Session J21: Nonlinear Dynamics: Reduced-Order Modeling I

Space-time POD as a unifying framework for modal decomposition

Modal decompositions, which seek to extract meaningful structures from flow data or governing equations, are widely used in fluid mechanics to explore physical processes and develop reduced-complexity models.  Many such methods are currently in use, but the connections between them and the appropriate choice of method for a given application are not always clear.  In this presentation, we show how a space-time formulation of proper orthogonal decomposition (POD) provides a framework for understanding most available modal decomposition methods.  In particular, we show how standard POD, spectral POD (SPOD), dynamic mode decomposition (DMD), resolvent analysis, and Hankel singular value decomposition can all be understood as special cases of space-time POD obtained in certain limits.  Moreover, we discuss how this unified viewpoint is helpful, e.g., we show how it motivates alternatives to Hankel singular value decomposition with improved convergence properties.

    

Efficient and precise determination of manifold coordinates for systems with complex spatiotemporal dynamics

Nonlinear dissipative partial differential equations (PDEs) are ubiquitous in describing phenomena in physics and engineering that display out-of-equilibrium dynamics, complex nonlinear behaviors, and even spatiotemporal chaos. The long-time dynamics of these formally infinite-dimensional systems are known to collapse onto finite-dimensional invariant manifolds. The task of identifying the manifold dimension/coordinates of these systems from simulation or experimental data is generally a nontrivial task. We address this challenge by combining a novel autoencoder architecture with implicit and weight regularization. Compared to existing methods, our framework does not require combinatorically expensive calculations of distance between data points and yields a sharp distinction between nonlinear degrees of freedom that are important and those that are not. Using our method, we successfully estimate the manifold dimension for a zoo of datasets including human handwriting data, an embedded Lorenz attractor, the Kuramoto-Sivashinsky Equation at domain sizes of L=22, 44, 66, and 88, 2D Kolmogorov flow and turbulent plane Couette flow. Finally, we demonstrate that our architecture provides a natural workflow for downstream reduced-order modeling and forecasting tasks.   

Real-time reduced order modeling of deterministic partial differential equations using time dependent basis

A new methodology using time dependent basis is presented for solving deterministic partial differential equations. In this method, a multi-dimensional variable is represented by a set of time-dependent orthonormal modes in each dimension and a core tensor. The best update for the modes at every time step is found as the solution evolves. To achieve lower computational cost for non-polynomial non-linearity in the equation an adaptive sparse interpolation algorithm is implemented. Thus a computational speedup is achieved using the discrete empirical interpolation method (DEIM). This method is implemented to solve the incompressible and compressible Navier-Stokes equations. The error convergence properties of the method for different reduction orders are compared. We also demonstrate the development of the modes according to the evolution of the flow physics. Due to the low rank representation of the solution at every time instant, the method also provides advantages in data storage for a large number of time steps. This advantage is notably evident in higher dimensions (d>2).   

Spectral Proper Orthogonal Decomposition via Dynamic Mode Decomposition for Non-Sequential Pairwise Data

Spectral proper orthogonal decomposition (SPOD) is a valuable tool to identify spatio-temporally coherent structures in statistically stationary turbulence. It can be used with particle image velocimetry (PIV) flow data, but current algorithms require sequential data with uniform time step, which is thus limited by camera speeds. Dynamic mode decomposition (DMD), a method that estimates structures that optimally capture the time dynamics of the data, relaxes the requirement for sequential data, but does not produce orthogonal, energy-ordered modes. The present work attempts to compute SPOD modes from pairs of PIV snapshots with a small time step but with large gaps between pairs, by using the DMD to estimate, segment-wise, sequential series of data that can then be used to estimate the SPOD modes, with the aim of dealiasing/resolving frequencies between the gap and pair Nyquist limits. This method is tested on several examples, including a numerical simulation of a turbulent jet. Results show this method can accurately estimate the SPOD mode shapes and dealias the SPOD spectrum with a resolved frequency that is 40 times higher than the Nyquist limit associated with the longer time step between pairs.   

Optimal waveform for fast entrainment of airfoil wakes

Phase synchronization analysis for time-periodic flows can be characterized using phase reduction analysis.  Phase reduction method reduces high-dimensional flow physics to single scalar phase dynamics.  The synchronization of phase dynamics to external forcing and optimal waveform of forcing could be obtained from phase sensitivity analysis.  We assess the receptivity of the wake through the spatial phase sensitivity fields for laminar flows over symmetric airfoils at high angles of attack.  We discuss the framework to obtain the optimal waveform for promoting synchronization of post-stall airfoil wakes to periodic actuation inputs.  We investigate the influence of the angle of attack on phase coupling functions and find optimal waveforms based on the spatial phase-sensitivity fields.  We observe that phase synchronization becomes harder to achieve for high angle of attack and the corresponding optimal actuation waveform becomes non-sinusoidal.  The present analysis for phase synchronization of time-varying flows reveals critical insights for modifying unsteady vortex dynamics with special care toward the time-adaptive nature of actuation.

This work is supported by the US Air Force Office of Scientific Research (FA9550-21-1-0178) and the US National Science Foundation (2129639).   

Compact manifold representation of airfoil wake-vortex gust interaction

Airfoil wake dynamics can be described by a collection of dynamically important modes. 

This translates to the existence of a low-dimensional representation of the underlying dynamics. 

This study considers the complex dynamics of vortex gust-airfoil wake interaction around a NACA0012 airfoil over a range of angles of attack. 

While undisturbed flows in the present example can be easily characterized in a low-dimensional space through linear techniques, it is very challenging to obtain a compact representation of the wakes under the influence of strong vortex gusts.

To extract a universal manifold for a variety of complex wakes, we develop a physics-guided nonlinear autoencoder. 

The current technique provides a physically-interpretable low-dimensional manifold that covers a variety of wake arrangements, which is difficult for conventional autoencoder-based techniques.

Furthermore, the present latent manifold can be leveraged to assess how complex wakes are affected and modified by disturbances in a low-dimensional space.

This approach achieves significant compression of unsteady aerodynamic flows and identifies the relevant latent space coordinates to optimally describe the seemingly complex dynamics with an extremely small number of variables.

    

Real-time Flow Field Measurement under Predetermined Control by Plasma Acutuator using Sparse Processing PIV

Flow field with predetermined control by the pair of plasma actuators (PA) was measured in real time using sparse processing particle image velocimetry (SPPIV), where SPPIV is a method to estimate the entire flow field from limited results of sparsely located particle-image-velocimetry (PIV)-analysis interrogation window. In this study, the PIV measurement was conducted for cylinder model under the following conditions; the freestream velocity, the diameter, and the position to attach the pair of PAs were 5 m/s, 30 mm, and a point on each side at 90° from the centerline in the freestream direction, respectively. Several proper orthogonal decomposition (POD) modes were used and the velocity field was estimated by SPPIV. The results of the analysis showed that the flow velocity field can reconstruct a large-scale structure with the upper two modes, and can be measured with very high estimation accuracy even with a small number of sensors (e.g., five). We also added information on the control input term to the estimation model and investigated changes in estimation accuracy under control by the pair of PAs. In the future, feedback control with SPPIV and PAs as the observer and the actuator is being planned, and the technologies for this have been sufficiently investigated.   

DMD-based Superresolution Measurement of Time-resolved Large-scale Turbulent Structures of a Supersonic Jet based on Acoustic and Dual Planar PIV Data

The present study proposes a framework of the superresolution measurement using total least square dynamic mode decomposition (TLS-DMD), Kalman filter (KF) and Rauch-Tung-Striebel (RTS) smoother applied to a Mach 1.2 nearly perfectly expanded supersonic jet. Dual planar particle image velocimetry (PIV) was utilized and paired velocity fields of the flow at a temporal resolution of 5,000Hz were obtained. High frequency acoustic data were simultaneously obtained at 200,000Hz. The reconstruction of the flow throughout the framework allows us to obtain dynamic modes that evolve along the reconstructed flow at the same temporal resolution as the acoustic data. Although some modes decay or diverge due to measurement noise, most of the resulting modes are stable and useful for reconstructing the large-scale turbulent structures that cause jet noise. The proposed method works well obtaining high temporal-resolution velocity fields of the main modes responsible for the characteristic acoustic spectrum of perfectly expanded supersonic flow.

    

Applicability of the Bayesian estimation-based spatial superresolution measurement to the velocity fields of a subsonic jet acquired by simultaneous dual PIV

The spatial superresolution framework based on proper orthogonal decomposition (POD) and Bayesian estimation was designed for the velocity field reconstruction of a subsonic jet. The framework was evaluated using the experimental data obtained from simultaneous dual Particle Image Velocimetry measurements of a subsonic jet with different magnifications. The framework was first tested using the artificial low-resolution (LR) velocity field generated by average pooling the high-resolution (HR) experimental velocity field. The conventional bicubic interpolation was employed for performance comparison. The framework successfully reconstructed the high-spatial-resolution velocity field from the coarse velocity field, with the estimation error of 6% at the resolution ratio of LR:HR = 1:3, using 2000 POD modes. Its error improved with the POD mode increase, where the fine structures in high POD modes are started to be reconstructed. Framework was then applied to the experimental velocity field datasets, which its resolution ratio is 1:3. Its estimation error was 77% when using top 5 POD modes and improved to 46% when using 2000 POD modes. Since the bicubic interpolation error was 110%, the framework outperformed bicubic interpolation in all modes.   

Author not Attending

The use of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. In this approach to turbulence decomposition, which has recently been termed Spectral POD (SPOD), Fourier modes are considered as solutions to the corresponding Fredholm integral equation of the second kind along homogeneous-periodic or homogeneous coordinates. In the present work, the notion that the POD modes formally converge to Fourier modes for increasing domain length is challenged. Numerical results indicate that the discrepancy between POD and Fourier modes along locally translationally invariant coordinates is coupled to the Taylor macro/micro scale ratio (MMSR) of the kernel in question. Increasing discrepancies are observed for smaller MMSRs, which are characteristic of low Reynolds number flows. It is observed that the asymptotic convergence rate of the eigenspectrum matches the corresponding convergence rate of the exact analytical Fourier spectrum of the kernel in question - even for extremely small domains and small MMSRs where the corresponding DFT spectra suffer heavily from windowing effects. These results indicate that the accumulated discrepancies between POD and Fourier modes play a role in producing the spectral convergence rates expected from Fourier transforms of translationally invariant kernels on infinite domains.   

On-the-fly reduced order modeling of finite-time nonlinear sensitivities

We present an on-the-fly reduced order modeling framework for computing the evolution of finite-time nonlinear sensitivities in dynamical systems. Unlike solving a linearized system that assumes infinitesimal perturbations around a base trajectory, this framework places no limitation on the size of the perturbation as nonlinear interactions are considered. We propose a model-driven low-rank approximation that leverages a time-dependent basis by extracting correlations between sensitivities on-the-fly. To this end, we derive forward low-rank evolution equations for an orthonormal state basis, correlation matrix, and orthonormal parametric basis. The resulting equations are Jacobian-free and leverage the same nonlinear solver that is used to compute the evolution of the base state. For nonlinear sensitivities with arbitrarily time dependent base state, we demonstrate that low-rank structure often exists, and can be extracted in real time by solving forward evolution equations. This is in direct contrast to traditional reduced order modeling techniques that leverage a static basis extracted from high-fidelity data during an offline stage. In this work, we demonstrate the efficacy of the method for a number of applications including transition for compressible flow.   

Interpretability of the latent space of autoencoders

Autoencoders are machine-learning methods that enable a reduced-order representation of data. They consist of an encoder, which compresses the data in a latent space, and a decoder, which decompresses the data back to the original space. If only linear operations are performed during the encoding and decoding phases, an autoencoder can learn the principal components of the data. On the other hand, if nonlinear activations functions are employed, an autoencoder learns a nonlinear model of the data in the latent space. The interpretability of the latent space, however, is not yet fully established. In this work, we physically interpret the latent space with simple tools from differential geometry. The interpretation is employed on canonical turbulent flows, i.e., the Kolmogorov flow and the minimal flow unit. The results show that the autoencoder learns the optimal submanifold in which the reduced-order dynamics is well represented. This work opens opportunities for extracting physical insight from the latent space and for nonlinear model reduction.