati diversi problemi di natura computazionale, per i quali è stato necessario trovare soluzioni innovative. Per questa ragione, una parte consistente di questo lavoro di tesi è stata dedicata ad implementare nuove condizioni al contorno diffusive, un nuovo modello di iniettore, una nuova mesh angolare ed un innovativo operatore collisionale per il modello di Bhatnagar-Gross-Krook che conservi esattamente massa, quantità di moto ed energia. Inoltre, è stato sviluppato un metodo innovativo di rimappatura, accurato al terzo ordine, che preserva la positività della soluzione e possiede bassa diffusione numerica.

In this Doktorarbeit the Lattice Boltzmann scheme, a heuristic method for the simulation of flows in complicated boundaries, is investigated. Its theory is renewed by emphasizing the entropy maximization principle, and new means for the modelling of geometries (including moving boundaries) and the visual representation of evoluting flows are presented. An object oriented implemen- tation is given with communication between objects realized by an interpreter object and communication from outside realized via interprocess communica- tion. Within the new theoretical apprach the applicability of existing Lattice Boltzmann schemes to model thermal flows for arbitrary temperatures is reex- amined
In dieser Doktorarbeit wird das Gitter-Boltzmann-Schema, eine heuristische Methode fuer die Simulation von Stroemungen innerhalb komplexer Raender, untersucht. Die zugrundeliegende Theorie wird unter neuen Gesichtspunkten, insbesondere dem Prinzip der Entropiemaximierung, betrachtet. Des weiteren werden neuartige Methoden fuer die Modellierung der Geometrie (einschl. beweglicher Raender) und der visuellen Darstellung aufgezeigt. Eine objektorientierte Implementierung wird vorgestellt, wobei die Kommunikation zwischen den Objekten über Interpreter-Objekte und die Kommunikation mit der Aussenwelt ueber Interprozess-Kommunikation gehandhabt wird. Mit dem neuen theoretischen Ansatz wird die Gueltigkeit bestehender Gitter-Boltzmann-Schemata fuer die Anwendung auf Stroemungen mit nicht konstanter Temperatur untersucht

Dans cette thèse, nous nous sommes intéressés à des écoulements complexes où les régimes hydrodynamique et raréfiés coexistent. On retrouve ce type d'écoulements dans des applications industrielles comme les pompes à vide ou encore les rentrées de capsules spatiales dans l'atmosphère, lorsque la distance entre les molécules de gaz devient si grande que le comportement microscopique des molécules doit être pris en compte. Pour ce faire, nous étudions 2 modèles de l'équation de Boltzmann, le modèle BGK et le modèle ES-BGK. Dans un premier temps, nous développons une nouvelle condition au bord permettant une transition continue de la solution du régime raréfié vers le régime hydrodynamique. Cette nouvelle condition permettant de préserver l'asymptotique vers les équations d'Euler compressible est ensuite incluse dans une méthode de frontière immergée pour traiter, à une précision raisonnable (ordre 2), le cas de solides immergés dans un écoulement, sur grilles cartésiennes. L'utilisation de grillescartésiennes permet une parallélisation aisée du code de simulation numérique afin d'obtenir une réduction considérable du temps de calcul, un des principaux inconvénients des modèles cinétiques. Par la suite, une approche dites aux grilles locales en vitesses est présentée réduisant également le temps de calcul de manière importante (jusqu'à 80%). Des simulations 3D sont également présentées montrant l'efficacité des méthodes. Enfin, le transport passive de particules solides dans un écoulement raréfié est étudié avec l'introduction d'un modèle de type Vlasov couplé au modèle cinétique. Grâce à une résolution basée sur des méthodes de remaillage, la pollution de dispositif optiques embarqués sur des satellites dues à des particules issues de la combustion incomplète dans les moteurs contrôlant d'altitude est étudiée
This work is devoted to the study of complex flows where hydrodynamic and rarefled regimes coexist. This kind of flows are found in vacuum pumps or hypersonic re-entries of space vehicles where the distance between gas molecules is so large that their microscopicbehaviour differ from the average behaviour of the flow and has be taken into account. We then consider two modelsof the Boltzmann equation viable for such flows: the BGK model dans the ES-BGK model.We first devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefled regime to the hydrodynamic regime. We then describe how this boundary condition (and boundary conditions in general) can be enforced with second order accuracy on an immersed body on Cartesian grids preserving the asymptotic limit towards compressible Euler equations. We exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models. A new approach considering local velocity grids is then presented showing important gain on the computational time (up to 80%). 3D simulations are also presented showing the efficiency of the methods. Finally, solid particle transport in a rarefied flow is studied. The kinetic model is coupled with a Vlasov-type equation modeling the passive particle transport solved with a method based on remeshing processes. As application, we investigate the realistic test case of the pollution of optical devices carried by satellites due to incompletely burned particles coming from the altitude control thrusters

Cette thèse introduit et étudie une nouvelle classe de schémas de Boltzmann sur réseau appelés schémas à vitesse relative. Les schémas de Boltzmann sur réseau visent à approcher des problèmes de nature macroscopique en mimant la dynamique microscopique d’équations cinétiques du type Boltzmann. L’algorithme calcule des distributions de particules évoluant au travers de deux phases de transport et de relaxation, les particules se déplaçant en les noeuds d’un réseau cartésien en espace. Les schémas de Boltzmann à plusieurs temps de relaxation (ou schéma MRT de d’Humières), dont la relaxation im- plique un ensemble de moments combinaison linéaire polynomiale des distributions, constituent le cadre initial de la thèse. Les schémas à vitesse relative sont une extension de ces schémas de d’Humières. Ils sont inspirés du schéma cascade de Geier apportant davantage de stabilité que les schémas de d’Hu- mières pour des régimes peu visqueux des équations de Navier-Stokes. La différence avec ces schémas se situe au niveau de la relaxation : elle utilise un ensemble de moments relatifs à un paramètre champ de vitesse fonction du temps et de l’espace. Cette différence se matérialise par une matrice de tran- sition des moments fixes (les schémas de d’Humières correspondent à un paramètre champ de vitesse nul) aux moments mobiles. La structure algébrique de cette matrice est étudiée. Le schéma cascade est ensuite traduit comme un schéma à vitesse relative pour un nouvel ensemble de polynômes définissant les moments. L’étude de la consistance des schémas à vitesse relative par la méthode des équations équivalentes est un point central de la thèse. Les équations limites pour un nombre arbitraire de dimen- sions et de vitesses sont dérivées et illustrées sur des exemples tels que le D2Q9 pour les équations de Navier-Stokes. Ces équations équivalentes sont également un outil pour prédire la stabilité des schémas grâce à l’analyse des termes de diffusion et dispersion. La dernière partie traite de la stabilité suivant le choix du paramètre champ de vitesse. Nous sommes particulièrement intéressés en les deux choix de paramètre nul (d’Humières) et la vitesse du fluide (cascade). Le schéma D2Q9 pour les équations de Navier-Stokes est étudié numériquement par une méthode de Von Neumann puis appuyé sur des cas tests non linéaires. La stabilité des schémas relatifs à la vitesse du fluide est dépendante du choix des polynômes définissant les moments. L’amélioration la plus notable se produit si les polynômes du schéma cascade sont choisis. Nous étudions enfin les stabilités théorique et numérique d’un schéma bidimensionnel minimal. Le contexte physique est la simulation d’une équation d’advection diffusion linéaire. Le choix de la vitesse d’advection comme paramètre champ de vitesse annule certains termes de dispersion des équations équivalentes contrairement aux schémas de d’Humières. Ceci se traduit par un meilleur comportement en termes de stabilité pour de grandes vitesses, appuyé théoriquement à l’aide d’une notion de stabilité à poids
In this PhD thesis, a new class of lattice Boltzmann schemes called relative velocity schemes is introduced and studied. The purpose of lattice Boltzmann schemes is to approximate problems of macroscopic nature using the microscopic dynamic of Boltzmann type kinetic equations. They compute particle distributions through two phases of transport and relaxation, the particles moving on the nodes of a cartesian lattice. The multiple relaxation times schemes---MRT of d'Humières---, whose relaxation uses a set of moments, linear combinations of the particle distributions, constitutes the initial framework of the thesis. The relative velocity schemes extend the MRT d'Humières schemes. They originate from the cascaded automaton of Geier which provides more stability for the low viscosity regime of the Navier-Stokes equations. Their difference with the d'Humières schemes is carried by the relaxation : a set of moments relative to a velocity field parameter function of space and time is used. This difference is represented by a shifting matrix sending the fixed moments---The d'Humières schemes are associated with a zero velocity field parameter---On the relative moments. The algebraic structure of this matrix is studied. The cascaded automaton is then interpreted as a relative velocity scheme for a new set of polynomials defining the moments. The consistency study of the relative velocity schemes with the equivalent equations method is a keypoint of the thesis. These equations are derived for an arbitrary number of dimensions and velocities. They are then illustrated on examples like the D2Q9 scheme for the Navier-Stokes equations. These equivalent equations are also a tool to predict the stability behaviour of the schemes by analysing their diffusion and dispersion terms. In a last part, the stability according to the velocity field parameter is studied. Two cases especially interest us : a parameter equal to zero---D'Humières schemes---And equal to the fluid velocity---Cascaded automaton. The D2Q9 scheme for the Navier-Stokes equations is numerically studied with a linear Von Neumann analysis and some non linear test cases. The stability of the relative velocity schemes depends on the choice of the polynomials defining the moments. The most important improvement occurs if the polynomials of the cascaded automaton are chosen. We finally study the theoretical and numerical stability of a minimal bidimensional scheme for a linear advection diffusion equation. If the velocity field parameter is chosen equal to the advection velocity, some dispersion terms of the equivalent equations vanish unlike the d'Humières scheme. This implies a better stability behaviour for high velocities, characterized thanks to theoretical weighted stability notion

Lors de la rentrée atmosphérique, l’écoulement raréfié de l’air autour de l’objet rentrant est régi par un modèle cinétique dérivé de l’équation de Boltzmann ; celui-ci décrit l’évolution d’une fonction de distribution des particules de gaz dans l’espace des phases, de dimension 6 dans le cas général. La simulation numérique déterministe de cet écoulement requiert donc le traitement d’une quantité considérable de données, soit un espace mémoire et un temps de calcul importants. Nous étudions dans ce travail différents moyens de réduire le coût de ces calculs. La première approche est une méthode permettant d’optimiser la taille de la grille de vitesses discrètes employée dans le calcul par une prédiction de l’allure des fonctions de distribution dans l’espace des vitesses, en supposant un faible déséquilibre thermodynamique du gaz. La seconde approche consiste à essayer d’exploiter les propriétés de préservation asymptotique des schémas Galerkin Discontinu, déjà établies dans le cadre du transport linéaire des neutrons, qui permettent de tenir compte des effets de la couche limite cinétique sans que celle-ci soit résolue par le maillage, alors que les méthodes classiques (comme les Volumes Finis) imposent l’utilisation de maillages très raffinés en zone de proche paroi. Dans une dernière partie, nous comparons les performances respectives de ces schémas Galerkin Discontinu et de quelques schémas Volumes Finis, appliqués au modèle BGK sur un cas simple, en étudiant en particulier leur comportement près des parois et les conditions aux limites numériques
During the atmospheric re-entry of a space engine, the rarefied air flow around the body is determined by a kinetic model derived from the Boltzmann equation, which describes the evolution of a distribution function of gas molecules in the phase space, this means a 6-dimensional space in the general case. Consequently, a deterministic numerical simulation of this flow requires large computational ressources, both in memory storage and CPU time. The aim of this work is to reduce those ressources, using two different approaches. The first one is a method allowing to optimize the size of the discrete velocity grid used for the computation by a prediction of the shape of the distributions in the velocity space, assuming that the gas is close to thermodynamic equilibrium. The second approach is an attempt to use the asymptotic preservation properties of Discontinuous Galerkin schemes, already established for neutron transport, which allow to take into account the effects of kinetic boundary layers even if they are not resolved by the mesh, while classical methods (such as Finite Volumes) require very refined meshes along the direction normal to the walls. In a last part, we compare the performances of these Discontinuous Galerkin schemes with some classical Finite Volumes schemes, applied to the BGK equation in a simple case, and pay particular attention to their near-wall behavior and numerical boundary conditions

Les phénomènes de transport en milieux poreux sont étudiés depuis près de deux siècles, cependant les travaux concernant les milieux fortement poreux sont encore relativement peu nombreux. Les modèles couramment utilisés pour les poreux classiques (lits de grains par exemple) sont peu applicables pour les milieux fortement poreux (les mousses par exemple), un certain nombre d’études ont été entreprises pour combler ce manque. Néanmoins, les résultats expérimentaux et numériques caractérisant les pertes de charge dans les mousses sont fortement dispersés. Du fait des progrès de l’imagerie 3D, une tendance émergente est la détermination des paramètres des lois d’écoulement à partir de simulations directes sur des géométries reconstruites. Nous présentons ici l’utilisation d’une nouvelle approche cinétique pour résoudre localement les équations de Navier-Stokes et déterminer les propriétés d’écoulement (perméabilité, dispersion, ...)
A novel kinetic scheme satisfying an entropy condition is developed, tested and implemented for the simulation of practical problems. The construction of this new entropic scheme is presented. A classical hyperbolic system is approximated by a discrete velocity vector kinetic scheme (with the simplified BGK collisional operator), but applied to an inviscid compressible gas dynamics system with a small Mach number parameter, according to the approach of Carfora and Natalini (2008). The numerical viscosity is controlled, and tends to the physical viscosity of the Navier-Stokes system. The proposed numerical scheme is analyzed and formulated as an explicit finite volume flux vector splitting (FVS) scheme that is very easy to implement. It is close in spirit to Lattice Boltzmann schemes, but it has the advantage to satisfy a discrete entropy inequality under a CFL condition and a subcharacteristic stability condition involving a cell Reynolds number. The new scheme is proved to be second-order accurate in space. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical benchmark tests. Some physical problems have been studied in order to show the usefulness of both schemes. The LB code was successfully used to determine the longitudinal dispersion of metallic foams, with the use of a novel indicator. The entropic code was used to determine the permeability tensor of various porous media, from the Fontainebleau sandstone (low porosity) to a redwood tree sample (high porosity). These results are pretty accurate. Finally, the entropic framework is applied to the advection-diffusion equation as a passive scalar

Lattice Boltzmann Method has been quite successful for incompressible flows. Its extension for compressible (especially supersonic and hypersonic) flows has attracted lot of attention in recent time. There have been some successful attempts but nearly all of them have either resulted in complex or expensive equilibrium function distributions or in extra energy levels. Thus, an efficient Lattice Boltzmann Method for compressible fluid flows is still a research idea worth pursuing for. In this thesis, a new Lattice Boltzmann Method has been developed for compressible flows, by using the concept of a relaxation system, which is traditionally used as semilinear alternative for non-linear hypebolic systems in CFD. In the relaxation system originally introduced by Jin and Xin (1995), the non-linear flux in a hyperbolic conservation law is replaced by a new variable, together with a relaxation equation for this new variable augmented by a relaxation term in which it relaxes to the original nonlinear flux, in the limit of a vanishing relaxation parameter. The advantage is that instead of one non-linear hyperbolic equation, two linear hyperbolic equations need to be solved, together with a non-linear relaxation term. Based on the interpretation of Natalini (1998) of a relaxation system as a discrete velocity Boltzmann equation, with a new isotropic relaxation system as the basic building block, a Lattice Boltzmann Method is introduced for solving the equations of inviscid compressible flows. Since the associated equilibrium distribution functions of the relaxation system are not based on a low Mach number expansion, this method is not restricted to the incompressible limit. Free slip boundary condition is introduced with this new relaxation system based Lattice Boltzmann method framework. The same scheme is then extended for curved boundaries using the ghost cell method. This new Lattice Boltzmann Relaxation Scheme is successfully tested on various bench-mark test cases for solving the equations of compressible flows such as shock tube problem in 1-D and in 2-D the test cases involving supersonic flow over a forward-facing step, supersonic oblique shock reflection from a flat plate, supersonic and hypersonic flows past half-cylinder.

碩士
國立臺灣大學
應用力學研究所
102
An accurate and direct algorithm for solving the classical Boltzmann equation and the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. In time domain, we use asymptotic-preserving method for solving two-dimensional Riemann problem by the classical Boltzmann equation and the semiclassical Boltzmann equation with very small relaxation time. After using asymptotic-preserving, we use fourth-order Runge-Kutta method to discrete time domain. In space domain, we use fifth-order weighted essentially non-oscillatory scheme to evolve the flux term. The discrete ordinate method is applied to remove the microscopic velocity dependency of the distribution function that renders the Boltzmann BGK equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. Computational examples of two-dimensional Riemann problems for rarefied gas flows at very small relaxation time are presented. By using WENO scheme, the results show good resolution in capturing the main flow features while using grids with few good points.

碩士
國立臺灣大學
應用力學研究所
103
This study is aimed at solving the semi-classical Boltzmann-BGK equation to figure out the characteristics of gas flow, especially for rarefied gases. The coupling transformation of both the unsteady one dimensional Sod shock tube and the unsteady two dimensional shock wave impinging upon a square cylinder were investigated numerically. In addition, in order to reduce the computational amount, an appropriated mechanism is applied in this study. To deal with the discontinuity existing in problem, the solution of the semi-classical Boltzmann-BGK equation, namely the velocity distribution function, was divided into two parts with the help of a smoothed dirac delta function. Modified semi-classical Boltzmann-BGK equations were derived and solved for them over the whole computational domain then; the sum of the two parts gives the velocity distribution function in the buffer region. Consequently no more interface conditions need considering and the simulation is largely simplified. Three types of the smoothing functions – linear, cosine, and hypertangent, were tested and the conversation effect in buffer zone were examined in this thesis. As far as numerical discretization is concerned, the discrete coordinate method is employed for the velocity space and a high resolution scheme, either Total Variation Diminishing (TVD) or Weighted Essentially Non Oscillatory (WENO), was utilized for the physical space. Finally the asymptotic preserving scheme is taken in this study as well, which makes the relaxation time independent of collision term of semi-classical Boltzmann-BGK equation, resulting in a significant reduction in the computational amount. Finally the flow fields of quantum gas described by Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein statistics were all simulated. From the test examples of one dimensional unsteady Sod shock wave tube and two dimensional unsteady shock wave impinging upon a square cylinder, the investigation shows a use of a smoothing function and a high resolution scheme combined with the asymptotic preserving scheme technique does help reducing the computational amount.

碩士
國立臺灣大學
應用力學研究所
100
A Multi Relaxation Time Semiclassical Lattice Boltzmann Method based on the Uehling-Uhlenbeck Boltzmann-BGK equation （Uehling-Uhlenbeck Boltzmann Bhatnagar-Gross-Krook Equation）and Multi Relaxation Time Lattice Boltzmann Method（MRT-LBM）is presented. The method is directly derived by projecting the kinetic governing equation onto the tensor Hermite polynomials and various hydrodynamic approximation orders can be achieved. Simulations of the lid　driven cavity flows based on D2Q9 lattice model for several Reynolds numbers and three different particles that obey Bose-Einstein and Fermi-Dirac and Maxwell-Boltzmann statistics are shown to illustrate the method. The results indicate distinct characteristics of the effects of quantum statistics.

Title: Non-newtonian fluid flow simulation using lattice Boltzmann method Author: Bc. Pavel Kuriščák Department: Mathematical Institute, Charles University Supervisor: RNDr. Ing. Jaroslav Hron Ph.D. Supervisor's e-mail address: Jaroslav.Hron@mff.cuni.cz Abstract: The aim of this thesis is to find and estabilish a modification to the Lattice Boltzmann Method, allowing it to simulate non-newtonian behaviour of fluids. In the theoretical part of thesis, there is introduced a derivation, based on the work of [22], that is capable of arriving to macroscopical Navier-Stokes equa- tions completely a priori from the Boltzmann equation, utilizing the Hermite basis expansion. This derivation is afterwards applied to the method suggested by [11], that uses the changed equilibrium distribution to fine-tune the local fluid viscosity according to the non-newtonian model. In the last part of thesis, this method is implemented in the form of lattice kinetic scheme and tested on three sample problems. Keywords: Lattice Boltzmann Method, non-newtonian fluids, Hermite expansion, lattice kinetic scheme

Turbulence is an open and challenging problem for mathematical approaches, physical modeling and numerical simulations. Numerical solutions contribute significantly to the understand of the nature and effects of turbulence. The focus of this thesis is the development of appropriate numerical methods for the computer simulation of turbulent flows. Many of the existing approaches to turbulence utilize analogies from kinetic theory. Degond & Lemou (J. Math. Fluid Mech., 4, 257-284, 2002) derived a k-✏ type turbulence model completely from kinetic theoretic framework. In the first part of this thesis, a numerical method is developed for the computer simulation based on this model. The Boltzmann equation used in the model has an isotropic, relaxation collision operator. The relaxation time in the collision operator depends on the microscopic turbulent energy, making it difficult to construct an efficient numerical scheme. In order to achieve the desired numerical efficiency, an appropriate change of frame is applied. This introduces a stiff relaxation source term in the equations and the concept of asymptotic preserving schemes is then applied to tackle the stiffness. Some simple numerical tests are introduced to validate the new scheme. In the second part of this thesis, alternative approaches are sought for more efficient numerical techniques. The Lattice Boltzmann Relaxation Scheme (LBRS) is a novel method developed recently by Rohan Deshmukh and S.V. Raghuram Rao for simulating compressible flows. Two different approaches for the construction of implicit sub grid scale -like models as Implicit Large Eddy Simulation (ILES) methods, based on LBRS, are proposed and are tested for Burgers turbulence, or Burgulence. The test cases are solved over a largely varying Reynolds number, demonstrating the efficiency of this new ILES-LBRS approach. In the third part of the thesis, as an approach towards the extension of ILES-LBRS to incompressible flows, an artificial compressibility model of LBRS is proposed. The modified framework, LBRS-ACM is then tested for standard viscous incompressible flow test cases.

博士
國立臺灣大學
應用力學研究所
93
The Boltzmann equation is a nonlinear, integral, and differential equation with many variables. It is difficult to be solved mathematically, so the collision term is usually replaced with a collision model. This will make it easier to deal with. In this paper, the velocity space will be discreted by applying discrete ordinate method. The relation between velocity space and distribution function is eliminated, so the distribution function can be represented as proper discrete velocity points. Therefore, the motion equation of distribution function, which is continuous in physical space, velocity space, and time, is an integral and differential equation, and by discrete ordinate method it becomes differential equations, which are continuous in physical space and time only and point-wise in velocity space. After this kind of treatment, the difficulties of numerical calculating will be greatly reduced. In this paper, the WENO scheme in conjuction with discrete ordinate method was applied to solve the model Blotzmann equation, and the implicit WENO scheme for the model Blotzmann equation was developed to solve the steady solutions of rarefied gas flows. First, the accuracy of the present scheme was verified by calculating the case of 1-D shock tube problem, which applied discrete ordinate method to discretize the velocity space of Blotzmann model equation and WENO scheme. The result of this case was also compared with results of other high resolution schemes. Because it is difficult to describe the behaviors of collisions between different species of gas molecule, the collision frequency of different species of gas molecule was first developed and substituted into Blotzmann model equation to solve the binary gas mixture flow problem. The suitability was verified by comparing the result of 1-D shock tube case with the analytic solution of Euler’s equation in low Knudsen number condition. The collision frequency developed in this paper can surely describe the behaviors of gas molecules via the result. In cases of 2-D flow problems, the external flows of cylinder and NACA 0012 airfoil were studied. For gas flow past cylinder, the characters of flow field in different Mach number and Knudsen number condition were investigated, and especially for low Knudsen number cases, the results were compared with calculating results of Euler’s equation. It showed that they are correspondent by comparing the characters of bow shock and wake. The convergence rates of different high resolution and implicit schemes were also investigated. The convergence behavior of the implicit WENO scheme developed in this paper is better than others. For gas flow past NACA 0012 airfoil, the calculating results were compared with results of experiment. It showed that the results of WENO scheme are of higher accuracy for the case with angle of attack.

This thesis presents results of a Ph.D. research in Energetics, carried out at the Department of Mechanics and Aeronautics of the University of Rome ”La Sapienza” and at the ”Istituto delle Applicazioni del Calcolo Mauro Picone” (National Research Council) of Rome. The main topic of the research has been focused on the analysis development and applications of a thermal model in the context of the kinetic schemes. In the last decade lattice kinetic theory, and most notably the Lattice Boltzmann Method (LBM), have met with significant success for the numerical simulation of a large variety of fluid flows, including real-world engineering applications. The Lattice Boltzmann Equation (LBE) is a minimal form of the Boltzmann kinetic equation, which is the evolution equation for a continuous one-body distribution function f(~x; ~v; t), wherein all details of molecular motion are removed except those that are strictly needed to represent the hydrodynamic behaviour at the macroscopic scale. The result is a very elegant and simple evolution equation for a discrete distribution function, or discrete population fi(~x; t) = f (~x; ~ci; t), which describes the probability to find a particle at lattice position ~x at time t, moving with speed ~ci. In a hydrodynamic simulation by using the LBE, one solves the only-two-steps evolution equations of the distribution functions of fictitious fluid particles: they move synchronously along rectilinear trajectories on a lattice space and then relax towards the local equilibrium because of the collisions. With respect to the more conventional numerical methods commonly used for the study of fluid flow situations, the kinetic nature of LBM introduces several advantages, including fully parallel algorithms and easy implementation of interfacial dynamics and complex boundaries, as in single and multi-phase flow i porous media. In addition, the convection operator is linear, no Poisson equation for the pressure must be resolved and the translation of the microscopic distribution function into the macroscopic quantities consists of simple arithmetic calculations. However, whereas LBE techniques shine for the simulation of isothermal, quasi incompressible flows in complex geometries, and LBM has been shown to be useful in applications involving interfacial dynamics and complex boundaries, the application to fluid flow coupled with non negligible heat transfer, turned out to be much more difficult. The LBE thermal models fall into three categories: the multispeed approach, the passive scalar approach and the doubled populations approach. The so-called multi-speed approach, which is a straightforward extension of the LBE isothermal models, makes theoretically possible to express both heat flux and temperature in terms of higher-order kinetic moments of the particle distribution functions fi(~x; t). It implies that higher-order velocity terms are involved in the formulation of equilibrium distribution and additional speeds are required by the corresponding lattices. The latter is arguably the major source of numerical instabilities of thermal lattice kinetic equations; in addition, it can seriously impair the implementation of the boundary conditions, a vital issue for the practical applications. The passive scalar and the doubled populations approaches are based on the idea of dispensing with the explicit representation of heat flux in terms of kinetic moments of the particle distribution function f(~x; ~v; t). A successful strategy consists of solving the temperature equation independently of LBE, possibly even with totally different numerical techniques. If the viscous heat dissipation and compression work done by the pressure are negligible, the temperature evolution equation is the same of a passive scalar and this approach enhances the numerical stability; the coupling to LBE is made by expressing the fluid pressure as the gradient of an external potential. Clearly, this strategy represents a drastic departure from a fully kinetic approach, and lacks some elegance. A more elegant possibility is to double the degrees of freedom and express thermal energy density and heat flux still as kinetic moments, but of a separate ’thermal’ distribution g(~x; ~v; t). Two sets of discrete distribution functions are used, dedicated to density and momentum fields, and temperature and heat flux fields, respectively. The advantage of this latter approach is that no kinetic moment beyond the first order is ever needed, since heat flux (third order vector moment of f) is simply expressed as the first order vector moment of g: as a result, disruptive instabilities conventionally attributed to the failure of reproducing higher-order moments in a discrete lattice are potentially avoided/mitigated. With respect to the previous approaches, the method is able to include viscous heating effects, and the boundary conditions are easily implemented because both f and g live in the same lattice, where additional speeds are not necessary. The price to pay is doubling of the storage requirements. As far as the thermal boundary conditions are concerned, LBE techniques usually handle the Dirichlet-type constraints; in contrast, the Neumann-type constraints are either limited to insulated walls or obtained imposing the temperature gradient at the wall through a strategy of transfer to a Dirichlet-type condition. For a wide class of real phenomena, the fixed temperature condition is clearly inadequate. Examples are represented by the cooling of devices, where the problem is characterized by an imposed thermal power to be removed, or by the air behavior in building rooms, where the temperature of the external walls is a direct consequence of the heat flux administered to the walls. In this framework, a General Purpose Thermal Boundary Condition (GPTBC) has been proposed, discussed and validated for an existing double population model. This thermal boundary condition is based on a counterslip approach as applied to the thermal energy. The incoming unknown thermal populations are assumed to be equilibrium distribution functions with a counterslip thermal energy density, which is determined so that suitable constraints are verified. The GPTBC proposed here can simulate explicitly either imposed wall temperature (Dirichlet-type constraint) or imposed wall heat fluxes (Neumann-type constraint), which allows LBM to be used for successful simulation of many types of heat transfer and fluid flows applications. Thus, the method can become an effective and alternative easy-to-apply tool, as well as the athermal LBE counterpart, especially for all those situations wherein the use of the usual theoretical approaches may fail, e.g., due to the complexity of the geometry. The validity of the developed GPTBC is demonstrated through its application to different flow configurations. With regard to channel flows, thermal Couette and Poiseuille flows has been simulated. The results obtained in case of Couette flows, show the model, together with the GPTBC, working over a wide range of physical parameters and allowing strong temperature gradients and heat dissipation effects to be detected. With regard to applications of the scheme to pressure gradient driven flows (Poiseuille flow), two different set-up are discussed. In LBE techniques, the most common set-up to simulate (nearly) incompressible flows consists of driving the flow with a constant force (i.e. a forcing term acting on the discrete populations), representing the constant pressure gradient, and applying periodic boundary conditions at inlet and outlet of the channel. In practical applications, one is often confronted with open flows, with prescribed inlet flow speed, and outlet pressure, or both prescribed inlet and outlet pressure values. In this case the common solution in LBE techniques, in which not pressure but only density values can be handled, is to force the flow by means of a density difference, between inlet and outlet sections, or by imposing velocity and density profiles. This strategy proves viable for athermal flows, so long as relative density changes (¢½=½) can be kept within a few percent, because the velocity profile maintains a parabolic behaviour. If heat transfer takes place, the temperature profile can change, in virtue of the nonuniform density along the channel; more specifically, one simulates the energy equation, taking in account the contribution of the term ¡p@xiui. In this case, the model has been shown to capture the expansion cooling effect, which gradually increases along the stream wise direction, and the opposing viscous heating effect. In order to come closer to the request of handling nearly incompressible flow and prescribed inlet/outlet boundary conditions, a different arrangement has been investigated. The idea is to impose boundary conditions in terms of inlet profile, with outlet variables left free to assume values coming from the run, still using a suitable amount of forcing. This hybrid formulation provides results in excellent agreement with theoretical solutions, for velocity, temperature and heat flux fields, as well as for Nusselt number behaviour, for a hydrodynamically fully developed flow; it also captures the effect of the coexistence of both a hydrodynamically and thermally developing flow, in the near inlet region, with an entry-length region depending on Prandtl number. With regard to applications to flows in enclosed spaces, the scheme has been used to simulate different cases of natural convection flow, which today represents an active subfield in heat transfer research. This great interest is due to the several fields in which natural convection is involved and to its importance in many engineering applications, e.g. heat transfer in buildings, solar energy collection, heat removal in micro electronics, cooling of nuclear reactors, dispersion of fire fumes in buildings and tunnels, ventilation of rooms. Compared with this great applicative interest, natural convection research is characterized by several theoretical and practical issues. The buoyancy-induced heat and momentum transfer in enclosures, also in simple geometries, strongly depends on geometric and physical conditions. Several regimes and complex phenomena of successive transitions can take place. Standard simulation techniques CFD cannot predict the behaviour of natural convection systems with high geometric complexity, or where viscous heating effects and/or non trivial conditions, related to the rheological law, are non negligible. As said, alternative approaches can be useful and required. Two flow configurations has been investigated for a wide range of Rayleigh number. Firstly, laminar flows in a square cavity, with vertical walls differently heated, have been discussed and results have been found in excellent agreement as compared with benchmark solutions, for both motion and heat transfer aspects. Then, laminar flows in a square cavity, with vertical walls heated and cooled by means of a constant uniform heat flux, which is a flow configuration never investigated by means of lattice Boltzmann methods, have been simulated; results have been found in excellent agreement as compared with those of previous works, obtained from a theoretical analysis. The study shows that the double population model provides reliable results over a wide range of physical parameters and in different situation of engineering interest. The new GPTBC provides good results for both imposed wall temperature and imposed wall heat fluxes conditions, beyond the adiabatic condition of previous schemes. These significant improvements, in the context of the kinetic schemes, can be added to the advantages specific to these methods, and primarily to Lattice Boltzmann Models, which make them competitive tools, with respect to the usual theoretical approaches and to the standard numerical techniques, for the simulation of complex hydrodynamic phenomena, from fully developed turbulence to phase transitions to granular flows. The thermal lattice Boltzmann method can become an effective and alternative tool, as well as the athermal counterpart, for successful simulation of many types of heat transfer and fluid flow processes, especially for all situations where complex phenomena take place.