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In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadiness to the transition to turbulence, is of critical importance for both academic and industrial applications. Different tools from dynamical systems theory can be used for this purpose. In this review, we present a concise theoretical and numerical framework focusing on practical aspects of the computation and stability analyses of steady and time-periodic solutions, with emphasis on high-dimensional systems such as those arising from the spatial discretization of the Navier-Stokes equations. Using a matrix-free approach based on Krylov methods, we extend the capabilities of the open-source high-performance spectral element-based time-stepper Nek5000. The numerical methods discussed are implemented in nekStab, an open-source and user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex three-dimensional geometries. The performance and accuracy of the methods are illustrated and examined using standard benchmarks from the fluid mechanics literature. Thanks to its flexibility and domain-agnostic nature, the methodology presented in this work can be applied to develop similar toolboxes for other solvers, most importantly outside the field of fluid mechanics.

Many unsteady flows exhibiting complex dynamics are nevertheless characterized by emergent large-scale coherence in space and time. Reduced-order models based on Galerkin projection of the governing equations onto an orthogonal modal basis approximate the flow as a low-dimensional dynamical system with linear and quadratic terms. However, these Galerkin models often fail to reproduce the true dynamics, in part because they ignore important nonlinear interactions with unresolved flow scales. Here, we use a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart-Landau equation. The leading order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modeling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows, Comment: 46 pages, 14 figures

Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.

A major goal for reduced-order models of unsteady fluid flows is to uncover and exploit latent low-dimensional structure. Proper orthogonal decomposition (POD) provides an energy-optimal linear basis to represent the flow kinematics, but converges slowly for advection-dominated flows and tends to overestimate the number of dynamically relevant variables. We show that nonlinear correlations in the temporal POD coefficients can be exploited to identify the underlying attractor, characterized by a minimal set of driving modes and a manifold equation for the remaining modes. By viewing these nonlinear correlations as an invariant manifold reduction, this least-order representation can be used to stabilize POD-Galerkin models or as a state space for data-driven model identification. In the latter case, we use sparse polynomial regression to learn a compact, interpretable dynamical system model from the time series of the active modal coefficients. We demonstrate this perspective on a quasiperiodic shear-driven cavity flow and show that the dynamics evolve on a torus generated by two independent Stuart-Landau oscillators. These results emphasize the importance of nonlinear dimensionality reduction to reveal underlying structure in complex flows., Comment: 33 pages, 14 figures

This work develops a low-dimensional nonlinear stochastic model of symmetry-breaking coherent structures from experimental measurements of a turbulent axisymmetric bluff body wake. Traditional model reduction methods decompose the field into a set of modes with fixed spatial support but time-varying amplitudes. However, this fixed basis cannot resolve the mean flow deformation due to variable Reynolds stresses, a central feature of Stuart's nonlinear stability mechanism, without the further assumption of weakly nonlinear interactions. Here, we introduce a parametric modal basis that depends on the instantaneous value of the unsteady aerodynamic center of pressure, which quantifies the degree to which the rotational symmetry of the wake is broken. Thus, the modes naturally interpolate between the unstable symmetric state and the nonlinear equilibrium. We estimate the modes from experimental measurements of the base pressure distribution by reducing the symmetry via phase alignment and averaging conditioned on the center of pressure. The amplitude dependence of the symmetric mode deviates significantly from the polynomial scaling predicted by weakly nonlinear analysis, confirming that the parametric basis is crucial for capturing the effect of strongly nonlinear interactions. We also introduce a second model term capturing axisymmetric fluctuations associated with the mean-field deformation. We then apply the Langevin regression system identification method to construct a stochastically forced nonlinear model for these two generalized mode coefficients. The resulting model reproduces empirical power spectra and probability distributions, suggesting a path towards developing interpretable low-dimensional models of globally unstable turbulent flows from experimental measurements., Comment: 23 pages, 10 figures

Understanding how the brain responds to sensory inputs is challenging: brain recordings are partial, noisy, and high dimensional; they vary across sessions and subjects and they capture highly nonlinear dynamics. These challenges have led the community to develop a variety of preprocessing and analytical (almost exclusively linear) methods, each designed to tackle one of these issues. Instead, we propose to address these challenges through a specific end-to-end deep learning architecture, trained to predict the brain responses of multiple subjects at once. We successfully test this approach on a large cohort of magnetoencephalography (MEG) recordings acquired during a one-hour reading task. Our Deep Recurrent Encoding (DRE) architecture reliably predicts MEG responses to words with a three-fold improvement over classic linear methods. To overcome the notorious issue of interpretability of deep learning, we describe a simple variable importance analysis. When applied to DRE, this method recovers the expected evoked responses to word length and word frequency. The quantitative improvement of the present deep learning approach paves the way to better understand the nonlinear dynamics of brain activity from large datasets.

Many physical systems characterized by nonlinear multiscale interactions can be effectively modeled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behavior are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behavior to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using both forward and adjoint Fokker-Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can effectively learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by colored noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply the proposed method to experimental measurements of a turbulent bluff body wake and show that the statistical behavior of the center of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise., Comment: 30 pages, 13 figures

PySINDy is a Python package for the discovery of governing dynamical systems models from data. In particular, PySINDy provides tools for applying the sparse identification of nonlinear dynamics (SINDy) (Brunton et al. 2016) approach to model discovery. In this work we provide a brief description of the mathematical underpinnings of SINDy, an overview and demonstration of the features implemented in PySINDy (with code examples), practical advice for users, and a list of potential extensions to PySINDy. Software is available at https://github.com/dynamicslab/pysindy.

dentifying accurate and yet interpretable low-order models from data has gained a renewed interest over the past decade. In the present work, we illustrate how the combined use of dimensionality reduction and sparse system identification techniques allows us to obtain an accurate model of the chaotic thermal convection in a two-dimensional annular thermosyphon. Taking as guidelines the derivation of the Lorenz system, the chaotic thermal convection dynamics simulated using a high-fidelity computational fluid dynamics solver are first embedded into a low-dimensional space using dynamic mode decomposition. After having reviewed the physical properties the reduced-order model should exhibit, the latter is identified using SINDy, an increasingly popular and flexible framework for the identification of nonlinear continuous-time dynamical systems from data. The identified model closely resembles the canonical Lorenz system, having a similar structure and exhibiting the same physical properties. Finally, extensions to other flow configurations with or without control are discussed.

With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has been become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems., Comment: To appear in "Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics", eds. A. Gelfgat, Springer

We propose a general dynamic reduced-order modeling framework for typical experimental data: time-resolved sensor data and optional non-time-resolved PIV snapshots. This framework contains four steps. First, the sensor signals are lifted to a dynamic feature space. Second, we identify a sparse human-interpretable nonlinear dynamical system for the feature state based on the sparse identification of nonlinear dynamics (SINDy). Third, if PIV snapshots are available, a local linear mapping from the feature state to velocity fields is shown to be orders of magnitudes more accurate than optimal modal expansions of the same order. Fourth, a generalized feature-based modal decomposition identifies coherent structures that are most dynamically correlated with the linear and nonlinear interaction terms in the sparse model, adding interpretability. Steps 1 and 2 define a black-box model. Optional steps 3 and 4 lift the black-box dynamics to a 'gray-box' model of the coherent structures, if non-time-resolved full-state data is available. This gray-box modeling strategy is successfully applied to the transient and post-transient laminar cylinder wake, and compares favorably with a POD model. We foresee numerous applications of this highly flexible modeling strategy, including estimation, prediction and control. Moreover, the feature space may be based on intrinsic coordinates, which are unaffected by a key challenge of modal expansion: the slow change of low-dimensional coherent structures with changing geometry and varying parameters.

The shear-induced deformation of a capsule with a stiff nucleus, a model of eukaryotic cells, is studied numerically. The membrane of the cell and of its nucleus are modelled as a thin and impermeable elastic material obeying a Neo-Hookean constitutive law. The membranes are discretised by a Lagrangian mesh and their governing equations are solved in spectral space using spherical harmonics, while the fluid equations are solved on a staggered grid using a second-order finite differences scheme. The fluid-structure coupling is obtained using an immersed boundary method. The numerical approach is presented and validated for the case of a single capsule in a shear flow. The variations induced by the presence of the nucleus on the cell deformation are investigated when varying the viscosity ratio between the inner and outer fluids, the membrane elasticity and its bending stiffness. The deformation of the eukaryotic cell is smaller than that of the prokaryotic one. The reduction in deformation increases for larger values of the capillary number. The eukaryotic cell remains thicker in its middle part compared to the prokaryotic one, thus making it less flexible to pass through narrow capillaries. For a viscosity ratio of 5, the deformation of the cell is smaller than in the case of uniform viscosity. In addition, for non-zero bending stiffness of the membrane, the deformation decreases and the shape is closer to an ellipsoid. Finally, we compare the results obtained modeling the nucleus as an inner stiffer membrane with those obtained using a rigid particle.

Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a numerical method for two-fluid systems subject to surface tension and depletion forces among the suspended droplets. The algorithm is based on an efficient solver for the incompressible two-phase Navier-Stokes equations, and uses a mass-conserving level set method to capture the fluid interface. The four novel ingredients proposed here are, firstly, an interface-correction level set (ICLS) method; global mass conservation is achieved by performing an additional advection near the interface, with a correction velocity obtained by locally solving an algebraic equation, which is easy to implement in both 2D and 3D. Secondly, we report a second-order accurate geometric estimation of the curvature at the interface and, thirdly, the combination of the ghost fluid method with the fast pressure-correction approach enabling an accurate and fast computation even for large density contrasts. Finally, we derive a hydrodynamic model for the interaction forces induced by depletion of surfactant micelles and combine it with a multiple level set approach to study short-range interactions among droplets in the presence of attracting forces.

Previous studies have shown that intermediate surface tension has a counterintuitive destabilizing effect on 2-phase planar jets. Here, the transition process in confined 2D jets of two fluids with varying viscosity ratio is investigated using DNS. Neutral curves for persistent oscillations are found by recording the norm of the velocity residuals in DNS for over 1000 nondimensional time units, or until the signal has reached a constant level in a logarithmic scale - either a converged steady state, or a "statistically steady" oscillatory state. Oscillatory final states are found for all viscosity ratios (0.1-10). For uniform viscosity (m=1), the first bifurcation is through a surface tension-driven global instability. For low viscosity of the outer fluid, there is a mode competition between a steady asymmetric Coanda-type attachment mode and the surface tension-induced mode. At moderate surface tension, the Coanda-type attachment dominates and eventually triggers time-dependent convective bursts. At high surface tension, the surface tension-dominated mode dominates. For high viscosity of the outer fluid, persistent oscillations appear due to a strong convective instability. Finally, the m=1 jet remains unstable far from the inlet when the shear profile is nearly constant. Comparing this to a parallel Couette flow (without inflection points), we show that in both flows, a hidden interfacial mode brought out by surface tension becomes temporally and absolutely unstable in an intermediate Weber and Reynolds regime. An energy analysis of the Couette setup shows that surface tension, although dissipative, induces a velocity field near the interface which extracts energy from the flow through a viscous mechanism. This study highlights the rich dynamics of immiscible planar uniform-density jets, where several self-sustained and convective mechanisms compete depending on the exact parameters.

Although major advances have been achieved over the past decades for the reduction and identification of linear systems, deriving nonlinear low-order models still is a chal- lenging task. In this work, we develop a new data-driven framework to identify nonlinear reduced-order models of a fluid by combining dimensionality reductions techniques (e.g. proper orthogonal decomposition) and sparse regression techniques from machine learn- ing. In particular, we extend the sparse identification of nonlinear dynamics (SINDy) algorithm to enforce physical constraints in the regression, namely energy-preserving quadratic nonlinearities. The resulting models, hereafter referred to as Galerkin regression models, incorporate many beneficial aspects of Galerkin projection, but without the need for a full-order or high-fidelity solver to project the Navier-Stokes equations. Instead, the most parsimonious nonlinear model is determined that is consistent with observed mea- surement data and satisfies necessary constraints. Galerkin regression models also readily generalize to include higher-order nonlinear terms that model the effect of truncated modes. The effectiveness of Galerkin regression is demonstrated on two different flow configurations: the two-dimensional flow past a circular cylinder and the shear-driven cavity flow. For both cases, the accuracy of the identified models compare favorably against reduced-order models obtained from a standard Galerkin projection procedure. Present results highlight the importance of cubic nonlinearities in the construction of accurate nonlinear low-dimensional approximations of the flow systems, something which cannot be readily obtained using a standard Galerkin projection of the Navier-Stokes equations. Finally, the entire code base for our constrained sparse Galerkin regression algorithm is freely available online.

Transition from steady state to intermittent chaos in the cubical lid-driven flow is investigated numerically. Fully three-dimensional stability analyses have revealed that the flow experiences an Andronov-Poincar\'e-Hopf bifurcation at a critical Reynolds number $Re_c$ = 1914. As for the 2D-periodic lid-driven cavity flows, the unstable mode originates from a centrifugal instability of the primary vortex core. A Reynolds-Orr analysis reveals that the unstable perturbation relies on a combination of the lift-up and anti lift-up mechanisms to extract its energy from the base flow. Once linearly unstable, direct numerical simulations show that the flow is driven toward a primary limit cycle before eventually exhibiting intermittent chaotic dynamics. Though only one eigenpair of the linearized Navier-Stokes operator is unstable, the dynamics during the intermittencies are surprisingly well characterized by one of the stable eigenpairs., Comment: Accepted for publication in Fluid Dynamics Research

We introduce a modified and simplified version of the pre-existing fully parallelized three-dimensional Navier--Stokes flow solver known as TPLS. We demonstrate how the simplified version can be used as a pedagogical tool for the study of computational fluid dynamics and parallel computing. TPLS is at its heart a two-phase flow solver, and uses calls to a range of external libraries to accelerate its performance. However, in the present context we narrow the focus of the study to basic hydrodynamics and parallel computing techniques, and the code is therefore simplified and modified to simulate pressure-driven single-phase flow in a channel, using only relatively simple Fortran 90 code with MPI parallelization, but no calls to any other external libraries. The modified code is analysed in order to both validate its accuracy and investigate its scalability up to 1000 CPU cores. Simulations are performed for several benchmark cases in pressure-driven channel flow, including a turbulent simulation, wherein the turbulence is incorporated via the large-eddy simulation technique. The work may be of use to advanced undergraduate and graduate students as an introductory study in computational fluid dynamics, while also providing insight for those interested in more general aspects of high-performance computing., Comment: 22 pages, 9 figures