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.
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