Entropic Lattice Boltzmann Models for Fluid Dynamics

Abstract

The lattice Boltzmann method (LBM) is a modern and highly successful approach to computational fluid dynamics based on a fully discrete kinetic equation and offers an attractive alternative to direct discretizations of the macroscopic continuum equations. However, the original single relaxation-time formulation (LBGK) was plagued by numerical instabilities and prevented the simulation of highly turbulent flows unless prohibitively high resolutions were employed. A number of improvements and variations of LBM attempted to alleviate this issue. In particular, the multiple relaxation-times methods (MRT) take advantage of the additional degrees of freedom in the LBM kinetic system in order to stabilize the solution and improve accuracy. However, the introduction of additional tunable constants must be chosen appropriately depending on the physics and the flow at hand and are not universal, and thus, the problem of stability could not be solved consistently. With the inception of the entropic lattice Boltzmann method (ELBM) which reintroduced the discrete-time equivalent of Boltzmann's H-theorem, the application range has grown widely not only for incompressible turbulent flows, but also for thermal flows, two-phase systems with non-ideal equations of state and has made the simulation of compressible and high Mach number flows possible. The ELBM, however, comes at the price of introducing a fluctuating viscosity. The main result of this thesis is the development of a new class of entropic MRT models which combine the salient advantages from MRT and ELBM while trying to circumvent their respective disadvantages. J. W. Gibbs' seminal prescription for constructing optimal states by maximizing the entropy under pertinent constraints is used to derive a novel lattice kinetic theory and the the notion of modifying the viscosity to stabilize sub-grid simulations is challenged in this kinetic framework. By exploiting the degrees of freedom of MRT and by compliance to the discrete-time H-theorem, the advantages of both MRT and ELBM can be retained and the problems of parameter tuning and fluctuating viscosity can be solved. The resulting models are accurate, adaptive, parameter-free, efficient and, more importantly, very stable The entropic MRT models are studied extensively for two- and three- dimensional benchmark simulations at turbulent conditions and compared to experiments, theory and other numerical methods. While the entropic MRT models yield the same results as the LBGK in the resolved limit, and converge towards the Navier-Stokes equations with second-order accuracy, sufficiently accurate results are obtained also at coarse resolutions without the use of sub-grid turbulence models. This made the simulation of high Reynolds number flows with complex boundaries possible which are most relevant for engineering applications. Due to the general nature of the entropic MRT methods, a number of extensions beyond the incompressible low Mach number regimes were explored and extensions for thermal flows and two-phase flows are presented. Moreover, a novel model for miscible binary mixtures is presented in combination with the entropic MRT collision step. The salient features of this model are the lack of interpolation and the use of thermal multi-speed lattices to account for the different sound speed of the two unequally heavy substances involved. Independent adjustment of the diffusion coefficient and kinematic viscosity of the mixture is assured and the diffusion follows the Stefan-Maxwell model. To conclude it is shown that entropy maximisation principle can be extended to multiple relaxation-time lattice Boltzmann models to create more stable and computationally efficient LBM models. This principle of entropy maximization is applied without nominal modification of viscosity and is extended to not only high Reynolds number flows but also thermal, mixture and multi-phase flows. This principle is also shown to be lattice independent and generic nature thus opening a possibility of new class of LB models that are more robust to under-resolution and have a wide range of applications.