The three-dimensional lid-driven cavity flow is investigated at Reynolds Number(Re)=10,000 for a wide range of spanwise-aspect ratios of 3:1:1, 0.5:1:1, and 1:1:1 using the Partially Averaged Navier-Stokes(PANS) turbulence closure model. The PANS turbulence model is a variable resolution turbulence closure model, where the unresolved-to-total ratios of kinetic energy (fk) and dissipation (fe), serve as resolution control parameters. This study focuses on two main aspects of PANS: (i) the evaluation of Turbulent transport models and (ii) the effect of grid spacing on accuracy of the numerical solution. PANS calculations are tested against LES and experimental results of Jordan (1994), in terms of both qualitative and quantitative quantities. The main coclusions are are: (i) for a given fk value, the Zero-Transport model is superior to the Maximum-Transport model for unresolved dissipation, (ii) both models are adequate for unresolved kinetic energy, and (iii) for a given grid size, the results depend heavily on grid spacing especially for larger fk values.

The three-dimensional lid-driven cavity flow is investigated at Reynolds Number(Re)=10,000 for a wide range of spanwise-aspect ratios of 3:1:1, 0.5:1:1, and 1:1:1 using the Partially Averaged Navier-Stokes(PANS) turbulence closure model. The PANS turbulence model is a variable resolution turbulence closure model, where the unresolved-to-total ratios of kinetic energy (fk) and dissipation (fe), serve as resolution control parameters. This study focuses on two main aspects of PANS: (i) the evaluation of Turbulent transport models and (ii) the effect of grid spacing on accuracy of the numerical solution. PANS calculations are tested against LES and experimental results of Jordan (1994), in terms of both qualitative and quantitative quantities. The main coclusions are are: (i) for a given fk value, the Zero-Transport model is superior to the Maximum-Transport model for unresolved dissipation, (ii) both models are adequate for unresolved kinetic energy, and (iii) for a given grid size, the results depend heavily on grid spacing especially for larger fk values.

In this thesis the solutions of the two-dimensional (2D) and three-dimensional (3D) lid-driven cavity problem are obtained by solving the steady Navier-Stokes equations at high Reynolds numbers. In 2D, we use the streamfunction-vorticity formulation to solve the problem in a square domain. A numerical method is employed to discretize the problem in the x and y directions with a spectral collocation method. The problem is coded in the MATLAB programming environment. Solutions at high Reynolds numbers are obtained up to $Re=25000$ on a fine grid of 131 * 131. The same method is also used to obtain the numerical solutions for the steady separated corner flow at high Reynolds numbers are generated using a for various domain sizes, at various Reynolds number which are much higher than those obtained by other researchers.Finally, the numerical solutions for the three-dimensional lid-driven cavity problem are obtained by solving the velocity-vorticity formulation of the Navier-Stokes equations for various Reynolds numbers. A spectral collocation method is employed to discretize the y and z directions and finite difference method is used to discretize the x direction. Numerical solutions are obtained for Reynolds number up to 200.

The present work deals with the development of a direct simulation strategy for solving the motion of spherical particles in non-Newtonian liquids. The purely viscous (non-elastic) non-Newtonian liquids are described by Bingham and thixotropy models. Validation of the strategy is performed for single phase (lid driven cavity flow) and two phase flows (sphere sedimentation). Lid driven cavity flow results illustrate the flow evolution of thixotropic liquid and subtle differences between thixotropic rheology and pseudo Bingham rheology. Single sphere sedimentation in Bingham liquid is shown to be influenced by the yield stress of the liquid. Time-dependent properties such as aging prominently affect the settling of a sphere in thixotropic liquid. The hydrodynamic interactions between two spheres are also studied at low and moderate Reynolds numbers. In thixotropic liquid, an intriguing phenomenon is observed where the separation distance between the spheres first increases and then rapidly decreases. / Chemical Engineering

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Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods. / Ph. D.

Cette étude se concentre sur la compréhension de la physique des instabilités dans différents écoulements de cisaillement, particulièrement la cavité entraînée et la cavité thermocapillaire, où l'écoulement d'un fluide incompressible est assuré soit par le mouvement d’une ou plusieurs parois, soit par des contraintes d’origine thermique.Un code spectral a été validé sur le cas très étudié de la cavité entrainée par une paroi mobile. Il est démontré dans ce cas que l'écoulement transit d'un régime stationnaire à un instationnaire au-delà d'une valeur critique du nombre de Reynolds. Ce travail est le premier à donner une interprétation physique de l'évolution non monotonique du nombre de Reynolds critique en fonction du facteur d'aspect. Lorsque le fluide est entraîné par deux parois mobiles, la cavité entraînée possède un plan de symétrie particulièrement sensible. Des solutions asymétriques peuvent être observés en plus de la solution symétrique au-dessus d'une certaine valeur du nombre de Reynolds. La transition oscillatoire entre la solution symétrique et les solutions asymétriques est expliquée physiquement par les forces en compétition. Dans le cas asymétrique, l'évolution de la topologie permet à l'écoulement de rester stationnaire avec l'augmentation du nombre de Reynolds. Lorsque l'équilibre est perdu une instabilité se manifeste par l'apparition d'un régime oscillatoire dans l'écoulement asymétrique.Dans une cavité thermocapillaire rectangulaire avec une surface libre, Smith et Davis prévoient deux types d'instabilités convectives thermiques: des rouleaux longitudinaux stationnaires et des ondes hydrothermales instationnaires. L'apparition de ses instabilités a été mis en évidence à plusieurs reprises expérimentalement et numériquement. Alors que les applications impliquent souvent plus d'une surface libre, il semble qu'il y ait peu de connaissances sur l'écoulement thermocapillaire entraînée avec deux surfaces libres. Un film liquide libre soumis à des contraintes thermocapillaires possède un plan de symétrie particulier comme dans le cas de la cavité entrainée par deux parois mobiles. Une étude de stabilité linéaire avec deux profils de vitesse pour le film liquide libre est présentée avec différents nombres de Prandtl. Au-delà d'un nombre de Marangoni critique, il est découvert que ces états de base sont sensibles à quatre types d'instabilités convectives thermiques qui peuvent conserver ou briser la symétrie du système. Les mécanismes qui permettent de prédire ces instabilités sont également découverts et interpréter en fonction de la valeur du nombre de Prandtl du fluide. La comparaison avec les travaux de Smith et Davis est faite. Une simulation numérique directe permet de valider les résultats obtenus avec l'étude de stabilité de linéaire. / This study focuses on the understanding of the physics of different instabilities in driven cavities, specifically the lid-driven cavity and the thermocapillarity driven cavity where flow in an incompressible fluid is driven either due to one or many moving walls or due to surface stresses that appear from surface tension gradients caused by thermal gradients. A spectral code is benchmarked on the well-studied case of the lid-cavity driven by one moving wall. In this case, It is shown that the flow transit form a steady regime to unsteady regime beyond a critical value of the Reynolds number. This work is the first to give a physical interpretation of the non-monotonic evolution of the critical Reynolds number versus the size of the cavity. When the fluid is driven by two facing walls moving in the same direction, the cavity possesses a plane of symmetry particularly sensitive. Thus, asymmetrical solutions can be observed in addition to the symmetrical solution above a certain value of the Reynolds number. The oscillatory transition between the symmetric solution and asymmetric solutions is explained physically by the forces in competition. In the asymmetric case, the change of the topology allows the flow to remain steady with increasing the Reynolds number. When the equilibrium is lost, an instability manifests by the appearance of an oscillatory regime in the asymmetric flow. In a rectangular cavity thermocapillary with a free surface, Smith and Davis found two types of thermal convective instabilities: steady longitudinal rolls and unsteady hydrothermal waves. The appearance of its instability has been highlighted repeatedly experimentally and numerically. While applications often involve more than a free surface, it seems that there is little knowledge about the thermocapillary driven flow with two free surfaces. A free liquid film possesses a particular plane of symmetry as in the case of the two-sided lid-driven cavity. A linear stability analysis for the free liquid film with two velocity profiles is presented with various Prandtl numbers. Beyond a critical Marangoni number, it is observed that these basic states are sensitive to four types of thermal convective instabilities, which can keep or break the symmetry of the system. Mechanisms that predict these instabilities are discovered and interpreted according to the value of the Prandtl number of the fluid. Comparison with the work of Smith and Davis is made. A direct numerical simulation is done to validate the results obtained with the linear stability analysis.

Abstract An experimental study is conducted to determine the velocity fields, from development to steady state, in a square enclosure due to movement of a constant velocity lid using Particle Image Velocitmetry (PIV). Experiments were conducted with water, seeded with hollow glass sphere particles 10 microns in diameter, at three different lid velocities leading to Reynolds numbers in the high laminar to transitional range. Driven Cavity Flow is a classic fluid dynamics case often used for benchmarking of computational codes. Previous work has primarily focused on improving computational codes, experimental work is lacking and focused on obtaining steady state readings. The test cavity is 1 inch (25.4mm) high by 1 inch (25.4 mm) wide leading to an aspect ratio of 1.0. The depth is taken to be 5 (127mm) inches to reduce the three dimensional effects. Readings are taken from development to steady state allowing for a full spectrum of flow characteristics. PIV technique is successful in capturing the development of driven cavity flow. Circulation is shown to increase strength with time and Reynolds number. PIV capture and processing settings are determined. Keywords: Driven Cavity Flow, Particle Image Velocimeter (PIV)

Numerous efforts have contributed to the study of phase-change problems for over a century|both analytical and numerical. Among those numerical approximations applied to solve phase-transition problems, phase-field models attract more and more attention because they not only capture two important effects, surface tension and supercooling, but also enable explicitly labeling the solid and liquid phases and the position of the interface. In the research of this dissertation, a phase-field model has been employed to simulate 2-D dendrite growth of pure nickel without a flow, and 2-D ice crystal growth in a high-Reynolds-number lid-driven-cavity flow. In order to obtain the details of ice crystal structures as well as the flow field behavior during freezing for the latter simulation, it is necessary to solve the phase-field model without convection and the equations of motion on two different scales. To accomplish this, a heterogeneous multiscale method is implemented for the phase-field model with convection such that the phase-field model is simulated on a microscopic scale and the equations of motion are solved on a macroscopic scale. Simulations of 2-D dendrite growth of pure nickel provide the validation of the phase-field model and the study of dendrite growth under different conditions, e.g., degree of supercooling, interface thickness, kinetic coefficient, and shape of the initial seed. In addition, simulations of freezing in a lid-driven-cavity flow indicate that the flow field has great effect on the small-scale dendrite structure and the flow eld behavior on the large scale is altered by freezing inside it.

This thesis focuses on the phenomenon of aliasing and its mitigation with two explicit filters, i.e., Shuman and Padé filters. The Shuman filter is applied to velocity components of the Navier--Stokes equations. A derivation of this filter is presented as an approximation of a 1-D “pure math” mollifier and extend this to 2D and 3D. Analysis of the truncation error and wavenumber response is conducted with a range of grid spacings, Reynolds numbers and the filter parameter, β. Plots of the relationship between optimal filter parameter β and grid spacing, L2-norm error and Reynolds number to suggest ways to predict β are also presented. In order to guarantee that the optimal β is obtained under various stationary flow conditions, the power spectral density analysis of velocity components to unequivocally identify steady, periodic and quasi-periodic behaviours in a range of Reynolds numbers between 100 and 2000 are constructed. Parameters in Pade filters need not be changed. The two filters are applied to velocities in this paper on perturbed sine waves and a lid-driven cavity. Comparison is based on execution time, error and experimental results.