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This work introduces a novel adaptive mesh refinement (AMR) method that utilizes dominant balance analysis (DBA) for efficient and accurate grid adaptation in computational fluid dynamics (CFD) simulations. The proposed method leverages a Gaussian mixture model (GMM) to classify grid cells into active and passive regions based on the dominant physical interactions in the equation space. Unlike traditional AMR strategies, this approach does not rely on heuristic-based sensors or user-defined parameters, providing a fully automated and problem-independent framework for AMR. Applied to the incompressible Navier-Stokes equations for unsteady flow past a cylinder, the DBA-based AMR method achieves comparable accuracy to high-resolution grids while reducing computational costs by up to 70%. The validation highlights the method's effectiveness in capturing complex flow features while minimizing grid cells, directing computational resources toward regions with the most critical dynamics. This modular and scalable strategy is adaptable to a range of applications, presenting a promising tool for efficient high-fidelity simulations in CFD and other multiphysics domains., Comment: 22 pages, 12 figures

In the realm of big data, discerning patterns in nonlinear systems affected by external control inputs is increasingly challenging. Our approach blends the coarse-graining strengths of centroid-based unsupervised clustering with the clarity of sparse regression in a unique way to enhance the closed-loop feedback control of nonlinear dynamical systems. A key innovation in our methodology is the employment of cluster coefficients via a cluster decomposition of time-series measurement data. This approach transcends the conventional emphasis on the proximity of time series measurements to cluster centroids, offering a more nuanced representation of the dynamics within phase space. Capturing the evolving dynamics of these coefficients enable the construction of a robust, deterministic model for the observed states of the system. This model excels in capturing a wide range of dynamics, including periodic and chaotic behaviors, under the influence of external control inputs. Demonstrated in both the low-dimensional Lorenz system and the high-dimensional scenario of a flexible plate immersed in fluid flow, our model showcases its ability to pinpoint critical system features and its adaptability in reaching any observed state. A distinctive feature of our control strategy is the novel hopping technique between cluster states, which successfully averts lobe switching in the Lorenz system and accelerates vortex shedding in fluid-structure interaction systems while maintaining the mean aerodynamic characteristics., Comment: 35 pages, 12 figures

Vortex-dominated flows are ubiquitous in engineering, and the ability to efficiently manipulate the dynamics of these vortices has broad applications, from wake shaping to mixing enhancement. However, the strongly nonlinear behavior of the vortex dynamics makes this a challenging task. In this work, we investigate the control of vortex dynamics by using a change of coordinates from the Biot-Savart equations into well-known invariants, such as the Hamiltonian, linear, and angular impulses, which are Koopman eigenfunctions. We then combine the resulting model with model predictive control to generate control laws that force the vortex system using "virtual cylinders". The invariant model is beneficial as it provides a linear, global description of the vortex dynamics through a recently developed Koopman control scheme for conserved quantities and invariants. The use of this model has not been well studied in the literature in the context of control. In this paper, we seek to understand the effect of changing each invariant individually or multiple invariants simultaneously. We use the 4-vortex system as our primary test bed, as it is the simplest configuration that exhibits chaotic behavior. We show that by controlling to specific invariant quantities, we can modify the transition from chaotic to quasiperiodic states. Finally, we computationally demonstrate the effectiveness of invariant control on a toy example of tracer mixing in the 4-vortex system.

The coupling interactions between deformable structures and unsteady fluid flows occur across a wide range of spatial and temporal scales in many engineering applications. These fluid-structure interactions (FSI) pose significant challenges in accurately predicting flow physics. In the present work, two multi-layer network approaches are proposed that characterize the interactions between the fluid and structural layers for an incompressible laminar flow over a two-dimensional compliant flat plate at a 35-degrees angle of attack. In the first approach, the network nodes are formed by wake vortices and bound vortexlets, and the edges of the network are defined by the induced velocity between these elements. In the second approach, coherent structures (fluid modes), contributing to the kinetic energy of the flow and structural modes, contributing to the kinetic energy of the compliant structure constitute the network nodes. The energy transfers between the modes are extracted using a perturbation approach. Furthermore, the network structure of the FSI system is simplified using the community detection algorithm in the vortical approach and by selecting dominant modes in the modal approach. Network measures are used to reveal the temporal behavior of the individual nodes within the simplified FSI system. Predictive models are then built using both data-driven and physics-based methods. Overall, this work sets the foundation for network-theoretic reduced-order modeling of fluid-structure interactions, generalizable to other multi-physics systems., Comment: 20 pages, 10 figures

We stabilize two-dimensional turbulent Kolmogorov flow by selectively altering the time rate of change of inviscid invariants (energy and enstrophy) of the flow. This method has earlier been demonstrated to modify the two-dimensional unforced decaying turbulent flow to reach steady states. However, Kolmogorov flow exhibits flow unsteadiness over a range of spatial and temporal scales which are driven by constant external forcing at wave number $k_f$. The energy injected at scale ($k \approx k_f$) is distributed to large ($k > k_f$) and small ($k < k_f$) wave numbers through interactions of different scales called direct and inverse energy cascading, respectively. The selective modification strategy autonomously identifies additional forcing inputs into the governing equations which results in the transformation of Kolmogorov flow into a non-trivial steady state. We demonstrate that this method is highly effective in altering the inverse energy cascading, a particularly intricate challenge in fluid dynamics. Intriguingly, we have also found that the steady states achieved through this approach resemble an invariant solution of the original Kolmogorov flow, highlighting the potential significance of our method., Comment: 20 pages, 8 figures

Aeroelastic structures, from insect wings to wind turbine blades, experience transient unsteady aerodynamic loads that are coupled to their motion. Effective real-time control of flexible structures relies on accurate and efficient predictions of both the unsteady aeroelastic forces and airfoil deformation. For rigid wings, classical unsteady aerodynamic models have recently been reformulated in state-space for control and extended to include viscous effects. Here we further extend this modeling framework to include the deformation of a flexible wing in addition to the quasi-steady, added mass, and unsteady viscous forces. We develop low-order linear models based on data from direct numerical simulations of flow past a flexible wing at low Reynolds number. We demonstrate the effectiveness of these models to track aggressive maneuvers with model predictive control while constraining maximum wing deformation. This system identification approach provides an interpretable, accurate, and low-dimensional representation of an aeroelastic system that can aid in system and controller design for applications where transients play an important role.

The network of interactions among fluid elements and coherent structures gives rise to the incredibly rich dynamics of vortical flows. These interactions can be described with the use of mathematical tools from the emerging field of network science, which leverages graph theory, dynamical systems theory, data science, and control theory. The blending of network science and fluid mechanics facilitates the extraction of the key interactions and communities in terms of vortical elements, modal structures, and particle trajectories. Phase-space techniques and time-delay embedding enable network-based analysis in terms of visibility, recurrence, and cluster transitions leveraging available time-series measurements. Equipped with the knowledge of interactions and communities, the network-theoretic approach enables the analysis, modeling, and control of fluid flows, with a particular emphasis on interactive dynamics. In this article, we provide a brief introduction to network science and an overview of the progress on network-based strategies to study the complex dynamics of fluid flows. Case studies are surveyed to highlight the utility of network-based techniques to tackle a range of problems from fluid mechanics. Towards the end of the paper, we offer an outlook on network-inspired approaches., Comment: 35 pages, 17 figures

In two-dimensional decaying homogeneous isotropic turbulence, kinetic energy and enstrophy are respectively transferred to larger and smaller scales. In such spatiotemporally complex dynamics, it is challenging to identify the important flow structures that govern this behavior. We propose and numerically employ two flow modification strategies that leverage the inviscid global conservation of energy and enstrophy to design external forcing inputs which change these quantities selectively and simultaneously, and drive the system towards steady-state or other late-stage behavior. One strategy employs only local flow-field information, while the other is global. We observe various flow structures excited by these inputs and compare with recent literature. Energy modification is characterized by excitation of smaller wavenumber structures in the flow than enstrophy modification., Comment: 15 pages, 11 figures

We focus on a convolutional neural network (CNN), which has recently been utilized for fluid flow analyses, from the perspective on the influence of various operations inside it by considering some canonical regression problems with fluid flow data. We consider two types of CNN-based fluid flow analyses; 1. CNN metamodeling and 2. CNN autoencoder. For the first type of CNN with additional scalar inputs, which is one of the common forms of CNN for fluid flow analysis, we investigate the influence of input placements in the CNN training pipeline. As an example, estimation of force coefficients of an inclined flat plate and two side-by-side cylinders in laminar flows is considered. We find that care should be taken for the placement of additional scalar inputs depending on the problem setting and the complexity of flows that users handle. We then discuss the influence of various parameters and operations on the CNN performance, with the utilization of autoencoder (AE). A two-dimensional turbulence is considered for the demonstration of AE. The results of AE highly rely on the decaying nature. Investigation on the influence of padding operation at a convolutional layer is also performed. The zero padding shows reasonable ability compared to other methods which account for the boundary conditions assumed in the numerical data. Moreover, the effect of the dimensional reduction/extension methods inside CNN is also examined. The CNN model is robust against the difference in dimension reduction operations, while it is sensitive to the dimensional extension methods. The findings of this paper will help us better design a CNN architecture for practical fluid flow analysis, Comment: 24 pages, 20 figures

Fluid flows play a central role in scientific and technological development, and many of these flows are characterized by a dominant oscillation, such as the vortex shedding in the wake of nearly all transportation vehicles. The ability to control vortex shedding is critical to improve the aerodynamic performance of these unsteady fluid flow systems. This goal requires precise characterization of how perturbations affect the long-time phase of the oscillatory flow, as well as the ability to control transient behaviors. In this work, we develop an energy-efficient flow control strategy to rapidly alter the oscillation phase of time-periodic fluid flows, leveraging theory developed for periodic biological systems. First, we perform a phase-sensitivity analysis to construct a reduced-order model for the response of the flow oscillation to impulsive control inputs at various phases. Next, we introduce two control strategies for real-time phase control based on the phase-sensitivity function: 1) optimal phase control, obtained by solving the Euler-Lagrange equations as a two-point boundary value problem, and 2) model-predictive control (MPC). Our approach is demonstrated for two unsteady flow systems, the incompressible laminar flow past a circular cylinder and the flow past an airfoil. We show that effective phase control may be achieved with several actuation strategies, including blowing and rotary control. Moreover, our control approach uses realistic measurements of the lift force on the body, rather than requiring high-dimensional measurements of the full flow field., Comment: 23 pages, 11 figures

Dynamic mode decomposition (DMD) provides a principled approach to extract physically interpretable spatial modes from time-resolved flow field data, along with a linear model for how the amplitudes of these modes evolve in time. Recently, DMD has been extended to work with more realistic data that is under-resolved either in time or space, or with data collected in the same spatial domain over multiple independent time windows. In this work, we develop an extension to DMD to synthesize globally consistent modes from velocity fields collected independently in multiple partially overlapping spatial domains. We propose a tractable optimization to identify modes that span multiple windows and align their phases to be consistent in the overlapping regions. First, we demonstrate this approach on data from direct numerical simulation, where it is possible to split the data into overlapping domains and benchmark against ground-truth modes. We consider the laminar flow past a cylinder as an example with distinct frequencies, along with the spatially developing mixing layer, which exhibits a frequency spectrum that evolves continuously as the measurement window moves downstream. Next, we analyze experimental velocity fields from PIV in six overlapping domains in the wake of a cross-flow turbine. On the numerical examples, we demonstrate the robustness of this approach to increasing measurement noise and decreasing size of the overlap regions. In all cases, it is possible to obtain a phase-aligned, composite reconstruction of the full time-resolved flow field from the phase-consistent modes over the entire domain., Comment: 20 pages, 11 figures

We propose a novel model-free self-learning cluster-based control strategy for general nonlinear feedback flow control technique, benchmarked for high-fidelity simulations of post-stall separated flows over an airfoil. The present approach partitions the flow trajectories (force measurements) into clusters, which correspond to characteristic coarse-grained phases in a low-dimensional feature space. A feedback control law is then sought for each cluster state through iterative evaluation and downhill simplex search to minimize power consumption in flight. Unsupervised clustering of the flow trajectories for in-situ learning and optimization of coarse-grained control laws are implemented in an automated manner as key enablers. Re-routing the flow trajectories, the optimized control laws shift the cluster populations to the aerodynamically favorable states. Utilizing limited number of sensor measurements for both clustering and optimization, these feedback laws were determined in only $O(10)$ iterations. The objective of the present work is not necessarily to suppress flow separation but to minimize the desired cost function to achieve enhanced aerodynamic performance. The present control approach is applied to the control of two and three-dimensional separated flows over a NACA 0012 airfoil with large-eddy simulations at an angle of attack of $9^\circ$, Reynolds number $Re = 23,000$ and free-stream Mach number $M_\infty = 0.3$. The optimized control laws effectively minimize the flight power consumption enabling the flows to reach a low-drag state. The present work aims to address the challenges associated with adaptive feedback control design for turbulent separated flows at moderate Reynolds number., Comment: 32 pages, 18 figures

A network community-based reduced-order model is developed to capture key interactions amongst coherent structures in high-dimensional unsteady vortical flows. The present approach is data-inspired and founded on network-theoretic techniques to identify important vortical communities that are comprised of vortical elements that share similar dynamical behavior. The overall interaction-based physics of the high-dimensional flow field is distilled into the vortical community centroids, considerably reducing the system dimension. Taking advantage of these vortical interactions, the proposed methodology is applied to formulate reduced-order models for the inter-community dynamics of vortical flows, and predict lift and drag forces on bodies in wake flows. We demonstrate the capabilities of these models by accurately capturing the macroscopic dynamics of a collection of discrete point vortices, and the complex unsteady aerodynamic forces on a circular cylinder and an airfoil with a Gurney flap. The present formulation is found to be robust against simulated experimental noise and turbulence due to its integrating nature of the system reduction.

A networked oscillator based analysis is performed for periodic bluff body flows to examine and control the transfer of kinetic energy. Spatial modes extracted from the flow field with corresponding amplitudes form a set of oscillators describing unsteady fluctuations. These oscillators are connected through a network that captures the energy exchanges amongst them. To extract the network of interactions among oscillators, amplitude and phase perturbations are impulsively introduced to the oscillators and the ensuing dynamics are analyzed. Using linear regression techniques, a networked oscillator model is constructed that reveals energy transfers and phase interactions among the modes. The model captures the nonlinear interactions amongst the modal oscillators through a linear approximation. A large collection of system responses are aggregated into a network model that captures interactions for general perturbations. The networked oscillator model describes the modal perturbation dynamics better than the empirical Galerkin reduced-order models. A model-based feedback controller is then designed to suppress modal amplitudes and the resulting wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two- dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady vortical flows., Comment: 33 pages, 16 figures

We examine discrete vortex dynamics in two-dimensional flow through a network-theoretic approach. The interaction of the vortices is represented with a graph, which allows the use of network-theoretic approaches to identify key vortex-to-vortex interactions. We employ sparsification techniques on these graph representations based on spectral theory for constructing sparsified models and evaluating the dynamics of vortices in the sparsified setup. Identification of vortex structures based on graph sparsification and sparse vortex dynamics are illustrated through an example of point-vortex clusters interacting amongst themselves. We also evaluate the performance of sparsification with increasing number of point vortices. The sparsified-dynamics model developed with spectral graph theory requires reduced number of vortex-to-vortex interactions but agrees well with the full nonlinear dynamics. Furthermore, the sparsified model derived from the sparse graphs conserves the invariants of discrete vortex dynamics. We highlight the similarities and differences between the present sparsified-dynamics model and the reduced-order models., Comment: 23 pages, 13 figures

The present paper reports on our effort to characterize vortical interactions in complex fluid flows through the use of network analysis. In particular, we examine the vortex interactions in two-dimensional decaying isotropic turbulence and find that the vortical interaction network can be characterized by a weighted scale-free network. It is found that the turbulent flow network retains its scale-free behavior until the characteristic value of circulation reaches a critical value. Furthermore, we show that the two-dimensional turbulence network is resilient against random perturbations but can be greatly influenced when forcing is focused towards the vortical structures that are categorized as network hubs. These findings can serve as a network-analytic foundation to examine complex geophysical and thin-film flows and take advantage of the rapidly growing field of network theory, which complements ongoing turbulence research based on vortex dynamics, hydrodynamic stability, and statistics. While additional work is essential to extend the mathematical tools from network analysis to extract deeper physical insights of turbulence, an understanding of turbulence based on the interaction-based network-theoretic framework presents a promising alternative in turbulence modeling and control efforts., Comment: 11 pages, 5 figures