Data-driven Modeling for Fluid Dynamics and Control

Abstract

One of the most critical tasks in fluid dynamics and control is to build simple, low-order, and accurate models. The models are essential for understanding dynamics and control. However, in many cases, the models are either unknown or too complicated to be useful. As an example, fluid flows are governed by Navier-Stokes equations (NSE), which remain intractable for real-time applications. Meanwhile, with increasing computational power and advances in experimental and numerical methods, researchers have access to much more data about dynamical systems. For instance, computational fluid dynamics (CFD) produces tons of data, but the data have not been fully utilized.

Data-driven modeling addresses these challenges by learning dynamical system models from data. This thesis focuses on data-driven modeling methods for applications in fluid dynamics and control. First, we propose an evaluation criterion to quantify the accuracy of dynamic mode decomposition (DMD), a data-driven algorithm for extracting spatial and temporal features about dynamical systems from data. DMD is a numerical approximation to the linear Koopman operator associated with a dynamical system. By exploiting this connection, the accuracy criterion is purely data-driven and physically meaningful. It also applies to other variants of DMD algorithms and assists in model selection.

Second, fast algorithms are developed for online dynamic mode decomposition (ODMD). Given real-time measurement about a dynamical system, this algorithm efficiently updates an adaptive model upon each new snapshot. It reduces both the computational time and memory requirements by order of magnitudes compared with existing methods. ODMD algorithm can be modified to gradually forget old data, which enables faster tracking of dynamics. ODMD also extends to both linear and nonlinear system identification, where control is included.

Finally, we study the input-output response of a separated flow past a flat plate. The analysis is based on the frequency-domain transfer function of the linearized NSE about the mean flow. The control input is body forcing, and the output is the flow field. This analysis sheds light on the optimal control placement and reveals that the trailing edge separation bubble is most sensitive to streamwise body force (control) in upstream of the separation point.