High-Order Numerical Algorithms for Steady and Unsteady Simulation of Viscous Compressible Flow

该项目的目标是为复杂区域中的三维、高速非定常粘性流体流动产生一种高效的高阶方法。尽管过去几年在相关研究领域取得了重大进展，但没有一个现有的求解器能够实现所有这些目标。为了实现这一目标，可以确定两个主要目标。首先，介绍了非结构网格高阶谱差分方法的发展。更具体地说，该格式将扩展到三维，并针对混合单元网格进行制定。后者对于高雷诺数流动的有效解析是必要的。此外，在分析和验证可行的粘性离散技术后，主要研究人员和他的同事将把该格式推广到混合网格上的Navier-Stokes方程。第二个主要目标是提高高阶方法的效率和稳健性。对于许多具有实际工程意义的问题，特别是高速粘性流动，三阶或更高空间阶的数值格式还不够有效。为了保持自由度数与二阶方法相当，网格单元的数目必须随着阶数的增加而减少。为了在没有振荡的情况下迅速地解决不连续问题，将引入自适应的h-p。对于稳态问题和时变问题，都需要一种快速求解技术来求解所产生的非线性方程组。为了实现这一目标，h-p多重网格策略将与定常和非定常流动的高效时间推进和松弛格式相结合。计算模拟已经彻底改变了工程设计过程，并且到目前为止是广泛的高科技行业中最重要的单一工具。这反过来又在许多对社会和科学非常重要的现实生活应用中产生了巨大影响，从航空航天到健康再到环境。这项拟议的研究旨在推进数值算法的技术水平，这将是美国在工业和科学应用的计算模拟方面保持领先地位所必需的。在航空航天工业中，自从喷气运输出现以来，模拟方法在很大程度上使远程商用飞机的燃油效率翻了一番。进一步改进的必要性对于航空运输的经济和环境影响都是至关重要的，大型喷气式飞机在一次飞行中释放其起飞重量(约400吨)的二氧化碳排放。欧盟已经认识到通过各种欧洲和国家倡议(如德国的Mega Flow和Mega Design计划)推进模拟技术的重要性。波音747机翼的模拟和优化重新设计可以节省“吨”的燃料、旅行时间和减少排放。计算机模拟对于许多其他应用的进步同样至关重要。例如，模拟通过跳动的心脏及其瓣膜的血液流动，帮助医生了解和修复故障的心脏；模拟空中交通管制模式可以更快地将该行业推向更安全的天空，减少空中碰撞的发生，减少改变飞行路线绕过天气系统的时间；对计算机芯片周围和快速旋转的磁盘驱动器表面上的气流进行建模，提高功能和可靠性。今天，对巨大的全球天气系统进行建模的能力可以改变正在与海啸、洪水和飓风的破坏作斗争的国家的命运；天体物理学和许多对国家安全具有重要意义的研究都依赖于科学和数学的这一新方面。所有这些的核心是更先进的数值算法的发展，这些算法提供了计算模拟的基石。拟议的研究旨在将这些扩展到更高效的高阶3D方法，虽然该研究的直接应用是与航空航天相关的流体流动问题，但改进的流体流动模拟能力立即可以转移到其他工业和科学应用，包括上述以及许多其他应用，如汽车和海洋行业、风能和广泛的环境问题。即使是城市地区建筑物周围的流动模式，也可以用来评估风在高楼丛生的峡谷中循环时，细菌或化学颗粒将如何传播。为了保持美国在迅速发展的全球经济中的技术领导地位和竞争力，计算模拟及其基本数值方法各方面的进一步进展至关重要。这些发展和技术的结果将发挥越来越重要的作用，并对环境、科学、经济和社会产生重大影响，因为这些结果将渗透到今天使用它们的许多应用中，以及未来需要计算各种物理的复杂非线性偏微分方程式的应用中。

The goal of this project if to produce an efficient high-order methodology for 3D, high-speed unsteady viscous fluid flow in complex domains. No existing solver achieves all these goals, despite the fact that significant progress in relevant research areas has been made during the last few years. To achieve this goal, two major objectives may be identified. Firstly, the development of the high-order Spectral Difference (SD) Method for unstructured grids will be addressed. More specifically, the scheme will be extended to three dimensions, and formulated for mixed-element grids. The latter is necessary for efficient resolution of flows at high Reynolds numbers. Furthermore, upon analyzing and validating viable viscous discretization techniques, the scheme will be extended by the principal investigator and his colleagues to the Navier-Stokes equations on mixed grids. The second major objective is to improve the efficiency and robustness of the high-order methodology. Numerical schemes of third or higher spatial order are not yet efficient enough for many problems of practical engineering interest, in particular high-speed viscous flow. To keep the number of degrees of freedom comparable to that of a second order method, the number of grid cells must be reduced as the order is increased. To resolve discontinuities sharply without oscillations h-p adaptively will be introduced. For both steady-state and time- dependent problems a fast solution technique is necessary to solve the arising systems of nonlinear equations. In order to meet this goal, an h-p multigrid strategy will be combined with efficient time stepping and relaxation schemes for both steady and unsteady flows.Computational simulation has revolutionized the engineering design process and is by now the single most important tool in a wide range of high technology industries. That in turn has had a huge impact in many real life applications of great importance to society and science, from aerospace to health to the environment. The proposed research is targeted at advancing the state of the art of numerical algorithms which will be needed for the U.S. to remain at the forefront of computational simulation for both industrial and scientific applications. In the Aerospace Industry simulation methods have been largely responsible for doubling the fuel efficiency of long range commercial aircraft since the advent of jet transport. The need for further improvement is crucial for both the economics and environmental impact of air transport, a jumbo jet releases its take-off weight (around 400 tons) in carbon dioxide emissions during a single flight. The European Union has recognized the importance of advancing simulation techniques with a variety of European wide and national initiatives (such as the German Mega Flow and Mega Design programs.) Simulations and optimized redesign of the 747 wing can save tons" of fuel, travel time and reduce emissions. Computer simulation is equally crucial to advances in numerous other applications. For example, simulations of blood flow through a beating heart and its valves, helps doctors understand and repair the malfunctioning heart; simulation of air traffic control patterns can move that industry faster toward safer skies, reduce the occurrence of mid-air collisions and the time to reroute flights around weather systems; modeling of air flow around computer chips and over the surface of a fast spinning disk drive improve function and reliability. Today's ability to model huge global weather system can change the fate of nations struggling with tsunami's, floods, the devastation of hurricanes; Astrophysics and many studies of importance to National Security all rely on this new aspect of science and mathematics. At the heart of all these is the development of more advanced numerical algorithms which provide the cornerstone of computational simulation. The proposed research intends to extend these to more efficient high-order 3D methods, While the immediate application of the research is to fluid flow problems related to aerospace, an improved fluid flow simulation capability is immediately transferable to other industrial and scientific applications, including those mentioned above as well as many others such as the automotive and marine industries, wind energy, and a wide range of environmental issues. Even the flow patterns in urban areas around buildings can be used to evaluate how germ or chemical particles would be spread by winds circulating through the 'canyons' of clustered tall buildings. Further advances in all aspects of computational simulation and its underlying numerical methology is essential in order to maintain the Unites States' technological leadership and competitiveness in the rapidly evolving global economy. The results of these developments and techniques will play an increasingly pivotal role and have major implications environmentally, scientifically economically and socially as the results filter down into the many applications using them today, as well as future applications that will need to evaluate complex nonlinear partial differential equations for a variety of physics.