Dissertations / Theses on the topic 'Equations in fluid flow study'

Sedimenting suspensions exist in a varity of natural phenomena and industrial applications. It is already observed in experiments that the dilute fibre suspensions experience a hydrodynamics instability under gravity at low Reynolds numbers. Initially well-mixed suspensions become inhomogeneous and anisotropic due to this instability.The main goal of this work is to understand the instability in a dilute fibre suspension by means of an Eulerian approach which is based on the Navier-Stokes equations coupled to Fokker-Planck equation for the PDF of fibres.Using a linear stability analysis, we show that inertia and hydrodynamic translational diffusion damp perturbations at long wavelengths and short wavelengths, respectively, leading to a wavenumber selection. For small, but finite Reynolds number of the fluid bulk motion, the most unstable wavenumber is a finite value which increases with Reynolds number, and where the diffusion narrows the range of unstable wavenumbers. With periodic boundary conditions, numerical simulations of the full non-linear evolution in time of a normal mode perturbation show that the induced flow may either die or saturate on a finite amplitude. The character of this long time behaviour is dictated by the wavenumber and the presence or absence of the translational and rotational diffusivities.In a simulation domain confined by vertical walls, a series of alternating structures of risers and streamers emerge continuously from the walls until they meet in the middle of the domain. For moderate times, this agrees qualitatively with experimental and theoretical results. Moreover, our simulation in a vessel of infinite height obtained an increasing wavelength evolution due to the congregation of the streamers or risers. In the end, there is constantly only one streamer left, and it drifts randomly to one side of the container until the evolution reaches a steady state. It is also found that the perturbations added to the initial conditions can induce more high density regions whose sizes and velocities are strongly linked to the initial perturbations of the number density or the flow field. In addition, the maximum number of streamers increases with Reynolds number, volume fraction and channel width.
QC 20120625

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new â mostly linearâ theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows.
Ph. D.

The solutions of nonlinear ordinary or partial differential equations are important in the study of fluid flow and heat transfer. In this thesis we apply the Homotopy Analysis Method (HAM) and obtain solutions for several fluid flow and heat transfer problems. In chapter 1, a brief introduction to the history of homotopies and embeddings, along with some examples, are given. The application of homotopies and an introduction to the solutions procedure of differential equations (used in the thesis) are provided. In the chapters that follow, we apply HAM to a variety of problems to highlight its use and versatility in solving a range of nonlinear problems arising in fluid flow. In chapter 2, a viscous fluid flow problem is considered to illustrate the application of HAM. In chapter 3, we explore the solution of a non-Newtonian fluid flow and provide a proof for the existence of solutions. In addition, chapter 3 sheds light on the versatility and the ease of the application of the Homotopy Analysis Method, and its capability in handling non-linearity (of rational powers). In chapter 4, we apply HAM to the case in which the fluid is flowing along stretching surfaces by taking into the effects of "slip" and suction or injection at the surface. In chapter 5 we apply HAM to a Magneto-hydrodynamic fluid (MHD) flow in two dimensions. Here we allow for the fluid to flow between two plates which are allowed to move together or apart. Also, by considering the effects of suction or injection at the surface, we investigate the effects of changes in the fluid density on the velocity field. Furthermore, the effect of the magnetic field is considered. Chapter 6 deals with MHD fluid flow over a sphere. This problem gave us the first opportunity to apply HAM to a coupled system of nonlinear differential equations. In chapter 7, we study the fluid flow between two infinite stretching disks. Here we solve a fourth order nonlinear ordinary differential equation. In chapter 8, we apply HAM to a nonlinear system of coupled partial differential equations known as the Drinfeld Sokolov equations and bring out the effects of the physical parameters on the traveling wave solutions. Finally, in chapter 9, we present prospects for future work.
Ph.D.
Department of Mathematics
Sciences
Mathematics PhD

We consider the stabilized finite element method for solving the incompressible Navier-Stokes equations and the magnetohydrodynamic (MHD) equations in two dimensions. The well-known instabilities arising from the application of standard Galerkin finite element method are eliminated by using the stabilizing subgrid method (SSM), the streamline upwind Petrov-Galerkin (SUPG) method, and the two-level finite element method (TLFEM). The domain is discretized into a set of regular triangular elements. In SSM, the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level finite element method with a stabilizing subgrid of a single node is described and its applications to the Navier-Stokes equations and MHD equations are displayed. This constitutes the main original contribution of this thesis. Numerical approximations employing the proposed algorithms are presented for some benchmark problems. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. The stabilized finite element method of SUPG type is applied to the unsteady Navier-Stokes equations together with a finite element discretization in the time domain. Thus, oscillations in the solution and the need of very small time increment are avoided in obtaining stable solutions.

We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.

Thesis (Ph. D.)--University of Sydney, 2000.
Title from title screen (viewed Apr. 23, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the Dept. of Mechanical Engineering, Faculty of Engineering. Includes bibliography. Also available in print form.

A weakly nonlinear stability analysis was conducted for the flow induced in an incompressible, Newtonian, viscous fluid lying between two infinite parallel plates which form a channel. The plates are oscillating synchronously in simple harmonic motion. The disturbed velocity of the flow was written in the form of a series in powers of a parameter which is a measure of the distance away from the linear theory neutral conditions. The individual terms of this series were decomposed using Floquet theory and Fourier series in time. The equations at second order and third order in were derived, and solutions for the Fourier coefficients were found using pseudospectral methods for the spatial variables. Various alternative methods of computation were applied to check the validity of the results obtained. The Landau equation for the amplitude of the disturbance was obtained, and the existence of equilibrium amplitude solutions inferred. The values of the coefficients in the Landau equation were calculated for the nondimensional channel half-widths h for the cases h = 5, 8, 10, 12, 14 and 16. It was found that equilibrium amplitude solutions exist for points in wavenumber Reynolds number space above the smooth portion of the previously determined linear stability neutral curve in all the cases examined. Similarly, Landau coefficients were calculated on a special feature of the neutral curve (called a ???finger???) for the case h = 12. Equilibrium amplitude solutions were found to exist at points inside the finger, and in a particular region outside near the top of the finger. Traces of the x-components of the disturbance velocities have been presented for a range of positions across the channel, together with the size of the equilibrium amplitude at these positions. As well, traces of the x-component of the velocity of the disturbed flow and traces of the velocity of the basic flow have been given for comparison at a particular position in the channel.

Within the eld of hydraulics there is a growing trend towards the use of computer based models, which have proven to be an invaluable tool in engineering. A range of commercial packages is available which encompass different mathematical models and a variety of solution strategies. A number of problems can be identified with the software currently available, and as a result, research continues into developing better numerical techniques for computational hydraulics. The issues most often addressed by researchers consider the application of faster and more accurate numerical methods, many of which were originally developed for gas dynamics problems. There has been a growing trend in favour of Riemann based methods constructed within the finite volume framework. Such methods are noted for their good conservation and shock capturing capabilities. However, the computational cost of employing theses algorithms can lead to excessively long run times, particularly when higher order mathematical models are used. This often is as a result of stability constraints placed upon explicit schemes, which require the smallest possible time step permitted throughout the grid, to be applied globally. One possibility for improving this situation is to use local time stepping, whereby individual cells are advanced by their own maximum allowable time steps. To incorporate this concept into a transient model requires the development of a suitable integration strategy, to ensure that the solution remains accurate in time. Two such strategies developed for the Euler equations are considered within this thesis for application to the Saint Venant equations of open channel flow. Both techniques have been demonstrated to reduce run times and improve the quality of solutions in the regions of discontinuities. The investigation considers the the first order scheme of Roe, together with a second order extension constructed using a ux limiter approach. he eects of using an upwind based source term treatment, specifically developed for Roe's scheme, are also considered, and the source term calculations are incorporated into the LTS framework. Results are presented for a series of steady state and transient test cases, which illustrate how local time stepping can lead to reduced run times and improved solution accuracy. The results also highlight the benets of using an upwind source term treatment, particularly when variations in the channel geometry occur.

[ES] El principal desafío en los motores turbina de gas empleados en aviación reside en aumentar la eficiencia del ciclo termodinámico manteniendo las emisiones contaminantes por debajo de las rigurosas restricciones. Ésto ha conllevado la necesidad de diseñar nuevas estrategias de inyección/combustión que operan en puntos de operación peligrosos por su cercanía al límite inferior de apagado de llama. En este contexto, el concepto Lean Direct Injection (LDI) ha emergido como una tecnología prometedora a la hora de reducir los óxidos de nitrógeno (NOx) emitidos por las plantas propulsoras de los aviones de nueva generación. En este contexto, la presente tesis tiene como objetivos contribuir al conocimiento de los mecanismos físicos que rigen el comportamiento de un quemador LDI y proporcionar herramientas de análisis para una profunda caracterización de las complejas estructuras de flujo de turbulento generadas en el interior de la cámara de combustión. Para ello, se ha desarrollado una metodología numérica basada en CFD capaz de modelar el flujo bifásico no reactivo en el interior de un quemador LDI académico mediante enfoques de turbulencia U-RANS y LES en un marco Euleriano-Lagrangiano. La resolución numérica de este problema multi-escala se aborda mediante la descripción completa del flujo a lo largo de todos los elementos que constituyen la maqueta experimental, incluyendo su paso por el swirler y entrada a la cámara de combustión. Ésto se lleva a cabo través de dos códigos CFD que involucran dos estrategias de mallado diferentes: una basada en algoritmos de generación y refinamiento automático de la malla (AMR) a través de CONVERGE y otra técnica de mallado estático más tradicional mediante OpenFOAM. Por un lado, se ha definido una metodología para obtener una estrategia de mallado óptima mediante el uso del AMR y se han explotado sus beneficios frente a los enfoques tradicionales de malla estática. De esta forma, se ha demostrado que la aplicabilidad de las herramientas de control de malla disponibles en CONVERGE como el refinamiento fijo (fixed embedding) y el AMR son una opción muy interesante para afrontar este tipo de problemas multi-escala. Los resultados destacan una optimización del uso de los recursos computacionales y una mayor precisión en las simulaciones realizadas con la metodología presentada. Por otro lado, el uso de herramientas CFD se ha combinado con la aplicación de técnicas de descomposición modal avanzadas (Proper Orthogonal Decomposition and Dynamic Mode Decomposition). La identificación numérica de los principales modos acústicos en la cámara de combustión ha demostrado el potencial de estas herramientas al permitir caracterizar las estructuras de flujo coherentes generadas como consecuencia de la rotura de los vórtices (VBB) y de los chorros fuertemente torbellinados presentes en el quemador LDI. Además, la implementación de estos procedimientos matemáticos ha permitido tanto recuperar información sobre las características de la dinámica de flujo como proporcionar un enfoque sistemático para identificar los principales mecanismos que sustentan las inestabilidades en la cámara de combustión. Finalmente, la metodología validada ha sido explotada a través de un Diseño de Experimentos (DoE) para cuantificar la influencia de los factores críticos de diseño en el flujo no reactivo. De esta manera, se ha evaluado la contribución individual de algunos parámetros funcionales (el número de palas del swirler, el ángulo de dichas palas, el ancho de la cámara de combustión y la posición axial del orificio del inyector) en los patrones del campo fluido, la distribución del tamaño de gotas del combustible líquido y la aparición de inestabilidades en la cámara de combustión a través de una matriz ortogonal L9 de Taguchi. Este estudio estadístico supone un punto de partida para posteriores estudios de inyección, atomización y combus
[CA] El principal desafiament als motors turbina de gas utilitzats a la aviació resideix en augmentar l'eficiència del cicle termodinàmic mantenint les emissions contaminants per davall de les rigoroses restriccions. Aquest fet comporta la necessitat de dissenyar noves estratègies d'injecció/combustió que radiquen en punts d'operació perillosos per la seva aproximació al límit inferior d'apagat de flama. En aquest context, el concepte Lean Direct Injection (LDI) sorgeix com a eina innovadora a l'hora de reduir els òxids de nitrogen (NOx) emesos per les plantes propulsores dels avions de nova generació. Sota aquest context, aquesta tesis té com a objectius contribuir al coneixement dels mecanismes físics que regeixen el comportament d'un cremador LDI i proporcionar ferramentes d'anàlisi per a una profunda caracterització de les complexes estructures de flux turbulent generades a l'interior de la càmera de combustió. Per tal de dur-ho a terme s'ha desenvolupat una metodología numèrica basada en CFD capaç de modelar el flux bifàsic no reactiu a l'interior d'un cremador LDI acadèmic mitjançant els enfocaments de turbulència U-RANS i LES en un marc Eulerià-Lagrangià. La resolució numèrica d'aquest problema multiescala s'aborda mitjançant la resolució completa del flux al llarg de tots els elements que constitueixen la maqueta experimental, incloent el seu pas pel swirler i l'entrada a la càmera de combustió. Açò es duu a terme a través de dos codis CFD que involucren estratègies de mallat diferents: una basada en la generación automàtica de la malla i en l'algoritme de refinament adaptatiu (AMR) amb CONVERGE i l'altra que es basa en una tècnica de mallat estàtic més tradicional amb OpenFOAM. D'una banda, s'ha definit una metodologia per tal d'obtindre una estrategia de mallat òptima mitjançant l'ús de l'AMR i s'han explotat els seus beneficis front als enfocaments tradicionals de malla estàtica. D'aquesta forma, s'ha demostrat que l'aplicabilitat de les ferramente de control de malla disponibles en CONVERGE com el refinament fixe (fixed embedding) i l'AMR són una opció molt interessant per tal d'afrontar aquest tipus de problemes multiescala. Els resultats destaquen una optimització de l'ús dels recursos computacionals i una major precisió en les simulacions realitzades amb la metodologia presentada. D'altra banda, l'ús d'eines CFD s'ha combinat amb l'aplicació de tècniques de descomposició modal avançades (Proper Orthogonal Decomposition and Dynamic Mode Decomposition). La identificació numèrica dels principals modes acústics a la càmera de combustió ha demostrat el potencial d'aquestes ferramentes al permetre caracteritzar les estructures de flux coherents generades com a conseqüència del trencament dels vòrtex (VBB) i dels raigs fortament arremolinats presents al cremador LDI. A més, la implantació d'estos procediments matemàtics ha permès recuperar informació sobre les característiques de la dinàmica del flux i proporcionar un enfocament sistemàtic per tal d'identificar els principals mecanismes que sustenten les inestabilitats a la càmera de combustió. Finalment, la metodologia validada ha sigut explotada a traves d'un Diseny d'Experiments (DoE) per tal de quantificar la influència dels factors crítics de disseny en el flux no reactiu. D'aquesta manera, s'ha avaluat la contribución individual d'alguns paràmetres funcionals (el nombre de pales del swirler, l'angle de les pales, l'amplada de la càmera de combustió i la posició axial de l'orifici de l'injector) en els patrons del camp fluid, la distribució de la mida de gotes del combustible líquid i l'aparició d'inestabilitats en la càmera de combustió mitjançant una matriu ortogonal L9 de Taguchi. Aquest estudi estadístic és un bon punt de partida per a futurs estudis de injecció, atomització i combustió en cremadors LDI.
[EN] Aeronautical gas turbine engines present the main challenge of increasing the efficiency of the cycle while keeping the pollutant emissions below stringent restrictions. This has led to the design of new injection-combustion strategies working on more risky and problematic operating points such as those close to the lean extinction limit. In this context, the Lean Direct Injection (LDI) concept has emerged as a promising technology to reduce oxides of nitrogen (NOx) for next-generation aircraft power plants In this context, this thesis aims at contributing to the knowledge of the governing physical mechanisms within an LDI burner and to provide analysis tools for a deep characterisation of such complex flows. In order to do so, a numerical CFD methodology capable of reliably modelling the 2-phase nonreacting flow in an academic LDI burner has been developed in an Eulerian-Lagrangian framework, using the U-RANS and LES turbulence approaches. The LDI combustor taken as a reference to carry out the investigation is the laboratory-scale swirled-stabilised CORIA Spray Burner. The multi-scale problem is addressed by solving the complete inlet flow path through the swirl vanes and the combustor through two different CFD codes involving two different meshing strategies: an automatic mesh generation with adaptive mesh refinement (AMR) algorithm through CONVERGE and a more traditional static meshing technique in OpenFOAM. On the one hand, a methodology to obtain an optimal mesh strategy using AMR has been defined, and its benefits against traditional fixed mesh approaches have been exploited. In this way, the applicability of grid control tools available in CONVERGE such as fixed embedding and AMR has been demonstrated to be an interesting option to face this type of multi-scale problem. The results highlight an optimisation of the use of the computational resources and better accuracy in the simulations carried out with the presented methodology. On the other hand, the use of CFD tools has been combined with the application of systematic advanced modal decomposition techniques (i.e., Proper Orthogonal Decomposition and Dynamic Mode Decomposition). The numerical identification of the main acoustic modes in the chamber have proved their potential when studying the characteristics of the most powerful coherent flow structures of strongly swirled jets in a LDI burner undergoing vortex breakdown (VBB). Besides, the implementation of these mathematical procedures has allowed both retrieving information about the flow dynamics features and providing a systematic approach to identify the main mechanisms that sustain instabilities in the combustor. Last, this analysis has also allowed identifying some key features of swirl spray systems such as the complex pulsating, intermittent and cyclical spatial patterns related to the Precessing Vortex Core (PVC). Finally, the validated methodology is exploited through a Design of Experiments (DoE) to quantify the influence of critical design factors on the non-reacting flow. In this way, the individual contribution of some functional parameters (namely the number of swirler vanes, the swirler vane angle, the combustion chamber width and the axial position of the nozzle tip) into both the flow field pattern, the spray size distribution and the occurrence of instabilities in the combustion chamber are evaluated throughout a Taguchi's orthogonal array L9. Such a statistical study has supposed a good starting point for subsequent studies of injection, atomisation and combustion on LDI burners.
Belmar Gil, M. (2020). Computational study on the non-reacting flow in Lean Direct Injection gas turbine combustors through Eulerian-Lagrangian Large-Eddy Simulations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/159882
TESIS

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Tese (Doutoramento)
IPEN/T
Escola Politécnica, Universidade de Sao Paulo - POLI/USP

In this work, a code has been developed that solves the Navier-Stokes equations using the finite volume method with unstructured triangular grids. A cell-centred, finite volume method is used and the pressure-velocity coupling is treated using both the SMTLE and the MAC algorithms. The major advantage of using triangular grids is their applicability to complex geometry. A special treatment is developed to ensure good quality triangular elements around the boundaries. The numerical simulation of incompressible flow at low Reynolds number is studied in this thesis. A code for generating triangular grids using the tri-tree algorithm has been written and an adaptive finite volume method developed for calculating laminar fluid flow. The grid is locally adapted at each time step, with grid refinement and derefinement dependent on the vorticity magnitude. The resulting grids have fine local resolution and are economical in reducing the numerical simulation time. The discretised equations are solved by using an iterative point by point Gauss-Seidel solver. For calculating the values of velocity and pressure at vertices of triangular grids, special interpolation schemes (averaged linear-interpolation and scattered interpolation) are used to increase the accuracy. To avoid the well known checkerboard error problems, i. e., the oscillations occurring in the pressure field, third derivative terms in pressure, first introduced by Rhie-chow (1983), are added to the mass flux velocity. Convective terms are approximated using a QUICK (Quadratic Upstream Interpolation for Convective Kinematics) differencing scheme which has been developed here in for unstructured grids. Three cases of two-dimensional viscous incompressible fluid flow have been investigated: the first is channel flow, in which the numerical results are compared with the analytical solution; the second case is the backward-facing step flow; and the third case is flow past circular cylinders at low Reynolds number (Re). The numerical results obtained for the last two cases are compared with published data. The evolution of vortex shedding is presented for the case of unidirectional flow past a circular cylinder at Re=200. In addition, drag and lift force coefficients are calculated and compared for single and multiple cylinders in unidirectional flow.

In this thesis, the two-dimensional, transient, laminar flow of viscous and incompressible fluids is solved by using the dual reciprocity boundary element method (DRBEM). Natural convection and mixed convection flows are also solved with the addition of energy equation. Solutions of natural convection flow of nanofluids and micropolar fluids in enclosures are obtained for highly large values of Rayleigh number. The fundamental solution of Laplace equation is used for obtaining boundary element method (BEM) matrices whereas all the other terms in the differential equations governing the flows are considered as nonhomogeneity. This is the main advantage of DRBEM to tackle the nonlinearities in the equations with considerably small computational cost. All the convective terms are evaluated by using the DRBEM coordinate matrix which is already computed in the formulation of nonlinear terms. The resulting systems of initial value problems with respect to time are solved with forward and central differences using relaxation parameters, and the fourth-order Runge-Kutta method. The numerical stability analysis is developed for the flow problems considered with respect to the choice of the time step, relaxation parameters and problem constants. The stability analysis is made through an eigenvalue decomposition of the final coefficient matrix in the DRBEM discretized system. It is found that the implicit central difference time integration scheme with relaxation parameter value close to one, and quite large time steps gives numerically stable solutions for all flow problems solved in the thesis. One-and-two-sided lid-driven cavity flow, natural and mixed convection flows in cavities, natural convection flow of nanofluids and micropolar fluids in enclosures are solved with several geometric configurations. The solutions are visualized in terms of streamlines, vorticity, microrotation, pressure contours, isotherms and flow vectors to simulate the flow behaviour.

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.

QC 20161010

Definition des caracteristiquees du jet parietal. Historique du sujet. Theorie de glauert permettant l'existence d'une solution semblable selon laquelle la forme des profils de vitesse dans chaque section du jet reste inchangee le long du jet. Etude de la stabilite lineaire et non lineaire. Resolution de l'equation d'orr-sommerfeld en theorie temporelle et en theorie spatiale. Etude experimentale: mesure des vitesses en presence d'un corps bidimensionnel ou tridimensionnel, mesures de temperature

Calcul de l'ecoulement d'un fluide newtonien dans le domaine voisin de l'equateur de la sphere : determination des profils de vitesse et de la fonction de courant pour des nombres de reynolds moderes. Le calcul de la vitesse de deformation permet de proposer un coefficient de frottement caracteristique des viscosimetres a chute de bille. Verification experimentale

Thesis (M.S.)--University of Nevada, Reno, 2007.
"May, 2007." Includes bibliographical references (leaves 73-77). Online version available on the World Wide Web. Library also has microfilm. Ann Arbor, Mich. : ProQuest Information and Learning Company, [2007]. 1 microfilm reel ; 35 mm.

The purpose of this thesis is to prove the existence of a unique solution to a system of partial differential equations which models the flow of a compressible barotropic fluid under periodic boundary conditions. The equations come from modifying the compressible Navier-Stokes equations. The proof utilizes the method of successive approximations. We will define an iteration scheme based on solving a linearized version of the equations. Then convergence of the sequence of approximate solutions to a unique solution of the nonlinear system will be proven. The main new result of this thesis is that the density data is at a given point in the spatial domain over a time interval instead of an initial density over the entire spatial domain. Further applications of the mathematical model are fluid flow problems where the data such as concentration of a solute or temperature of the fluid is known at a given point. Future research could use boundary conditions which are not periodic.

A numerical high order difference method is developed for solution of the incompressible Navier-Stokes equations. The solution is determined on a staggered curvilinear grid in two dimensions and by a Fourier expansion in the third dimension. The description in curvilinear body-fitted coordinates is obtained by an orthogonal mapping of the equations to a rectangular grid where space derivatives are determined by compact fourth order approximations. The time derivative is discretized with a second order backward difference method in a semi-implicit scheme, where the nonlinear terms are linearly extrapolated with second order accuracy.

An approximate block factorization technique is used in an iterative scheme to solve the large linear system resulting from the discretization in each time step. The solver algorithm consists of a combination of outer and inner iterations. An outer iteration step involves the solution of two sub-systems, one for prediction of the velocities and one for solution of the pressure. No boundary conditions for the intermediate variables in the splitting are needed and second order time accurate pressure solutions can be obtained.

The method has experimentally been validated in earlier studies. Here it is validated for flow past a circular cylinder as an example of a physical test case and the fourth order method is shown to be efficient in terms of grid resolution. The method is applied to external flow past a parabolic body and internal flow in an asymmetric diffuser in order to investigate the performance in two different curvilinear geometries and to give directions for future development of the method. It is concluded that the novel formulation of boundary conditions need further investigation.

A new iterative solution method for prediction of velocities allows for larger time steps due to less restrictive convergence constraints.

Numerical solutions of the Navier-Stokes Equations became more popular in recent decades with increasingly accessible and powerful computational resources. Simulations in reconstructed or artificial pore geometries are often performed to gain insight into microscopic fluid flow structures or are used for upscaling quantities of interest, like hydraulic conductivity. A physically adequate representation of pore-scale flow fields requires analysis of large domains. We solve the incompressible NSE in artificial ordered and random pore-space structures. A simple cubic and face-centred packings of spheres placed in a square duct are analysed. For the fluid flow simulations of random media, packings of spheres, icosahedra, and cubes forming unconsolidated sediments are generated using a rigid body simulation software. The Direct Numerical Simulation method is used for the solution of the NSE implemented in the open-source computational fluid dynamics software OpenFOAM. The influence of the number of spheres in ordered packings, the mesh type, and the mesh resolution is investigated for fluid flow up to Reynolds numbers of 100 based on the spheres' diameter. The random media mesh generation method relies on approximate surface reconstruction. The resulting tetrahedral meshes are then used for steady-state simulations and refined based on an a-posteriori error estimator. The fluid flow simulation results can further be used twofold: 1) They provide homogenized hydro-mechanical properties of the analysed medium for the larger meso and macro groundwater flow simulations. A concept of one-way binding for large-scale simulations is presented. 2) Visualisation: A post-processing image rendering technique was employed in interactive and still image visualisation environments allowing better overview over local fluid flow structures. The ogs FEM code for the solution of large-scale groundwater processes was inspected for computational efficiency. The conclusions drawn from this analysis formed the~basis for the implementation of the~new version of the code---ogs6. The improvements include comparison of linear algebra software realisations and an implementation of optimized memory access patterns in FEM-local assembler part.

This thesis describes a system for dealing with free surface fluid simulations, and the components needed in order to construct such a system. It builds upon recent research, but in a computer graphics context the amount of available literature is limited and difficult to implement. Because of this, the text aims at providing a solid foundation of the mathematics needed, at explaining in greater detail the steps needed to solve the problem, and lastly at improving some aspects of the animation process as it has been described in earlier works.

The aim of the system itself is to provide visually plausible renditions of animated fluids in three dimensions in a manner that allows it to be usable in a visual effects production context.

The novel features described include a generalized interaction layer providing greater control to artists, a new way of dealing with moving objects that interact with the fluid and a method for adding source and drain capabilities.

Kyoto University (京都大学)
0048
新制・課程博士
博士(エネルギー科学)
甲第11956号
エネ博第126号
新制||エネ||31(附属図書館)
23745
UT51-2006-B135
京都大学大学院エネルギー科学研究科エネルギー基礎科学専攻
(主査)教授 三島 嘉一郎, 教授 代谷 誠治, 教授 芹澤 昭示
学位規則第4条第1項該当

Two problems in the study of elastic filaments are considered.First, a reliable numerical algorithm is developed that candetermine the shape of a static elastic rod under a variety ofconditions. In this algorithm the governing equations are writtenentirely in terms of local coordinates and are discretized usingfinite differences. The algorithm has two significant advantages:firstly, it can be implemented for a wide variety of the boundaryconditions and, secondly, it enables the user to work with generalconstitutive relationships with only minor changes to thealgorithm. In the second problem a model is presented describingthe dynamics of an elastic tube conveying a fluid. First weanalyze instabilities that are present in a straight rod or tubeunder tension subject to increasing twist in the absence of afluid. As the twist is increased beyond a critical value, thefilament undergoes a twist-to-writhe bifurcation. A multiplescales expansion is used to derive nonlinear amplitude equationsto examine the dynamics of the elastic rod beyond the bifurcationthreshold. This problem is then reinvestigated for an elastic tubeconveying a fluid to study the effect of fluid flow on thetwist-to-writhe instability. A linear stability analysisdemonstrates that for an infinite rod the twist-to-writhethreshold is lowered by the presence of a fluid flow. Amplitudeequations are then derived from which the delay of bifurcation dueto finite tube length is determined. It is shown that the delayedbifurcation threshold depends delicately on the length of the tubeand that it can be either raised or lowered relative to thefluid-free case. The amplitude equations derived for the case of aconstant average fluid flux are compared to the case where theflux depends on the curvature. In this latter case it is shownthat inclusion of curvature results in small changes in some ofthe coefficients in the amplitude equations and has only a smalleffect on the post-bifurcation dynamics.

Fokker-Planck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, both theoretically and computationally, centered around them. In part one, we consider the optimal transport for finite discrete states, which are on a finite but arbitrary graph. By defining a discrete 2-Wasserstein metric, we derive Fokker-Planck equations on finite graphs as gradient flows of free energies. By using dynamical viewpoint, we obtain an exponential convergence result to equilibrium. This derivation provides tools for many applications, including numerics for nonlinear partial differential equations and evolutionary game theory. In part two, we introduce a new stochastic differential equation based framework for optimal control with constraints. The framework can efficiently solve several real world problems in differential games and Robotics, including the path-planning problem. In part three, we introduce a new noise model for stochastic oscillators. With this model, we prove global boundedness of trajectories. In addition, we derive a pair of associated Fokker-Planck equations.

"Compressible fluid dynamics of flow through plain-faced long orifices was investigated with the hope of gaining insight into the fluid dynamics of tip leakage flow. The Reynolds number range investigated was greater than 10*. Measurements were made of the discharge coefficient as a function of back pressure ratio for a sharp-edged orifice and long orifices with an l/d from 1/2 to 8. The discharge coefficient measurements indicate the mass flow rate in an orifice with an l/d of approximately 2 is the largest and the flow rate in a sharp-edged orifice is the smallest for pressure ratios greater than 0.27. The mass flow rate in a sharp-edged orifice is largest for pressure ratios below 0.27. To visualize the flow in a long orifice and model centerline pressure variation, a water table study was performed. The results demonstrate that the flow separates from the sharp corner at the orifice entrance, it accelerates to a maximum Mach number, and then the pressure increases. For back pressures above 0.50, a pressure decrease follows the initial pressure increase. If the maximum Mach number is supersonic, oblique shocks will exist. At the higher back pressures that produce supersonic maximum Mach numbers (0.50 P_B/P₀ < 0.70), the oblique shocks reflect from the centerline as ""Mach reflections"" and the flow is subsonic after the pressure increase. The maximum Mach number for a back pressure ratio of 0.50 is approximately 1.5. At lower back pressure ratios (P_B/P₀ <0.70), the oblique shocks reflect from the centerline in a ""regular"" manner and the flow remains supersonic on the centerline once supersonic speeds are reached. The flow in a long orifice is relatively constant for all back pressure ratios below approximately 0.30. The maximum Mach number for pressure ratios below 0.30 is approximately 1.8. One-dimensional analyses were used to model the flow in long orifices with maximum Mach numbers less than 1.3. Higher discharge coefficients of long orifices compared to sharp-edged orifices are due to pressure rises in the orifices caused by mixing and shock waves. These increases in the discharge coefficients are partly offset by friction and boundary layer blockage. For maximum Mach numbers greater than 1.3, the flow in long orifices is believed to become significantly two-dimensional because of supersonic effects."
Master of Science

For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.