Session K08: Nonlinear Dynamics: Coherent Structures (8:45am - 9:30am CST)

Revealing self-similar structure with a data-driven wavelet decomposition

A hallmark of turbulence is the range of physical scales it comprises and the cascade of kinetic energy down the hierarchy of scales. Kolmogorov hypothesized the existence of an intermediate range of self-similar scales, and it was later argued that spatial intermittency is a key ingredient accounting for some of the shortcomings of Kolmogorov's theory. Here we describe a method, inspired by wavelets, that adaptively decomposes a dataset into an energetic hierarchy of structures localized in scale and space. We call the resulting basis a ``data-driven wavelet decomposition". The method reflects the inherent structure of the dataset it acts on, whether it has no structure, structure dominated by a single scale, or multi-scale structure. In particular, from homogeneous isotopic turbulence data we retrieve spatially localized, self-similar, hierarchical structures. We emphasize that self-similarity is not built into the analysis, rather, it emerges from the data. This decomposition provides a starting point for the characterization of localized hierarchical turbulent structures in a wide variety of fluid flows, which we think of as the building blocks of turbulence.   

Navigating unsteady flows: From finite-time Lyapunov exponents to finite-horizon energy-optimal trajectories

Energy-efficient trajectory planning is essential for mobile sensors in adaptive environmental sensing and monitoring arrays. Most active sensing platforms operate in unsteady background flows, such as ocean currents, hurricanes, and windy urban environments. Existing literature has shown that globally energy-efficient paths of mobile sensors tend to utilize the Lagrangian coherent structures (LCS) in unsteady flows. However, the connection between finite-horizon energy-efficient trajectories and LCS remains elusive. To explore this connection, we used a finite-horizon model predictive control algorithm to generate energy-efficient trajectories in a double gyre flow field, which is a canonical model for chaotic mixing in the ocean. In particular, we performed an exhaustive search through optimization hyperparameters, including the prediction time horizon, sampling time, mobility constraints on the sensors’ velocity relative to the flow, state and control penalty weights, and terminal cost. We uncover a strong connection between the energy spent along a trajectory and the presence of background coherent structures, the presence of periodic orbits around the desired target, and links between the gyre oscillating frequency and the Pareto optimal landscape of the control law.   

Extraction of finite-time coherent sets in 3D Rayleigh-Benard Convection using the dynamic Laplacian

Turbulent convection flows in nature are often organized in regular large-scale patterns, which evolve slowly relative to to the typical convective timescale, and are arranged on spatial scales that are much larger than the layer height. Prominent examples are cloud streets in the atmosphere and granulation patterns in solar convection. This order in a fully developed turbulent flow is sometimes called turbulent superstructure in convection. Large-scale structure formation in turbulent Rayleigh-Benard convection recently became accessible in direct numerical simulations, which resolve all relevant scales of turbulence in horizontally extended domains with a large aspect ratio. Using DNS output we apply the dynamic Laplacian approach [Froyland, 2015] to identify these turbulent superstructures as finite-time coherent sets in both quasi-2D and fully three-dimensional settings. A modest number of trajectories are meshed and a finite-element based method is employed to numerically estimate the dynamic Laplacian [Froyland & Junge, 2018]. The coherent sets are encoded in the leading eigenvectors of the dynamic Laplacian, and the individual coherent sets are ``decoded'' using the recently developed Sparse Eigenbasis Approximation (SEBA) algorithm [Froyland, Rock, & Sakellariou, 2019]   

Finding unstable periodic orbits: a hybrid approach with the use of polynomial optimization

A novel hybrid method of computing unstable periodic orbits (UPOs) for polynomial ODE systems is presented. Turbulent coherent structures can correspond to UPOs. This new technique combines the use of polynomial optimization with collocation and numerical continuation methods. Our method requires no a priori knowledge of trajectories. Instead, the UPO search procedure is initiated by first constructing suitably constrained auxiliary functions with polynomial optimization. In previous work (Lakshmi et al. SIAM J. Appl. Dyn. Syst., 19, 763-787, (2020)), the sublevel sets of such functions have been shown to localize UPOs. We will show how auxiliary functions can also be used to implement a simple yet effective control strategy to stabilize UPOs. This enables one to formulate a family of controlled ODE systems parameterized by a parameter k. The original ODE system is recovered as k tends to zero. Solutions that are highly unstable for k = 0 may be less unstable for other k. Periodic orbits can be converged for the controlled ODE systems with a collocation method. Numerical continuation of the obtained orbits in k allows one to find UPOs for the original system. The potential of this new method is illustrated by presenting the results of applying it to selected ODEs.   

Travelling waves in the asymptotic suction boundary layer

The asymptotic suction boundary layer (ASBL) is a flow that develops over a flat bottom plate in the presence of suction through that plate, resulting in a constant boundary layer thickness. As such, it shares certain properties with parallel shear flows and spatially developing boundary layers. Travelling-wave solutions with large-scale low-speed areas that extend into the free stream, reminiscent of large-scale low-momentum zones that influence mixing and extreme events in turbulent boundary layers, have been found in the ASBL. Here, we continue such a solution with respect to Reynolds number and domain size up to domains of size $(L_x/\delta, H/\delta, L_z/\delta) = (14.5\pi,20,7.25\pi)$. It appears through a saddle-node bifurcation in Reynolds number and generally disappears through saddle-node bifurcations in domain size. In short domains, the structure is known to localise in spanwise direction, we find that it does not do so in the streamwise direction. We study the spatial structure of its dominant instabilities as a function of domain size, leading to a phenomenological description of breakdown scenarios of travelling-wave type free-stream coherent structures in the ASBL.   

Phase-consistent dynamic mode decomposition from multiple overlapping spatial domains

We develop an extension to dynamic mode decomposition (DMD) to synthesize globally consistent modes from velocity fields collected independently in multiple partially overlapping spatial domains. Using mathematical optimization, we introduce the notion of phase-consistency for data measurements that ensure consistency of underlying physics for various fluid flow problems. The proposed data-assimilation technique improves the quality of experimental PIV measurements and is robust to experimental noise. We validate our approach using data from direct numerical simulations of laminar flow past a cylinder with distinct frequencies as well as the spatially developing mixing layer, which exhibits a frequency spectrum that evolves continuously as the measurement window moves downstream. Then we demonstrate the utility of the approach on experimental velocity fields from PIV in six overlapping domains in the wake of a cross-flow turbine. The approach may dramatically lower the acquisition cost for PIV in fluids, making real-time control a possibility, even for turbulent flows with many size and time scales.   

Minimal representations of flow trajectories reveal Lagrangian Coherent Structures

What is the most compact piece of information through which one could fully describe the kinematics of a flow trajectory? Inspired by recent progress in the unsupervised learning of dynamical systems, we employ a deep variational autoencoder (VAE) to compress flow trajectories through a low-dimensional latent space and reconstruct the trajectories using only information in this latent space. We find that by imposing certain constraints on the structure of this low-dimensional space and given only the relative motion of trajectories (i.e. not including their absolute position in the flow) our framework learns to encode trajectories into their respective Lagrangian Coherent Structures (LCSs) as the most efficient minimal representation of their kinematics. We discuss this work along with possible extensions to the analysis of transient LCSs.   

Multi-Scale Proper Orthogonal Decomposition: Identifying Small Scale Structures based on a Large Scale Structure

Proper Orthogonal Decomposition (POD) is a technique to extract modes from a scalar or vector field by optimizing the mean square of the variable being examined. Outputs of POD are a set of orthogonal modes $\Phi_j$ with their associated temporal coefficients $a_j(t)$ and energy levels $\lambda_j$ (Berkooz et Al., 1993, Taira et Al., 2017). Using multi-resolution dynamic mode decomposition (MR-DMD) (Kutz et Al., 2015) and conditional POD (Berkooz et Al., 1993) as inspiration, multi-scale POD (MS-POD) can identify the small scale structures based on the sign and strength of a large scale structure. Due to the non-linear interaction between large and small scale structures in fluids, energy of the small scale structures are influenced by the large scale structures in a phenomenon known as amplitude modulation (Mathis et Al., 2009). Filters such as Gaussian, top-hat and spectral filters have been used to understand small scale structures (Mathis et Al., 2009, Saxton-Fox et Al.,2019), however, all of these filters have various shortcomings. In this talk, the MS-POD algorithm will be presented along with results from two sample data sets. Finally, the promising outlook of MS-POD will be discussed in relation to large and small scale coherent structure interaction in turbulence.   

Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces

Koopman analysis is becoming increasingly popular in a variety of scientific contexts. Algorithms to approximate the Koopman operator, such as the dynamic mode decomposition (DMD) and sparsity-promoting variants therein are being applied to many fluid problems routinely. However, even with a rich dictionary of nonlinear observables, its nonlinear variants, e.g., extended/kernel dynamic mode decomposition (EDMD/KDMD) are less popular and seldom applied to large-scale fluid dynamic systems primarily due to the difficulty in discerning the Koopman invariant subspace among numerous resulting Koopman triplets: eigenvalues, eigenvectors, and modes. To address these issues, we propose an algorithm based on mode-by-mode error analysis and multi-task feature learning to extract the most informative Koopman invariant subspace by removing redundant and spurious Koopman triplets from EDMD/KDMD. Effectiveness of the algorithm is demonstrated on several classical problems in fluid mechanics.   

Temporally dependent coherent structures: Effects of deterministic wall boundary conditions on quasi-streamwise vortices in drag reduced turbulent pipe flow

Wall oscillations are an active drag reduction mechanism which imparts a deterministic spanwise boundary layer in a mean streamwise flow. The oscillation takes the form of sinewave in time spanwise velocity boundary which imparts phase dependence on turbulent statistics. We present the estimated conditionally averaged quasi-streamwise vortices about the wall phase optimal ejection event. Vortices are presented as a function of wall normal location and wall phase. Visualized by the square of swirling strength, comparisons to the unperturbed estimated vortices are made. Observations about the effect of spanwise boundary layer on near wall streamwise structures are made. Turbulent Fluctuations are separated from the total estimated field and presented to make distinctions between the linear and non-linear structures of near wall turbulence. We show that drag reduction has the effect of weakening and distorting the average quasi-streamwise vortex in drag reduced turbulent pipe flow.