Bulletin of the American Physical Society
76th Annual Meeting of the Division of Fluid Dynamics
Sunday–Tuesday, November 19–21, 2023; Washington, DC
Session T31: NLD Coherent Structures II 
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Chair: Sutanu Sarkar, University of California, San Diego Room: 156 
Monday, November 20, 2023 4:25PM  4:38PM 
T31.00001: Time delay embeddings to uncover unstable periodic orbits and exact coherent structures in chaotic fluid systems Prerna M Patil, Eurika Kaiser, Nathan Kutz, Steven L Brunton The datadriven modeling of dynamical systems is rapidly developing, especially for fluid systems. However, many leading algorithms require highdimensional, fullstate training data, while for many realworld systems only very limited or partial measurements are available. Time delay embeddings provide a principled approach to reconstruct an attractor from such limited time series measurements. In this work, we evaluate the use of longtime delay embeddings to capture progressively sophisticated unstable periodic orbits (UPOs) of a chaotic system; in fluid dynamics, these orbits correspond to exact coherent structures (ECSs). The behavior of a chaotic attractor can be understood as the combination of all its periodic orbits. This is particularly relevant when studying the turbulent behavior of fluid flows, which can be represented by an infinite number of these orbits. While an infinite number of orbits are needed to fully describe the system, the dominant dynamics can be captured by the shortest, or ‘fundamental’, orbits. By finding enough of these orbits, we can make predictions about the statistical behavior of turbulence. 
Monday, November 20, 2023 4:38PM  4:51PM 
T31.00002: How to compute periodic orbits and equilibria of NavierStokes without suffering from chaos Omid Ashtari, Zheng Zheng, Tobias M Schneider Unstable nonchaotic invariant solutions of the NavierStokes equations capture transitional turbulent flow dynamics, yet numerically identifying equilibria and periodic orbits using common shooting methods has remained challenging. We thus propose a class of alternative computational methods for identifying the nonchaotic building blocks of fluid turbulence. These are variational methods unaffected by exponential error amplification associated with timemarching a chaotic system. Technically, we use adjoints to solve an optimization problem whose global minima represent invariant solutions. For incompressible 3D shear flows we treat the nonlocal pressure constraint within the adjoint formulation via an adaptation of the KleiserSchumann influence matrix method. We compute multiple equilibria and periodic orbits of different canonical shear flows, highlighting the robustness of the methodology. 
Monday, November 20, 2023 4:51PM  5:04PM 
T31.00003: A flow complexity estimation method based on modified persistent homology method Huixuan Wu, Zhongquan Zheng, Jerry Zhou This research focuses on the recurrent flow patterns of a wake flow downstream of two cylinders in tandem. A modified persistent homology method is employed in this research. In the traditional persistent homology computation, input data is treated as isolated points in a highdimensional space. In contrast, this study introduces a filtration process that considers only the topological connections that are local minima, eliminating duplicated edges present in the usual VietorisRips filtration. 
Monday, November 20, 2023 5:04PM  5:17PM 
T31.00004: Lagrangian Coherent Set Detection with Topological Advection Spencer Smith, Rida Ilahi Lagrangian coherent structures (LCSs) determine the transport properties and mixing dynamics of general aperiodic fluid flows, much as invariant manifolds and periodic orbits do for autonomous or periodic systems. Due to the prevalence of LCSs in nature and industry, there exist many successful techniques for detecting them in data. However, these approaches typically require very fine trajectory data to reconstruct velocity fields and compute CauchyGreentensorrelated quantities (such as FTLE fields). We use topological techniques to help detect coherent trajectory sets in relatively sparse 2D advection problems. In particular, we use a new, computationally efficient algorithm which evolves topological loops (material curves) forward in time due to the movement of sets of advected particles. Starting with simple loops, and their future state  given by the topological advection algorithm, we systematically combine them to form the boundary of LCSs. We then show how this approach effectively and efficiently reveals the coherent sets of trajectories in models flows. 
Monday, November 20, 2023 5:17PM  5:30PM Author not Attending 
T31.00005: Abstract Withdrawn

Monday, November 20, 2023 5:30PM  5:43PM 
T31.00006: Connecting value functions to flow map operators for path planning in flow fields Kartik Krishna, Steven L Brunton, Zhuoyuan Song Path planning through unsteady fluid flow fields is a challenging problem with important applications in robotics, navigation, and sensor deployment. It is possible to formulate this task as an optimization problem which involves solving for value functions over unsteady flow fields. Value functions play a fundamental role at the heart of all of optimal control, including recent formulations in reinforcement learning and model predictive control. However, the highly nonlinear and multiscale nature of fluid flow fields often make it challenging to solve for, and interpret value functions. In this work, we establish a strong connection between the fluid coherent structures and material separatrices, which can be quantified through the finite time Lyapunov exponent (FTLE) field, and value functions computed over the domain. In particular, we show that FTLE ridges tend to demarcate sharp transitions in the value function, illuminating regions where active transport may be achieved at a much lower cost. Further, we explore mathematical connections between the particle flow map operator underlying the FTLE field and solutions of the Hamilton Jacobi Bellman equation, which is central to optimal control and reinforcement learning. We demonstrate this approach on simple unsteady flow fields to develop intuition. These findings will lay the theoretical foundation for principled performance analyses of motion planning policies and deployment strategies for intelligent mobile sensors in fluid flows. 
Monday, November 20, 2023 5:43PM  5:56PM 
T31.00007: Stabilizing twodimensional turbulent Kolmogorov flow via selective modification of inviscid invariants Aditya G Nair, Gaurav Kumar We devise a methodology to stabilize twodimensional turbulent Kolmogorov flow by carefully adjusting the rates of change of inviscid invariants, particularly energy, and enstrophy. Our technique automatically incorporates additional forces into the primary governing equation of the flow. This adjustment leads to a change in the way energy and enstrophy evolve over time, effectively guiding the turbulent flow to reach a stable state with a steadystate control input. When we remove the added control term, the flow moves closer to this invariant solution of Kolmogorov flow. The stable states serve as effective initial points for finding equilibrium solutions of the flow using Newton's method. This research presents a significant advancement in our ability to control and manipulate turbulent flow dynamics. 
Monday, November 20, 2023 5:56PM  6:09PM 
T31.00008: A timedomain preconditioner for the resolvent and harmonic resolvent analyses Alberto Padovan, Ricardo Frantz, JeanChristophe Loiseau, Daniel J Bodony Resolvent analysis is a frequencydomain formalism used to study the inputoutput dynamics of fluid flows in the proximity of timeinvariant base flows (e.g., a steady solution of the NavierStokes equation, or the temporal mean of a turbulent flow). The harmonic resolvent analysis is analogous, except that the base flow is periodic in time. Both formulations require computing a singular value decomposition of the frequencydomain operator that maps harmonic forcing inputs to posttransient harmonic outputs. In practice, this involves solving large (or extremely large, in the case of the harmonic resolvent) frequencydomain algebraic systems of equations. Furthermore, solving these equations in the frequency domain usually requires computational functionality that is not readily avilable in inhouse or opensource timestepping CFD codes. 
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