Improved Methods for Incompressible Viscous Flow Simulation

这项工作的目的是研究四个具有高潜在影响的基本新思想，以提高流体流动模拟的准确性和效率。这些想法都是从根本上解决流动模拟中众所周知的挑战的新方法，建立在坚实的数学基础之上，并且受到更精确的物理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)研究用分段连续单元近似速度的有限元方法中改进质量守恒的数值方法；3)发展基于增强物理的流动问题方案，除能量外，还强制执行离散守恒定律（例如Navier-Stokes方程中的螺旋度，磁流体动力学中的交叉螺旋度）。4)改进湍流近似反褶积模型的数值方法。在大多数涉及水、油和/或大多数其他液体流动的工程应用中，模拟不可压缩粘性流体流动是一项重要的子任务。准确有效地模拟这些流程的能力可以改进工程设计，缩短设计的周转时间，并且当在计算机模型而不是物理模型上进行测试时，还可以显着节省成本。然而，用于执行这些模拟的现代计算方法在许多感兴趣的问题上仍然不可靠。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控制大气流动、海洋流动和气候建模的相关方程系统产生影响。该项目的更广泛影响包括在该研究领域培养研究生和本科生，编写一本关于流体模拟模型的书，并向高中生推广。

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.