Low-complexity Stochastic Modeling and Control of Turbulent Shear Flows

自然界和工程应用中的大多数流动都是复杂和无序的(湍流)。飞机、舰船和潜艇周围的湍流耗散动能增加了对它们运动的阻力(阻力)；维持飞机处于巡航状态所需的燃料中，约有一半用于克服湍流施加的阻力。同样，在风电场中，湍流会降低叶片的气动效率，从而减少能量捕获。了解和控制湍流流动在这些应用中扮演着重要的角色，并可能严重影响美国的经济、国家安全和环境。该奖项的影响范围更广，从经济和环境效益到风力发电场、输送管道和车辆性能的改善。主要调查员计划在数学及其应用研究所组织一次关于流体建模和控制的讲习班。这个研讨会的目的是向来自工程、数学和物理界的学生、研究人员和专业人员的跨学科观众展示控制工程和系统理论的实用性。这个项目的智力价值在于研究的新颖性和跨学科性质。湍流的二阶统计量可以通过实验或通过数值模拟获得。这些统计数据对于理解流动物理的基本原理和开发低复杂性模型具有重要意义。这样的模型将被用于控制设计，以抑制湍流。由于实验或数值计算的限制，通常情况下，只有有限数量的流场分量之间的某些时空关联可用。因此，以与已知动力学一致的方式完成流场的统计签名是有意义的。这个反问题的方法依赖于一个由随机强迫线性化的Navier-Stokes方程所支配的模型。在这里，强迫的统计数据是未知的，并试图解释给定的相关性。识别合适的随机强迫将允许PI完成速度场的相关数据。虽然系统动力学对允许的关联施加了线性约束，但这样的反问题允许强迫关联的许多解。PI将使用核范数最小化来获得低复杂度的相关结构。这种复杂性转化为将用于生成所识别的强迫统计的时空过滤器的维度。

Most flows in nature and in engineering applications are complex and disordered (turbulent). Dissipation of kinetic energy by turbulent flow around airplanes, ships, and submarines increases resistance to their motion (drag); about half of the fuel required to maintain the aircraft at cruise conditions is used to overcome the drag force imposed by the turbulent flow. Similarly, in wind farms, turbulence reduces the aerodynamic efficiency of the blades, thereby decreasing the energy capture. Understanding and controlling turbulent fluid flows plays an important role in these applications, and may critically impact US economy, national security, and the environment. The broader impacts of this award range from economic and environmental benefits to improved performance of wind farms, transporting pipes, and vehicles. The Principle Investigators (PIs) plan to organize a workshop on modeling and control of fluids at the Institute for Mathematics and its Applications. This workshop will be aimed at showcasing utility of control engineering and systems theory to an interdisciplinary audience of students, researchers, and professionals from the engineering, mathematics, and physics communities.The intellectual merit of this project's effort lies in the novelty and interdisciplinary nature of the research. Second-order statistics of turbulent flows can be obtained either experimentally or via numerical simulations. The statistics are relevant in understanding fundamentals of flow physics and for the development of low-complexity models. Such models will be used for control design in order to suppress turbulence. Due to experimental or numerical limitations it is often the case that only certain spatio-temporal correlations between a limited numbers of flow field components are available. Thus, it is of interest to complete the statistical signature of the flow field in a way that is consistent with the known dynamics. The approach to this inverse problem relies on a model governed by stochastically forced linearized Navier-Stokes equations. Here, the statistics of forcing are unknown and sought to explain the given correlations. Identifying suitable stochastic forcing will allow the PIs to complete the correlation data of the velocity field. While the system dynamics impose a linear constraint on the admissible correlations, such an inverse problem admits many solutions for the forcing correlations. The PIs will use nuclear norm minimization to obtain correlation structures of low complexity. This complexity translates into dimensionality of spatio-temporal filters that will be used to generate the identified forcing statistics.