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了动能，增加了它们运动的阻力（阻力）；在巡航状态下维持飞机所需的燃料中，大约有一半用于克服湍流所施加的阻力。同样，在风力发电厂，湍流降低了叶片的空气动力学效率，从而减少了能量捕获。理解和控制湍流流体流动在这些应用中起着重要作用，并可能对美国经济、国家安全和环境产生重大影响。该奖项的广泛影响包括从经济和环境效益到改善风电场、运输管道和车辆的性能。主要研究人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。本次研讨会旨在向跨学科的学生、研究人员和来自工程、数学和物理社区的专业人士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。湍流的二阶统计量既可以通过实验得到，也可以通过数值模拟得到。统计数据与理解流动物理的基本原理和开发低复杂性模型有关。这些模型将用于控制设计，以抑制湍流。由于实验或数值的限制，通常只有有限数量的流场分量之间的某些时空相关性是可用的。因此，以一种与已知动力学相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纳维-斯托克斯方程控制的模型。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。确定合适的随机强迫将允许pi完成速度场的相关数据。当系统动力学对可容许的相关性施加线性约束时，这样的逆问题允许许多强制相关性的解。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确定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.