The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of fluids, from water moving around a naval vessel to blood flowing through the arteries of the cardiovascular system. Furthermore, the secrets of turbulence are widely believed to be locked within the Navier-Stokes equations. Despite the enormous applicability of the Navier-Stokes equations, the underlying behavior of solutions to the partial differential system remains little understood. Indeed, one of the Clay Mathematics Institute's famed Millenium Prize Problems involves the establishment of existence and smoothness results for Navier-Stokes solutions, and turbulence is considered, in the words of famous physicist Richard Feynman, to be "the last great unsolved problem of classical physics." Numerical simulation has proven to be a very useful tool in the analysis of the Navier-Stokes equations. Simulation of incompressible flows now plays a major role in the industrial design of automobiles and naval ships, and simulation has even been utilized to study the Navier-Stokes existence and smoothness problem. In spite of these successes, state-of-the-art incompressible flow solvers are not without their drawbacks. For example, standard turbulence models which rely on the existence of an energy spectrum often fail in non-trivial settings such as rotating flows. More concerning is the fact that most numerical methods do not respect the fundamental geometric properties of the Navier-Stokes equations. These methods only satisfy the incompressibility constraint in an approximate sense. While this may seem practically harmless, conservative semi-discretizations are typically guaranteed to balance energy if and only if incompressibility is satisfied pointwise. This is especially alarming as both momentum conservation and energy balance play a critical role in flow structure development. Moreover, energy balance is inherently linked to the numerical stability of a method. In this dissertation, novel B-spline discretizations for the generalized Stokes and Navier-Stokes equations are developed. The cornerstone of this development is the construction of smooth generalizations of Raviart-Thomas-Nedelec elements based on the new theory of isogeometric discrete differential forms. The discretizations are (at least) patch-wise continuous and hence can be directly utilized in the Galerkin solution of viscous flows for single-patch configurations. When applied to incompressible flows, the discretizations produce pointwise divergence-free velocity fields. This results in methods which properly balance both momentum and energy at the semi-discrete level. In the presence of multi-patch geometries or no-slip walls, the discontinuous Galerkin framework can be invoked to enforce tangential continuity without upsetting the conservation and stability properties of the method across patch boundaries. This also allows our method to default to a compatible discretization of Darcy or Euler flow in the limit of vanishing viscosity. These attributes in conjunction with the local stability properties and resolution power of B-splines make these discretizations an attractive candidate for reliable numerical simulation of viscous incompressible flows. / text

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. / text

The fluid/gravity correspondence relates solutions of the incompressible Navier-Stokes equation to metrics which solve the Einstein equations. We propose propose two possible approaches to establish this correspondence: perturbative expansion for shear modes and large mean curvature expansion for algebraically special metrics. / Physics

published_or_final_version / abstract / toc / Mechanical Engineering / Doctoral / Doctor of Philosophy

The two-dimensional, viscous flow around an elliptic cylinder undergoing prescribed unsteady motions is analyzed. The fluid is taken to be incompressible. Departing from the conventional vorticity-stream function approach, the Biot-Savart law of induced velocities is utilized to account for the contribution to the velocity field of the different vorticity fields comprising the flow. These include the internal vorticity due to the rotation of the body, the free vorticity in the fluid surrounding the body, and the bound vorticity distributed along the body contour. In order to apply the method, the body must be assumed to be replaced by fluid of the same density as the undisturbed surroundings. The replacement fluid must have a rigid motion exactly the same as the actual body motion. This can be achieved by placing suitable distributed vorticity fields within and on the surface of the body. The bound vorticity on the body surface is in the form of a vortex sheet, and its distribution is governed by a Fredholm integral equation of the second kind. The equation is derived in detail. It is solved numerically. The motion of the free vorticity in the flow field is governed by the Navier-Stokes equations written in terms of vorticity. The descretized vorticity transport equation is derived for a control volume and is solved numerically using an explicit method with a forward-difference for the time derivative, and a central-difference for the diffusive terms. An upwind method is used for convection terms. The results obtained using the present method are compared with a number of special cases available in the literature. Viscous flows around a circular cylinder rotating in any arbitrary fashion possess an exact solution, as presented in Chapter 2. Two cases of this flow are chosen for comparison. In the first case the circular cylinder is initially given an impulsive twist such that it rotates with a constant velocity about its axis. In the second case, the angular velocity of the circular cylinder increases with time exponentially. For a Reynolds number of 100, based on the cylinder radius and the internal vorticity, the exact solutions are compared with the numerical results. Viscous flow around an elliptic cylinder of .0996 aspect ratio rotating with a constant angular velocity is another special case, available in the literature, which is chosen for comparison. For this case the Reynolds number, based on the cylinder semi-major-axis and internal vorticity is 202. The agreement in all above-mentioned cases is excellent. Finally, viscous flow around an elliptic cylinder of .25 aspect ratio undergoing a combined translation and pitching oscillation is presented. A Reynolds number of 500, based on the semi-major-axis and body translational velocity, is chosen for this case. No similar case has been reported until now. This case, however, is only one of the many cases that can be handled by the present method.

No description available.

No description available.

No description available.

No description available.

Thesis (Ph.D.)--Tufts University, 1992. / Submitted to the Dept. of Mechanical Engineering. Adviser: Vincent P. Manno. Includes bibliographical references (leaves 183-193). Access restricted to members of the Tufts University community. Also available via the World Wide Web;