arXiv subjects

Michael D. Graham

Publications and source records attributed to Michael D. Graham.

At least 19 recordsLinked to original sources

arXiv2026

Flapping instability of elastic disks in Stokes flows

Fluid-structure interactions at low Reynolds number can lead to a much richer phenomenology than previously expected. Here, we study the dynamics of a freely suspended, thin elastic disk in a shear flow, where the plane of the disk is initially parallel to the flow plane. Using a combination of experiments and simulations, we demonstrate that beyond a critical flow strength the disk deforms, performing flapping dynamics, in which the disk curves up and down periodically relative to the horizontal shear plane. The bifurcation diagram obtained by simulation reveals several oscillatory solutions, including a wiggling motion that is predicted by a linear stability analysis. The flapping dynamics is shown to be a subcritical instability whose key ingredient is the finite extensibility of the disk. The behavior we observe has implications for emerging investigations on the flow dynamics of sheet-like particles, such as 2D polymers and 2D crystalline materials immersed in viscous fluids.

physics.flu-dyn↗

arXiv2026

Structures of elastoinertial turbulence in pipe flow

Elastoinertial turbulence (EIT) is a self-sustaining chaotic state resulting from the interplay between inertia and elasticity in the flow of dilute polymeric solutions, and its emergence is believed to limit the achievable drag reduction in turbulence flow using polymer additives. In the present study, we introduce a viscoelastic variant of spectral proper orthogonal decomposition (VESPOD) that decomposes velocity and polymeric stress fields of EIT together into well-defined orthogonal oscillating modes such that the decomposition is optimal in the terms of the total mechanical energy of the flow. Using this technique, we investigate the dominant coherently evolving structures underlying the dynamics of EIT in axisymmetric pipe flow. By analyzing distinct peaks in the leading eigenvalue of the VESPOD eigenvalue spectrum, we find that the dynamics of EIT in pipe flow is dominated by three distinct families of traveling waves, where the higher wavenumber structures of each family are simple harmonics of their respective fundamental waves. The radial velocity fields of the traveling waves are characterized by the formation of large-scale structures spanning the pipe radial direction. However, the polymeric stress fields corresponding to them are characterized by the formation of thin inclined sheets of high stress fluctuations at the critical layers of the respective waves, i.e.~ the locations where the wave speed of the VESPOD mode matches the mean streamwise velocity. Additionally, these sheets exhibit nested structures, where the polymeric sheets of faster waves are confined by those of the immediately slower waves.

physics.flu-dyn↗

arXiv2025

Identifying efficient routes to laminarization: an optimization approach

The nonlinear and chaotic nature of turbulent flows poses a major challenge for designing effective control strategies to maintain or induce low-drag laminar states. Traditional linear methods often fail to capture the complex dynamics governing transitions between laminar and turbulent regimes. In this work, we introduce the concept of the minimal seed for relaminarization-the closest point to a reference state in the turbulent region of the state space that triggers a direct transition to laminar flow without a chaotic transient. We formulate the identification of this optimal perturbation as a fully nonlinear optimization problem and develop a numerical framework based on a multi-step penalty method to compute it. Applying this framework to a nine-mode model of a sinusoidal shear flow, we compute the minimal seeds for both transition to turbulence and relaminarization. While both of these minimal seeds lie infinitesimally close to the laminar-turbulent boundary-the edge of chaos-they are generally unrelated and lie in distant and qualitatively distinct regions of state space, thereby providing different insights into the flow's underlying structure. We find that the optimal perturbation for triggering transition is primarily in the direction of the mode representing streamwise vortices (rolls), whereas the optimal perturbation for relaminarization is distributed across multiple modes without strong contributions in the roll or streak directions. By analyzing trajectories originating from these minimal seeds, we find that both transition and laminarization behavior are controlled by the stable and unstable manifolds of a periodic orbit on the edge of chaos. The laminarizing trajectory obtained from the minimal seed for relaminarization provides an efficient pathway out of turbulence and can inform the design and evaluation of flow control strategies aimed at inducing laminarization.

physics.flu-dyn↗

arXiv2025

Blending data and physics for reduced-order modeling of systems with spatiotemporal chaotic dynamics

While data-driven techniques are powerful tools for reduced-order modeling of systems with chaotic dynamics, great potential remains for leveraging known physics (i.e. a full-order model (FOM)) to improve predictive capability. We develop a hybrid reduced order model (ROM), informed by both data and FOM, for evolving spatiotemporal chaotic dynamics on an invariant manifold whose coordinates are found using an autoencoder. This approach projects the vector field of the FOM onto the invariant manifold; then, this physics-derived vector field is either corrected using dynamic data, or used as a Bayesian prior that is updated with data. In both cases, the neural ordinary differential equation approach is used. We consider simulated data from the Kuramoto-Sivashinsky and complex Ginzburg-Landau equations. Relative to the data-only approach, for scenarios of abundant data, scarce data, and even an incorrect FOM (i.e. erroneous parameter values), the hybrid approach yields substantially improved time-series predictions.

cs.LG↗

arXiv2024

Red blood cell partitioning and segregation through vascular bifurcations in a model of sickle cell disease

The impact of cell segregation and margination in blood disorders on microcirculatory hemodynamics within bifurcated vessels are physiologically significant, yet poorly understood. This study presents a comprehensive computational investigation of red blood cell (RBC) suspension dynamics, with a focus on a model of sickle cell disease (SCD) as an example of a disorder associated with subpopulations of aberrant RBCs. The findings reveal how cell margination influences cellular partitioning and distributions as well as vessel wall shear stress (WSS) at vascular bifurcations. Normal RBCs, which migrate toward the channel center, exhibit the Zweifach-Fung effect, preferentially entering high-flow-rate branches. In contrast, sickle cells, which marginate near the vessel wall, demonstrate an anti-Zweifach-Fung effect, favoring lower-flow-rate branches due to their position within the cell-free layer (CFL). The upstream segregation of cells remains downstream through the bifurcation, where sickle cells accumulate along the outer branch walls. This accumulation of sickle cells increases the frequency of high WSS events via direct physical interactions, particularly on the outer side of high-velocity branches, potentially contributing to the vascular damage and endothelial disruption observed in many disorders that affect RBCs. In geometrically asymmetric bifurcations, cells preferentially enter branches with larger radii, underscoring the influence of geometric complexity on microcirculatory blood flow. These findings provide insights into microvascular hemodynamics in SCD and other blood disorders.

physics.bio-ph↗

arXiv2024

Microcirculatory blood flow with aberrant levels of red blood cell aggregation

Recent clinical results indicate that aberrant erythrocyte aggregation in hematological disorders is accompanied by endothelial damage and glycocalyx disruption, but the underlying biophysical mechanisms remain unclear. This study uses direct computational modeling to explore how red blood cell (RBC) aggregation impacts shear stress in small blood vessels, highlighting the increased risk of vascular damage. RBC aggregation creates a heterogeneous distribution, leading to variations in the cell-free layer thickness and fluctuating wall shear stress, especially near vessel walls. This effect aligns with experimental findings on endothelial disruption linked to RBC clustering near the wall, potentially reducing the protective glycocalyx layer. The power spectral density analysis of wall shear stress fluctuations reveals that, with RBC aggregation, there is a distinct peak near frequency f = 0.04, indicating increased fluctuations due to aggregated RBC clusters traveling close to the vessel wall. The presence of aberrant cells in blood disorders, modeled here by sickle cells, further amplifies these effects, as aggregation-enhanced margination drives sickle cells closer to vessel walls, exacerbating shear stress fluctuations and increasing the likelihood of vascular injury and inflammation. Simulations show that curved vascular geometry, with curvature accentuating RBC clustering near vessel walls, intensifies aggregation-induced wall shear stress fluctuations and increases the risk of vascular damage, particularly in sickle cell disease where sickle cells marginate closer to the wall.

physics.flu-dyn↗

arXiv2024

On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions

This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time-evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary learning (EDMD-DL), when combined with a state space projection, is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We highlight how this projection step introduces nonlinearity into the evolution equations, enabling significantly improved EDMD-DL predictions. With this projection, EDMD-DL leads to a nonlinear dynamical system on the state space, which can be represented in either discrete or continuous time. This system has a natural structure for neural networks, where the state is first expanded into a high dimensional feature space followed by a linear mapping which represents the discrete-time map or the vector field as a linear combination of these features. Inspired by these observations, we implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow, showing comparable performance across methods in terms of short-time trajectory prediction, reconstruction of long-time statistics, and prediction of rare events. These results highlight the equivalence of the EDMD-DL implementation with a state space projection to a neural ODE representation of the dynamics. We also show that these methods provide comparable performance to a non-Markovian approach in terms of prediction of extreme events.

nlin.CD↗

arXiv2024

Elastoinertial turbulence: Data-driven reduced-order model based on manifold dynamics

Elastoinertial turbulence (EIT) is a chaotic state that emerges in the flows of dilute polymer solutions. Direct numerical simulation (DNS) of EIT is highly computationally expensive due to the need to resolve the multi-scale nature of the system. While DNS of 2D EIT typically requires $O(10^6)$ degrees of freedom, we demonstrate here that a data-driven modeling framework allows for the construction of an accurate model with 50 degrees of freedom. We achieve a low-dimensional representation of the full state by first applying a viscoelastic variant of proper orthogonal decomposition to DNS results, and then using an autoencoder. The dynamics of this low-dimensional representation are learned using the neural ODE method, which approximates the vector field for the reduced dynamics as a neural network. The resulting low-dimensional data-driven model effectively captures short-time dynamics over the span of one correlation time, as well as long-time dynamics, particularly the self-similar, nested traveling wave structure of 2D EIT in the parameter range considered.

physics.flu-dyn↗

arXiv2024

Data-driven prediction of large-scale spatiotemporal chaos with distributed low-dimensional models

Complex chaotic dynamics, seen in natural and industrial systems like turbulent flows and weather patterns, often span vast spatial domains with interactions across scales. Accurately capturing these features requires a high-dimensional state space to resolve all the time and spatial scales. For dissipative systems the dynamics lie on a finite-dimensional manifold with fewer degrees of freedom. Thus, by building reduced-order data-driven models in manifold coordinates, we can capture the essential behavior of chaotic systems. Unfortunately, these tend to be formulated globally rendering them less effective for large spatial systems. In this context, we present a data-driven low-dimensional modeling approach to tackle the complexities of chaotic motion, Markovian dynamics, multi-scale behavior, and high numbers of degrees of freedom within large spatial domains. Our methodology involves a parallel scheme of decomposing a spatially extended system into a sequence of local `patches', and constructing a set of coupled, local low-dimensional dynamical models for each patch. Here, we choose to construct the set of local models using autoencoders (for constructing the low-dimensional representation) and neural ordinary differential equations, NODE, for learning the evolution equation. Each patch, or local model, shares the same underlying functions (e.g., autoencoders and NODEs) due to the spatial homogeneity of the underlying systems we consider. We apply this method to the Kuramoto-Sivashinsky equation and 2D Kolmogorov flow, and reduce state dimension by up to two orders of magnitude while accurately capturing both short-term dynamics and long-term statistics.

nlin.CD↗

arXiv2024

Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow

The simulation of turbulent flow requires many degrees of freedom to resolve all the relevant times and length scales. However, due to the dissipative nature of the Navier-Stokes equations, the long-term dynamics are expected to lie on a finite-dimensional invariant manifold with fewer degrees of freedom. In this study, we build low-dimensional data-driven models of pressure-driven flow through a circular pipe. We impose the `shift-and-reflect' symmetry to study the system in a minimal computational cell (e.g., smallest domain size that sustains turbulence) at a Reynolds number of 2500. We build these models by using autoencoders to parametrize the manifold coordinates and neural ODEs to describe their time evolution. Direct numerical simulations (DNS) typically require on the order of O(105) degrees of freedom, while our data-driven framework enables the construction of models with fewer than 20 degrees of freedom. Remarkably, these reduced order models effectively capture crucial features of the flow, including the streak breakdown. In short-time tracking, these models accurately track the true trajectory for one Lyapunov time, while at long-times, they successfully capture key aspects of the dynamics such as Reynolds stresses and energy balance. Additionally, we report a library of exact coherent states (ECS) found in the DNS with the aid of these low-dimensional models. This approach leads to the discovery of seventeen previously unknown solutions within the turbulent pipe flow system, notably featuring relative periodic orbits characterized by the longest reported periods for such flow conditions.

physics.flu-dyn↗

arXiv2024

Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows

The Koopman operator enables the analysis of nonlinear dynamical systems through a linear perspective by describing time evolution in the infinite-dimensional space of observables. Here this formalism is applied to shear flows, specifically the minimal flow unit plane Couette flow and a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force (Moehlis et al., New J. Phys. 6, 56 (2004)). We identify a finite set of observables and approximate the Koopman operator using neural networks, following the method developed by Constante-Amores et al., Chaos, 34(4), 043119 (2024). Then the time evolution is determined with a method here denoted as "Projected Koopman Dynamics". Using a high-dimensional approximate Koopman operator directly for the plane Couette system is computationally infeasible due to the high state space dimension of direct numerical simulations (DNS). However, the long-term dynamics of the dissipative Navier-Stokes equations are expected to live on a lower-dimensional invariant manifold. Motivated by this fact, we perform dimension reduction on the DNS data with a variant of the IRMAE-WD (implicit rank-minimizing autoencoder-weight decay) autoencoder (Zeng et al., MLST 5(2), 025053 (2024)) before applying the Koopman approach. We compare the performance of data-driven state space (neural ordinary differential equation) and Projected Koopman Dynamics approaches in manifold coordinates for plane Couette flow. Projected Koopman Dynamics effectively captures the chaotic behavior of Couette flow and the nine-dimensional model, excelling in both short-term tracking and long-term statistical analysis, outperforming other frameworks applied to these shear flows.

physics.flu-dyn↗

arXiv2024

Data-driven low-dimensional model of a sedimenting flexible fiber

The dynamics of flexible filaments entrained in flow, important for understanding many biological and industrial processes, are computationally expensive to model with full-physics simulations. This work describes a data-driven technique to create high-fidelity low-dimensional models of flexible fiber dynamics using machine learning; the technique is applied to sedimentation in a quiescent, viscous Newtonian fluid, using results from detailed simulations as the data set. The approach combines an autoencoder neural network architecture to learn a low-dimensional latent representation of the filament shape, with a neural ODE that learns the evolution of the particle in the latent state. The model was designed to model filaments of varying flexibility, characterized by an elasto-gravitational number $\mathcal{B}$, and was trained on a data set containing the evolution of fibers beginning at set angles of inclination. For the range of $\mathcal{B}$ considered here (100-10000), the filament shape dynamics can be represented with high accuracy with only four degrees of freedom, in contrast to the 93 present in the original bead-spring model used to generate the dynamic trajectories. We predict the evolution of fibers set at arbitrary angles and demonstrate that our data-driven model can accurately forecast the evolution of a fiber at both trained and untrained elasto-gravitational numbers.

physics.flu-dyn↗

arXiv2024

Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches

Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a 'pure' Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step -- this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state space approach (neural ODEs). We consider systems encompassing two and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state space approach offers superior performance compared to the 'pure' Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state space approach.

cs.LG↗

arXiv2024

Nested traveling wave structures in elastoinertial turbulence

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyze simulations of two dimensional EIT in channel flow using Spectral Proper Orthogonal Decomposition (SPOD), discovering a family of traveling wave structures that capture the sheetlike stress fluctuations that characterize EIT. The frequency dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well defined traveling structures. The structure of the dominant traveling mode exhibits shift reflect symmetry similar to the viscoelasticity modified Tollmien Schlichting (TS) wave, where the velocity fluctuation in the traveling mode is characterized by large scale regular structures spanning the channel and the polymer stress field is characterized by thin, inclined sheets of high polymer stress localized at the critical layers near the channel walls. The traveling structures corresponding to the higher frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next lower frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the "walls" for the structure of mode $\kappa+1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self similarity between the structures. The physical idea behind this theory is that the sheetlike localized stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.

physics.flu-dyn↗

arXiv2023

Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow

Data-driven reduced-order models of the dynamics of complex flows are important for tasks related to design, understanding, prediction, and control. Many flows obey symmetries, and the present work illustrates how these can be exploited to yield highly efficient low-dimensional data-driven models for chaotic flows. In particular, incorporating symmetries both guarantees that the reduced order model automatically respects them and dramatically increases the effective density of data sampling. Given data for the long-time dynamics of a system, and knowing the set of continuous and discrete symmetries it obeys, the first step in the methodology is to identify a "fundamental chart", a region in the state space of the flow to which all other regions can be mapped by a symmetry operation, and a set of criteria indicating what mapping takes each point in state space into that chart. We then find a low-dimensional coordinate representation of the data in the fundamental chart with the use of an autoencoder architecture that also provides an estimate of the dimension of the invariant manifold where data lie. Finally, we learn dynamics on this manifold with the use of neural ordinary differential equations. We apply this method, denoted "symmetry charting" to simulation data from two-dimensional Kolmogorov flow in a chaotic bursting regime. This system has a continuous translation symmetry, and discrete rotation and shift-reflect symmetries. With this framework we observe that less data is needed to learn accurate data-driven models, more robust estimates of the manifold dimension are obtained, equivariance of the NSE is satisfied, better short-time tracking with respect to the true data is observed, and long-time statistics are correctly captured.

cs.LG↗

arXiv2023

Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and $L_2$ regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework's ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a "collective weight variable" incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.

cs.LG↗

arXiv2023

Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds

The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolve towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function space approach (approximating the Koopman operator) and a state space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require O(103) and O(105) degrees of freedom respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state space and function space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.

physics.flu-dyn↗

arXiv2023

Flow instabilities in circular Couette flow of wormlike micelle solutions with a reentrant flow curve

In this work, we numerically investigate flow instabilities of inertialess circular Couette flow of dilute wormlike micelle solutions. Using the reformulated reactive rod model (RRM-R) [Hommel and Graham, JNNFM 295 (2021) 104606], which treats micelles as rigid Brownian rods undergoing reversible scission and fusion in flow, we study the development and behavior of both vorticity banding and finger-like instabilities. In particular, we focus on solutions that exhibit reentrant constitutive curves, in which there exists some region where the shear stress, $τ$, has a multivalued relation to shear rate, $\dotγ$. We find that the radial dependence of the shear stress in circular Couette flow allows for solutions in which parts of the domain lie in the region of the flow curve where $\partial τ/\partial \dotγ > 0$, while others lie in the region where $\partial τ/\partial \dotγ < 0$; this mixed behavior can lead to complex flow instabilities that manifest as finger-like structures of elongated and anisotropically-oriented micelles. In 3D simulations we find that the initial instability is 2D in origin, and 3D finger-like structures arise through the axial instability of 2D sheets. Finally, we show that the RRM-R can capture vorticity banding in narrow-gap circular Couette flow and that vorticity bands are linearly stable to perturbations.

physics.flu-dyn↗