Improved Methods for Incompressible Viscous Flow Simulation

这项工作的目的是调查四个根本性的新想法，具有很高的潜在影响，以提高流体流动模拟的准确性和效率。 这些想法中的每一个都是解决流动模拟中众所周知的挑战的一种全新方法，建立在坚实的数学基础上，并且受到物理上更精确的模型和数值方法将产生更准确结果的想法的激励。提出的主要研究思路是：1）发展新的Navier-Stokes方程的速度-涡度-螺旋度公式，2）研究改进有限元方法中质量守恒的数值方法，即用分段连续单元近似速度，3）开发用于流动问题的增强的基于物理的方案，除了能量之外，还执行离散守恒定律。（如Navier-Stokes方程中的螺旋度，磁流体力学中的交叉螺旋度）; 4）改进湍流近似解卷积模型的数值方法。不可压缩粘性流体流动的模拟是涉及水、油和/或大多数其他液体流动的大多数工程应用中的重要子任务。 准确有效地模拟这些流动的能力可以改进工程设计，缩短设计周转时间，并且在计算机模型而不是物理模型上进行测试时还可以显著节省成本。 然而，现代计算方法进行这些模拟仍然不可靠的许多问题的利益。 该项目将通过开发/改进具有坚实数学基础的方法来改进最先进的方法，更好地加强模拟中的物理保真度（即避免非物理解决方案），并提高模拟技术的效率。此外，本文开发的模型和方法将有可能对管理大气流动、海洋流动和气候建模的相关方程系统产生影响。 这个项目的更广泛的影响包括培训研究生和本科生在这一研究领域，一本书的写作模型流体模拟，并推广到高中学生。

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.