Development of a hierarchical scale-adaptive large-eddy simulation method for the study of turbulence

研究表明，大气湍流对我们的社会造成了巨大的经济损失。例如，美国航空公司每年支付超过1000万美元来补偿与颠簸相关的航班改道。2018年，恶劣天气给加拿大造成了超过19亿美元的损失。另一方面，风能是可再生能源的主要来源。2018年，加拿大风能装机容量超过1.2万兆瓦。加拿大在数学、流体力学和数值天气预报方面一直处于世界领先地位。数学发现揭示了湍流如何在汽车、飞机、风力涡轮机、桥梁和建筑物上产生阻力；以及它如何通过对大尺度大气和海洋的影响来决定天气。我的研究计划致力于从数学上描述湍流的混乱本质是如何在所有尺度上组织成漩涡的。特别是，我将使用我开发的数学技术来建立一个完全确定的湍流模型。这一创新的湍流模型将被用于研究大气物理、风工程和航空航天中常见的公开问题。该研究计划将影响计算科学(非线性偏微分方程的自适应数值方法)和流体力学物理(湍流建模)。原则上，现有的湍流模型是随机的，并不是为探测快速增强的涡流而设计的。这意味着计算资源的使用效率低下。第一个目标扩展了我们的小波方法，其中涡旋伸展机制取代了随机湍流模型。然后，我们可以从第一原理推导出有效的方程，这些方程只关注湍流的局部显著影响，而不需要描述复杂环境的每一个微小细节。第二个和第三个目标告诉计算机模型，目前的数据中缺少哪些方面的物理因素可能会加剧涡旋。该建议有两个主要策略：1)将小波的尺度自适应性与湍流的涡动力学表示相结合，求解有效的湍流方程；2)建立一个模拟湍流的数据库，直接捕捉大气湍流的影响，然后使用这些数据来学习如何更好地设计建筑物、风电场、汽车、飞机、船只等。这个研究项目将在流体物理的数学理论和工程应用之间的交叉点培养2名博士、3名硕士和5名理科学生。他们将学习数学创新，以创造新的技术和实际应用。我们创建的数据库和计算机模型以及在这一研究项目中获得的知识将使加拿大在应对新出现的挑战方面走在前列。这一提议有助于国际上努力开发下一代人工智能计算模型。

Studies indicate that atmospheric turbulence is responsible for substantial financial losses in our society. For instance, US aviation pays over $10 million per year to compensate for turbulence related rerouting of flights. In 2018, severe weather caused losses of over $1.9 billion in Canada. On the other hand, wind is a major source of renewable energy. Canada's wind energy capacity surpassed 12,000 megawatts in 2018. Canada has been a world leader in mathematics, fluid mechanics, and numerical weather prediction. Mathematical discoveries reveal how turbulence creates the drag on cars, airplanes, wind turbines, bridges, and buildings; and how it dictates the weather through its influence on large- scale atmosphere and oceans. My research program is dedicated to describing, mathematically, how the chaotic nature of turbulence gets organized into vortices across all scales. In particular, I will use the mathematical techniques that I develop to build a fully deterministic turbulence model. This innovative turbulence model will be applied to investigate common open problems in the physics of atmosphere, wind engineering, and aerospace. The research program will impact both computational sciences (adaptive numerical methods for nonlinear partial differential equations) and physics of fluid mechanics (turbulence modeling). In principle, existing turbulence models are stochastic and not designed to detect rapidly intensified vortices. This means that computational resources are being used inefficiently. The first objective extends our wavelet method, where the vortex stretching mechanism replaces the stochastic turbulence model. We can then derive effective equations from first principles, which focus only on the locally significant influence of turbulence, without needing to describe every little detail of the complex surroundings. The second and the third objective teach the computer model what aspects of the physics is absent in the current data which could intensify a vortex.    The proposal has two main strategies: 1) solve the effective equations of turbulence by combining the scale-adaptivity of wavelets with the vortex dynamics representation of turbulence; and 2) generate a database of simulated turbulent flows, which directly capture only the influence of atmospheric turbulence, and then use such data to learn how to design better buildings, wind farms, cars, airplanes, ships, etc.  This research program will train 2 PhD, 3 MSc, and 5 BSc students at the intersection between mathematical theory and engineering applications of the physics of fluids. They will learn mathematical innovations for creating new techniques and practical applications. The database and computer models we create and the knowledge gained in this research project will keep Canada at the forefront to deal with emerging challenges. This proposal contributes to an international effort to develop the next generation of artificially intelligent computational models.