Go With the Flow: Developing Computational Fluid Dynamics Simulations According to the Navier-Stokes Equations

Danielle Shepherd

2014

Although air flow and fluid flow occur around us everyday, how much is understood about these complicated phenomena? This project sought to explore fluid flow, according to the Navier-Stokes equations, through computational fluid dynamics and computer simulation. For this project, two simulations were created; the first used an artificial compressibility term to update the fluid where the second inverted a Laplacian matrix. The first simulation, while it produced an accurate steady state, artificially progressed to achieve it. As a result, the second simulation was created. This simulation could use either LU decomposition or a direct method to obtain the inverse matrix. The direct method employed Mathematica to invert the Laplacian to double precision and then stored the data in a binary file, which could later be imported into the simulation. This technique reduced round-off error and computation time, allowing for the accurate modeling of larger systems. All the simulations displayed a parabolic profile result during steady state, as predicted by experiments and mathematical derivation. The development and shedding of vortices, within 10 hours of simulation time on a circa 2010 iMac computer, also helped affirm the accuracy of the simulation results. Implementing the exact inverse Laplacian provides a new technique for computational fluid dynamics simulation, as no other method uses the exact inverse matrix. With this advancement, the simulations may improve our knowledge of fluids and how objects interact with their flow.