Time-Dependent Low-Rank Input-Output Operator for Forced Linearized Dynamics with Unsteady Base Flows

TL;DR

This paper introduces a novel reduced-order modeling approach using time-dependent bases to efficiently analyze forced linearized dynamics in flows with arbitrarily changing base states, enabling rapid stability and response assessments.

Contribution

The paper develops a time-dependent low-rank solution operator framework using f-OTD for arbitrary time-dependent base flows, extending existing methods to more complex unsteady flow scenarios.

Findings

f-OTD effectively captures flow response structures.

Enables fast transient stability analysis.

Facilitates rapid computation of flow responses to multiple impulses.

Abstract

Understanding the linear growth of disturbances due to external forcing is crucial for flow stability analysis, flow control, and uncertainty quantification. These applications typically require a large number of forward simulations of the forced linearized dynamics, often in a brute-force fashion. When dealing with simple steady-state or periodic base flows, there exist powerful and cost-effective solution operator techniques. Once these solution operators are constructed, they can be used to determine the response to various forcings with negligible computational cost. However, these methods do not apply to problems with arbitrarily time-dependent base flows. This paper develops and investigates reduced-order modeling with time-dependent bases (TDBs) to build low-rank solution operators for forced linearized dynamics with arbitrarily time-dependent base flows. In particular, we use forced optimally time-dependent decomposition (f-OTD), which extracts the time-dependent correlated structures of the flow response to various excitations. Several demonstrations are included to illustrate the utility of the f-OTD low-rank approximation for performing global transient stability analysis. Additionally, we demonstrate the application of f-OTD in computing the post-transient response of linearized Navier-Stokes equations to a large number of impulses, which has applications in flow control.