Academic literature on the topic 'Discrete Velocity Boltzmann Schemes'
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Journal articles on the topic "Discrete Velocity Boltzmann Schemes"
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Hsu, C. T., S. W. Chiang, and K. F. Sin. "A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods." Communications in Computational Physics 11, no. 4 (April 2012): 1397–414. http://dx.doi.org/10.4208/cicp.150510.150511s.
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AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.
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Mischler, Stéphane. "Convergence of Discrete-Velocity Schemes for the Boltzmann Equation." Archive for Rational Mechanics and Analysis 140, no. 1 (November 1, 1997): 53–77. http://dx.doi.org/10.1007/s002050050060.
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Buet, C. "Conservative and Entropy Schemes for Boltzmann Collision Operator of Polyatomic Gases." Mathematical Models and Methods in Applied Sciences 07, no. 02 (March 1997): 165–92. http://dx.doi.org/10.1142/s0218202597000116.
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We propose two discrete velocity models derived from the Boltzmann equation of Larsen–Borgnakke type for polyatomic gases. These two models are natural extensions of previously discussed discrete velocity models used for monoatomic gases. These two models have the same properties as the continuous one, which are conservation of mass, momentum and energy, discrete Maxwellians as equilibrium states and H-theorems.
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Diaz, Manuel A., Min-Hung Chen, and Jaw-Yen Yang. "High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation." Communications in Computational Physics 18, no. 4 (October 2015): 1012–49. http://dx.doi.org/10.4208/cicp.171214.210715s.
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AbstractHigh-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.
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MATTILA, KEIJO K., DIOGO N. SIEBERT, LUIZ A. HEGELE, and PAULO C. PHILIPPI. "HIGH-ORDER LATTICE-BOLTZMANN EQUATIONS AND STENCILS FOR MULTIPHASE MODELS." International Journal of Modern Physics C 24, no. 12 (November 13, 2013): 1340006. http://dx.doi.org/10.1142/s0129183113400068.
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The lattice Boltzmann (LB) method, based on mesoscopic modeling of transport phenomena, appears to be an attractive alternative for the simulation of complex fluid flows. Examples of such complex dynamics are multiphase and multicomponent flows for which several LB models have already been proposed. However, due to theoretical or numerical reasons, some of these models may require application of high-order lattice-Boltzmann equations (LBEs) and stencils. Here, we will present a derivation of LBEs from the discrete-velocity Boltzmann equation (DVBE). By using the method of characteristics, high-order accurate equations are conveniently formulated with standard numerical methods for ordinary differential equations (ODEs). In particular, we will derive implicit LB schemes due to their stability properties. A simple algorithm is presented which enables implementation of the implicit schemes without resorting to, e.g. change of variables. Finally, some numerical experiments with high-order equations and stencils together with two specific multiphase models are presented.
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Wang, Liang, Xuhui Meng, Hao-Chi Wu, Tian-Hu Wang, and Gui Lu. "Discrete effect on single-node boundary schemes of lattice Bhatnagar–Gross–Krook model for convection-diffusion equations." International Journal of Modern Physics C 31, no. 01 (December 20, 2019): 2050017. http://dx.doi.org/10.1142/s0129183120500175.
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The discrete effect on the boundary condition has been a fundamental topic for the lattice Boltzmann method (LBM) in simulating heat and mass transfer problems. In previous works based on the anti-bounce-back (ABB) boundary condition for convection-diffusion equations (CDEs), it is indicated that the discrete effect cannot be commonly removed in the Bhatnagar–Gross–Krook (BGK) model except for a special value of relaxation time. Targeting this point in this paper, we still proceed within the framework of BGK model for two-dimensional CDEs, and analyze the discrete effect on a non-halfway single-node boundary condition which incorporates the effect of the distance ratio. By analyzing an unidirectional diffusion problem with a parabolic distribution, the theoretical derivations with three different discrete velocity models show that the numerical slip is a combined function of the relaxation time and the distance ratio. Different from previous works, we definitely find that the relaxation time can be freely adjusted by the distance ratio in a proper range to eliminate the numerical slip. Some numerical simulations are carried out to validate the theoretical derivations, and the numerical results for the cases of straight and curved boundaries confirm our theoretical analysis. Finally, it should be noted that the present analysis can be extended from the BGK model to other lattice Boltzmann (LB) collision models for CDEs, which can broaden the parameter range of the relaxation time to approach 0.5.
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Mieussens, Luc. "Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries." Journal of Computational Physics 162, no. 2 (August 2000): 429–66. http://dx.doi.org/10.1006/jcph.2000.6548.
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Aristov, V. V., O. V. Ilyin, and O. A. Rogozin. "Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods." Journal of Computational Science 40 (February 2020): 101064. http://dx.doi.org/10.1016/j.jocs.2019.101064.
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Buet, C. "A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics." Transport Theory and Statistical Physics 25, no. 1 (January 1996): 33–60. http://dx.doi.org/10.1080/00411459608204829.
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Wu, Junlin, Zhihui Li, Aoping Peng, and Xinyu Jiang. "Numerical Simulations of Unsteady Flows From Rarefied Transition to Continuum Using Gas-Kinetic Unified Algorithm." Advances in Applied Mathematics and Mechanics 7, no. 5 (July 21, 2015): 569–96. http://dx.doi.org/10.4208/aamm.2014.m523.
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AbstractNumerical simulations of unsteady gas flows are studied on the basis of Gas-Kinetic Unified Algorithm (GKUA) from rarefied transition to continuum flow regimes. Several typical examples are adopted. An unsteady flow solver is developed by solving the Boltzmann model equations, including the Shakhov model and the Rykov model etc. The Rykov kinetic equation involving the effect of rotational energy can be transformed into two kinetic governing equations with inelastic and elastic collisions by integrating the molecular velocity distribution function with the weight factor on the energy of rotational motion. Then, the reduced velocity distribution functions are devised to further simplify the governing equation for one- and two-dimensional flows. The simultaneous equations are numerically solved by the discrete velocity ordinate (DVO) method in velocity space and the finite-difference schemes in physical space. The time-explicit operator-splitting scheme is constructed, and numerical stability conditions to ascertain the time step are discussed. As the application of the newly developed GKUA, several unsteady varying processes of one- and two-dimensional flows with different Knudsen number are simulated, and the unsteady transport phenomena and rarefied effects are revealed and analyzed. It is validated that the GKUA solver is competent for simulations of unsteady gas dynamics covering various flow regimes.
Dissertations / Theses on the topic "Discrete Velocity Boltzmann Schemes"
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Håkman, Olof. "Boltzmann Equation and Discrete Velocity Models : A discrete velocity model for polyatomic molecules." Thesis, Karlstads universitet, Institutionen för matematik och datavetenskap (from 2013), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-76143.
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In the study of kinetic theory and especially in the study of rarefied gas dynamics one often turns to the Boltzmann equation. The mathematical theory developed by Ludwig Boltzmann was at first sight applicable in aerospace engineering and fluid mechanics. As of today, the methods in kinetic theory are extended to other fields, for instance, molecular biology and socioeconomics, which makes the need of finding efficient solution methods still important. In this thesis, we study the underlying theory of the continuous and discrete Boltzmann equation for monatomic gases. We extend the theory where needed, such that, we cover the case of colliding molecules that possess different levels of internal energy. Mainly, we discuss discrete velocity models and present explicit calculations for a model of a gas consisting of polyatomic molecules modelled with two levels of internal energy.I studiet av kinetisk teori och speciellt i studiet av dynamik för tunna gaser vänder man sig ofta till Boltzmannekvationen. Den matematiska teorien utvecklad av Ludwig Boltzmann var vid första anblicken tillämpbar i flyg- och rymdteknik och strömningsmekanik. Idag generaliseras metoder i kinetisk teori till andra områden, till exempel inom molekylärbiologi och socioekonomi, vilket gör att vi har ett fortsatt behov av att finna effektiva lösningsmetoder. Vi studerar i denna uppsats den underliggande teorin av den kontinuerliga och diskreta Boltzmannekvationen för monatomiska gaser. Vi utvidgar teorin där det behövs för att täcka fallet då kolliderande molekyler innehar olika nivåer av intern energi. Vi diskuterar huvudsakligen diskreta hastighetsmodeller och presenterar explicita beräkningar för en modell av en gas bestående av polyatomiska molekyler modellerad med två lägen av intern energi.
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Morris, Aaron Benjamin. "Investigation of a discrete velocity Monte Carlo Boltzmann equation." Thesis, [Austin, Tex. : University of Texas, 2009. http://hdl.handle.net/2152/ETD-UT-2009-05-127.
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Bernhoff, Niclas. "On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation." Doctoral thesis, Karlstads universitet, Fakulteten för teknik- och naturvetenskap, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-2373.
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We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models. Existence of solutions of weakly non-linear half-space problems for general DVMs are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found. Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution. The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.
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Späth, Peter. "Renewed Theory, Interfacing, and Visualization of Thermal Lattice Boltzmann Schemes." Doctoral thesis, Universitätsbibliothek Chemnitz, 2000. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200000648.
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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- aminedIn 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
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Février, Tony. "Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112316/document.
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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 poidsIn 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
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Jobic, Yann. "Numerical approach by kinetic methods of transport phenomena in heterogeneous media." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4723/document.
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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
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Hegermiller, David Benjamin. "A new method to incorporate internal energy into a discrete velocity Monte Carlo Boltzmann Equation solver." Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-08-4328.
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A new method has been developed to incorporate particles with internal structure into the framework of the Variance Reduction method [17] for solving the discrete velocity Boltzmann Equation. Internal structure in the present context refers to physical phenomena like rotation and vibration of molecules consisting of two or more atoms. A gas in equilibrium has all modes of internal energy at the same temperature as the translational temperature. If the gas is in a non-equilibrium state, translational temperature and internal temperatures tend to proceed towards an equilibrium state during equilibration, but they all do so at different relaxation rates. In this thesis, rotational energy of a distribution of molecules is modeled as a single value at a point in a discrete velocity space; this represents the average rotational energy of molecules at that specific velocity. Inelastic collisions are the sole mechanism of translational and rotational energy exchange, and are governed by a modified Landau-Teller equation. The method is tested for heat bath simulations, or homogeneous relaxations, and one dimensional shock problems. Homogeneous relaxations demonstrate that the rotational and translational temperatures equilibrate to the correct final temperature, which can be predicted by conservation of energy. Moreover, the rates of relaxation agree with the direct simulation Monte Carlo (DSMC) method with internal energy for the same input parameters. Using a fourth order method for convecting mass along with its corresponding internal energy, a one dimensional Mach 1.71 normal shock is simulated. Once the translational and rotational temperatures equilibrate downstream, the temperature, density and velocity, predicted by the Rankine-Hugoniot conditions, are obtained to within an error of 0.5%. The result is compared to a normal shock with the same upstream flow properties generated by the DSMC method. Internal vibrational energy and a method to use Larsen Borgnakke statistical sampling for inelastic collisions is formulated in this text and prepared in the code, but remains to be tested.text
Books on the topic "Discrete Velocity Boltzmann Schemes"
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Succi, Sauro. Lattice Relaxation Schemes. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199592357.003.0014.
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In Chapter 13, it was shown that the complexity of the LBE collision operator can be cut down dramatically by formulating discrete versions with prescribed local equilibria. In this chapter, the process is taken one step further by presenting a minimal formulation whereby the collision matrix is reduced to the identity, upfronted by a single relaxation parameter, fixing the viscosity of the lattice fluid. The idea is patterned after the celebrated Bhatnagar–Gross–Krook (BGK) model Boltzmann introduced in continuum kinetic theory as early as 1954. The second part of the chapter describes the comeback of the early LBE in optimized multi-relaxation form, as well as few recent variants hereof.
Book chapters on the topic "Discrete Velocity Boltzmann Schemes"
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"Discrete-Velocity Models and Lattice Boltzmann Methods for Convection-Radiation Problems." In Progress in Computational Physics Volume 3: Novel Trends in Lattice-Boltzmann Methods, edited by Mapundi K. Banda and Mohammed Seaid, 53–90. BENTHAM SCIENCE PUBLISHERS, 2013. http://dx.doi.org/10.2174/9781608057160113030006.
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UKAI, S. "ON THE HALF–SPACE PROBLEM FOR THE DISCRETE VELOCITY MODEL OF THE BOLTZMANN EQUATION." In Series on Advances in Mathematics for Applied Sciences, 160–74. WORLD SCIENTIFIC, 1998. http://dx.doi.org/10.1142/9789812816481_0005.
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Schürrer, F. "Discrete Models of the Boltzmann Equation in Quantum Optics and Arbitrary Partition of the Velocity Space." In Series on Advances in Mathematics for Applied Sciences, 259–98. WORLD SCIENTIFIC, 2003. http://dx.doi.org/10.1142/9789812796905_0010.
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Tuck, Adrian F. "Radiative and Chemical Kinetic Implications." In Atmospheric Turbulence. Oxford University Press, 2008. http://dx.doi.org/10.1093/oso/9780199236534.003.0009.
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The laws governing the dynamical behaviour of atoms and molecules are quantum mechanical, and specify that their internal energy states are discrete, with only definite photon energies inducing transitions between them, subject to selection rules. These energy levels appear as spectra in different regions of the electromagnetic spectrum: pure rotational lines in the microwave or far infrared, ‘rovibrational’ (rotation + vibration) lines in the middle and near infrared, while electronic transitions, sometimes with associated rotational and vibrational structure (‘rovibronic’) occur from the near infrared through the visible to the ultraviolet. An important feature of these spectra in the atmosphere is that they do not appear as single sharp lines, but are collisionally broadened about the central energy into ‘line shapes’ which frequently overlap with other transitions, both from the same molecule and from others. One of the primary dynamical quantities involved in the processes broadening these line shapes is the relative velocity of the molecules with which the photon absorbing and emitting molecules are colliding. These are primarily N2 and O2 in the atmosphere; if they have an overpopulation of fast moving molecules relative to a Maxwell–Boltzmann distribution, as we have suggested, the line shapes will be affected. Molecules such as carbon dioxide, water vapour, and ozone are all active in the infrared via rovibrational transitions, with water vapour being light enough and so having sufficiently rapid rotation that it has rotational bands appearing in the far infrared rather than the microwave. Nitrous oxide, N2O, and methane, CH4, are also active, but make smaller contributions because of their lower abundances. Molecular nitrogen and molecular oxygen, because they are homonuclear diatomic molecules, do not absorb or emit via electric dipole allowed transitions in the atmospherically important regions of the electromagnetic spectrum. Molecular oxygen, having a triplet ground state, does have weak forbidden and magnetic dipole transitions which, however, play only a very small role in the radiative balance. It should be noted that the translational energy of molecules in a large system like the atmosphere is effectively continuous rather than quantized.
Conference papers on the topic "Discrete Velocity Boltzmann Schemes"
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Chen, Leitao, Laura Schaefer, and Xiaofeng Cai. "An Accurate Unstructured Finite Volume Discrete Boltzmann Method." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-87136.
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Unlike the conventional lattice Boltzmann method (LBM), the discrete Boltzmann method (DBM) is Eulerian in nature and decouples the discretization of particle velocity space from configuration space and time space, which allows the use of an unstructured grid to exactly capture complex boundary geometries. A discrete Boltzmann model that solves the discrete Boltzmann equation (DBE) with the finite volume method (FVM) on a triangular unstructured grid is developed. The accuracy of the model is improved with the proposed high-order flux schemes and interpolation scheme. The boundary treatment for commonly used boundary conditions is also formulated. A series of problems with both periodic and non-periodic boundaries are simulated. The results show that the new model can significantly reduce numerical viscosity.
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Malkov, E. A., S. O. Poleshkin, and M. S. Ivanov. "Discrete velocity scheme for solving the Boltzmann equation with the GPGPU." In 28TH INTERNATIONAL SYMPOSIUM ON RAREFIED GAS DYNAMICS 2012. AIP, 2012. http://dx.doi.org/10.1063/1.4769532.
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Hsu, C. T., S. W. Chiang, and K. F. Sin. "A Novel Dynamics Lattice Boltzmann Method for Gas Flows." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31237.
Full textAbstract:
The lattice Boltzmann method (LBM), where discrete velocities are specifically assigned to ensure that a particle leaves one lattice node always resides on another lattice node, has been developed for decades as a powerful numerical tool to solve the Boltzmann equation for gas flows. The efficient implementation of LBM requires that the discrete velocities be isotropic and that the lattice nodes be homogeneous. These requirements restrict the applications of the currently-used LBM schemes to incompressible and isothermal flows. Such restrictions defy the original physics of Boltzmann equation. Much effort has been devoted in the past decades to remove these restrictions, but of less success. In this paper, a novel dynamic lattice Boltzmann method (DLBM) that is free of the incompressible and isothermal restrictions is proposed and developed to simulate gas flows. This is achieved through a coordinate transformation featured with Galilean translation and thermal normalization. The transformation renders the normalized Maxwell equilibrium distribution with directional isotropy and spatial homogeneity for the accurate and efficient implementation of the Gaussian-Hermite quadrature. The transformed Boltzmann equation contains additional terms due to local convection and acceleration. The velocity quadrature points in the new coordinate system are fixed while the correspondent points in the physical space change from time to time and from position to position. By this dynamic quadrature nature in the physical space, we term this new scheme as the dynamic quadrature scheme. The lattice Boltzmann method (LBM) with the dynamic quadrature scheme is named as the dynamic lattice Boltzmann method (DLBM). The transformed Boltzmann equation is then solved in the new coordinate system based on the fixed quadrature points. Validations of the DLBM have been carried with several benchmark problems. Cavity flows problem are used. Excellent agreements are obtained as compared with those obtained from the conventional schemes. Up to date, the DLBM algorithm can run up to Mach number at 0.3 without suffering from numerical instability. The application of the DLBM to the Rayleigh-Bernard thermal instability problem is illustrated, where the onset of 2D vortex rolls and 3D hexagonal cells are well-predicted and are in excellent agreement with the theory. In summary, a novel dynamic lattice Boltzmann method (DLBM) has been proposed with algorithm developed for numerical simulation of gas flows. This new DLBM has been demonstrated to have removed the incompressible and isothermal restrictions encountered by the traditional LBM.
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Kang, Shin K., and Yassin A. Hassan. "A Comparative Study of Interface Schemes in the Immersed Boundary Method for a Moving Solid Boundary Problem Using the Lattice Boltzmann Method." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-30908.
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For moving boundary problems, previous body-conformal grid methods require frequent re-meshing as the boundary moves, thus increasing computational cost. An immersed boundary method (IBM) is an attractive method to resolve the problem since it is based on the fixed, non-body-conformal grids. In the IBM, force density terms are used so that no-slip boundary condition is satisfied on the boundary. On the other hand, lattice Boltzmann methods (LBMs) have been used as an alternative of Navier-Stokes equation method due to their efficiency to parallelize and simplicity to implement. The common feature of the IBM and the LBM of using non-body-conformal grids motivated the use of the IBM in the lattice Boltzmann method frame, which is usually called an immersed boundary-lattice Boltzmann method (IB-LBM). Besides, a split-forcing property in the LBM, due to its kinetic nature, facilitates the use of direct-forcing IBM. For the evaluation of boundary force density term, we need to adopt an interpolation scheme because the boundary, in general, does not match computational nodes. The interpolation schemes can be classified into diffuse and sharp interface schemes. The former usually uses the discrete delta function to evaluate the boundary force on the prescribed boundary points, while the latter uses interpolation from neighboring fluid nodes to evaluate the boundary force on the computation node either inside or outside closest to the boundary. In the diffuse scheme, the boundary force density terms evaluated on the boundary points should be distributed onto neighboring computational nodes using the discrete delta functions so that the boundary effect may exert on computational process. The objective of this study is to compare two interface schemes simultaneously for a moving boundary problem under the IB-LBM and to understand advantages and disadvantages of each scheme. We considered a problem of flow induced by inline oscillation of a circular cylinder since both experimental and body-conformal grid method results are available for this problem. Velocity results from both schemes showed overall good agreement with experimental data. However, the sharp interface scheme showed spurious oscillations in the surface force coefficient and pressure fields, although after filtering or smoothing, the force coefficients showed good agreement with the body-fitted results. In contrast, the diffuse interface scheme produced smooth variations in the surface force coefficient but over-predicted the absolute values especially at phase angles with the high magnitude of accelerations. These results can be attributed to the use of discrete delta functions. We could reduce the over-prediction by considering the effect of the diffuse area.
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Yang, L. M., C. Shu, and J. Wu. "Numerical Simulation of Microflows by a DOM With Streaming and Collision Processes." In ASME 2016 5th International Conference on Micro/Nanoscale Heat and Mass Transfer. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/mnhmt2016-6494.
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Inspired from the idea of developing lattice Boltzmann method (LBM), a discrete ordinate method (DOM) with streaming and collision processes is presented for simulation of microflows in this work. The current method is quite different from the conventional discrete ordinate method (DOM), unified gas kinetic scheme (UGKS) and discrete unified gas kinetic scheme (DUGKS), in which the finite volume method (FVM) or the finite difference method (FDM) is usually utilized to discretize the discrete velocity Boltzmann equation (DVBE). Due to the application of FVM or FDM, the evaluation of the flux of distribution function at the cell interface becomes an essential step for these approaches. Besides that, for the UGKS and DUGKS, not only the flux of distribution functions but also the conservative variables at the cell interface are needed to be computed. These processes require a lot of computational efforts. In contrast, for the developed method, it only needs interpolations within the cell to perform the streaming process. Thus, the computational efficiency can be improved accordingly. To compare the accuracy and efficiency of present scheme with those of DSMC and/or UGKS, several numerical examples including the Couette flow, pressure driven Poiseuille flow and thermal transpiration flow are simulated. Numerical results showed that the solution accuracy of current scheme is comparable to that of DSMC and UGKS. However, as far as the computational efficiency is concerned, the present scheme is more efficient than UGKS.
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Suga, K., S. Takenaka, T. Ito, M. Kaneda, T. Kinjo, and S. Hyodo. "Lattice Boltzmann Flow Simulation in Micro-Nano Transitional Porous Media." In 2010 14th International Heat Transfer Conference. ASMEDC, 2010. http://dx.doi.org/10.1115/ihtc14-22283.
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In order to simulate heat and mass transfer in porous media whose scales range micron to nanometers, this work intends to provide a scheme for flow field simulation for such porous media. Since Navier-Stokes equations are no longer applicable to high Knudsen number (Kn) flow regimes, the conventional lattice Boltzmann method (LBM) cannot be applicable to flows in nanoscale porous media. Hence, a modified lattice Boltzmann method is applied for computing flows in micro-nano porous media in the transitional flow regimes at moderately high Knudsen numbers. The lattice Boltzmann equation applied is an extended version using an effective relaxation time associated with Kn and a regularization procedure coupled with Maxwell’s diffuse-scattering boundary condition for walls. For the flow field where the representative molecular mean free path varies (effectively the Kn varies locally), the locally defined Kn is introduced. In order to verify the LBM scheme, the results are compared with those of the molecular dynamics (MD) simulations by the Leonard-Jones potential. The flow fields considered are in modeled nano-porous media whose porosity is around 0.9. The results of micro-nanoscale porous media flows at Knudsen numbers: Kn = 0.04–0.24 show reasonable agreement in both the simulation methods and confirm the reliability of the presently applied LBM. Interestingly, in complex flow geometry, the advantage of higher order discrete velocity models of the LBM is not notable. Therefore, it is concluded that conventional discrete velocity models, say the D2Q9 and D3Q19 models are reasonably enough for flows in micro-nanoscale porous media.
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Li, Like, Chen Chen, Renwei Mei, and James F. Klausner. "Conjugate Interface Heat and Mass Transfer Simulation With the Lattice Boltzmann Equation Method." In ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels collocated with the ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/icnmm2014-21864.
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An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation (LBE) method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including: (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, and (iii) steady heat conduction inside a circular domain with two different solid materials. The accuracy and order-of-convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration) and heat (mass) flux are examined in detail and compared with those obtained from the “half lattice division” treatment in the literature. The present analysis and numerical results show that the half lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved interfaces, the present treatment yields second-order accurate interior and interfacial temperatures (concentrations) and first-order accurate interfacial heat (mass) flux. An increase of order-of-convergence by one degree is obtained for each of these three quantities compared with the half lattice division scheme.
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Frandsen, Jannette B. "A Lattice Boltzmann Bluff Body Model for VIV Suppression." In 25th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2006. http://dx.doi.org/10.1115/omae2006-92271.
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In this paper, the suitability of a mesoscopic approach involving a single phase Lattice Boltzmann (LB) model is examined. In contrast, to continuum based numerical models, where only space and time are discrete, the discrete variables of the LB model are space, time and particle velocity. With reference to the Boltzmann equation of classical kinetic theory, the distribution of fluid molecules is represented by particle distribution functions. The LB method simulates fluid flow by tracking particle distributions. It is notable that the formulation avoids the need to include the Poisson equation. An elastic-collision scheme with no-slip walls is prescribed. The central idea behind proposing the present formulation is many fold. One goal is to capture smaller scales naturally, postponing the need of applying empirical turbulence models. Another goal is to get further insight into nonlinearities in steep and breaking free surfaces to improve current continuum mechanics solutions. Although the long term goal is to predict bluff-body high Reynolds number flows and breaking water waves, the present study is limited to laminar flow simulations and continuous free surfaces. The case studies presented include bluff bodies embedded in Reynolds number flows in the order of 100–200. The free surface test cases represent bore propagation past a single and multiple structures. The 2-D uniform grid solutions are compared with findings reported in the literature. Vortex patterns are studied when single or several objects are located in the bluff-body wakes. From a mitigation point of view, the model presents an easy means of re-arranging bluff bodies to study optimum solutions for VIV suppression with/without a free surface.
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Bazargan, Majid, and Mostafa Varmazyar. "Modeling of Free Convection Heat Transfer to a Supercritical Fluid in a Square Enclosure by the Lattice Boltzmann Method." In ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences. ASMEDC, 2009. http://dx.doi.org/10.1115/ht2009-88463.
Full textAbstract:
During the last decade a number of numerical computations based on the finite volume approach have been reported studying various aspects of heat transfer near the critical point. In this paper, a Lattice Boltzmann Method (LBM) has been developed to simulate laminar free convection heat transfer to a supercritical fluid in a square enclosure. The LBM is an ideal mesoscopic approach to solve nonlinear macroscopic conservation equations due to its simplicity and capability of parallelization. The Lattice Boltzmann Equation (LBE) represents the minimal form of the Boltzmann kinetic equation. The LBE is a very elegant and simple equation, for a discrete density distribution function and is the basis of the LBM. For the mass and momentum equations, an LBM is used while the heat equation is solved numerically by a finite volume scheme. In this study, inter-particle forces are taken into account for non-ideal gases in order to simulate the velocity profile more accurately. The laminar free convection cavity flow has been extensively used as a benchmark test to evaluate the accuracy of the numerical code. It is found that the numerical results of this study are in good agreement with the experimental and numerical results reported in the literature. The results of the LBM–FVM combination are found to be in excellent agreement with the FVM–FVM combination for the Navier-Stokes and heat transfer equations.
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Frandsen, Jannette B. "A Mesoscopic Model Approach to Passively Control Vortex Wakes Using Single/Multiple Bodies." In ASME 2006 Pressure Vessels and Piping/ICPVT-11 Conference. ASMEDC, 2006. http://dx.doi.org/10.1115/pvp2006-icpvt-11-93759.
Full textAbstract:
In this paper, vortex patterns are studied when single or several objects are located in the bluff-body wakes. The suitability of a mesoscopic approach involving a single phase Lattice Boltzmann (LB) model is examined. The central idea behind proposing the present formulation is to capture smaller scales naturally, postponing the need of applying empirical turbulence models. In contrast, to continuum mechanics based numerical models, where only space and time are discrete, the discrete variables of the LB model are space, time and particle velocity. With reference to the Boltzmann equation of classical kinetic theory, the distribution of fluid molecules is represented by particle distribution functions. It is notable that the formulation avoids the need to include the Poisson equation. An elastic-collision scheme with no-slip walls is prescribed. Although the long term goal is to predict bluff-body high Reynolds number flows, the present study is limited to laminar flow simulations. The case studies include sharp edge bodies embedded in Re flows in the order of 100–250. The 2-D uniform grid solutions are compared with findings reported in the literature and promising agreements have been found. This study is important to a variety of applications, in particular, the wind, ocean and coastal engineering communities. From a mitigation point of view, the model presents an easy means of re-arranging bluff bodies to study optimum solutions for VIV suppression. It is notable that the CPUs are favorable for the multiple bluff body solutions compared to current published continuum mechanics models.