Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition

We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow.Comment: 35 pages, 20 figure

Symplectic Model Reduction of Hamiltonian Systems

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure

Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning

In this paper, we put forth a long short-term memory (LSTM) nudging framework for the enhancement of reduced order models (ROMs) of fluid flows utilizing noisy measurements. We build on the fact that in a realistic application, there are uncertainties in initial conditions, boundary conditions, model parameters, and/or field measurements. Moreover, conventional nonlinear ROMs based on Galerkin projection (GROMs) suffer from imperfection and solution instabilities due to the modal truncation, especially for advection-dominated flows with slow decay in the Kolmogorov width. In the presented LSTM-Nudge approach, we fuse forecasts from a combination of imperfect GROM and uncertain state estimates, with sparse Eulerian sensor measurements to provide more reliable predictions in a dynamical data assimilation framework. We illustrate the idea with the viscous Burgers problem, as a benchmark test bed with quadratic nonlinearity and Laplacian dissipation. We investigate the effects of measurements noise and state estimate uncertainty on the performance of the LSTM-Nudge behavior. We also demonstrate that it can sufficiently handle different levels of temporal and spatial measurement sparsity. This first step in our assessment of the proposed model shows that the LSTM nudging could represent a viable realtime predictive tool in emerging digital twin systems