Дисертації з теми "Lattice Boltzmann scheme"

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

Nous nous intéressons au problème d’extraction du fouillis de mer dans des images radar marin. Le parti pris est de développer des méthodes de traitement d’image permettant de s’affranchir au mieux d’hypothèses sur la nature du fouillis de mer et du signal d’intérêt. D’une part, nous proposons un algorithme basé sur une approche variationnelle originale : un modèle multiphasique à interface diffuse. Les résultats obtenus montrent que l’algorithme est efficace lorsque le signal d’intérêt a un rapport signal-sur-fouillis suffisamment grand. D’autre part, nous nous intéressons à l’implémentation de schémas de Boltzmann sur réseau pour des problèmes de convection-diffusion à vitesse d’advection non constante et un terme source non nul. Nous décrivons le calcul de la consistance obtenue par analyse asymptotique à l’échelle acoustique et avec un opérateur de collision à temps de relaxation multiples, et étudions la stabilité de ces schémas dans un cas particulier. Les résultats obtenus montrent que les schémas proposés permettent de supprimer le bruit résiduel et de renforcer le signal d’intérêt sur l’image obtenue grâce à la première méthode. Enfin, nous proposons une méthode d’apprentissage permettant de s’affranchir d’hypothèses sur la nature du signal d’intérêt. En effet, en complément de l’algorithme par approche variationnelle, nous proposons un algorithme basé sur le traitement pulse-Doppler lorsque le signal d’intérêt est exo-clutter et a un rapport signal-sur-fouillis faible. Les résultats obtenus à partir du double auto-encodeur que nous proposons, étant comparables aux résultats fournis par chacune des deux méthodes, permettent de valider cette approche
We focus on the problem of sea clutter extraction in marine radar images. The aim is to develop image processing methods allowing us to avoid assumptions about the nature of the sea clutter and the signal of interest. On the one hand, we propose an original algorithm based on a variational approach : a multiphase model with diffuse interface. The results obtained show that the algorithm is efficient when the signal of interest has a sufficiently large signal-to-clutter ratio. On the other hand, we focus on the implementation of lattice Boltzmann schemes for convection-diffusion problems with non-constant advection velocity and non-zero source term. We describe the computation of the consistency obtained by asymptotic analysis at the acoustic scale and with a multiple relaxation time collision operator, and study the stability of these schemes in a particular case. The obtained results show that the proposed schemes allow removing the residual noise and to enhance the signal of interest on the image obtained with the first method. Finally, we propose a learning method allowing us to avoid assumptions on the nature of the signal of interest. Indeed, in addition to the variational approach, we propose an algorithm based on pulse-Doppler processing when the signal of interest is exo-clutter and has a low signal-to-clutter ratio. The results obtained from the proposed double auto-encoder, being comparable to the results provided by each of the two methods, allow validating this approach

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

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.

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
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.

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.

The lattice Boltzmann method (LBM) has emerged as a highly efficient model for the simulation of incompressible flows in the last few decades. Its extension to compressible flows is hindered by the fact that the equilibrium distributions rep- resent truncated Taylor series expansions in Mach number, limiting the strategy to low Mach number flows. Numerous efforts have been undertaken to extend this method to compressible flows recently. Some of these approaches have resulted in complicated or expensive equilibrium distribution functions. A few approaches are limited to subsonic flows while a few others have too many tuning parameters without clear guidelines to x their values. In this context, it is worth exploring newer avenues to develop an efficient lattice Boltzmann method for compressible fluid flows. In this thesis, we utilize a novel interpretation of the discrete velocity Boltzmann relaxation systems to develop new lattice Boltzmann methods for compressible flows. In these new lattice Boltzmann relaxation schemes (LBRS), the equilib- rium distributions are free from the low-Mach number expansions. In fact, the equilibrium distributions are simple algebraic combinations of the conserved vari- ables and the fluxes. This novel LB method is tested for the 1-D and 2-D Euler equations using a D1Q3 and a D2Q9 model respectively. Various bench-mark test cases have been considered to demonstrate the robustness of the new scheme. This strategy can be easily extended to other hyperbolic systems of conservation laws representing the shallow-water flows and the ideal magnetohydrodynamics. In this work, the extension of LBRS to the shallow-water equations in both one and two-dimensions and to the 1-D ideal MHD equations is demonstrated through bench-mark test problems. Finally, extension of this new method to parabolic equations is demonstrated by applying it to the viscous Burgers equation. The formulation of LBRS for the viscous case is based on the interpretation of the diffusion term in the resulting relaxation system as a physical diffusion term. Two novel approaches to extend the new scheme to viscous Burgers equation are presented.

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.