Bulletin of the American Physical Society
77th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 24–26, 2024; Salt Lake City, Utah
Session X11: Nonlinear Dynamics: Model Reduction |
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Chair: Oliver T. Schmidt, University of California San Diego Room: 155 A |
Tuesday, November 26, 2024 8:00AM - 8:13AM |
X11.00001: Discovery of Nonlinear Flow Physics Using Optimal Modal Momentum and Energy Budget Analysis Oliver T. Schmidt, Brandon Chung Yuen Yeung, Tianyi Chu We propose an orthogonal modal decomposition method for data-driven energy budget analysis, which identifies pairs of modes that represent the acceleration and convective terms of the spectral Navier-Stokes equations. These modes are triadically consistent and optimally correlated. The method extracts coherent structures involved in three-wave interactions by focusing on spectral momentum transfer, optimizing third-order space-time flow statistics. It distinguishes between two interacting components: a catalyst and a momentum donor, which together contribute to a tertiary component, the recipient. The resulting modes maximize the covariance between the projections of the convective and recipient terms onto their respective modes. This method extends bispectral mode decomposition (BMD) by incorporating the exact quadratic nonlinearity of the Navier-Stokes equations. Similar to classical proper orthogonal decomposition (POD), it provides ranked, orthonormal bases for the convective and recipient terms that are jointly optimal. Applications include numerical data of a canonical unsteady cylinder wake and experimental data of a turbulent wind turbine wake by Biswas & Buxton (2024, JFM). |
Tuesday, November 26, 2024 8:13AM - 8:26AM |
X11.00002: Building dynamical stability into data-driven quadratic reduced-order models Mai Peng, Alan A Kaptanoglu, Christopher J Hansen, Jake Stevens-Haas, Krithika Manohar, Steven L Brunton Quadratically nonlinear reduced-order models (ROMs) are commonly used for approximating the dynamics of fluids, plasmas, and many other physical systems. However, it is challenging to a-priori guarantee the local or global dynamical stability of reduced-order models built from data. For instance, a minimal requirement for physically-motivated ROMs is long-time boundedness for any initial condition, yet many ROMs in the literature still fail this basic requirement. For quadratically nonlinear systems with energy-preserving nonlinearities, the Schlegel and Noack trapping theorem (Schlegel and Noack 2015) provides necessary and sufficient conditions for long-time boundedness to hold. This analytic theorem was subsequently incorporated into system identification and machine learning techniques in order to produce a-priori bounded models directly from data (Goyal 2023, Kaptanoglu 2021, Ouala 2023). |
Tuesday, November 26, 2024 8:26AM - 8:39AM |
X11.00003: Data-driven stability analysis of chaotic systems in latent spaces Elise Özalp, Luca Magri The spatio-temporal dynamics of turbulent and chaotic systems are inherently unstable, which makes the design of accurate reduced-order models for forecasting challenging. We employ a data-driven framework to separate observations into spatial and temporal components, using a convolutional autoencoder (CAE) to compute a latent space representing the chaotic dynamics. An echo state network (ESN) predicts the temporal evolution of the latent representation. With the CAE-ESN, we perform stability analysis in the latent manifold from data only. The Lyapunov spectrum, Kaplan-Yorke dimension, and covariant Lyapunov vectors are computed to analyse the chaotic properties and geometric structure of the latent manifold. The analysis is performed on the Kuramoto-Sivashinsky equation, where it produces a latent space that preserves key properties of the chaotic system, thus retaining the geometric structure of the attractor in the latent space. Finally, the method is applied to a direct numerical simulation of the turbulent 2D Kolmogorov flow. This work opens new opportunities for analysing the stability properties of chaotic systems from data only. |
Tuesday, November 26, 2024 8:39AM - 8:52AM |
X11.00004: Proper latent decomposition (PLD) Daniel Kelshaw, Luca Magri The dynamics of fluids can be modelled on latent spaces, which are nonlinear manifolds that can be inferred by autoencoders. The latent space is, however, difficult to interpret, which calls for decomposition methods. Linear decomposition methods, such as POD, are not suitable for nonlinear manifolds. To generalise POD to nonlinear manifolds, we introduce the Proper Latent Decomposition (PLD) both mathematically and algorithmically. This approach allows us to explore the underlying flow structures in flows, without imposing a linear ansatz like in POD, and find a coordinate chart (Principal Geodesic Modes) on the latent space. We showcase the PLD on three systems: a laminar wake past a bluff body and the chaotic Kolmogorov flow. We find that the proposed methodology can extract physical modes and can provide a physically motivated reduced-order representation of turbulent flows, which is expressive and compact. This work provides a framework for decomposing turbulent flows with nonlinear methods, which opens opportunities for interpretability and reduced-order modelling. |
Tuesday, November 26, 2024 8:52AM - 9:05AM |
X11.00005: Mori-Zwanzig Mode Decomposition: Transient Flows Michael Woodward, Yifeng Tian, Alessandro Gabbana, Yen Ting Lin, Daniel Livescu Many flows in nature contain transient dynamics, where the fluid system moves from one equilibrium state to another. Flows of this type represent significant challenges for current modal decomposition techniques, such as Dynamic Mode Decomposition. We investigate the data-driven Mori- Zwanzig Mode Decomposition (MZMD) to reconstruct and perform future state prediction of transient dynamics. We first consider a 2D transient flow over a cylinder; where the flow transitions from an unstable equilibrium point to a heteroclinic orbit representing the von Karman vortex street. We then perform MZMD analysis of a more complex multiphase flow in 3D, simulating the growth and departure of vapor bubbles from a heated orifice in a quiescent liquid. We show that MZMD significantly improves upon DMD for reconstructing and performing future state predictions by extracting transient modes that improve the ability to resolve such transient dynamics, by increasing the spectral complexity at a similar computational cost. This is achieved by the introduction of Mori-Zwanzig memory terms, which account for the effects the unresolved dynamics have on the resolved variables (observables). The improvement saturates after a certain number of memory terms are considered, with a finite decaying memory effect. |
Tuesday, November 26, 2024 9:05AM - 9:18AM |
X11.00006: ABSTRACT WITHDRAWN
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Tuesday, November 26, 2024 9:18AM - 9:31AM |
X11.00007: CUR-Based Implicit Integration of Random Partial Differential Equations on Low-Rank Manifolds Mohammad Hossein Naderi, Hessam Babaee This work presents a cost-effective approach for the implicit time integration of parametric partial differential equations (PDEs) on low-rank matrix manifolds using time-dependent bases (TDBs). This enables the low-rank approximation of stiff PDEs, providing a computationally efficient solution. The key to the cost-effectiveness of the proposed methodology is the utilization of a CUR matrix decomposition for low-rank approximation, which avoids costly nonlinear solves. By evaluating the PDE only at selected points, the method achieves substantial speedups compared to full-order model integration. Moreover, the presented framework incorporates rank adaptivity, allowing for the adjustment of the approximation rank over time. This rank adaptivity facilitates error control and ensures that only the necessary modes are retained to achieve the desired accuracy. The efficacy of the implicit TDB method is demonstrated through several analytical and PDE examples, including the stochastic Burgers' and Gray-Scott equations. |
Tuesday, November 26, 2024 9:31AM - 9:44AM |
X11.00008: Finding Approximate Solutions to the Navier-Stokes Equations Using Automatic Differentiation Joseph L Holey, Mohammed Alhashim, Michael P Brenner Oseen’s equation models the flow past an object by replacing the standard non-linear convection term in the Navier-Stokes equations with the interaction of the velocity field and the velocity of the object itself. Near the body this approximation is only accurate for Re < 1 but by using an effective viscosity proportional to Re-3/4 the drag could be accurately computed for axisymmetric flow around a wide range of ellipsoidal bodies up to Re ~100. We seek to extend these results to higher Re and different flows past objects by utilising the power of automatic differentiation in jax to optimise over a wide range of linear terms. We hope that this will give a better qualitative understanding of the driving mechanisms of the full equations. |
Tuesday, November 26, 2024 9:44AM - 9:57AM |
X11.00009: A low dimensional model to predict secondary mean flow in a square duct Ahmed I El-Nadi, Barbara Lopez-Doriga, Ricardo Vinuesa, Scott T. M. Dawson Turbulent flows through square and rectangular ducts have mean flowfields that include secondary (non-streamwise) mean velocity components, typically consisting of pairs of counterrotating streamwise vortices near each corner (Prandtl’s secondary flow of the second kind). While such secondary flows are typically small in magnitude, they are important for accurate characterization of turbulent statistics, yet are also typically difficult for turbulence models to accurately predict. This project aims to determine the simplest possible model that can correctly characterize the qualitative features of such secondary mean flows. To do this, we develop a nonlinear, physics-based Galerkin projection model, using basis functions obtained from analysis of the linearized Navier-Stokes equations about a laminar baseflow (using either stability or resolvent modes). We demonstrate that a model of this type obtained using only streamwise-constant modes is sufficient for predicting a physically realistic secondary mean, provided a white noise forcing term is included when simulating the model. We further discuss what these findings suggest about the physical origins of such secondary mean flows. |
Tuesday, November 26, 2024 9:57AM - 10:10AM |
X11.00010: Mapping flow fields using LCS constrained regression Kartik Krishna, Steve Brunton, Zhuoyuan Song Mapping of unsteady flow fields from Lagrangian trajectory measurements (for example collected from ocean floats) is an important and challenging task from the perspective of navigation and guidance of mobile sensors operating within them. Some of the challenges of mapping are that we require algorithms that require a small amount of training data and make minimal assumptions of the flow field. In order for the method to generalize, we propose the use of regression on Lagrangian trajectories using a library of "Gaussian vortices'' to reconstruct the flow field. Furthermore, we enforce that the estimated flow field obeys the Lagrangian coherent structures (LCS) through linear constraints. This is useful in situations where we have knowledge of large scale coherent structures and want to resolve the entire flow field at the smaller scales. We demonstrate the effectiveness of our strategy on examples such as the double-gyre and show that by enforcing the LCS, we can reduce the amount of training data needed. Moreover, our method is not restricted to enforcing the LCS but the formulation can potentially also be used to enforce solid boundaries such as landmasses within the flow field. |
Tuesday, November 26, 2024 10:10AM - 10:23AM |
X11.00011: Analytical approach to identifying a bifurcation point in reduced-nonlinear dynamical systems formed by shift mode and oscillation modes Yuto Nakamura, Shintaro Sato, Naofumi Ohnishi The reduced-order models for fluid flow provide a simple tool for analyzing nonlinear aspects of the Navier-Stokes equations. Bifurcation is a classical nonlinear phenomenon, and analyzing dynamical systems of reduced-order models that capture it facilitates our understanding of bifurcation. In this study, we analyze a reduced-order model based on the Galerkin projection for the Hopf bifurcation of the flow around a cylinder, which consists of a steady-state solution referred to as the shift mode and the oscillatory modes of the wake vortex street. The dynamical system of the reduced-order model is linearized at the equilibrium points. Eigenvalue analysis for a linearized dynamical system yields eigenvalues as a function of Reynolds number, and the bifurcation point is identified analytically. In the ROM constructed by changing the Reynolds number of the shift mode and the oscillation modes, the bifurcation Reynolds number is investigated using our analytical approach. Our understanding of the Hopf bifurcation is advanced by examining the Reynolds number of the modes used in the dynamical system and its bifurcation point. Furthermore, it provides valuable insights into constructing practical ROMs that accurately capture the Hopf bifurcation. |
Tuesday, November 26, 2024 10:23AM - 10:36AM |
X11.00012: Characterization of perturbative nonlinear systems using symmetry methods Christopher Pezanosky, Scott D Ramsey, Jesse Canfield, Len Margolin, Darrin Visarraga This work studies the application of analytic symmetry methods to peturbative nonlinear oscillators featured prominently in dynamical systems. Examining the differential equations governing the behavior of such systems, we use the method of approximate Lie groups to develop perturbative solutions as 'deformations' of the symmetric, unperturbed system. Additionally, we connect the concept of the Renormalization Group method, which is used to improve the global nature of approximate solutions, to fundamental Lie symmetries of the equation. Namely, invariance of the governing equation with respect to translation of the independent variable is seen to allow for the construction of 'renormalized' solutions. In this way, we are able to systematically build uniformly valid, approximate solutions to perturbative nonlinear equations by exploiting their approximate and Renormalization Group symmetries. These solutions are then compared to those obtained by standard perturbation methods using a combination of analytic and numerical techniques. Ultimately, we seek to identify how the context and symmetries of the underlying system inform the relative merits of the corresponding solution methods. |
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