Browsing by Subject "Navier-Stokes equations"

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A discontinuous Petrov-Galerkin methodology for incompressible flow problems
(2013-08) Roberts, Nathan Vanderkooy; Demkowicz, Leszek; Moser, Robert deLancey
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.
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.
A new incompressible Navier-Stokes method is developed for unstructured general hybrid meshes which contain all four types of elements in a single computational domain, namely tetrahedra, pyramids, prisms, and hexahedra. Various types of general hybrid meshes are utilized and appropriate numerical flux computation schemes are presented. The artificial compressibility method with a dual time-stepping scheme is used for the time-accurate solution of the incompressible Navier-Stokes equations. The Spalart-Allmaras turbulence model is also presented in the dual time-stepping form and is solved in a strongly coupled manner with the incompressible Navier-Stokes equations. The developed scheme is applied to the study of the inflow turbulence effect on the hydrodynamic forces exerted on a circular cylinder. In order to accommodate possible structural and mesh motion, the method is extended to the arbitrary Lagrangian-Eulerian (ALE) frame of reference. The geometric conservation law is satisfied with the proposed ALE scheme in moving mesh simulations. The developed ALE scheme is applied to the vortex induced vibration of a cylinder. A strong coupling of fluid and structure interaction based on the predictor-corrector method is presented. The superior stability property of the strong coupling is demonstrated by a comparison with the weak coupling. Finally, the developed methods are parallelized for distributed memory machines using partitioned general hybrid meshes and an efficient parallel communication scheme to minimize CPU time.
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.
The vanishing viscosity limit for incompressible fluids in two dimensions
(2005) Kelliher, James Patrick; Vishik, Mikhail
The 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.