Bulletin of the American Physical Society
74th Annual Meeting of the APS Division of Fluid Dynamics
Volume 66, Number 17
Sunday–Tuesday, November 21–23, 2021; Phoenix Convention Center, Phoenix, Arizona
Session H31: Nonlinear Dynamics: Model Reduction & Turbulence III |
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Chair: Scott Dawson, Illinois Institute of Technology Room: North 232 ABC |
Monday, November 22, 2021 8:00AM - 8:13AM |
H31.00001: On the role of nonlinear correlations in reduced-order modeling Jared Callaham, Steven L Brunton, Jean-Christophe Loiseau A major goal for reduced-order models of unsteady fluid flows is to uncover and exploit latent low-dimensional structure. Proper orthogonal decomposition (POD) provides an energy-optimal linear basis to represent the flow kinematics, but converges slowly for advection-dominated flows and tends to overestimate the number of dynamically relevant variables. We show that nonlinear correlations in the temporal POD coefficients can be exploited to identify the underlying attractor, characterized by a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD-Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolve on a torus generated by two independent Stuart-Landau oscillators. These results emphasize the importance of nonlinear dimensionality reduction to reveal underlying structure in complex flows. |
Monday, November 22, 2021 8:13AM - 8:26AM |
H31.00002: Galerkin Reduced Order Models for Compressible Flows with Differentiable Programming SURYAPRATIM CHAKRABARTI, Arvind T Mohan, Daniel Livescu, Datta V Gaitonde Incorporating active predictive control strategies in dynamical systems such as turbulent flows is an active research topic. Reduced Order Models (ROM) facilitate this goal since they can provide accurate and computationally cheap forecasts. Widely employed Galerkin projection-based ROMs (GP-ROM) can suffer from instabilities and inaccuracies over long time horizons, despite the use of calibration techniques. Here, we utilize the Neural Galerkin Projection (NeuralGP) procedure; a differentiable programming-based approach that blends the strengths of a purely data-driven neural network-based technique to the physics-driven GP-ROM formulation. In NeuralGP, the structure of the ROM ODE can be specified and the ROM coefficients are learned directly from the data. We demonstrate this for a transonic flow over a buffeting NACA0012 airfoil where the NeuralGP implicitly learns stable ROM coefficients and accurately predicts over significantly longer time horizons, compared to the GP-ROM. We then compare the GP-ROM and Neural GP coefficients to study their stability properties. Further, to facilitate full state estimation we also demonstrate the use of a deep learning-based compressed sensing tool that reproduces the flow state from a few randomly placed sensors. |
Monday, November 22, 2021 8:26AM - 8:39AM |
H31.00003: RONS: Reduced-order nonlinear solutions for PDEs with conserved quantities Mohammad M Farazmand, William Anderson Reduced-order models where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time-dependent variables have thus far been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent variables. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems, and in particular fluid dynamics. We demonstrate the efficacy of RONS on three fluid flows: an advection-diffusion equation, the propagation of surface waves, and vortex dynamics in ideal fluids. |
Monday, November 22, 2021 8:39AM - 8:52AM |
H31.00004: A new algorithm for computing global resolvent modes in a CPU and memory efficient manner Ali Farghadan, Eduardo Martini, André Cavalieri, Aaron S Towne Resolvent (or input-output) analysis has proven to be a useful tool for understanding and modeling turbulent flows. In particular, the leading singular vectors of the resolvent operator provide insight into energy amplification mechanisms and coherent structures. However, standard algorithms for computing singular modes of the resolvent operator scale poorly with problem size, hindering application to many flows of interest, especially three-dimensional ones. Recent methods using randomized singular value decomposition (RSVD) have reduced computational cost but still contain bottlenecks for large systems. We combine the RSVD algorithm with an efficient time-stepping method that exploits the direct and adjoint time-domain equations underlying the resolvent system to overcome these bottlenecks. We show that our algorithm scales linearly with problem size, drastically reducing CPU and memory costs for large systems. Moreover, our algorithm simultaneously computes the resolvent modes for a range of frequencies. In this talk, we illustrate the efficacy of our algorithm by applying it to two- and three-dimensional jets. |
Monday, November 22, 2021 8:52AM - 9:05AM |
H31.00005: Parametrized reduced-order modeling of nonlinear separated flows Urban Fasel, Scott T Dawson, Steven L Brunton Flight and renewable energy systems, ranging from micro air vehicles to wind turbines, encounter large operating regimes. Efficient reduced-order unsteady aerodynamic models that accurately capture the dynamics over large operating regimes are essential for prediction and control. In this work, we consider viscous, separated flows over a rigid flat plate and identify interpretable models that accurately capture the transient and post-transient lift and drag dynamics over large nonlinear flow regimes. The identified models are based on data gathered from Reynolds number 100 direct numerical simulations. We use a variant of the sparse identification of nonlinear dynamics (SINDy) algorithm to find a parametrized, low-order model that is valid over a wide range of angles of attack including the critical angle of attack at which a bifurcation occurs. We show that the parametrized model can accurately capture the bifurcation location, limit cycle amplitudes and phases, and transient decay rates of the rigid flat plate flow over a range of operating conditions. |
Monday, November 22, 2021 9:05AM - 9:18AM |
H31.00006: Hankel singular vectors are space-time POD modes Peter K Frame, Aaron S Towne We shed light on a connection between space-time proper orthogonal decomposition (POD) and the singular value decomposition (SVD) of a particular block Hankel matrix. The block Hankel matrix formed using successive vector-valued snapshots of the flow state, and the SVD of this matrix, lie at the heart of several popular methods for modeling and analyzing fluid systems. We show that the left singular vectors (and singular values) of this Hankel matrix correspond to a discrete approximation of continuous space-time POD modes (and eigenvalues), which are the solution of an optimization problem defined over a time window of interest. Furthermore, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively. These connections provide insight into the construction of the Hankel matrix, including the weighting in which the SVD modes are optimal, and the impact of the time steps between rows and columns of the matrix on the convergence of the approximation. These insights lead to a modified matrix, depending on the weight and time step, for which the SVD modes are optimal in the desired weighting. |
Monday, November 22, 2021 9:18AM - 9:31AM |
H31.00007: Phase sensitivity analysis of post-stall airfoil wakes. Vedasri Godavarthi, Yoji Kawamura, Kunihiko Taira Phase reduction analysis of time-periodic flows reveals perturbation dynamics with respect to the base flow scalar phase variable. Synchronization properties of the unsteady flows can be derived from the high-fidelity phase sensitivity fields (Z ) obtained from the adjoint-based phase reduction analysis. We use this technique to determine the influence of perturbations around post-stall airfoils on their phase dynamics. We consider symmetric airfoils of different thicknesses at post-stall angles of attack (α) for Re = 100. We observe that the influence of α on Z is significant compared to that of thickness. For lower α, the phase can be altered with less actuation input as reflected in the large magnitudes of Z around the airfoil. This follows from the wake dynamics as perturbing the large vortices formed at higher α requires larger actuation input. The Z fields are found to scale with the lift force with high sensitivity around the leading and trailing edges. The present approach can support open loop unsteady flow control strategies by accelerating or decelerating vortex shedding over different bodies. |
Monday, November 22, 2021 9:31AM - 9:44AM |
H31.00008: Charts and atlases for nonlinear data-driven dynamics on a manifold Michael D Graham, Daniel Floryan We present a method that learns minimal-dimensional dynamical models directly from time series data. The key enabling assumption is that the data, although nominally high-dimensional, lives on a lower-dimensional manifold. Our method is based on the description of a manifold as an atlas of charts—overlapping patches that can be invertibly mapped to low-dimensional Euclidean spaces. We learn the dynamics in each chart's low-dimensional Euclidean coordinates, and sew these local models together to produce a dynamical system for the global dynamics on the manifold. Our examples—ranging from simple low-dimensional periodic dynamics to complex non-periodic bursting dynamics of the Kuramoto-Sivashinsky equation—will bear out three major benefits of our atlas-of-charts-based framework: (1) the ability to obtain dynamical models of the lowest possible dimension, which previous methods are incapable of; (2) a divide-and-conquer approach that leads to computational benefits including scalability, the ability to adapt models locally, and an embarrassingly parallelizable algorithm; and (3) the ability to separate state space into region of distinct behaviors. |
Monday, November 22, 2021 9:44AM - 9:57AM |
H31.00009: Objective discovery of fluid dynamical regimes with unsupervised machine learning Bryan Kaiser, Juan A Saenz, Maike Sonnewald, Daniel Livescu Significant advances in the understanding and modeling of dynamical systems has been enabled by the identification of processes that locally and approximately dominate system behavior, or dynamical regimes. The conventional regime identification method involves tedious and ad hoc parsing of data to judiciously obtain scales to ascertain which governing equation terms are dominant in each fluid dynamical regime. Surprisingly, no objective and universally applicable criterion exists to robustly identify dynamical regimes in an unbiased manner, neither for conventional nor for machine learning-based methods of analysis. Here, we formally define dynamical regime identification as an optimization problem by using a verification criterion, and we show that an unsupervised learning framework can automatically and credibly identify regimes. This eliminates reliance upon conventional analyses, with vast potential to accelerate discovery. Our verification criterion also enables unbiased comparison of regimes identified by different methods. In addition to diagnostic applications, the verification criterion and learning framework are immediately useful for data-driven dynamical process modeling. Automation of this kind of approximate mechanistic analysis is necessary to search for new dynamical insights from increasingly large data streams. |
Monday, November 22, 2021 9:57AM - 10:10AM |
H31.00010: Optimal bounds in Taylor--Couette flow Anuj Kumar In this presentation, we find optimal bounds on mean quantities, such as the energy dissipation rate, torque and the Nusselt number in Taylor--Couette flow using the well-known background method. The main result of our study is that we can obtain the analytical expression of the dependence of these optimal bounds on the radius ratio, which is the geometrical parameter in the problem. First, we find the optimal bounds by numerically solving the Euler--Lagrange equations obtained from a variational formulation of the background method. We then obtain suboptimal but analytical bounds on the mean quantities using analysis techniques that employ a definition of the background flow with two boundary layers (near the inner and the outer cylinder), whose relative thicknesses were optimized to obtain this suboptimal bound. Crucially, the optimal bounds have the same dependence on the radius ratio as the suboptimal bounds in the limit of high Reynolds number. We compare the geometrical dependence of optimal bounds on mean quantities with the geometrical dependence observed in the DNS and experiments of turbulent Taylor–Couette flow. Further, in this study, we probed into the structure of the optimal perturbed flow in the Taylor--Couette problem and using the insights gained here, we rigorously dismiss the applicability of the background method to certain flow problems. |
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