| Born in 1977 to Lucy and Gingold & Monaghan, the smoothed particle hydrodynamics (SPH) method has been extended to a broad range of problems, including astrophysics, computational fluid dynamics (CFD), molecular dynamics, solid mechanics, and parallel tree codes. Over the course of the past two decades, SPH has been shown to be quite robust, yielding more-than-reasonable answers to challenging physical problems. Its advantages over more common, time-tested techniques, such as the finite element method (FEM), are its lack of a grid, its inherently transient nature, and the ease with which difficult physics can be incorporated directly into the method.; At the heart of the SPH method is the smoothing, or kernel, function W, which allows the physical observables at any position to be determined from the properties of surrounding positions. To date, many kernel functions have been conjectured, and every researcher seems to have his/her own favorite W. In Chapter 2 of this thesis, we describe and analyze several of the more popular kernels (both exponential and polynomial), and we present a new kernel function W BK, which is able to easily emulate any and all of the existing kernel functions currently in use.; Chapter 3 outlines the SPH fluid mechanics formulation as derived from the Navier-Stokes equation. Alternative forms of the SPH equations are discussed, and special attention is paid to the momentum conservation through anti-symmetrization of the SPH equations. We also prove that the two most common methods for the determination of the pseudo-particle density, summation over neighbors and the SPH farm of the continuity equation, are equivalent.; Boundary conditions are the subject matter of Chapter 4. When the SPH method was first developed, it was exclusively used for astrophysical simulations, and the existence of solid boundaries was irrelevant. However, now that SPH has been adapted for confined engineering systems, boundary conditions are of the utmost importance. We discuss the invention of “ghost” particles, and using the new WBK, we derive the mathematical form for the ghost particle contributions to density and the equation of motion.; Chapter 5 summarizes our application of the SPH formulation to the Navier-Stokes equations in the following systems: (1) Couette (drag) flow of a single-component fluid; (2) Poiseuille (pressure-driven) flow of a single-component fluid; (3) combined (Couette + Poiseuille) flow of a single-component fluid; (4) Couette flow of two immiscible fluids; (5) Poiseuille flow of two immiscible fluids; and (6) drag-induced rotational flow of a single-component fluid.; In this thesis, we have demonstrated that SPH is able to generate accurate flow profiles with only several hundred particles. Also, in the case of two immiscible fluids, we found that SPH does not require any additional boundary conditions at the fluid-fluid interface to preserve shear-stress continuity.; We believe that the success of SPH in the CFD area, at least for the near future, is linked to techniques like FEM, where the advantages of both methodologies can be exploited. However, we sense that SPH will eventually be the best scheme for the simulation of complex systems, e.g., solid particles entrained in fluid flows. |
