Erosion has a long history in science and is used in many different fields today, for example in geology for coastal erosion and in the oil industry for pipe erosion. It is very difficult to study erosion both analytically. Numerically it is difficult due to moving and shape-changing boundaries. Here we develop a numerical model in 3D using the Lattice-Boltzmann method, which is good at simulating complex moving boundaries, and erosion capabilities are implemented. Both laminar and turbulent flow can be modelled with this program. Using an experimentally derived model for the mass change due to erosion in clay and mud-type objects, one can derive equations predicting that the volume of a sphere should, due to erosion, scale as V ∼ −t2. This is also observed with simulations. The shapes of a double sphere with different orientations and a cube in laminar flow we find to have similar power law exponent P, P = 2±0.1. But a cube eroding in Re = 800 had no power law behaviour, meaning that the current analytical framework is incomplete. The possibility of a more general framework is presented for future research. Different Reynolds number also affected the power law behaviour and the shape change over time for the different solids. Very little research has been made for erosion of planetesimals, but it has been argued that erosion can be relevant to their fate. Using the same erosion model, an equation of the erosion time is found for laminar flows and for a sphere. Simulation results find that the equation works within an order of magnitude for turbulent flows, a double sphere and a cube. This gives an estimate of the erosion time t∗ of planetesimals to be t∗ ∼ 1s, given a size of radius equal to 10cm and 1km, an orbital eccentricity e > 10−2 and a distance at r = 1 a.u. Implying that orbits for planetesimals with low eccentricity might be favoured.

The fluid-structure interactions of wall-mounted slender structures, such as cilia, filaments, flaps, and flags, play an important role in a broad range of physical processes: from the coherent waving motion of vegetation, to the passive flow control capability of hair-like surface coatings. While these systems are ubiquitous, their coupled nonlinear response exhibits a wide variety of behaviours that is yet to be fully understood, especially when multiple structures are considered. The purpose of this work is to investigate, via numerical simulation, the fluid-structure interactions of arrays of slender structures over a range of input conditions. A direct modelling approach, whereby the individual structures and their dynamics are fully resolved, is realised via a lattice Boltzmann-immersed boundary model, which is coupled to two different structural solvers: an Euler-Bernoulli beam model, and a finite element model. Results are presented for three selected test cases - which build in scale from a single flap in a periodic array, to a small finite array of flaps, and finally to a large finite array - and the key behaviour modes are characterised and quantified. Results show a broad range of behaviours, which depend on the flow conditions and structural properties. In particular, the emergence of coherent waving motions are shown to be closely related to the natural frequency of the array. Furthermore, this behaviour is associated with a lock-in between the natural frequency of the array and the predicted frequency of the fluid instabilities. The original contributions of this work are: the development and application of a numerical tool for direct modelling of large arrays of slender structures; the characterisation of the behaviour of slender structures over a range of input conditions; and the exposition of key behaviour modes of slender structures and their relation to input conditions.

Les méthodes de Boltzmann sur réseaux (LBM) ont été appliquées avec beaucoup de succès aux écoulements hydrodynamiques en milieux poreux. Cependant, la limitation de ces méthodes aux écoulements hydrodynamiques et isothermes, les rendent insuffisantes pour simuler des écoulements de gaz dans des milieux micro-poreux. Dans ce cas, il est en effet fréquent que le libre parcours moyen des molécules du gaz, soit du même ordre de grandeur que la taille des pores dans lesquels il s’écoule. De tels écoulements ne seront alors plus en régime hydrodynamique, mais dans des régimes qualifiés de glissement et de transitionnel ; régimes pour lesquels les LBM standards ne sont plus valides. D’autre part, le caractère isotherme des LBM les rendent inutilisables, par exemple dans le cas où le gaz subit une détente à travers le milieu. Il est nécessaire, pour décrire de tels écoulements et phénomènes, de se placer au niveau cinétique. La démarche proposée repose sur la décomposition de la fonction de distribution sur la base des polynômes d’Hermite et l’emploi de la quadrature de Gauss-Hermite associée à cette projection. L’aspect systématique de ce développement amène naturellement à considérer divers ordres d’approximation de l’équation de Boltzmann-BGK sous diverses quadratures. Il résulte alors de ces différentes approximations toute une famille de discrétisations de l’équation de Boltzmann-BGK, dont les LBM classiques ne sont qu’un membre. La détermination de l’approximation la plus adaptée est réalisée par analyse systématique des résultats obtenus aux différents ordres d’approximation. Ces méthodes sont testées avec succès dans des cas modèles. / The lattice Boltzmann method (LBM) have been applied very successfully to hydrodynamic flows in porous media. However, the limitation of these methods to isothermal and hydrodynamic flows, make them inadequate to simulate gas flows in micro-porous media. Indeed, in these conditions, the mean free path of the molecules could be of the same magnitude order as the pore size in which gas flows. Such flows will not be in hydrodynamic regime, but in regimes qualified of, slip or transitional ; for which the LBM are no longer valid. On the other hand, the isothermal character of LBM make them unusable, for example, in the case where the gas undergoes expansion through the media. It is then necessary, to take the kinetic point of view to describe such flows and phenomena. The proposed approach is based on the decomposition of the distribution function on the Hermite polynomials basis and the use of Gauss-Hermite quadrature associated with this projection. The systematic nature of this development naturally leads to consider different order of approximation of the Boltzmann-BGK equation in various quadratures. It then follows from these various approximations, a family of discretizations of the Boltzmann-BGK equation, whose classical LBM are a member. Determining the most suitable approximation is achieved by systematic analysis of the results obtained with different approximation orders. These methods are successfully tested in model cases.

The lattice Boltzmann (LB) method is a versatile way to model complex fluids with hydrodynamic interactions through solving the Navier-Stokes equations. It is well-known that the role of hydrodynamic interactions is ignorable in studying the Boltzmann equilibrium of colloidal (Brownian) particles. However, full hydrodynamic interactions play an important role in their dynamics. In the LB framework for moving colloids, the “bounce-back on links” method is used to calculate the hydrodynamic forces. In this thesis, three kinds of colloidal complex fluids with full hydrodynamic interactions are simulated by lattice Boltzmann methods: colloids in a binary fluid, magnetic colloids in a single fluid and magnetic colloids in a binary fluid. First, we have done extensive simulations of nanoparticles in a binary fluid, following up previous work[1] which predicted formation of a “bijel” (bicontinuous interfacially jammed emulsion gel) in symmetric fluid quenches. Our work in this thesis focuses on the analysis of the dynamics after nanoparticles become arrested on the fluid-fluid interfaces under conditions varying from a symmetric quench to a strongly asymmetric quench. Although these new simulations extend the time window studied by a factor of two, slow domain growth is still observed. Our new analyses address the mechanics of the slow residual dynamics which involves cooperative motion of the nanoparticles at the fluid-fluid interfaces. The second topic is the LB simulation of colloidal ferrofluids to see the effect of full hydrodynamic interactions among magnetic colloids. The main focus is on how the hydrodynamic interaction affects both the equilibrium dynamics of these dipolar systems and also their transient dynamics to form clusters. Numerically, magnetic colloids are implemented with the long-range dipolar interactions described by Ewald summation. To check the effect of full hydrodynamic interactions, Brownian dynamics without any hydrodynamic interaction has been done for comparison: Monte Carlo results are also reported. We confirm that our LB generates the Boltzmann distribution for static equilibrium properties, by comparison with these methods. However, the equilibrium dynamics is altered: hydrodynamic interactions make the structural relaxations slower in both the short-time and the long-time regime. This slow relaxation rate is also found for transient motions. The third topic addresses magnetic colloids in a binary fluid. In contrast with the preceding two systems which correspond directly to laboratory experiments, this last system is so far only predicted by the LB results in this thesis. To explore this hypothetical new material by the LB method, the basic structures are investigated in terms of both domain growth morphology and the arrangement of magnetic colloids. Under conditions varying from a symmetric quench to an asymmetric quench, a chainlike arrangement is observed for dipoles jammed on the surfaces, but the basic morphology of domains is still maintained regardless of the dipolar strength. In addition, applying external field affects the morphology of domains and the stability of domain structures.

The main sources of airframe noise in commercial aircrafts are the landing gear and the highlift devices. Among the high-lift devices, the slat deserves special attention since it represents a distributed source along the wing span. During approach and landing the noise generated by the slat can be comparable to the engine generated noise. For the design of quieter high-lift systems, it is important to understand the physics responsible for the slat noise generation. The objective of the work described in this thesis is to correlate the dynamics of large scale turbulent structures at different airfoil configurations with the characteristics of the noise generated by these structures. Four different configurations were investigated, ranging two airfoil angles of attack and three slat positions relative to the main element. The unsteady flow data was provided by a Lattice-Boltzmann based computational code. The Proper Orthogonal Decomposition technique was used for the objective identification of large scale structures in the slat region. Two different metrics were considered for the eduction of the coherent structures: one based on the Turbulent Kinetic Energy of the structures; and one based on their correlation to the noise emitted by the slat. The results of the transient simulations showed good agreement with wind tunnel measurements, providing confidence on the relevance of the analysis. The noise spectra of three of the cases simulated were dominated by a series of narrowband peaks at low frequency, while the spectrum of the remaining case was broadband in nature. Analysis of the averaged flow showed a large variation of the size and shape of the recirculating zone inside the slat cove and on the reattachment position of the mixing layer, between the simulated cases. The results indicated that, as the reattachment point approximates the region of the gap between the slat and the main element, the noise emission power increases. The large scale structures most correlated to the noise were typically two-dimensional and their shape suggests they resulted from the growth of disturbances in the mixing layer due to the inflectional instability. The dynamics of the noise correlated structures at the frequencies of the peaks was consistent with the existence of an acoustic feedback mechanism acting inside the slat cove. Based on the observation of the educed structures a model to predict the frequencies of peaks was proposed, showing good agreement with the frequencies computed from the unsteady flow data. / As principais fontes de ruído não propulsivo em aeronaves comerciais são os trens-de-pouso e os dispositivos híper-sustentadores. Entre os dispositivos híper-sustentadores, o eslate se destaca por constituir uma fonte distribuída ao longo da envergadura da asa. Durante a fase de aproximação e aterrissagem, o eslate pode gerar ruído com níveis comparáveis ao gerado pelos motores. Para viabilizar projetos de aerofólios com eslates menos ruidosos, é importante compreender os fenômenos fluidodinâmicos responsáveis pela geração desse ruído. O trabalho descrito neste texto tem por objetivo verificar se existe correlação entre o comportamento de grandes estruturas turbulentas em diferentes configurações do aerofólio com as características do ruído aeroacústico gerado por elas. O escoamento em quatro configurações diferentes foi estudado, abrangendo dois ângulos de ataque e três posições do eslate em relação ao elemento principal. Os dados do escoamento para análise foram gerados através de simulações numéricas não estacionárias utilizando um código comercial baseado no Método Lattice-Boltzmann. O método da Decomposição Ortogonal Apropriada foi utilizado para a identificação das estruturas de grande escala baseada em critérios objetivos. Duas métricas distintas foram utilizadas, uma baseada na energia cinética turbulenta e outra baseada na correlação com as ondas acústicas geradas a partir do eslate. Os resultados das simulações transientes apresentaram boa concordância com resultados experimentais. O espectro de ruído de três casos simulados são dominados por picos de baixa frequência, enquanto o espectro do quarto caso é tipicamente de banda larga. A análise do escoamento indica uma tendência de aumento do ruído à medida que o recolamento se aproxima do bordo de fuga do eslate. As estruturas mais correlacionadas com o ruído são tipicamente bi-dimensionais e seu formato indica que são resultado do crescimento de perturbações na camada de mistura devido à instabilidade inflexional. A dinâmica das estruturas correlecionadas com o ruído na frequência dos picos é consistentes com a existência de uma retro-alimentação das perturbações da camada de mistura por ondas acústica na cova do eslate. Um modelo para previsão das frequências dos picos foi proposto a partir da observação das estruturas identificadas pela Decomposição Ortogonal Apropriada, mostrando boa concordância com as frequências observadas nos espectros calculados com base nos dados transientes das simulações.

L'objectif de cette thèse est de développer et d'analyser des techniques numériques basées sur la méthode e Boltzmann sur réseau (LBM) pour résoudre des systèmes non linéaires de type advection-réaction-diffusion provenant de la physique et de la biologie. Avec la LBM, des problèmes portant sur des quantités moyennées densité, potentiel, vitesse, etc) sont exprimés à l'échelle particulaire. Nous approchons la solution de l'équation e Boltzmann relative au comportement d'un champs de particules puis nous recomposons les quantités moyennées solutions des équations traitées. Dans un premier temps, nous développons un cadre général approprié permettant de traiter plusieurs types de systèmes non linéaires (paraboliques, elliptiques, ou couplées ' variables réelles ou complexes), avec des applications à des modèles tels que Burger-Fisher, écoulement de fluides en milieu poreux, Helmoltz, Patlar-Keller-Segel, ou encore Schrodinger. Pour chaque problème, nous analysons le comportement asymptotique de la méthode, quand le nombre de Knudsen tend vers zéro (par le développement de Chapman-Enskog) et nous effectuons l'analyse numérique de la convergence et de la stabilité de la méthode. Dans un deuxième temps, nous nous intéressons à un problème réaliste d'électrophysiologie cardio-vasculaire. Nous adaptons la méthode LBM développée pour approcher les solutions d'un système de type bidomaine permettant de simuler le comportement de potentiels électriques et les interactions ioniques ans la région du myocarde. L'étude et la modélisation d'un tel type de problème est un enjeu sanitaire majeur ans le traitement des pathologies liées par exemple à l'arythmie cardiaque. Notre but étant d'obtenir des comportements réalistes, nous introduisons au sein de ce système bidomaine des opérateurs de retard afin de tenir compte des temps de retard dans les transmissions de signaux. Une fois l'existence et l'unicité de la solution démontrées, nous proposons une série de simulations avec des paramètres physiques et biologiques réalistes afin de valider la méthode proposée. / In this thesis, we develop and analyze numerical techniques based on the lattice Boltzmann method LBM) for solving systems of nonlinear advection-diffusion-reaction equations from physics and biology. Wi BM, problems relating to averaged quantities (density, potential, velocities, etc.) are expressed at the particle scale. We approach the solution of Boltzmann equation relating to the behavior of a particle field and then we recompose the averaged quantities solutions of treated systems. In the first part, we develop an appropriate general framework to deal with several types of non-linear systems (parabolic, elliptic, or coupled, with real or complex variables), with applications to models such as Burger-Fisher, fluid flow in a porous medium, Helmoltz, Patlar-Keller-Segel, or Schrodinger. For each problem, we analyze the asymptotic behavior of the method, when the number of Knudsen tends to zero (by the development of Chapman-Enskog) and we perform the numerical analysis of convergence and stability of the method. In the second part, we have taken an interest in a realistic problem of cardio-vascular electrophysiology. We adapt the developed LBM method to approach e solutions of a bidomain type system for simulating the behavior of electrical potentials and ionic interactions in myocardial region. The study and modeling of this type of problem is a major health issue in the treatment of pathologies related, for example, to cardiac arrhythmia. Since our goal is to obtain realistic behaviors, we introduce time-delay operators into this coupled system in order to take into account delay in signal transmissions. Once the existence and uniqueness of solution have been demonstrated, we propose a series of simulations with realistic physical and biological parameters to validate the proposed method.

In this thesis work fluid flow and heat transfer in two-dimensional microchannels are studied numerically. A computer code based on Lattice Boltzmann Method (LBM) is developed for this purpose. The code is written using MATLAB and Jacket software and has the important feature of being able to run parallel on Graphics Processing Units (GPUs). The code is used to simulate flow and heat transfer inside micro and macro channels. Obtained velocity profiles and Nusselt numbers are compared with the Navier-Stokes based analytical and numerical results available in the literature and good matches are observed. Slip velocity and temperature jump boundary conditions are used for the micro channel simulations with Knudsen number values covering the slip flow regime. Speed of the parallel version of the developed code running on GPUs is compared with that of the serial one running on CPU and for large enough meshes more than 14 times speedup is observed.

Articular cartilage has a limited ability to heal due to its avascular, aneural, and alymphatic nature. Currently, there is a need for alternative therapies for diseases that affect articular cartilage such as osteoarthritis. Recently, it has been shown that tissue constructs, which resemble cartilage in structure and function, can be cultured in vitro in a cell-polymer bioreactor system. Bioreactors provide a three dimensional environment that promotes cell proliferation and matrix production. The primary objective of this study is to accurately simulate fluid mechanics using the lattice Boltzmann method for application to a cell-polymer bioreactor system. Lattice Boltzmann (LB) is a flexible computation technique that will allow for the simulation of a moving construct under various bioreactor conditions. The method predicts macroscopic hydrodynamics by considering virtual particle interactions. Derived from the Lattice Gas Automata, lattice Boltzmann allows for mass transfer, complex geometries, and particle dynamics. A primary goal is to characterize the accuracy of the LB implementation and eventually the shear stresses felt by a tissue construct in this dynamic environment. This information is important since recent studies show that chondrocytic function may depend on the mechanical stimuli produced by fluid flow. Hence, shear stress may affect the final mechanical properties of tissue constructs. In this study, numerical simulations are done first in 2D and then extended to 3D to test the LB implementation. Simulations of the rotating wall vessel (RWV) bioreactor are then undertaken. The results are benchmarked against computations done with a commercial CFD package, FLUENT, and compared with analytic solutions and experimental data.

In this dissertation, we explore direct-forcing immersed boundary methods (IBM) under the framework of the lattice Boltzmann method (LBM), which is called the direct-forcing immersed boundary-lattice Boltzmann method (IB-LBM). First, we derive the direct-forcing formula based on the split-forcing lattice Boltzmann equation, which recovers the Navier-Stokes equation with second-order accuracy and enables us to develop a simple and accurate formula due to its kinetic nature. Then, we assess the various interface schemes under the derived direct-forcing formula. We consider not only diffuse interface schemes but also a sharp interface scheme. All tested schemes show a second-order overall accuracy. In the simulation of stationary complex boundary flows, we can observe that the sharper the interface scheme is, the more accurate the results are. The interface schemes are also applied to moving boundary problems. The sharp interface scheme shows better accuracy than the diffuse interface schemes but generates spurious oscillation in the boundary forcing terms due to the discontinuous change of nodes for the interpolation. In contrast, the diffuse interface schemes show smooth change in the boundary forcing terms but less accurate results because of discrete delta functions. Hence, the diffuse interface scheme with a corrected radius can be adopted to obtain both accurate and smooth results. Finally, a direct-forcing immersed boundary method (IBM) for the thermal lattice Boltzmann method (TLBM) is proposed to simulate non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection-diffusion equation of temperature (Model 2). The proposed methods are validated through natural convection problems with stationary and moving boundaries. In terms of accuracy, the results obtained from the IBMs based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1. Overall, this study serves to establish the feasibility of the direct-forcing IB-LBM as a viable tool for computing various complex and/or moving boundary flow problems.

We explore the capability of lattice Boltzmann equation (LBE) method for complex fluid flows involving turbulence, mixing, and reaction. In the first study, LBE schemes for binary scalar mixing and multi-component reacting flow with reactions are developed. Simulations of initially non-premixed mixtures yield scalar probability distribution functions that are in good agreement with numerical data obtained from Navier-Stokes (NS) equation based computation. One-dimensional chemically-reacting flow simulation of a premixed mixture yields a flame speed that is consistent with experimentally determined value. The second study involves direct numerical simulation (DNS) and large-eddy simulation (LES) of decaying homogenous isotropic turbulence (HIT) with and without frame rotation. Three categories of simulations are performed: (i) LBE-DNS in both inertial and rotating frames; (ii) LBE-LES in inertial frame; (iii) Comparison of the LBE-LES vs. NS-LES. The LBE-DNS results of the decay exponents for kinetic energy k and dissipation rate ε, and the low wave-number scaling of the energy spectrum agree well with established classical results. The LBE-DNS also captures rotating turbulence physics. The LBE-LES accurately captures low-wave number scaling, energy decay and large scale structures. The comparisons indicate that the LBE-LES simulations preserve flow structures somewhat more accurately than the NS-LES counterpart. In the third study, we numerically investigate the near-field mixing features in low aspect-ratio (AR) rectangular turbulent jets (RTJ) using the LBE method. We use D3Q19 multiple-relaxation-time (MRT) LBE incorporating a subgrid Smagorinsky model for LES. Simulations of four jets which characterized by AR, exit velocity, and Reynolds number are performed. The investigated near-field behaviors include: (1) Decay of mean streamwise velocity (MSV) and inverse MSV; (2) Spanwise and lateral profiles of MSV; (3) Half-velocity width development and MSV contours; and (4) Streamwise turbulence intensity distribution and spanwise profiles of streamwise turbulence intensity. The computations are compared against experimental data and the agreement is good. We capture both unique features of RTJ: the saddle-back spanwise profile of MSV and axis-switching of long axis from spanwise to lateral direction. Overall, this work serves to establish the feasibility of the LBE method as a viable tool for computing mixing, combustion, and turbulence.