The research work of the present thesis was mainly aimed at exploiting one of the
strengths of the Lattice Boltzmann methods, namely, the ability to handle complicated
geometries to accurately simulate flows in complex geometries. In this thesis, we perform
a very detailed theoretical analysis of the finite volume unstructured lattice Boltzmann
method (ULBM) in three dimensions, considering the Bhatnagar-Gross-Krook (BGK)
relaxation time approximation for the collision operator, one of the more commonly used
by the community. Regarding this scheme, two time integration methods are considered
and through the Chapman-Enskog multi-scale expansion technique the dependence of the
kinetic viscosity on each scheme is investigated. Seeking for optimal numerical schemes
to eciently simulate a wide range of complex flows a variant of the finite element,
off-lattice Boltzmann method [5], which uses the characteristic based integration is also
implemented. Using the latter scheme, numerical simulations are conducted in flows of
different complexities: flow in a (real) porous network and turbulent flows in ducts with
wall irregularities.
From the simulations of flows in porous media driven by pressure gradients, anomalous
transport features of Lagrangian trajectories are investigated. Several statistical
properties of both Lagrangian and Eulerian velocities are also examined. Based on these
measurements, an eective model is considered to assess the role of the pressure gradient
on the transport of Lagrangian tracers. In the calculations simple statistical arguments
are used, based on the approach developed by [12, 64]. Such formalisms contain as
a fundamental tool the theory of Levy stable distributions[23, 14]. It is found that
the pressure gradient induces a superdiusive behavior along the mean motion while a
subdiusive scaling is observed in the orthogonal directions.
Numerical simulations of turbulent flows in ducts with irregular walls are also implemented,
as mentioned above. Preliminary results regarding the characterization of
the turbulent energy landscape motivated the development of a phenomenological model
to describe the spatial probability density function of the turbulent kinetic energy
uctuations.
Closely following the ideas recently devoloped in [68], the proposed model
combines recent ndings on the spatial proliferation mechanisms of turbulent spots [7],
with Townsend attached eddy hypothesis [63], [68]. Preliminary results obtained from
the numerical experiments are compared with predictions of the model. Also, some statistical
features of the turbulent kinetic energy production term are illustrated. Some
concluding remarks are given in the last part of the thesis in which main ndings of this
PhD thesis are discussed.