This manuscript demonstrates the well-posedness (existence, uniqueness, and regularity of solutions) of the Cauchy problem for simplified equations of nematic liquid crystal hydrodynamic flow in three dimensions for initial data that is uniformly locally L3(R3) integrable (L3U(R3)). The equations examined are a simplified version of the equations derived by Ericksen and Leslie. Background on the continuum theory of nematic liquid crystals and their flow is provided as are explanations of the related mathematical literature for nematic liquid crystals and the Navier–Stokes equations.

Two-dimensional flow in a viscous incompressible fluid, generated by a circular cylinder executing large-amplitude rectilinear oscillations in a plane perpendicular to its axis and parallel to one of the sides of a surrounding rectangular box filled with incompressible fluid is studied numerically. The circular cylinder moves back and forth through its own wake, resulting in an extremely complex flow field. For ease of implementing boundary conditions, a numerically generated body-fitted coordinate system is used. At each time step, the physical domain is doubly-connected, and a cut is introduced in order to map it into a rectangular computational domain. A body-fitted grid is generated by solving a pair of Laplace equations with a simple grid spacing control method which preserves the essential one-to-one property of the mapping. A finite difference/pseudo-spectral technique is used in this work to solve the Navier-Stokes equations in velocity-vorticity formulation. The time integration of the vorticity transport equation is handled by a fully explicit three-level Adams-Bashforth method. The two Poisson equations for the velocity components are 11-banded and block-diagonal in form, and are solved by a preconditioned biconjugate gradient routine. An integral constraint on the vorticity field is used to determine the boundary vorticity that simultaneously satisfies the no-slip and no-penetration conditions. The surface vorticity is uniquely determined by a general solution procedure developed in this study which is valid for flows over multiple solid bodies. With this approach, the physical process of vorticity generation on the solid boundary is properly simulated and the principle of vorticity conservation is satisfied. Results for various test cases and the complex vortex shedding phenomena generated by an oscillating circular cylinder are presented and discussed.

For simulations of highly complex geometries, frequently encountered in many fields of science and engineering, the process of generating a high-quality, body-fitted grid is very complicated and time-intensive. Thus, one of the principal goals of contemporary CFD is the development of numerical algorithms, which are able to deliver computationally efficient, and highly accurate solutions for a wide range of applications involving multi-physics problems, e.g. Fluid Structure Interaction (FSI). Immersed interface/boundary methods provide considerable advantages over conventional approaches, especially for flow problems containing moving boundaries.In the present work, a novel, robust, highly-accurate, Immersed Interface Method (IIM) is developed, which is based on a local Taylor-series expansion at irregular grid points enforcing numerical stability through a local stability condition. Various immersed methods have been developed in the past; however, these methods only considered the order of the local truncation error. The numerical stability of these schemes was demonstrated (in a global sense) by considering a number of different test-problems. None of these schemes used a concrete local stability condition to derive the irregular stencil coefficients. This work will demonstrate that the local stability constraint is valid as long as the DFL-number does not reach a limiting value. The IIM integrated into a newly developed Incompressible Navier-Stokes (INS) solver is used herein to simulate fully coupled FSI problems. The extension of the novel IIM to a higher-order method, the compressible Navier-Stokes equations and the Maxwell's equations demonstrate the great potential of the novel IIM.In the second part of this dissertation, the newly developed INS solver is employed to study the flow of a stalled airfoil and steady/unsteady stenotic flows. In this context, a new biglobal stability analysis approach based on solving an Initial Value Problem (IVP), instead of the traditionally used EigenValue Problem (EVP), is presented. It is demonstrated that this approach based on an IVP is computationally less expensive compared to EVP approaches while still capturing the relevant physics.

No description available.

Stably stratified flows are common in the environment such as in the atmospheric· boundary layer, the oceans, lakes and estuaries. Understanding mixing and dispersion in these flows is of fundamental importance in applications such as the prediction of pollution dispersion and for weather and climate prediction/models. Mixing efficiency in stratified flows is a measure of the proportion of the turbulent kinetic energy that goes into increasing the potential energy of the fluid by irreversible mixing. This can be important for parameterizing the effects of mixing in stratified flows. In this research, fully resolved direct numerical simulations (DNS) of the Navier-Stokes equations are used to study transient turbulent mixing events. The breaking of internal waves in the atmosphere could be a source of such episodic events in the environment. The simulations have been used to investigate the mixing efficiency (integrated over the duration of the event) as a function of the initial turbulence Richardson number Ri = N2L2/U2, where N is the buoyancy frequency, L is the turbulence length scale, and u is the turbulence velocity scale. Molecular effects on the mixing efficiency have been investigated by varying the Prandtl number Pr = V/K, where v is the viscosity and K is the scalar diffusivity. Comparison of the DNS results with grid turbulence experiments has been carried out. There is broad qualitative agreement between the experimental and DNS results.· However the experiments suggest a maximum mixing efficiency of 6% while our DNS gives values about five times higher. Reasons for this discrepancy are investigated. The mixing efficiency has also been determined using linear theory. It is found that the results obtained for the very stable cases converge on those obtained from DNS suggesting that strongly stratified flows exhibit linear behaviour. Lagrangian analysis of mixing is fundamental in understanding turbulent diffusion and mixing. Dispersion models such as that of Pearson, Puttock & Hunt (1983) are based on a Lagrangian approach. A particle-tracking algorithm (using a cubic spline interpolation scheme following Yeung &Pope, 1988) was developed and incorporated into the DNS code to enable an investigation into the fundamental aspects of mixing and diffusion from a Lagrangian perspective following fluid elements. From the simulations, the ensemble averaged rate of mixing as a function of time indicates clearly that nearly all the mixing in these flows occurs within times of order 3 Vu. The mean square vertical displacement statistics show how the stable stratification severely inhibits the vertical displacement of fluid elements but has no effect on displacements in the transverse direction. This is consistent with the Pearson, Puttock & Hunt model. The important link that asymptotic value of the mean square vertical displacement is a measure of the total irreversible mixing that has occurred in the flow is made. However the results show that the change in density of the fluid elements is only weakly correlated to the density fluctuations during the time when most of the mixing occurs, which contradicts a key modeling assumption of the PPH theory. Improvements to the parameterization of this mixing are investigated. Flow structures in stably stratified turbulence were examined using flow visualization software. The turbulence structure for strong stratification resembles randomly scattered pancakes that are flattened in the horizontal plane. It appears that overturning motions are the main mechanism by which mixing occurs in these flows. / Thesis (M.Sc.Eng.)-University of Natal, Durban, 2002.

No description available.

This study focuses on numerical modeling of fluid flow and solute transport in rough-walled rock fractures and fracture-matrix systems, with the main aim to investigate the impacts of fracture surface roughness on flow and transport processes in rock fractures. Both 2D and 3D fracture models were built from laser-scanned surface tomography of a real granite rock sample, to consider realistic features of surface tomography and potential asperity contacts. The flow was simulated by directly solving the Navier-Stokes equations (NSE) and the transport was modeled by solving the advection-dispersion equation (ADE) in the entire domain of fracture-matrix system, including matrix diffusion process. Such direct simulations provided detailed flow and concentration fields for quantitatively analysis of flow and transport behavior. The detailed analysis of surface roughness decomposition, complex flow patterns (i.e., channeling, transverse and eddy flows), effective advective flow apertures, effective transmissivity, effective dispersivity, residence time, transport resistance and specific surface area demonstrated significant impacts of realistic fracture surface roughness on fluid flow and solute transport processes in rock fractures. The results show that the surface roughness and shear displacement caused asperity contacts significantly enhance nonlinearity and complexity of flow and transport processes in rough-walled fractures and fracture-matrix systems. The surface roughness also causes invasion flows in intersected fractures which enhance solute mixing at fracture intersections. Therefore, the fracture surface roughness is an important source of uncertainty in application of such simplified models like cubic law (CL) for fluid flow and analytical solutions for solute transport in rock fractures. The research conducted advances our understanding of realistic flow and transport processes in natural fractured rocks. The results are useful for model validation/extension, uncertainty analysis/quantification and laboratory experiments design in the context of various applications related to fracture flow and transport. / Denna studie fokuserar på numerisk modellering av vätskeflöde och transport av lösta ämnen i frakturer med ojämna väggar samt fraktur-matrissystem, med det huvudsakliga syftet att undersöka effekterna av frakturernas ytjämnhet på flödes- och transportprocesser i bergsfrakturer. Både 2D och 3D modeller skapades utifrån laser skannad tomografi av ett verkligt bergartsprov av granit, för att överväga de realistiska egenskaperna hos ytan och potentiell skrovlighet. Flödet simulerades genom att lösa Navier-Stokes ekvationer (NSE) och transporten modellerades genom att lösa advektion-dispersion ekvation (ADE) i hela domänen av fraktur-matrissystemet, inklusive diffusions process i matrisen. Sådana direkta simuleringar resulterade i detaljerade flödes- och koncentrationsfält för att kvantitativt kunna analysera flödet och transportbeteendet. En detaljerad analys av upplösningen av ytjämnhet, komplexa flödesmönster (dvs kanalisering, tvärgående och virvelströmmar), effektiv advektiv flödesöppning, effektiv transmissivitet, effektiv dispersivitet, uppehållstid, transport motstånd och specifik yta visade signifikanta effekter av realistiska ojämna frakturväggar på vätskeflöde och lösta transportprocesser i bergssprickor. Resultaten visar att ytjämnhet och skjuvningssystemsorsakade asperitetskontakter avsevärt förbättrar olinjäritet och komplexitet av flödes- och transportprocesser i frakturer med ojämna väggar samt fraktur-matrissystem. Ytråheten orsakar också intrång av flöde i tvärgående frakturer vilket ökar blandingen av lösta ämnen i korsningarna. Därför är ytjämnhet av frakturerna en viktig källa till osäkerhet i tillämpningen av sådana förenklade modeller som kubisk lag (CL) för vätskeflöde och analytiska lösningar för transport av lösta ämnen i bergsfrakturer. Studien har ökat förståelsen för realistiska flödes- och transportprocesser i naturligt sprucket berg. Resultaten är användbara för modellvalidering/förlängning, osäkerhetsanalys/kvantifiering och design av laboratorieexperiment i samband med olika tillämpningar av flöde och transport i bergsfrakturer. / <p>QC 20161010</p>

No description available.

Wang Jing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 55-59). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Formulation of the Problem --- p.10 / Chapter 1.1 --- Reformulated Navier-Stokes Equations --- p.10 / Chapter 1.2 --- Linearized Problems --- p.15 / Chapter 2 --- Construction of the Approximate Solution --- p.19 / Chapter 2.1 --- Two-scale Asymptotic Expansions --- p.19 / Chapter 2.2 --- Determination of Each Inner and Boundary Terms --- p.22 / Chapter 2.3 --- Truncation Terms --- p.31 / Chapter 3 --- Estimates of the Error Term of the Approximate Solution and Main Results --- p.33 / Chapter 3.1 --- Error Equations --- p.33 / Chapter 3.2 --- Energy Estimates --- p.36 / Chapter 3.2.1 --- BasicL2 Estimates --- p.36 / Chapter 3.2.2 --- Tangential Derivatives Estimates --- p.38 / Chapter 3.2.3 --- Normal Derivatives Estimates --- p.49 / Chapter 3.3 --- Pointwise Estimates --- p.52 / Bibliography --- p.55

This thesis deals with the problem of the asymptotic behavior of solutions to several nonlinear equations from fluid dynamics on both mesoscopic and macroscopic levels, including Boltzmann equation, compressible Navier-Stokes equations and the system of viscous conservation laws with positive definite viscosity matrix. The main purpose is to study the asymptotic behavior of solutions to those equations towards linear and nonlinear waves, such as shock waves, rarefaction waves and contact discontinuities as either the times goes to infinity, or the viscosity and heat conductivity go to zero for the macroscopic equations or the mean free path goes to zero for the mesoscopic equations. Those limit processes are singular. For the system of viscous conservation laws, we show the large time asymptotic nonlinear stability of a superposition of viscous shock waves and viscous contact waves for the system of viscous conservation laws with small initial perturbations, provided that the strengths of these viscous waves are small and of the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a new estimate on the heat equation. For the Boltzmann equation, the main purpose is to study the asymptotic equivalence for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered-rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t = 0) as the mean free path epsilon &rarr; 0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. As a by-product, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat-conductivity (depending on the viscosity) is also obtained. / Zeng, Huihui. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 102-110). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.