Bulletin of the American Physical Society
75th Annual Meeting of the Division of Fluid Dynamics
Volume 67, Number 19
Sunday–Tuesday, November 20–22, 2022; Indiana Convention Center, Indianapolis, Indiana.
Session A20: Nonlinear Dynamics: Koopman and Related Approaches |
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Chair: Shaowu Pan, Rensselaer Polytechnic Institute Room: 207 |
Sunday, November 20, 2022 8:00AM - 8:13AM |
A20.00001: PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator Shaowu Pan, Eurika Kaiser, Nathan Kutz, Steven L Brunton PyKoopman is a Python package for the data-driven approximation of the Koopman operator in dynamical systems. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics and facilitates the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. In particular, PyKoopman provides tools for data-driven system identification for unforced and actuated systems that build on the equation-free dynamic mode decomposition (DMD) and its nonlinear variants including EDMD, KDMD, time delayed DMD, scalable KDMD, and a neural network version. In this work, we provide a brief description of the mathematical underpinnings of the Koopman operator, an overview and demonstration of the features implemented in PyKoopman (with code examples), practical advice for users, and a list of potential extensions to PyKoopman. Software is also available on Github. |
Sunday, November 20, 2022 8:13AM - 8:26AM |
A20.00002: A stochastic SPOD-Koopman model for broadband turbulent flows Tianyi Chu, Oliver T Schmidt A data-driven low-order model for broadband turbulent flows that uses spectral proper orthogonal decomposition (SPOD) modes as the modal basis is presented. A discrete Koopman operator obtained via extended dynamic mode decomposition (EDMD) is used to propagate the solution. The proposed stochastic two-level model governs a compound state consisting of modal expansion and forcing coefficients. This approach follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. Under the linear time-invariant assumption, the modal expansion coefficients are advanced by the modified Koopman operator in the first level. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to model nonlinear interactions between modes. Closure is achieved by a dewhitening filter that imprints the second-order statistics of the remaining residue onto the forcing coefficients. |
Sunday, November 20, 2022 8:26AM - 8:39AM |
A20.00003: Data-driven Mori-Zwanzig operators for boundary layer transition Michael Woodward, Yifeng Tian, Arvind Mohan, Yen Ting Lin, Michael Chertkov, Daniel Livescu Data-driven reduced-order modeling (ROM) of complex dynamical systems, such as those found in turbulent flows, is an active area of research, as it offers the potential to tackle notoriously challenging problems in engineering and the physical sciences. In this work, we apply and extend to inhomogeneous flows a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ) operators, in order to extract large scale coherent structures of the Markovian and memory terms and to develop reduced order models with memory effects. MZ is based on a generalized Koopman's description of dynamical systems and provides a mathematically rigorous procedure for constructing non-Markovian reduced-order models of resolved variables from high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. We apply this method to the flow over a cylinder as a test case, as well as to a spatially inhomogeneous flow corresponding to laminar-turbulent boundary-layer transition. We compare this data-driven MZ approach to Dynamic Mode Decomposition (DMD) and analyze the effects of incorporating the memory terms in prediction, such as measuring the generalization errors and visualizing large-scale spatiotemporal structures. |
Sunday, November 20, 2022 8:39AM - 8:52AM |
A20.00004: Identification of interscale dynamics and triadic interactions of coherent structures in turbulent wakes Daniel Foti Turbulent wakes are often characterized by dominant coherent structures over disparate scales. Coherent structure formation, evolution, and breakdown can be attributed to kinetic energy transport and transfer between spatio-temporal scales. Energy transfer is governed by triadic interactions, which are non-linear, non-local, and manifest as a triplet of wavenumbers or frequencies. The spatial and spectral qualities of a broad range coherent structure frequency scales are identified through Koopman mode decomposition and triple decomposition of the velocity in the wake of a square cylinder at Re=3900. The formulation results in a system of evolution of coherent kinetic energy equations based on specific identified scales coupled by interscale energy transfer terms. The frequency sum-zero condition of triadic interactions is implicitly enforced, and coherent kinetic energy based on interaction of two scales and energy budget of a single scale are explicitly quantified. Spectra reveal organization for both forward and reverse energy cascade. While turbulence production is concentrated in lower frequency scales related to vortex shedding, the transfer between scales is significant over a wide range. |
Sunday, November 20, 2022 8:52AM - 9:05AM |
A20.00005: Non-Intrusive Learning of Physics-Informed Spatio-Temporal Surrogate Models for Fluid Flow Prediction Sudeepta Mondal, Soumalya Sarkar Multi-physics simulations are often performed to capture the fine spatio-temporal scales governing the evolution of fluid flows. These simulations are often high-fidelity in nature, and can be computationally very expensive for data generation, thereby creating a computational bottleneck for practical engineering design problems. Data-driven spatio-temporal surrogate modeling has been a popular solution to tackle this computational bottleneck because these machine learning models can be orders of magnitude faster than the actual simulations. However, one key limitation of purely data-driven approaches is their lack of generalization to out-of-distribution inputs. In this paper, we propose a physics-informed spatio-temporal surrogate modeling framework constrained by the physics of the underlying nonlinear dynamical system representing the fluid flow process. The framework leverages state-of-the-art advancements in the field of Koopman autoencoders to learn the underlying fluid flow dynamics in a non-intrusive way, coupled with a spatio-temporal surrogate model which forecasts the flowfields in a specified time window for unknown operating conditions. We evaluate our framework on a prototypical fluid flow problem of interest: two-dimensional incompressible flow around a cylinder. |
Sunday, November 20, 2022 9:05AM - 9:18AM |
A20.00006: Lagrangian scale decomposition via the graph Fourier transform Theodore MacMillan, Nicholas T Ouellette Scale decomposition is ubiquitous in the analysis of complex fluid flows. Often with an eye towards reduced order modelling, considerable effort has been invested in the development of novel sets of basis functions for such decompositions. The most successful sets of basis functions have been developed in the Eulerian perspective, where traditional tools of calculus and newer methods from data science can be most easily leveraged. Here, we take advantage of recent interest in graph-based approaches to Lagrangian coherence as well as new methods for the scale analysis of complex networks to introduce a new scale decomposition that is instead fully Lagrangian and based on transport. To do so, we adapt a technique from network science know as the graph Fourier transform and develop a novel graph correlation function that allows us to quantitatively describe our Lagrangian decomposition as a function of scale. This method allows better interpretability of the dynamic consequences of kinematic coherence as well as the ability to perform traditional Eulerian tasks (such as decomposition, filtering, and compression) on Lagrangian quantities. We illustrate our techniques on examples drawn from coherent-structure analysis, ocean mixing, and cloud physics. |
Sunday, November 20, 2022 9:18AM - 9:31AM |
A20.00007: Optimally time-dependent mode analysis of vortex gust-airfoil wake interactions Yonghong Zhong, Alireza Amiri-Margavi, Hessam Babaee, Kunihiko Taira Identification of the transient dynamics is important for understanding the underlying physics of unsteady fluid flows. Such analysis is challenging for flows with unsteady base states. In this study, optimally time-dependent (OTD) modes, a set of orthonormal modes that traces the most amplified directions of the dynamics, are considered to study the primary transient flow features. We apply the OTD mode analysis to gust vortex-airfoil wake interactions, which exhibit strong transient amplification as the gust vortex convects over the airfoil. The OTD modes capture the transient instabilities related to the dynamic characteristics of the unsteady flow. Moreover, OTD modes with time-varying linear operators uncover transient modes induced by vortical interactions. This study provides physical insights into the time-evolving amplification modes for nonlinear, non-stationary flows with implications for active flow control of highly unsteady flows. |
Sunday, November 20, 2022 9:31AM - 9:44AM |
A20.00008: Pseudospectral behavior of the linear operator and its influence on modal amplitudes in the dynamic mode decomposition Het D Patel, Chi-An Yeh We investigate the pseudospectral behavior of the linear operator obtained from dynamic mode decomposition (DMD) and the extent to which it can effect the amplitudes of the DMD modes. While the amplitude is generally used to guide the selection of dominant modes, we observe that, when performing exact DMD with a large number of POD bases (for which the linear operator is projected onto), modes with physical structures do not align with high amplitudes. Meanwhile, performing DMD via explicitly forming the companion matrix from the snapshots does not suffer from such a misalignment, given that the data matrix has full rank. This draws us to examine the pseudospectra of the DMD linear operators obtained from both approaches. We find that, in exact DMD, increasing the number of POD bases also increases the nonnormality of the reduced-order linear operator. This increasing nonnormality can be characterized by the pseudospectral level as well as the eigenvalue condition number, computed by taking the inner product between the direct and adjoint eigenvectors. Meanwhile, the companion matrix exhibits much lower nonnormality, making it more robust against the perturbations due to numerical error. We demonstrate these using a large dataset of a turbulent wake flow over a NACA 0012 airfoil. The difference in the levels of nonormality of the linear operators leaves a cautionary note to the selection of DMD algorithms, as the nonnormality may also play a crucial role in the data-driven resolvent analysis. |
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