The objective of this work is to investigate four fundamentally new ideas, with high potential impact, for improving accuracy and efficiency in fluid flow simulations. Each of these ideas is a fundamentally new approach to well-known challenges in flow simulation, is built from a solid mathematical foundation, and is motivated by the idea that more physically accurate models and numerical methods will produce more accurate results. The main ideas proposed for study are 1) development of the new velocity-vorticity-helicity formulation of the Navier-Stokes equations, 2) investigation of numerical methods for improving mass conservation in finite element methods that approximate velocity with piecewise continuous elements, 3) development of enhanced physics based schemes for flow problems that enforce discrete conservation laws in addition to energy (e.g. helicity in Navier-Stokes, cross-helicity in magnetohydrodynamics), 4) improving numerical methods for approximate deconvolution models of turbulence.
Simulating incompressible viscous fluid flow is an important subtask in most every engineering application involving the flow of water, oil, and/or most other liquids. The ability to accurately and efficiently simulate these flows leads to improved engineering designs, improves turn-around time for designs, and also significant cost savings when testing is done on a computer model instead of a physical model. However, modern computational methods for performing these simulations remain unreliable on many problems of interest. This project will improve the state-of-the-art methods by developing/improving methods with a solid mathematical foundation, better enforcing the physical fidelity in simulations (i.e. avoiding non-physical solutions), and improving efficiency in the simulation techniques. Furthermore, the models and methods developed herein will have the potential to make an impact on the related systems of equations that govern atmospheric flow, oceanic flow, and climate modeling. Broader impacts for this project includes training of graduate and undergraduate students in this field of research, the writing of a book on models for fluid simulation, and outreach to high school students.