No description available.

Mathematical modeling of ocean waves is based on the formulation and solution of the appropriate equations of continuity, momentum and the choice of proper initial and boundary conditions. Under the influence of gravity, many free surface water waves can be modeled by the shallow water equations (SWE) with the assumption that the horizontal length scale of the wave is much greater than the depth scale and the wave height is much less than the fluid's mean depth. Furthermore, to describe three dimensional flows in the hydrostatic and Boussinesq limits, the multilayer SWE model is used, where the fluid is discretized horizontally into a set of vertical layers, each having its own height, density, horizontal velocity and geopotential. In this study, we used an explicit staggered finite volume method to solve single and multilayer SWE, with and without density stratification and bathymetry, to understand the dynamic of surface waves and internal waves. We implemented a two-dimensional version of the incompressible DYNAMICO method and compare it with a one-dimensional SWE. For multilayer SWE, we considered both two layer and a linear stratification of density, with very small density gradient, consistent with Boussinesq approximation. We used Lagrangian vertical coordinate which doesn't allow mass to flow across vertical layers. Numerical examples are presented to verify multilayer SWE model against single layer SWE, in terms of the phase speed and the steepness criteria of wave profile. In addition, the phase speed of the barotropic and baroclinic mode of two-layer SWE also verified our multilayer SWE model. We found that, for multilayer SWE, waves move slower than single layer SWE and get steeper than normal when they flow across bathymetry. A series of numerical experiment were carried out to compare 1-D shallow water solutions to 2-D solutions with and without density as well as to explain the dynamics of surface wave and internal wave. We found that, a positive fluctuations on free surface causes water to rise above surface level, gravity pulls it back and the forces that acquired during the falling movement causes the water to penetrate beneath it's equilibrium level, influences the generation of internal waves. Internal waves travel considerably more slowly than surface waves. On the other hand, a bumpy or a slicky formation of surface waves is associated with the propagation of internal waves. The interaction between these two waves is therefore demonstrated and discussed. / Thesis / Master of Science (MSc) / In the modelling of ocean wave, the formulation and solution of appropriate equations and proper initial and boundary conditions are required. The shallow water equations (SWE) are derived from the conservation of mass and momentum equations, in the case where the horizontal length scale of the wave is much greater than the depth scale and the wave height is much less than the fluid's mean depth. In multilayer SWE, the fluid is discretized horizontally into a set of vertical layers, each having its own height, density, horizontal velocity and geopotential. In this study, we used an explicit staggered finite volume method to solve single and multilayer SWE, with and without density stratification and bathymetry, to understand the dynamic of surface waves and internal waves. A series of numerical experiments were carried out to validate our multilayer model. It is found that, in the presence of density differences, surface waves for the multilayer SWE move slowly and get more steep than normal when they flow across bathymetry. Also, a positive fluctuations on free surface generates internal waves at the interior of ocean which propagate along the line of density gradient.

The solution to partial differential equations generally requires approximations that result in numerical error in the final solution. Of the different types of numerical error in a solution, discretization error is the largest and most difficult error to estimate. In addition, the accuracy of the discretization error estimates relies on the solution (or multiple solutions used in the estimate) being in the asymptotic range. The asymptotic range is used to describe the convergence of a solution, where an asymptotic solution approaches the exact solution at a rate proportional to the change in mesh spacing to an exponent equal to the formal order of accuracy. A non-asymptotic solution can result in unpredictable convergence rates introducing uncertainty in discretization error estimates. To account for the additional uncertainty, various discretization uncertainty estimators have been developed. The goal of this work is to evaluation discretization error and discretization uncertainty estimators based on Richardson extrapolation for computational fluid dynamics problems. In order to evaluate the estimators, the exact solution should be known. A select set of solutions to the 2D Euler equations with known exact solutions are used to evaluate the estimators. Since exact solutions are only available for trivial cases, two applications are also used to evaluate the estimators which are solutions to the Navier-Stokes equations: a laminar flat plate and a turbulent flat plate using the k-Ï SST turbulence model. Since the exact solutions to the Navier-Stokes equations for these cases are unknown, numerical benchmarks are created which are solutions on significantly finer meshes than the solutions used to estimate the discretization error and uncertainty. Metrics are developed to evaluate the accuracy of the error and uncertainty estimates and to study the behavior of each estimator when the solutions are in, near, and far from the asymptotic range. Based on the results, general recommendations are made for the implementation of the error and uncertainty estimators. In addition, a new uncertainty estimator is proposed with the goal of combining the favorable attributes of the discretization error and uncertainty estimators evaluated. The new estimator is evaluated using numerical solutions which were not used for development and shows improved accuracy over the evaluated estimators. / Master of Science

A numerical code for solving the time-dependent incompressible 3D Navier-Stokes equations with finite volumes on overlapping staggered grids in cylindrical and rectangular geometry is developed. In the code, written in FORTRAN, the momentum equation for the velocity is solved by projection method and Poisson equation for the pressure is solved by ADI implicit method in two directions combined with discrete fast Fourier transform in the third direction. A special technique for overcoming the singularity on the cylinder's axis is developed. This code, taking into account dependence upon temperature of the viscosity, density and surface tension of the liquid, is used to study the fluid motion in a cylinder with free cylindrical surface (under normal and zero-gravity conditions); and in a rectangular closed cell with a source of thermocapillary convection (bubble inside attached to one of the cell's faces). They are significant problems in crystal growth and in general experiments in fluid dynamics respectively. Nevertheless, the main study is dedicated to the liquid bridge problem.<p><p>The development of thermocapillary convection inside a cylindrical liquid bridge is investigated by using a direct numerical simulation of the 3D, time-dependent problem for a wide range of Prandtl numbers, Pr = 0.01 - 108. For Pr > 0.08 (e.g. silicon oils), above the critical value of temperature difference between the supporting disks, two counter propagating hydrothermal waves bifurcate from the 2D steady state. The existence of standing and traveling waves is discussed. The dependence of viscosity upon temperature is taken into account. For Pr = 4, 0-g conditions, and for Pr = 18.8, 1-g case with unit aspect ratio an investigation of the onset of chaos was numerically carried out. <p><p>For a Pr = 108 liquid bridge under terrestrial conditions ,the appearance and the development of thermoconvective oscillatory flows were investigated for different ambient conditions around the free surface.<p><p>Transition from 2D thermoconvective steady flow to a 3D flow is considered for low-Prandtl fluids (Pr = 0.01) in a liquid bridge with a non-cylindrical free surface. For Pr < 0.08 (e.g. liquid metals), in supercritical region of parameters 3D but non-oscillatory convective flow is observed. The computer program developed for this simulation transforms the original non-rectangular physical domain into a rectangular computational domain.<p><p>A study of how presence of a bubble in experimental rectangular cell influences the convective flow when carrying out microgravity experiments. As a model, a real experiment called TRAMP is numerically simulated. The obtained results were very different from what was expected. First, because of residual gravity taking place on board any spacecraft; second, due to presence of a bubble having appeared on the experimental cell's wall. Real data obtained from experimental observations were taken for the calculations.<p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

The advance in computational fluid dynamics in recent years has provided the opportunity for many fluid dynamic problems to be analyzed numerically. One such problem concerns the modeling of plunging water into a still water body, often encountered in pump stations. Air bubbles introduced into the system by the plunging jet can be a significant problem, especially when consumed into operating pumps. The classical approach to investigate the hydrodynamics of plunging jet in pump stations is by physical model studies. This approach is time consuming, tedious and costly. The availability of computational power today, along with appropriate numerical techniques, allows such phenomenon to be studied in a greater level of detail and more cost efficient. Despite the advantages of numerical studies, little attention has been devoted to solve the plunging jet and air transport problem numerically. In this current work, a 3-dimensional finite volume, Large Eddy Simulation (LES) code is developed to simulate these flow conditions. For turbulent flow, the large scale quantities were numerically resolved while the dynamic sub-grid scale model is used to model the small scale energy dissipations. The code also has the capability to handle free surface deformation, an important aspect in simulating the impact section of an impinging jet. Modeling of the air entrainment is performed numerically utilizing the information obtained from the hydrodynamics. Migration of air bubbles is modeled using the scalar transport equation, modified to account for the buoyancy of the bubbles. Instead of the typical Lagrangian schemes, which track individual air bubbles, air bubble dynamics are modeled in the form of concentrations. Modeling air bubbles in this manner is computational efficient and simpler to implement. For the air entrainment simulations, standard numerical boundaries conditions and empirical entrainment equations are used to provide the necessary boundary conditions. The developed model is compared with the literature, producing satisfactory results, suggesting that the code has an excellent potential of extending its application to practical industry practices.

The successful application of Computational Aeroacoustics (CAA) requires high accuracy numerical schemes with good dissipation and dispersion characteristics. Unstructured meshes have a greater geometrical flexibility than existing high order structured mesh methods. This work investigates the suitability of unstructured mesh techniques by computing a two-dimensionallinearised Euler problem with various discretisation schemes and different mesh types. The goal of the present work is the development of an unstructured numerical method with the high accuracy, low dissipation and low dispersion required to be an effective tool in the study of aeroacoustics. The suitability of the unstructured method is investigated using aeroacoustic test cases taken from CAA Benchmark Workshop proceedings. Comparisons are made with exact solutions and a high order structured method. The higher order structured method was based upon a standard central differencing spatial discretisation. For the unstructured method a vertex-based data structure is employed. A median-dual control volume is used for the finite volume approximation with the option of using a Green-Gauss gradient approximation technique or a Least Squares approximation. The temporal discretisation used for both the structured and unstructured numerical methods is an explicit Runge-Kutta method with local timestepping. For the unstructured method, the gradient approximation technique is used to compute gradients at each vertex, these are then used to reconstruct the fluxes at the control volume faces. The unstructured mesh types used to evaluate the numerical method include semi-structured and purely unstructured triangular meshes. The semi-structured meshes were created directly from the associated structured mesh. The purely unstructured meshes were created using a commercial paving algorithm. The Least Squares method has the potential to allow high order reconstruction. Results show that a Weighted Least gradient approximation gives better solutions than unweighted and Green-Gauss gradient computation. The solutions are of acceptable accuracy on these problems with the absolute error of the unstructured method approaching that of a high order structured solution on an equivalent mesh for specific aeroacoustic scenarios.

The world population is expected to increase to approximately 11 billion by 2100. The ageing population (aged 60 and over) is projected to exceed the number of children in 2047. This will be a situation without precedent. The number of citizens with disorders of old age like Dementia will rise to 115 million worldwide by 2050. The estimated cost of Dementia will also increase, from $604 billion in 2010, to $1,117 billion by 2030. At the same time, medical expertise, evidence-driven policymaking and commissioning of services are increasingly evolving the definitive architecture of comprehensive long-term care to account for these changes. Technological advances, such as those provided by computational science and biomedical engineering, will allow for an expansion in our ability to model and simulate an almost limitless variety of complex problems that have long defied traditional methods of medical practice. Numerical methods and simulation offer the prospect of improved clinically relevant predictive information, and of course optimisation, enabling more efficient use of resources for designing treatment protocols, risk assessment and urgently needed management of a long term care system for a wide spectrum of brain disorders. Within this paradigm, the importance of the relationship of senescence of cerebrospinal fluid transport to dementia in the elderly make the cerebral environment notably worthy of investigation through numerical and computational modelling. Hydrocephalus can be succinctly described as the abnormal accumulation (imbalance between production and circulation) of cerebrospinal fluid (CSF) within the brain. Using hydrocephalus as a test bed, one is able to account for the necessary mechanisms involved in the interaction between cerebral fluid production, transport and drainage. The current state of knowledge about hydrocephalus, and more broadly integrative cerebral dynamics and its associated constitutive requirements, advocates that poroelastic theory provides a suitable framework to better understand the disease. In this work, Multiple-network poroelastic Theory (MPET) is used to develop a novel spatio-temporal model of fluid regulation and tissue displacement in various scales within the cerebral environment. The model is discretised in a variety of formats, through the established finite difference method, finite difference – finite volume coupling and also the finite element method. Both chronic and acute hydrocephalus was investigated in a variety of settings, and accompanied by emerging surgical techniques where appropriate. In the coupled finite difference – finite volume model, a key novelty was the amalgamation of anatomically accurate choroid plexuses with their feeding arteries and a simple relationship relaxing the constraint of a unique permeability for the CSF compartment. This was done in order to account for Aquaporin-4 sensitisation. This model is used to demonstrate the impact of aqueductal stenosis and fourth ventricle outlet obstruction. The implications of treating such a clinical condition with the aid of endoscopic third (ETV) and endoscopic fourth ventriculostomy (EFV) are considered. It was observed that CSF velocity in the aqueduct, along with ventricular displacement, CSF pressure, wall shear stress and pressure difference between lateral and fourth ventricles increased with applied stenosis. The application of ETV reduced the aqueductal velocity, ventricular displacement, CSF pressure, wall shear stress and pressure difference within nominal levels. The greatest reversal of the effects of atresia come by opting for ETV rather than the more complicated procedure of EFV. For the finite difference model incorporating nonlinear permeability, qualitatively similar results were obtained in comparison to the pertinent literature, however, there was an overall amplification of ventriculomegaly and transparenchymal pressure difference using this model. A quantitative and qualitative assessment is made of hydrocephalus cases involving aqueductal stenosis, along with the effects to CSF reabsorption in the parenchyma and subarachnoid space. The finite element discretisation template produced for the n^th- dimensional transient MPET system allowed for novel insight into hydrocephalus. In the 1D formulation, imposing the breakdown of the blood-CSF barrier responsible for clearance resulted in an increase in ventricular displacement, transparenchymal venous pressure gradient and transparenchymal CSF pressure gradient, whilst altering the compliance proved to markedly alter the rate of change of displacement and CSF pressure gradient. The influence of Poisson's ratio was investigated through the use of the dual-grid solver in order to distinguish between possible over or under prediction of the ventricular displacement. In the 2D model based on linear triangles, the importance of the MPET boundary conditions is acknowledged, along with the quality of the underlying mesh. Interesting results include that the fluid content is highest in the periventricular region and the skull, whilst after longer time scales, the peak CSF content becomes limited to the periventricular region. Venous fluid content is heavily influenced by the Biot-Willis constant, whilst both the venous and CSF/ISF compartments show to be strongly influenced by breakdown in the blood-CSF barrier. Increasing the venous compliance effects the arterial, capillary and venous compartments. Decreasing the venous compliance shows an accumulation of fluid, possibly helping to explain why the ventricles can be induced to compress rather than expand under decreased compliance. Finally, a successful application of the 3D-MPET template is shown for simple geometries. It is envisaged that future observations into the biology of cerebral fluid flow (such as perivascular CSF-ISF fluid exchange) and its interaction with the surrounding parenchyma, will demand the evolution of the MPET model to reach a level of complexity that could allow for an experimentally guided exploration of areas that would otherwise prove too intricate and intertwined under conventional settings.

Este trabalho apresenta um estudo sobre a solução numérica da equação geral de advecção-difusão usando uma metodologia numérica conservativa. Para a discretização espacial, é usado o Método de Volumes Finitos devido à natureza conservativa da equação em questão. O método é configurado de modo a ter suas variáveis centradas em centro de célula e, para as variáveis, como a velocidade, centradas nas faces um método de interpolação de segunda ordem é utilizado para um ajuste numérico ao centro. Embora a implementação computacional tenha sido feita de forma paramétrica de maneira a acomodar outros esquemas numéricos, a discretização temporal dá ênfase ao Método de Crank-Nicolson. Tal método numérico, sendo ele implícito, dá origem a um sistema linear de equações que, aqui, é resolvido empregando-se o Método Multigrid-Multinível. A corretude do código implementado é verificada a partir de testes por soluções manufaturadas, de modo a checar se a ordem de convergência prevista em teoria é alcançada pelos métodos numéricos. Um jato laminar é simulado, com o acoplamento entre a equação de Navier-Stokes e a equação geral de advecção-difusão, em um domínio computacional tridimensional. O jato é uma forma de vericar se o algoritmo de geração de malhas adaptativas funciona corretamente. O módulo produzido neste trabalho é baseado no código computacional AMR3D-P desenvolvido pelos grupos de pesquisa do IME-USP e o MFLab/FEMEC-UFU (Laboratório de Dinâmica de Fluidos da Universidade Federal de Uberlândia). A linguagem FORTRAN é utilizada para o desenvolvimento da metodologia numérica e as simulações foram executadas nos computadores do LabMAP(Laboratório da Matemática Aplicada do IME-USP) e do MFLab/FEMEC-UFU. / This work presents a study about the numerical solution of variable coecients advectiondi usion equation, or simply, general advection-diusion equation using a conservative numerical methodology. The Finite Volume Method is choosen as discretisation of the spatial domain because the conservative nature of the focused equation. This method is set up to have the scalar variable in a cell centered scheme and the vector quantities, such velocity, are face centered and they need a second order interpolation to get adjusted to the cell center. The computational code is parametric, in which, any implicit temporal discretisation can be choosen, but the emphasis relies on Crank-Nicolson method, a well-known second order method. The implicit nature of aforementioned method gives a linear system of equations which is solved here by the Multilevel-Multigrid method. The correctness of the computational code is checked by manufactured solution method used to inspect if the theoretical order of convergence is attained by the numerical methods. A laminar jet is simulated, coupling the Navier-Stokes equation and the general advection-diusion equation in a 3D computational domain. The jet is a good way to check the corectness of adaptative mesh renement algorithm. The module designed here is based in a previous implemented code AMR3D-P designed by IME-USP and MFLab/FEMEC-UFU (Fluid Dynamics Laboratory, Federal University of Uberlândia). The programming language used is FORTRAN and the simulations were run in LabMAP(Applied Mathematics Laboratoy at IME-USP) and MFLab/FEMEC-UFU computers.

In this thesis study, Detached Eddy Simulation turbulence model is studied in two dimension mainly for flow over single element airfoils in high Reynolds numbers to gain experience with model before applying it to a three dimensional simulations. For this aim, Spalart-Allmaras and standard DES ,DES97, turbulence models are implemented to parallel, viscous, hybrid grid flow solver. The flow solver ,Set2d, is written in FORTRAN language. The Navier-Stokes equations are discretized by first order accurately cell centered finite volume method and solved explicitly by using Runge-Kutta dual time integration technique. Inviscid fluxes are computed using Roe flux difference splitting method. The numerical simulations are performed in parallel environment using domain decomposition and PVM library routines for inter-process communications. To take into account the effect of unsteadyness after the convergence is ensured by local time stepping technique for four order magnitude drop in density residual, global time stepping is applied for 20000 iterations. The solution algorithm is validated aganist the numerical and experimental studies for single element airfoils in subsonic and transonic flows. It is seen that Spalart-Allmaras and DES97 turbulence models give the same results in the non-seperated flows. Grey area is investigated by changing $C_{DES}$ coefficient. Modeled Stress Depletion which cause reduction of eddy viscosity is observed.

This thesis presents the numerical algorithms to carry out the face transformation. The main efforts are denoted to the finite volume method (FVM) with the Delaunay triangulation to solve the Laplace equations in the harmonic transformation undergone in face images. The advantages of the FVM with the Delaunay triangulation are: (1) Easy to formulate the linear algebraic equations, (2) Good to retain the geometric and physical properties, (3) less CPU time needed. The numerical and graphical experiments are reported for the face transformations from a female to a male, and vice versa. The computed sequential and absolute errors are and , where N is division number of a pixel into subpixels. Such computed errors coincide with the analysis on the splitting-shooting method (SSM) with piecewise constant interpolation in [Li and Bui, 1998c].