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

TL;DR

This work addresses efficient stability analysis for disturbances around arbitrarily time-dependent base flows by constructing a time-evolving, low-rank input-output operator via forced optimally time-dependent decomposition (f-OTD). By formulating the disturbance problem as a matrix differential equation and evolving a reduced subspace $\mathbf{V}(t)\approx \mathbf{U}(t)\mathbf{Y}(t)^\mathrm{T}$, the authors derive explicit evolution equations for $\mathbf{U}$ and $\mathbf{Y}$ and show how to initialize and rank the subspace without offline data. The approach yields a fast surrogate for multi-forcing scenarios, connects to resolvent analysis in steady settings, and is demonstrated across a toy model, a temporally evolving jet, Kolmogorov flow, and 2D decaying turbulence, highlighting both accuracy and computational savings. These results enable rapid transient stability assessments and forcing design for unsteady base flows, with potential extensions to rank adaptivity and rigorous links to resolvent theory.

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.

Figures (13)

• Figure 1: Toy model: (a) Singular values obtained from FOM and f-OTD for $r=1$. (b) Comparison of the reconstructed first sensitivity vector $\mathbf{v}_1(t) =[v_{11}(t), v_{12}(t), v_{13}(t) ]^T$ obtained from FOM and the rank-1 f-OTD approximation.
• Figure 2: Resolvent analysis (RA) and f-OTD for Burgers equation. (a) Comparison of the first $15$ dominant singular values of the resolvent operator with the asymptotic solution of f-OTD. (b) The first dominant response and forcing mode from the resolvent operator and f-OTD.
• Figure 3: Schematic of the temporally evolving jet. The evolution of the base flow at different time instances ($t_1=12$, $t_2 =15$, $t_3 =18$, $t_4=21$).
• Figure 4: Temporally evolving jet: comparison of the dominant normalized singular values obtained from FOM and f-OTD.
• Figure 5: Temporally evolving jet: comparison of the temporal evolution of the perturbation velocity at two probe locations of $(x_A,y_A) = (0.39, 0.59)$ and $(x_B,y_B) = (0.10, 0.12)$.