arXiv subjects

Michael D. Graham

Publications and source records attributed to Michael D. Graham.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2023

Data-driven low-dimensional dynamic model of Kolmogorov flow

Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flows processes and is challenging to model. The trajectory of the flow travels near relative periodic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32x32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. Further small improvements in the results occur at a dimension of nine. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.

cs.LG↗

arXiv2023

Self-similar, spatially localized structures in turbulent pipe flow from a data-driven wavelet decomposition

This study aims to extract and characterize structures in fully developed pipe flow at a friction Reynolds number of $\text{Re}_τ= 12\,400$. To do so, we employ data-driven wavelet decomposition (DDWD) [D.~Floryan and M.~D.~Graham, PNAS 118, e2021299118 (2021)], a method that combines features of proper orthogonal decomposition and wavelet analysis in order to extract energetic and spatially localized structures from data. We apply DDWD to streamwise velocity signals measured separately via a thermal anemometer at 40 wall-normal positions. The resulting localized velocity structures, which we interpret as being reflective of underlying eddies, are self-similar across streamwise extents of 40 wall units to one pipe radius, and across wall-normal positions from $y^+=350$ to $y/R=1$. Notably, the structures are similar in shape to Meyer wavelets. Projections of the data onto the DDWD wavelet subspaces are found to be self-similar as well, but in Fourier space; the bounds of self-similarity are the same as before, except streamwise self-similarity starts at a larger length scale of $450$ wall units. The evidence of self-similarity provided in this study lends further support to Townsend's attached eddy hypothesis, although we note that the self-similar structures are detected beyond the log layer and extend to large length scales.

physics.flu-dyn↗

arXiv2023

Effect of polymer additives on dynamics of water level in an open channel

The presence of a tiny amount of polymers (a few parts per million) in a fluid dramatically reduces turbulent drag. For this reason, polymer additives have been proposed to be used in flood remediation: in an open channel at a fixed flow rate, the decrease in friction due to polymer addition is expected to lead to a decrease in water height in the channel. However, in a recent field experiment, a counterintuitive transient increase in water height has been observed far downstream of polymer injection. We numerically investigate the effect of polymer additives on the water height in a long canal using the shallow water equations augmented with an evolution equation for polymer concentration that incorporates turbulent dispersion and polymer degradation. Just downstream of polymer injection, the water height decreases due to the decreased friction at a fixed volumetric flow rate. Further downstream, however, the height can increase, consistent with the experimental observation. We elucidate the mechanism of this unexpected rise in water height in the canal. We also suggest a technique to mitigate the water rise, as this is detrimental for practical applications.

physics.flu-dyn↗

arXiv2023

Predicting extreme events in a data-driven model of turbulent shear flow using an atlas of charts

Dynamical systems with extreme events are difficult to capture with data-driven modeling, due to the relative scarcity of data within extreme events compared to the typical dynamics of the system, and the strong dependence of the long-time occurrence of extreme events on short-time conditions.A recently developed technique [Floryan, D. & Graham, M. D. Data-driven discovery of intrinsic dynamics. Nat Mach Intell $\textbf{4}$, 1113-1120 (2022)], here denoted as $\textit{Charts and Atlases for Nonlinear Data-Driven Dynamics on Manifolds}$, or CANDyMan, overcomes these difficulties by decomposing the time series into separate charts based on data similarity, learning dynamical models on each chart via individual time-mapping neural networks, then stitching the charts together to create a single atlas to yield a global dynamical model. We apply CANDyMan to a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force [Moehlis, J., Faisst, H. & Eckhardt, B. A low-dimensional model for turbulent shear flows. New J Phys $\textbf{6}$, 56 (2004)], which undergoes extreme events in the form of intermittent quasi-laminarization and long-time full laminarization. We demonstrate that the CANDyMan method allows the trained dynamical models to more accurately forecast the evolution of the model coefficients, reducing the error in the predictions as the model evolves forward in time. The technique exhibits more accurate predictions of extreme events, capturing the frequency of quasi-laminarization events and predicting the time until full laminarization more accurately than a single neural network.

physics.flu-dyn↗

arXiv2023

Scattering-Informed Microstructure Prediction during Lagrangian Evolution (SIMPLE) -- A data-driven framework for modeling complex fluids in flow

An overarching challenge in rheology is to develop constitutive models for complex fluids for which we lack accurate first principles theory. A further challenge is that most experiments probing dynamical structure and rheology do so only in very simple flow fields that are not characteristic of the complex deformation histories experienced by material in a processing application. A recently-developed experimental methodology holds potential to overcome this challenge by incorporating a fluidic four-roll mill (FFoRM) into scanning small-angle x-ray scattering instrumentation (sSAXS) [Corona, P. T. et al. Sci. Rep. 8, 15559 (2018); Corona, P. T. et al. Phys. Rev. Mater 6, 045603 (2022)] to rapidly generate large data sets of scattering intensity for complex fluids along diverse Lagrangian flow histories. To exploit this uniquely rich FFoRM-sSAXS data, we propose a machine learning framework, Scattering-Informed Microstructure Prediction under Lagrangian Evolution (SIMPLE), which uses FFoRM-sSAXS data to learn an evolution equation for the scattering intensity and an associated tensorial differential constitutive equation for the stress. The framework incorporates material frame indifference and invariance to arbitrary rotations by data preprocessing. We use autoencoders to find an efficient reduced order model for the scattering intensity and neural network ordinary differential equations to predict the time evolution of the model coordinates. The framework is validated on a synthetic FFoRM-sSAXS data set for a dilute rigid rod suspension. The model accurately predicts microstructural evolution and rheology for flows that differ significantly from those used in training. SIMPLE is compatible with but does not require material-specific constraints or assumptions.

physics.flu-dyn↗

arXiv2023

Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learning

The high dimensionality and complex dynamics of turbulent flows remain an obstacle to the discovery and implementation of control strategies. Deep reinforcement learning (RL) is a promising avenue for overcoming these obstacles, but requires a training phase in which the RL agent iteratively interacts with the flow environment to learn a control policy, which can be prohibitively expensive when the environment involves slow experiments or large-scale simulations. We overcome this challenge using a framework we call "DManD-RL" (data-driven manifold dynamics-RL), which generates a data-driven low-dimensional model of our system that we use for RL training. With this approach, we seek to minimize drag in a direct numerical simulation (DNS) of a turbulent minimal flow unit of plane Couette flow at Re=400 using two slot jets on one wall. We obtain, from DNS data with $\mathcal{O}(10^5)$ degrees of freedom, a 25-dimensional DManD model of the dynamics by combining an autoencoder and neural ordinary differential equation. Using this model as the environment, we train an RL control agent, yielding a 440-fold speedup over training on the DNS, with equivalent control performance. The agent learns a policy that laminarizes 84% of unseen DNS test trajectories within 900 time units, significantly outperforming classical opposition control (58%), despite the actuation authority being much more restricted. The agent often achieves laminarization through a counterintuitive strategy that drives the formation of two low-speed streaks, with a spanwise wavelength that is too small to be self-sustaining. The agent demonstrates the same performance when we limit observations to wall shear rate.

physics.flu-dyn↗

arXiv2023

Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow

Because the Navier-Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterizing the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modeling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $\textit{Re}=400$, where a fully resolved solutions requires $\mathcal{O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short-times, the models track the true trajectory for multiple Lyapunov times, and, at long-times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; additionally, we find nine orbits that correspond to previously unknown solutions in the full system.

physics.flu-dyn↗

arXiv2022

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data

A common problem in time series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens theorem proves a time delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.

cs.LG↗

arXiv2022

Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems

In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or chaotic dynamics. We present a data-driven modeling method that accurately captures shocks and chaotic dynamics by proposing a novel architecture, stabilized neural ordinary differential equation (ODE). In our proposed architecture, we learn the right-hand-side (RHS) of an ODE by adding the outputs of two NN together where one learns a linear term and the other a nonlinear term. Specifically, we implement this by training a sparse linear convolutional NN to learn the linear term and a dense fully-connected nonlinear NN to learn the nonlinear term. This is in contrast with the standard neural ODE which involves training only a single NN for learning the RHS. We apply this setup to the viscous Burgers equation, which exhibits shocked behavior, and show better short-time tracking and prediction of the energy spectrum at high wavenumbers than a standard neural ODE. We also find that the stabilized neural ODE models are much more robust to noisy initial conditions than the standard neural ODE approach. We also apply this method to chaotic trajectories of the Kuramoto-Sivashinsky equation. In this case, stabilized neural ODEs keep long-time trajectories on the attractor, and are highly robust to noisy initial conditions, while standard neural ODEs fail at achieving either of these results. We conclude by demonstrating how stabilizing neural ODEs provide a natural extension for use in reduced-order modeling by projecting the dynamics onto the eigenvectors of the learned linear term.

cs.LG↗

arXiv2022

Data-driven discovery of intrinsic dynamics

Dynamical models underpin our ability to understand and predict the behavior of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and in the data-driven case the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behavior of the system. In truth, these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study, we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system's intrinsic state variables directly from time series data, and also learns predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together -- a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behavior. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system.

cs.LG↗

arXiv2022

Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning

Deep reinforcement learning (RL) is a data-driven method capable of discovering complex control strategies for high-dimensional systems, making it promising for flow control applications. In particular, the present work is motivated by the goal of reducing energy dissipation in turbulent flows, and the example considered is the spatiotemporally chaotic dynamics of the Kuramoto-Sivashinsky equation (KSE). A major challenge associated with RL is that substantial training data must be generated by repeatedly interacting with the target system, making it costly when the system is computationally or experimentally expensive. We mitigate this challenge in a data-driven manner by combining dimensionality reduction via an autoencoder with a neural ODE framework to obtain a low-dimensional dynamical model from just a limited data set. We substitute this data-driven reduced-order model (ROM) in place of the true system during RL training to efficiently estimate the optimal policy, which can then be deployed on the true system. For the KSE actuated with localized forcing ("jets") at four locations, we demonstrate that we are able to learn a ROM that accurately captures the actuated dynamics as well as the underlying natural dynamics just from snapshots of the KSE experiencing random actuations. Using this ROM and a control objective of minimizing dissipation and power cost, we extract a control policy from it using deep RL. We show that the ROM-based control strategy translates well to the true KSE and highlight that the RL agent discovers and stabilizes an underlying forced equilibrium solution of the KSE system. We show that this forced equilibrium captured in the ROM and discovered through RL is related to an existing known equilibrium solution of the natural KSE.

cs.LG↗

arXiv2022

Wrinkling and multiplicity in the dynamics of deformable sheets in uniaxial extensional flow

The processing of thin-structured materials in a fluidic environment, from nearly inextensible but flexible graphene sheets to highly extensible polymer films, arises in many applications. So far, little is known about the dynamics of such thin sheets freely suspended in fluid. In this work, we study the dynamics of freely suspended soft sheets in uniaxial extensional flow. Elastic sheets are modeled with a continuum model that accounts for in-plane deformation and out-of-plane bending, and the fluid motion is computed using the method of regularized Stokeslets. We explore two types of sheets: "stiff" sheets that strongly resist bending deformations and always stay flat, and "flexible" sheets with both in-plane and out-of plane deformability that can wrinkle. For stiff sheets, we observe a coil-stretch-like transition, similar to what has been observed for long-chain linear polymers under extension as well as elastic sheets under planar extension: in a certain range of capillary number (flow strength relative to in-plane deformability), the sheets exhibit either a compact or a highly stretched conformation, depending on deformation history. For flexible sheets, sheets with sufficiently small bending stiffness wrinkle to form various conformations. Here, the compact-stretched bistability still occurs, but is strongly modified by the wrinkling instability: a highly-stretched planar state can become unstable and wrinkle, after which it may dramatically shrink in length due to hydrodynamic screening associated with wrinkling. Therefore, wrinkling renders a shift in the bistability regime. In addition, we can predict and understand the nonlinear long-term dynamics for some parameter regimes with linear stability analysis of the flat steady states.

cond-mat.soft↗

arXiv2021

Constitutive modeling of dilute wormlike micelle solutions: shear-induced structure and transient dynamics

We present a reformulation of the 'reactive rod model' (RRM) of Dutta and Graham [Dutta, Sarit and Graham, Michael D., JNNFM 251 (2018)], a constitutive model for describing the behavior of dilute wormlike micelle solutions. The RRM treats wormlike micelle solutions as dilute suspensions of rigid Brownian rods undergoing reversible scission and growth in flow. Evolution equations for micelle orientation and stress contribution are coupled to a kinetic reaction equation for a collective micelle length, producing dynamic variations in the length and rotational diffusivity of the rods. This model has previously shown success in capturing many critical steady-state rheological features of dilute wormlike micelle solutions, particularly shear-thickening and -thinning, non-zero normal stress differences, and a reentrant shear stress-shear rate curve, and could fit a variety of steady state experimental data. The present work improves on this framework, which showed difficulty in capturing transient dynamics and high-shear behavior, by reformulating the kinetic equation for micelle growth on a more microstructural (though still highly idealized) basis. In particular, we allow for micelle growth associated with strong alignment of rods and breakage due to tensile stresses along the micelles. This new formulation captures both steady and transient shear rheology in good agreement with experiments. We also find good agreement with available steady state extensional rheology.

cond-mat.soft↗

arXiv2021

Tollmien-Schlichting route to elastoinertial turbulence in channel flow

Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity [Shekar et al., J.Fluid Mech. 893, A3 (2020)]. Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS attractor and to elastoinertial turbulence (EIT). At Reynolds number $Re=3000$, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch. At fixed Weissenberg number, $Wi$ and increasing Reynolds number from 3000-10000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. We show that this transition is directly tied to the strength of the TS critical layer fluctuations and can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation point of the Kelvin cat's eye structure of the TS wave exceeds $\frac{1}{2}$. At $Re=10000$, unlike $3000$, the Newtonian TS attractor evolves continuously into the EIT state as $Wi$ is increased from zero to about $13$. We describe how the structure of the flow and stress fields changes, highlighting in particular a "sheet-shedding" process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.

physics.flu-dyn↗

arXiv2021

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -- a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ~0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.

cs.LG↗

arXiv2021

Perspectives on viscoelastic flow instabilities and elastic turbulence

Viscoelastic fluids are a common subclass of rheologically complex materials that are encountered in diverse fields from biology to polymer processing. Often the flows of viscoelastic fluids are unstable in situations where ordinary Newtonian fluids are stable, owing to the nonlinear coupling of the elastic and viscous stresses. Perhaps more surprisingly, the instabilities produce flows with the hallmarks of turbulence -- even though the effective Reynolds numbers may be $O(1)$ or smaller. We provide perspectives on viscoelastic flow instabilities by integrating the input from speakers at a recent international workshop: historical remarks, characterization of fluids and flows, discussion of experimental and simulation tools, and modern questions and puzzles that motivate further studies of this fascinating subject. The materials here will be useful for researchers and educators alike, especially as the subject continues to evolve in both fundamental understanding and applications in engineering and the sciences.

physics.flu-dyn↗

arXiv2021

Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics

Deep reinforcement learning (RL) is a data-driven, model-free method capable of discovering complex control strategies for macroscopic objectives in high-dimensional systems, making its application towards flow control promising. Many systems of flow control interest possess symmetries that, when neglected, can significantly inhibit the learning and performance of a naive deep RL approach. Using a test-bed consisting of the Kuramoto-Sivashinsky Equation (KSE), equally spaced actuators, and a goal of minimizing dissipation and power cost, we demonstrate that by moving the deep RL problem to a symmetry-reduced space, we can alleviate limitations inherent in the naive application of deep RL. We demonstrate that symmetry-reduced deep RL yields improved data efficiency as well as improved control policy efficacy compared to policies found by naive deep RL. Interestingly, the policy learned by the the symmetry aware control agent drives the system toward an equilibrium state of the forced KSE that is connected by continuation to an equilibrium of the unforced KSE, despite having been given no explicit information regarding its existence. I.e., to achieve its goal, the RL algorithm discovers and stabilizes an equilibrium state of the system. Finally, we demonstrate that the symmetry-reduced control policy is robust to observation and actuation signal noise, as well as to system parameters it has not observed before.

cs.LG↗