Sparse Sensing and DMD-Based Identification of Flow Regimes and Bifurcations in Complex Flows

Abstract

We present a sparse sensing framework based on Dynamic Mode Decomposition (DMD) to identify flow regimes and bifurcations in large-scale thermo-fluid systems. Motivated by real-time sensing and control of thermal-fluid flows in buildings and equipment, we apply this method to a Direct Numerical Simulation (DNS) data set of a 2D laterally heated cavity. The resulting flow solutions can be divided into several regimes, ranging from steady to chaotic flow. The DMD modes and eigenvalues capture the main temporal and spatial scales in the dynamics belonging to different regimes. Our proposed classification method is data-driven, robust w.r.t measurement noise, and exploits the dynamics extracted from the DMD method. Namely, we construct an augmented DMD basis, with "built-in" dynamics, given by the DMD eigenvalues. This allows us to employ a short time-series of data from sensors, to more robustly classify flow regimes, particularly in the presence of measurement noise. We also exploit the incoherence exhibited among the data generated by different regimes, which persists even if the number of measurements is small compared to the dimension of the DNS data. The data-driven regime identification algorithm can enable robust low-order modeling of flows for state estimation and control.

Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations

“…We first rewrite the right-hand side of the ROM model (10) to isolate the linear viscous term as follows, (19) where D ∈ R r×r represents a constant, symmetric negative definite matrix, and the function F (·) represents the remainder of the ROM model, i.e., the part without damping.…”

“…This includes the case of parametric uncertainties in (9) that produce structured uncertainties in (19) . To treat this case, we use Lyapunov theory and propose a nonlinear closure model that robustly stabilizes the ROM in the sense of Lagrange.…”

“…Assumption 2 allows us to consider a general class of PDEs and their associated ROMs. Indeed, all we require is that the right-hand side of (10) can be decomposed as (19) , where a linear damping term can be extracted and the remaining nonlinear term F is bounded. This could allow for more general parametric dependencies and includes many structured uncertainties of the ROM, e.g., a bounded parametric uncertainty can be formulated in this manner.…”

“…There are many approaches to compute those basis functions for nonlinear systems. For example, some of the most common methods are proper orthogonal decomposition (POD) [18] , dynamic mode decomposition (DMD) [19] , and reduced-basis methods (RBM) [20] . Another challenge for nonlinear problems, in particular, is to model the influence of the discarded basis functions on the retained basis functions in the dynamical system for q ( t ).…”

“…We notice that this POD-ROM has mainly a linear term and two quadratic terms, so that it can be written in the form (19) , with…”