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

该项目的目标是为复杂域的三维高速非定常粘性流体流动提供一种高效的高阶方法。尽管在过去几年中相关研究领域取得了重大进展，但现有的求解器无法实现所有这些目标。为实现这一目标，可以确定两个主要目标。首先，研究了非结构网格高阶谱差法的发展。更具体地说，该方案将扩展到三维，并为混合元素网格制定。后者对于高雷诺数流动的有效分辨是必要的。此外，在分析和验证可行的粘性离散化技术后，该方案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。对于许多实际工程问题，特别是高速粘性流动问题，三阶或更高空间阶的数值格式还不够有效。为了保持自由度的数量与二阶方法相当，网格单元的数量必须随着阶数的增加而减少。为了在无振荡的情况下快速地解决不连续性，将引入自适应的hp。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非线性方程组的生成系统。为了实现这一目标，hp多网格策略将与有效的时间步进和松弛方案相结合，用于稳定和非稳定流动。计算仿真已经彻底改变了工程设计过程，并且是目前在广泛的高科技行业中最重要的工具。这反过来又对许多对社会和科学非常重要的现实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数值算法的艺术状态，这将需要美国保持在工业和科学应用的计算模拟的前沿。在航空航天工业中，自喷气式运输出现以来，模拟方法在很大程度上使远程商用飞机的燃油效率提高了一倍。对于航空运输的经济和环境影响来说，进一步的改进是至关重要的，一架大型喷气式飞机在一次飞行中释放的二氧化碳相当于起飞重量（约400吨）。欧盟已经认识到推进模拟技术与各种欧洲范围和国家倡议的重要性（如德国的Mega Flow和Mega Design计划）。747机翼的模拟和优化设计可以节省大量燃料，飞行时间和减少排放。计算机模拟对许多其他应用的进步同样至关重要。例如，模拟血液流经跳动的心脏及其瓣膜，可以帮助医生了解并修复故障的心脏；空中交通管制模式的模拟可以使航空业更快地走向更安全的天空，减少空中碰撞的发生，并减少围绕天气系统重新安排航班的时间；计算机芯片周围和快速旋转磁盘驱动器表面的气流建模提高了功能和可靠性。今天，模拟巨大的全球天气系统的能力可以改变与海啸、洪水、飓风破坏作斗争的国家的命运；天体物理学和许多对国家安全至关重要的研究都依赖于科学和数学的这个新方面。所有这些的核心是更先进的数值算法的发展，它提供了计算模拟的基石。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可立即转移到其他工业和科学应用中，包括上述应用，以及许多其他应用，如汽车和海洋工业，风能和广泛的环境问题。甚至城市地区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风在密集高层建筑的“峡谷”中循环传播。为了在快速发展的全球经济中保持美国的技术领先地位和竞争力，在计算模拟及其基础数值方法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关键的作用，并对环境，科学，经济和社会产生重大影响，因为这些结果过滤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.