Browsing by Subject "Navier-Stokes equations"
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Item A computational study of laminar and turbulent flows in rotating rectangular ducts(Texas Tech University, 1993-12) Asan, HabipRotating flow problems have long been of interest in fluid mechanics because they have a wide range of applications in engineering practice. The flow of fluid in the radial passageways of radial pumps or centrifugal compressors and the flow through the cooling passages of turbine blades are some of the examples of the rotating flow problems. Better understanding of the effect of the rotation on the flows mentioned above will help engineers to design more efficient rotary machines such as turbines and compressors. This work is concerned with fully developed incompressible laminar and turbulent flows through rectangular straight ducts rotating in an orthogonal mode. The Navier-Stokes equations are solved by the finite-volume method for low to high rotation rates. The finite-volume equations are formulated in strong conservative form on a general, nonorthogonal grid system. The resulting equations are solved by an implicit, time marching, pressure correction based algorithm. Solutions are obtained for aspect ratios 1, 2, and 3. For laminar flow, predictions have been performed for Reynolds number of 2000 and for turbulent flow the computations were carried out for a Reynolds number of 20000. Values of the Rossby number ranged from 0.1 to 3500 for laminar flow and from 0.3 to 140 for turbulent flow. The standard k-e model is used to model the turbulence. Low rotational speeds cause the formation of a pair of symmetric vortices on the cross-section. At higher rotational speeds, a more complex four-vortex structure develops. The transition point depends on the cross-sectional geometry. Moreover, over a range of Rossby number, either two- or four-vortex solutions are possible. The rotation leads to significant differences between the values of friction factor and Nusselt number on the suction and pressure sides of the duct. Some comparisons with available experimental and theoretical results were made.Item A validation study of a software implementation of the Gauge method for the incompressible Navier-stokes equations(2011-08) Gohlke, Jedidiah W.; Long, Kevin; Howle, Victoria E.; Kirby, Robert C.; Hoang, LuanThe incompressible Navier-Stokes equations model the relationship between the velocity and pressure of a fluid and can be used to describe the motion of a fluid. Because they are nonlinear, they are in most cases difficult or impossible to solve analytically. They must, therefore, be approximated numerically. One approach developed by Alexandre Chorin is to use a projection method to approximate the pressure and velocity. We will look at Chorin’s approach and then consider a modified approach called the gauge method, which mitigates some of the boundary condition difficulties in Chorin’s approach. We will then consider an improvement to the gauge method that gives us second order convergence in the velocity in both time and space and slightly better than order 3/2 convergence in the pressure in time. We will then present a code to automate the gauge method and consider some test cases to verify that the code converges at the appropriate rate.Item An analytical and numerical investigation of the Kuo tornado model(Texas Tech University, 2000-05) Vugrin, Eric D.The goal of this work is to present an analytical and numerical investigation of the Kuo tornado model. The derivations of the model and a more general solution are described. The effects of several parameters on the streamlines are examined, and numerical simulations and methods are presented to analyze the associated heat equation.Item Asymptotic expansions of the regular solutions to the 3D Navier-Stokes equations and applications to the analysis of the helicity(Texas A&M University, 2005-08-29) Hoang, Luan ThachA new construction of regular solutions to the three dimensional Navier{Stokes equa- tions is introduced and applied to the study of their asymptotic expansions. This construction and other Phragmen-Linderl??of type estimates are used to establish su??- cient conditions for the convergence of those expansions. The construction also de??nes a system of inhomogeneous di??erential equations, called the extended Navier{Stokes equations, which turns out to have global solutions in suitably constructed normed spaces. Moreover, in these spaces, the normal form of the Navier{Stokes equations associated with the terms of the asymptotic expansions is a well-behaved in??nite system of di??erential equations. An application of those asymptotic expansions of regular solutions is the analysis of the helicity for large times. The dichotomy of the helicity's asymptotic behavior is then established. Furthermore, the relations between the helicity and the energy in several cases are described.Item Compressible Fluid Flow Through an Orifice(Texas Tech University, 1973-05) Waller, Herschel NathanielNot Available.Item A discontinuous Petrov-Galerkin methodology for incompressible flow problems(2013-08) Roberts, Nathan Vanderkooy; Demkowicz, Leszek; Moser, Robert deLanceyIncompressible 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.Item Donaldson-Sullivan tornado model(Texas Tech University, 2000-05) Mickel, Church E.The purpose of this paper is to analytically and numerically explore the Generalize Donaldson-Sullivan Tornado Model. Essentially, the Donaldson-Sullivan tornado model is a stationary solution of the Navier-Stokes equation. This solution was derived in the late 1950's and early 1960's and was believed to model a tornado. As the solution is quite complicated and almost impossible to analytically investigate, a numerical investigation is called for. As one will see. the Donaldson-Sullivan Tornado Model shares some qualities with those of an actual tornado. However, one will find that there are many more properties of this solution that do not appear to emulate a tornado. The case ^ / 1 or the Generalized Donaldson-Sullivan Tornado Model is another stationary solution of the Navier-Stokes equation. Like the Donaldson-Sullivan Solution, this solution is quite complicated and requires the use of numerical techniques to efficiently explore its properties. As one will see. this solution possesses some extremely interesting properties. However, it is the b(>lief of the au;hor (.[" ;';.:.- i^n^,: : that these properties do not accurately represent the behavior of a tornado.Item High viscosity solutions of the Navier-Stokes equations modeling a tornado vortex(Texas Tech University, 2000-12) Baker, Joshua ThomasThis thesis will discuss several aspects of the Navier-Stokes system in cylindrical coordinates. We will highlight both the classical Donaldson-Sullivan solution and the recently discovered generalized Donaldson-Sullivan solution and will give their derivations. Our main focus will be on the following system and in particular the viscosity constant, v.Item Local feedback regularization of three-dimensional Navier-Stokes equations on bounded domains(Texas Tech University, 1997-05) Balogh, AndrasThe specific problem we consider here is inspired by recent advances in the control of nonlinear distributed parameter systems and its possible applications to hydrodynamics. The main objective is to investigate the extent to which the 3-dimensional Navier-Stokes system can be regularized using a particular, physically motivated, feedback control law. The specific choice of feedback mechanism is motivated by a work of O.A. Ladyzhenskaya [7] in which she introduces a modification of the Navier-Stokes equation on a three dimensional bounded domain and shows that the resulting perturbed system possesses global dynamics and, furthermore, this dynamics is stable. It is in this sense that we understand the system to be regularized.Item Lubrication as a damping force in the model of eye rotation under listing's constraint(2012-05) Karsli, Neslihan; Aulisa, Eugenio; Ibragimov, Akif; Ghosh, Bijoy K.In this work, the effect of lubrication forces to the rotation of the eye is investigated where the eye is modeled as a perfect sphere and the movement is restricted to Listing's Law. For the lubrication forces, the nondimensional forms of Navier-Stokes and continuity equations are found, the boundary conditions are applied and the torques are derived. Then, Listing's constraint is introduced to the model and a potential function and lubrication as a damping force has been added. Using lubrication as a damping force, an optimum potential function for the model is derived.Item Mass transfer in the cone and plate system and its applications(Texas Tech University, 1980-05) Lo, Jen-tsenMass transfer and electrode reaction for an electrochemical process in the cone and plate geometry was studied. Analytical expressions for the concentration profiles and current distribution were obtained for the following four cases: (i) unsteady state electrolysîs below limiting current conditions; (ii) unsteady state electrolysis at limiting current conditions; (iii) steady state electrolysis below the limiting current conditions; (iv) steady state electrolysis at limiting current conditions. The reaction order, the reaction rate constant, the electrode reaction transfer coefficient, the number of electrons involved, and the diffusion coefficient can be determined by solving for the current necessary to maintain a constant potential difference across the electrodes (potentiostatic method).Item A new incompressible Navier-Stokes method with general hybrid meshes and its application to flow/structure interactions(2005) Ahn, Hyung Taek; Dawson, Clinton N.; Kallinderis, Y.Item Prediction of flows around ship-shaped hull sections in roll using an unsteady Navier-Stokes solver(2008-08) Yu, Yi-Hsiang, 1976-; Kinnas, Spyros A.Ship-shaped hulls have often been found to be subject to excessive roll motions, and therefore, inhibit their use as a stable production platform. To solve the problem, bilge keels have been widely adopted as an effective and economic way to mitigate roll motions, and their effectiveness lies in their ability to damp out roll motions over a range of frequencies. In light of this, the present research focuses on roll motions of shipshaped hulls. A finite volume method based two-dimensional Navier-Stokes solver is developed and further extended into three dimensions. The present numerical scheme is implemented for modeling the flow around ship-shaped hulls in roll motions and for predicting the corresponding hydrodynamic loads. Also conducted are studies on the hydrodynamic performance of ship-shaped hull sections in prescribed roll motions and in transient decay motions. Systematic studies of the grid resolutions and the effects of free surface, hull geometries and amplitude of roll angle are performed. Predictions from the present method compare well to those of other methods, as well as to measurements from experiments. Non-linear effects, due to flow viscosity, were observed in small as well as in large roll amplitudes, particularly in the cases of hulls with sharp corners. The study also shows that it is inadequate to use a linear combination of added-mass and damping coefficients to represent the corresponding hydrodynamic loads. As a result, it also makes the calculation of the hull response in time domain inevitable. Finally, the capability of the present numerical scheme to apply to fully three-dimensional ship motion simulations is demonstrated.Item The vanishing viscosity limit for incompressible fluids in two dimensions(2005) Kelliher, James Patrick; Vishik, MikhailThe Navier-Stokes equations describe the motion of an incompressible fluid of constant density and constant positive viscosity. With zero viscosity, the Navier-Stokes equations become the Euler equations. A question of longstanding interest to mathematicians and physicists is whether, as the viscosity goes to zero, a solution to the Navier-Stokes equations converges, in an appropriate sense, to a solution to the Euler equations: the so-called “vanishing viscosity” or “inviscid” limit. We investigate this question in three settings: in the whole plane, in a bounded domain in the plane, and for radially symmetric solutions in the whole plane. Working in the whole plane and in a bounded domain, we assume a particular bound on the growth of the L p -norms of the initial vorticity (curl of the velocity) with p, and obtain a bound on the convergence rate in the vanishing viscosity limit. This is the same class of initial vorticities considered by Yudovich and shown to imply uniqueness of the solution to the Euler equations in a bounded domain lying in Euclidean space of dimension 2 or greater. For radially symmetric initial vorticities we obtain a more precise bound on the convergence rate as a function of the smoothness of its initial vorticity as measured by its norm in a Sobolev space or in certain Besov spaces. We also consider the questions of existence, uniqueness, and regularity of solutions to the Navier-Stokes and Euler equations, as necessary, to make sense of the vanishing viscosity limit. In particular, we investigate properties of the flow for solutions to the Euler equations in the whole plane. We construct a specific example of an initial vorticity for which there exists a unique solution to the Euler equations whose associated flow lies in no H¨older space of positive exponent for any positive time. This example is an adaptation of a bounded-vorticity example of Bahouri and Chemin’s, which they show has a flow lying in no H¨older space of exponent greater than an exponentially decreasing function of time.