<p>Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.</p>

In der vorliegenden Arbeit wird die Methode der Anwendung der hyperkomplexen Funktionentheorie zur Behandlung partieller Differentialgleichungen über beschränkten Gebieten unter Benutzung einer orthogonalen Zerlegung des Raumes L_2(U) verallgemeinert. Zum einen kann diese Zerlegung als direkte Zerlegung über dem Raum L_p(G),p>1, verallgemeinert werden, was die Untersuchung partieller Differentialgleichungen über allgemeinen Sobolev-Räumen W_p^k(G),p>1,k natürliche Zahl, ermöglicht. Dies wird am Beispiel des Stokes-Problems demonstriert. Zum anderen wird ein modifizierter Cauchy-Kern über unbeschränkten Gebieten eingeführt, deren Komplement eine nichtleere offene Menge enthält. Grundlegende Resultate der Cliffordanalysis über beschränkten Gebieten werden auf diese Situation verallgemeinert und eine orthogonale Zerlegung des Raumes L_2(G) bewiesen. Diese Resultate werden im weiteren dazu benutzt, das stationäre Stokes- bzw. Navier-Stokes-Problem in dem allgemeinen Fall eines unbeschränkten Gebietes zu untersuchen. Im weiteren wird gezeigt, dass sich die entwickelten Methoden auch auf partielle Differentialgleichungen höherer Ordnung anwenden lassen. Dies wird am Beispiel der biharmonischen Gleichung mit Randbedingungen, die Komponenten in Normalenrichtung und tangentieller Richtung besitzen, demonstriert. Am Ende beschäftigen wir uns mit der Verallgemeinerung der komplexen Methoden von Vekua. Dazu werden hyperkomplexe Verallgemeinerungen des komplexen Pi-Operators untersucht und auf die Lösung von hyperkomplexen Beltramigleichungen angewandt. / A modified Cauchy kernel is introduced over unbounded domains whose complement contain non-empty open sets. Basic results on Clifford analysis over bounded domains are now carried over to this more general context. In the end boundary value problems, e.g. for the Stokes-system or the Navier-Stokes-system, will be studied in the case of an unbounded domain without using weighted Sobolev spaces. In the latter part of this paper we deal with hypercomplex generalizations of the complex Pi-operator which turn out to have most of the useful properties of their complex origin. Afterwards the application of this operator to the solution of hypercomplex Beltrami equations will be studied.

Im Rahmen des Modellierungsseminars wurde die Kopplung einer Strömung mit der Ausbreitung einer reaktiven Substanz im Strömungsgebiet untersucht. Die Strömung wurde dabei durch die inkompressiblen Navier-Stokes-Gleichungen beschrieben. Zusätzlich wurde ein mathematisches Modell für die Ausbreitung der Substanz durch eine Diffusions-Konvektions-Gleichung bestimmt. Beide wurden durch die FEM- Sofware NAVIER berechnet und simuliert.

Thesis (M.S.) -- Mississippi State University. Department of Computational Engineering. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--Mississippi State University. Department of Engineering. / Title from title screen. Includes bibliographical references.

A computational approach to analyze the hydrodynamic performance of a hydrofoil or a marine propeller undergoing unsteady motion has been developed. In order to simulate heave and pitch motion of a hydrofoil, an unsteady boundary element method based modeling is performed. The wake of the hydrofoil is modeled by a continuous dipole sheet and determined in time by applying a force-free condition on its surface. An explicit vortex core model is adapted in this model to capture the rolling up shape and to avoid instability due to roll-up deformation of the wake. The numerical results of the developed model are compared with analytical results and those from the commercial Reynolds-Averaged Navier-Stokes solver (ANSYS/FLUENT). The results show close level of agreement with each other. The problem of flow around a marine propeller performing surge, roll and heave motion in an unbounded fluid is formulated and solved using both a vortex-lattice method and a boundary element method. A fully unsteady wake alignment algorithm is implemented into the vortex-lattice method in order to satisfy the force-free condition on the propeller wake surface. Finally, a comparative study of transient propeller forces on a propeller blade obtained from BEM and VLM (with or without fully aligned wake) is carried out and results are presented. In some cases, results from the presented methods are compared with those from RANS or other numerical methods available in the literature. / text

Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

Disertacijos santraukoje apžvelgiami Navjė-Stokso lygčių periodiniai laiko atžvilgiu uždaviniai srityse su cilindriniais išėjimais į begalybę. / In this PhD dissertation summary is considered time periodic Navier-Stokes equations in domains with cylindrical outlets to infinity.

In the thesis there is studied nonhomogenous boundary value problem for the stationary Navier-Stokes system in domains which may have two types of outlets to infinity: paraboloidal and layer type. The boundary is multiply connected. It consists of connected noncompact components, forming the outer boundary, and connected compact components, forming the inner boundary. We suppose that the fluxes over the components of the inner boundary are sufficiently small, while we do not impose any restrictions on fluxes over the infinite components of the outer boundary. We prove that the formulated problem admits at least one weak solution which, depending on the geometry of the domain, may have either finite or infinite Dirichlet integral. / Disertacijoje nagrinėjama stacionari Navjė-Stokso sistema su nehomogenine kraštine sąlyga srityse su išėjimais į begalybę. Bendru atveju išėjimai į begalybę gali būti tiek paraboloidiniai, tiek sluoksnio tipo. Srities kraštą sudaro baigtinis skaičius nekompaktiškų jungių komponenčių, kurios suformuoja išorininį kraštą, ir baigtinis skaičius kompaktiškų jungių komponenčių, kurios suformuoja vidinį srities kraštą. Darydami prielaidą, kad srautai per vidinio krašto komponentes yra pakankamai maži, o srautų dydžiui per išorinio krašto komponentes nedarant jokių apribojimų, įrodome suformuluoto uždavinio bent vieno sprendinio egzistavimą. Priklausomai nuo srities geometrijos, uždavinio sprendinys gali turėti tiek baigtinį, tiek begalinį Dirichlė integralą.

Disertacijoje nagrinėjama stacionari Navjė-Stokso sistema su nehomogenine kraštine sąlyga srityse su išėjimais į begalybę. Bendru atveju išėjimai į begalybę gali būti tiek paraboloidiniai, tiek sluoksnio tipo. Srities kraštą sudaro baigtinis skaičius nekompaktiškų jungių komponenčių, kurios suformuoja išorininį kraštą, ir baigtinis skaičius kompaktiškų jungių komponenčių, kurios suformuoja vidinį srities kraštą. Darydami prielaidą, kad srautai per vidinio krašto komponentes yra pakankamai maži, o srautų dydžiui per išorinio krašto komponentes nedarant jokių apribojimų, įrodome suformuluoto uždavinio bent vieno sprendinio egzistavimą. Priklausomai nuo srities geometrijos, uždavinio sprendinys gali turėti tiek baigtinį, tiek begalinį Dirichlė integralą. / In the thesis there is studied nonhomogenous boundary value problem for the stationary Navier-Stokes system in domains which may have two types of outlets to infinity: paraboloidal and layer type. The boundary is multiply connected. It consists of connected noncompact components, forming the outer boundary, and connected compact components, forming the inner boundary. We suppose that the fluxes over the components of the inner boundary are sufficiently small, while we do not impose any restrictions on fluxes over the infinite components of the outer boundary. We prove that the formulated problem admits at least one weak solution which, depending on the geometry of the domain, may have either finite or infinite Dirichlet integral.