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.

This thesis deals with the lattice-Boltzmann method (LBM) in combination with other methods to solve thermal flow problems. The three primary, current approaches for thermal lattice-Boltzmann method (TLBM) will be introduced. The three approaches are the multispeed approach by McNamara and Alder , the passive scalar approach by Shan, and the thermal distribution model proposed by He et al. Shi et al. simplified the thermal distribution model for incompressible thermal flows. The model proposed by Shi et al. was simulated and compared to a hybrid LBM and energy equation model proposed by Khiabani et al. The thermal lattice-Boltzmann method will be compared to the temperature fields generated by the energy equation of the hybrid method. To determine which method is better suited from computer simulations the two will be compared for computational demands, and the speed of both convergence and computation.

The dynamics of fiber orientation is of great interest for efforts to predict the microstructure and material properties of a suspension flow system. In this research a fiber-level, hybrid simulation method, LBM‒EBF (coupled lattice‒Boltzmann method with the external boundary force method) is undertaken to advance the current understanding of the hydrodynamic interaction induced rotational diffusion mechanism for rigid fibers in semidilute suspension of low Reynolds number flow. The LBM‒EBF simulations correctly predict the orbit constant distribution of fibers in a sheared semidilute suspension flow. It is demonstrated that an anisotropic, weak rotary diffusion model can fit the orbit constant distribution very well, but it can not describe the asymmetry in Stokes flow observed in semidilute suspension. The rotational diffusion process is then characterized with a three dimensional spatial tensor representation of the rotational diffusivity. A scalar measure of the rotational diffusion‒'scalar Folgar‒Tucker constant', C[subscript I], is extracted from this tensor. The study provides substantial numerical evidence that the range of C[subscript I] (0.0038 to 0.0165) obtained by Folgar&Tucker (J. reinf. plast. and comp, v.3, 1984) in a semidilute regime is overly diffusive, and that the correct magnitude is of O(10⁻⁴). The study reveals that the interactions among fibers become more frequent with either the decrease of fiber aspect-ratio, r[subscript p] (keeping nL³ constant, where n is the fiber number density, and L is the fiber length) or with the increase of nL³ (keeping r[subscript p] constant) in the semidilute regime, which in consequence causes an increase in C[subscript I]. The rheological properties of sheared semidilute suspension are also computed with direct LBM‒EBF simulations. The LBM‒EBF investigation is extended to characterize the fiber orientation in a linearly contracting channel similar to a paper machine 'headbox'. It is found that the rotational diffusion is the predominant term over the strain rate in the semidilute regime for a low Reynolds number flow, and it results in a decreasing trend of rotational Peclet number, Pe, along the contraction centerline. Lastly, in order to improve the numerical consistency of the existing LBM‒EBF approach, a modification to the body force term in the LB equation is suggested, which can recover the exact macroscopic hydrodynamics from the mesoscale.

To develop or engineer artificial tissues in tissue engineering, a detailed knowledge of the in vitro culture process including cell and tissue growth inside porous scaffolds, nutrient transport, and the shear stress acting on the cells is of great advantage. It has been shown that obtaining such information by means of experimental techniques is exceedingly difficult and in some ways impossible. Mathematical modelling and computational simulation based on computational fluid dynamics (CFD) has emerged recently to be a promising tool to characterize the culture process. However, due to the complicated structure of porous scaffolds, modelling and simulation of the in vitro cell culture process has been shown to be a challenging task. Furthermore, due to the cell growth during the culture process, the geometry of the scaffold structure is not constant, but changes with time, which makes the task even more challenging. To overcome these challenges, the research presented in this thesis is aimed at developing a CFD-based mathematical model and multi-time scale computational framework for culturing cell-scaffold constructs placed in perfusion bioreactors. To predict the three-dimensional (3D) cell growth in a porous tissue scaffold placed inside a perfusion bioreactor, a model is developed based on the continuity and momentum equations, a convection-diffusion equation and a suitable cell growth equation, which characterize the fluid flow, nutrient transport and cell growth, respectively. To solve these equations in a coupled fashion, an in-house FORTRAN code is developed based on the multiple relaxation time lattice Boltzmann method (MRT LBM), where the D3Q19 MRT LBM and D3Q7 MRT LBM models have been used for the fluid flow and mass transfer simulation, respectively. In the model cell growth equation, the transport of nutrients, i.e. oxygen and glucose, as well as the shear stress induced on the cells are considered for predicting the cell growth rate. In the developed model and computational framework, the influence of the dynamic strand surface on the local flow and nutrient concentration has been addressed by using a two-way coupling between the cell growth and local flow field and nutrient concentration, where a control-volume method within the LBM framework is applied. The simulation results provide quantification of the biomechanical environment, i.e. fluid velocity, shear stress and nutrient concentration inside the bioreactor. The final simulation applied the cell growth model to the culture of a three-zone tissue scaffold where the scaffold strands were initially seeded with cells. The prediction for the 3D cell growth rate indicates that the increase in the cell volume fraction is much higher in the front region of the scaffold due to the higher nutrient supply. The higher cell growth in the front zone reduces the permeability of the porous scaffold and significantly reduces the nutrient supply to the middle and rear regions of the scaffold, which in turn limit the cell growth in those regions. However, implementation of a bi-directional perfusion approach, which reverses the flow direction for second half of the culture period, is shown to significantly improve the nutrient transport inside the scaffold and increase the cell growth in the rear zone of the scaffold. The results in this study also demonstrate that the developed mathematical model and computational framework are capable of realistically simulating the 3D cell growth over extended culture periods. As such, they represent a promising tool for enhancing the growth of tissues in perfusion bioreactors.

Particle suspensions are common both in nature and in various technological applications. The complex nature of hydrodynamic interactions between particles and the solvent makes such analysis difficult that often requires numerical modeling to understand the behavior of particle suspensions. In this dissertation, we employ a hybrid computational model that integrates a lattice spring model for solid mechanics and a lattice Boltzmann model for fluid dynamics. We use this model to study several practical problems in which the dynamics of spherical and spheroidal particles and deformable capsules in dilute suspensions plays an important role. The results of our studies yield new information regarding the dynamics of solid particle in pressure-driven channel flows and disclose the nonlinear effects associated with fluid inertia leading to particle cross-stream migration. This information not only give us a fundamental insight into the dynamics of dilute suspensions, but also yield engineering guidelines for designing high throughput microfluidic devices for sorting and separation of synthetic particles and biological cells. We first demonstrate that spherical particles can be size-separated in ridged microchannels. Specifically, particles with different sizes follow distinct trajectories as a result of the nonlinear inertial effects and secondary flows created by diagonal ridges in the channel. Then, separation of biological cells by their differential stiffness is studied and compared with experimental results. Cells with different stiffness squeeze through narrow gaps between solid diagonal ridges and channel wall, and migrate across the microchannel with different rates depending on their stiffness. This deformability-based microfluidic platform may be valuable for separating diseased cells from healthy cells, as a variety of cell pathologies manifest through the change in mechanical cell stiffness. Finally, the dynamics of spheroid particles in simple shear and Poiseuille flows are studied. Stable rotational motion, cross-stream migration, and equilibrium trajectories of non-spherical particles in flow are investigated. Effects of particle and fluid inertia on dynamics of particles are disclosed. The dependence of equilibrium trajectory on particle shape reveals a potential application for shape based particle separation.

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