Low-complexity Stochastic Modeling and Control of Turbulent Shear Flows

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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.