KINETICALLY CONSISTENT THERMAL LATTICE BOLTZMANN MODELS

The lattice Boltzmann (LB) method has developed into a numerically robust and efficient technique for simulating a wide variety of complex fluid flows. Unlike conventional CFD methods, the LB method is based on microscopic models and mesoscopic kinetic equations in which the collective long-term behavior of pseudo-particles is used to simulate the hydrodynamic limit of a system. Due to its kinetic basis, the LB method is particularly useful in applications involving interfacial dynamics and complex boundaries, such as multiphase or multicomponent flows. However, most of the LB models, both single and multiphase, do not satisfy the energy conservation principle, thus limiting their ability to provide quantitatively accurate predictions for cases with substantial heat transfer rates. To address this issue, this dissertation focuses on developing kinetically consistent and energy conserving LB models for single phase flows, in particular.\ud Firstly, through a procedure similar to the Galerkin method, we present a mathematical formulation of the LB method based on the concept of projection of the distributions onto a Hermite-polynomial basis and their systematic truncation. This formulation is shown to be capable of approximating the near incompressible, weakly compressible, and fully compressible (thermal) limits of the continuous Boltzmann equation, thus obviating the previous low-Mach number assumption. Physically it means that this formulation allows a kinetically-accurate description of flows involving large heat transfer rates. The various higher-order discrete-velocity sets (lattices) that follow from this formulation are also compiled. The resulting higher-order thermal model is validated for benchmark thermal flows, such as Rayleigh-Benard convection and thermal Couette flow, in an off-lattice framework. Our tests indicate that the D2Q39-based thermal models are capable of modeling incompressible and weakly compressible thermal flows accurately. In the validation process, through a finite-difference-type boundary treatment, we also extend the applicability of higher-order la ttices to flow-domains with solid boundaries, which was previously restricted.\ud Secondly, we present various off-lattice time-marching schemes for solving the discrete Boltzmann equation. Specifically, the various temporal schemes are analyzed with respect to their numerical stability as a function of the maximum allowable time-step . We show that the characteristics-based temporal schemes offer the best numerical stability among all other comparable schemes. Due to this enhanced numerical stability, we show that the usual restriction no longer applies, enabling larger time-steps, and thereby reducing the computational run-time. The off-lattice scheme were also successfully extended to higher-order LB models.\ud Finally, we present the algorithm and single-core optimization techniques for a off-lattice, higher-order LB code. Using simple cache optimization techniques and a proper choice of the data-structure, we obtain a 5-7X improvement in performance compared to a naive, unoptimized code. Thereafter, the optimized code is parallelized using OpenMP. Scalability tests indicate a parallel efficiency of 80% on shared-memory systems with up to 50 cores (strong scaling). An analysis of the higher-order LB models also show that they are less memory-bound if the off-lattice temporal schemes are used, thus making them more scalable compared to the stream-collide type scheme