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Development Of A New Finite-Volume Lattice Boltzmann Formulation And Studies On Benchmark FlowsVilasrao, Patil Dhiraj 07 1900 (has links) (PDF)
This thesis is concerned with the new formulation of a finite-volume lattice Boltzmann equation method and its implementation on unstructured meshes. The finite-volume discretization with a cell-centered tessellation is employed. The new formulation effectively adopts a total variation diminishing concept. The formulation is analyzed for the modified partial differential equation and the apparent viscosity of the model. Further, the high-order extension of the present formulation is laid out. Parallel simulations of a variety of two-dimensional benchmark flows are carried out to validate the formulation.
In Chapter 1, the important notions of the kinetic theory and the most celebrated equation in the kinetic theory, ‘the Boltzmann equation’ are given. The historical developments and the theory of a discrete form of Boltzmann equation are briefly discussed. Various off-lattice schemes are introduced. Various methodologies adopted in the past for the solution of the lattice Boltzmann equation on finite-volume discretization are reviewed. The basic objectives of this thesis are stated.
In Chapter2,the basic formulations of lattice Boltzmann equation method with a rational behind different boundary condition implementations are discussed. The benchmark flows are studied for various flow phenomenon with the parallel code developed in-house. In particular, the new benchmark solution is given for the flow induced inside a rectangular, deep cavity.
In Chapter 3, the need for off-lattice schemes and a general introduction to the finite-volume approach and unstructured mesh technology are given. A new mathematical formulation of the off-lattice finite-volume lattice Boltzmann equation procedure on a cell-centered, arbitrary triangular tessellation is laid out. This formulation employs the total variation diminishing procedure to treat the advection terms. The implementation of the boundary condition is given with an outline of the numerical implementation. The Chapman-Enskog (CE) expansion is performed to derive the conservation equations and an expression for the apparent viscosity from the finite-volume lattice Boltzmann equation formulation in Chapter 4. Further, the numerical investigations are performed to analyze the apparent viscosity variation with respect to the grid resolution.
In Chapter 5, an extensive validation of the newly formulated finite-volume scheme is presented. The benchmark flows considered are of increasing complexity and are namely
(1) Posieuille flow, (2) unsteady Couette flow, (3) lid-driven cavity flow, (4) flow past a backward step and (5) steady flow past a circular cylinder. Further, a sensitivity study to the various limiter functions has also been carried out.
The main objective of Chapter6is to enhance the order of accuracy of spatio-temporal calculations in the newly presented finite-volume lattice Boltzmann equation formulation. Further, efficient implementation of the formulation for parallel processing is carried out. An appropriate decomposition of the computational domain is performed using a graph partitioning tool. The order-of-accuracy has been verified by simulating a flow past a curved surface. The extended formulation is employed to study more complex unsteady flows past circular cylinders.
In Chapter 7, the main conclusions of this thesis are summarized. Possible issues to be examined for further improvements in the formulation are identified. Further, the potential applications of the present formulation are discussed.
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Simulace proudění tekutiny okolo překážek Lattice Boltzmannovou metodou / Simulation of fluid flow around obstacles by Lattice Boltzmann MethodPrinz, František January 2020 (has links)
The task of this diploma thesis is the Lattice Boltzmann method (LBM). LBM is a mesoscopic method describing the particle motion in a fluid by the Boltzmann equation, where the distribution function is involved. The Chapman-Enskog expansion shows the connection with the macroscopic Navier-Stokes equations of conservation laws. In this process the Hermite polynoms are used. The Lattice Boltzmann equation is derived by the discretisation of velocity, space and time which is concluding to the numerical algorithm. This algorithm is applied at two problems of fluid flow: the two-dimensional square cavity and a flow arround obstacles. In both cases were the results of velocities compared to results calculated by finite volume method (FVM). The relative errors are in order of multiple 1 %.
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