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

本计画的目标是建立一套有效的高阶方法，以求解复杂区域内的三维高速非定常黏性流体流动。没有现有的求解器实现所有这些目标，尽管在过去几年中，在相关的研究领域已经取得了重大进展。为实现这一目标，可确定两个主要目标。首先，发展高阶谱差分（SD）方法的非结构网格将被解决。更具体地说，该计划将扩展到三维，并制定了混合单元网格。后者是必要的高雷诺数流动的有效解决方案。此外，在分析和验证可行的粘性离散技术后，该计划将由首席研究员和他的同事扩展到混合网格上的Navier-Stokes方程。第二个主要目标是提高高阶方法的效率和鲁棒性。 三阶或更高阶的数值格式对于许多实际工程问题，特别是高速粘性流动，还不够有效。为了保持自由度的数量与二阶方法的自由度相当，网格单元的数量必须随着阶数的增加而减少。为了解决不连续性急剧没有振荡的h-p自适应将被引入。对于稳态和时间相关的问题，快速解决技术是必要的，以解决所产生的非线性方程组。为了实现这一目标，一个h-p多重网格策略将与有效的时间步进和松弛格式相结合，用于定常和非定常流动。计算模拟已经彻底改变了工程设计过程，现在是高科技工业中最重要的工具。 这反过来又对许多对社会和科学具有重大意义的真实的生活应用产生了巨大影响，从航空航天到健康再到环境。 拟议的研究旨在推进美国在工业和科学应用中保持计算模拟前沿所需的数值算法的最新发展。 在航空航天工业中，自从喷气式运输机出现以来，模拟方法在很大程度上使远程商用飞机的燃油效率提高了一倍。进一步改进的必要性对于航空运输的经济和环境影响至关重要，大型喷气式飞机在单次飞行中释放其起飞重量（约400吨）的二氧化碳排放量。欧盟已经认识到推进模拟技术的重要性，并采取了各种欧洲范围和国家的举措（如德国Mega Flow和Mega Design计划）。 747机翼的模拟和优化重新设计可以节省数吨的燃料，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，模拟血液流过跳动的心脏及其瓣膜，有助于医生了解和修复故障的心脏;模拟空中交通管制模式可以使该行业更快地走向更安全的天空，减少空中碰撞的发生，并减少改变航班的时间。对计算机芯片周围和快速旋转磁盘驱动器表面上的气流进行建模，改善了功能和可靠性。 今天的能力，以模拟巨大的全球天气系统可以改变国家的命运与海啸的斗争，洪水，飓风的破坏;天体物理学和许多研究的重要性，以国家安全都依赖于这个新的方面的科学和数学。 在所有这些的核心是更先进的数值算法，提供计算模拟的基石的发展。拟议的研究旨在将这些扩展到更有效的高阶3D方法，虽然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可以立即转移到其他工业和科学应用，包括上述以及许多其他应用，如汽车和海洋工业，风能，以及广泛的环境问题。甚至城市建筑物周围的气流模式也可以用来评估细菌或化学颗粒如何通过密集高层建筑的“峡谷”循环的风传播。为了保持美国在快速发展的全球经济中的技术领先地位和竞争力，在计算模拟及其基本数值方法学的各个方面的进一步发展是必不可少的。这些发展和技术的结果将发挥越来越重要的作用，并在环境，科学，经济和社会方面产生重大影响，因为这些结果将渗透到今天使用它们的许多应用中，以及未来需要评估各种物理学的复杂非线性偏微分方程的应用。

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.