arXiv subjects

Beverley J. McKeon

Publications and source records attributed to Beverley J. McKeon.

At least 19 recordsLinked to original sources

arXiv2026

Data-driven linear analysis of dynamical systems via nonlinearity-subtracted dynamic mode decomposition

The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method --- nonlinearity-subtracted DMD (NSDMD) --- that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.

math.DS↗

arXiv2026

Data-driven linear analysis of turbulent flows

Mean-flow-based linear analyses of turbulent flows, such as resolvent analysis, provide valuable insight about flow structures and their dynamics that has been widely leveraged to model, control and understand the underlying flow physics. However, these analyses are computationally expensive for flows over complex geometries and require the use of specialized codes that are typically only available in research environments. On the other hand, data-driven modal decompositions, such as the dynamic mode decomposition (DMD), identify turbulent flow structures that, although statistically relevant, do not provide insight into the physical mechanisms driving their dynamics. Here we introduce a novel data-driven method -- nonlinearity-subtracted DMD (NSDMD) -- that leverages knowledge of the structure of the Navier--Stokes equations to ensure that the learned operator is a low-rank approximation of the underlying mean-flow-linearized dynamics. Specifically, the method uses snapshots of the nonlinear terms in the perturbation equations to explicitly account for the contribution of the nonlinear forcing to the dynamics. We demonstrate the use of NSDMD to perform data-driven resolvent analysis on direct numerical simulation (DNS) and large-eddy simulation (LES) datasets, starting with a minimal channel flow and scaling up to the flow over a full aircraft model. As a result, NSDMD allows performing linear analyses of turbulent flows as a post-processing step on simulation data obtained with any available high-fidelity computational fluid dynamics (CFD) code.

physics.flu-dyn↗

arXiv2026

On the Poisson-Source Basis of Logarithmic Wall-Pressure-Variance Growth

In high-Reynolds-number wall-bounded flows, the inner-scaled wall-pressure variance \ra{is often represented as a} logarithmic increase with frictional Reynolds number. We consider the two sources of the incompressible pressure--Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of quadratic velocity fluctuations. This paper establishes a link between the sources and the coefficients in \rtwo{a logarithmic} inner-scaled variance \rtwo{representation}. \rone{To leading order} we \rone{posit that} the \ra{linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient}. The illustrative dataset is direct numerical simulation (DNS) at \rtwo{frictional Reynolds number} $δ^+\approx 550$, although the principal contribution is the establishment of a mechanistic link to well-known high-$δ^+$ scalings of wall-bounded turbulence. Through consideration of the sources and the integral solution method of the Poisson equation, we find that the linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle. \rone{To leading order}, this contribution becomes $δ^+$ invariant under inner scaling, thus contributing an offset in the \rtwo{logarithmic representation}. The interfacial regions between uniform momentum zones characteristic of the inertial layer (vortical fissures) spatially localise strain and vorticity contributions and contain an increasingly large proportion of the strain and vorticity. We show that fissures act as a compact carrier for the source terms, with the nonlinear term especially prominent in these regions. By considering the inertial layer statistics, we link the changing nonlinear contribution to \rtwo{the} $\ln δ^+$ \rtwo{growth}, in agreement with previous empirical observations.

physics.flu-dyn↗

arXiv2026

Two-component inner--outer scaling model for the wall-pressure spectrum at high Reynolds number

Wall-pressure fluctuations beneath turbulent boundary layers drive noise and structural fatigue through interactions between fluid and structural modes. Conventional predictive models for the spectrum--such as the widely accepted Goody model (\textit{AIAA Journal} 42 (9), 2004, 1788--1794)--fail to capture the energetic growth in the {low-frequency range} that occurs at high Reynolds number, while at the same time over-predicting the variance. To address these shortcomings, two semi-empirical models are proposed for the wall-pressure spectrum in canonical turbulent boundary layers, pipes and channels for friction Reynolds numbers $δ^+$ ranging from 180 to 47 000. The models are based on consideration of two {spectral components} that represent the contributions to the wall pressure fluctuations from inner-scale motions and outer-scale motions. The first model expresses the pre-multiplied spectrum as the sum of two overlapping log-normal {components}: an inner-scaled term that is $δ^+$-invariant and an outer-scaled term whose amplitude broadens smoothly with $δ^+$. Calibrated against large-eddy simulations, direct numerical simulations, and recent high-$δ^+$ pipe data, it reproduces the {inner-scaled peak} and the emergence of an outer-scaled peak at large $δ^+$. The second model, developed around newly available pipe data, uses theoretical arguments to prescribe the spectral shapes of the inner and outer {components}. Embedding the $δ^+$-dependence in smooth asymptotic functions yields a formulation that varies continuously with $δ^+$ {and generalises beyond the calibration range}. Both models capture the full spectrum and {recover} the observed logarithmic growth of its variance, {providing a compact, physics-informed empirical representation} for more accurate engineering predictions of wall-pressure fluctuations.

physics.flu-dyn↗

arXiv2025

Spatio-temporal characterization of nonlinear forcing and response in turbulent channel flow

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a nonlinear forcing to the linear resolvent operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. A framework to quantify the triadic contributions to the forcing and response by each pair of triplets is developed and applied to data from direct numerical simulations of a turbulent channel at $Re_τ \approx 550$. The linear resolvent operator is incorporated to provide the missing link from energy transfer between modes to the effect on the spectral turbulent kinetic energy. The coefficients highlight the importance of interactions involving large-scale structures, providing a natural connection to the modeling assumptions in quasi-linear (QL) and generalized quasi-linear (GQL) analyses. Specifically, it is revealed that the QL and GQL reductions efficiently capture important triadic interactions in the flow, especially when including of a small number of wavenumbers into the GQL large-scale base flow. Additionally, spatio-temporal analyses of the triadic contributions to a single mode representative of the near-wall cycle demonstrate the spatio-temporal nature of the triadic interactions and the effect of the resolvent operator, which selectively amplifies certain forcing profiles. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.

physics.flu-dyn↗

arXiv2024

A resolvent-based perspective on the generation of Mach wave radiation from compressible boundary layers

We identify forcing mechanisms that separately amplify subsonic and supersonic features obtained from a linearized Navier-Stokes based model for compressible parallel boundary layers. Resolvent analysis is used to analyse the linear model, where the non-linear terms of the linearized equations act as a forcing to the linear terms. Considering subsonic modes, only the solenoidal component of the forcing to the momentum equations amplify these modes. When considering supersonic modes, we find that these are pressure fluctuations that radiate into the freestream. Within the freestream, these modes closely follow the trends of inviscid Mach waves. There are two distinct forcing mechanisms that amplify the supersonic modes: (i) the 'direct route' where the forcing to the continuity and energy equations and the dilatational component of the forcing to the momentum equations directly force the mode and (ii) the 'indirect route' where the solenoidal component of the forcing to the momentum equations force a response in wall-normal velocity, and this wall-normal velocity in-turn forces the supersonic mode. A majority of the supersonic modes considered are dominantly forced by the direct route. However, when considering Mach waves that are, like in direct numerical simulations, forced from the buffer layer of the flow, the indirect route of forcing becomes significant. We find that these observations are also valid for a streamwise developing boundary layer. These results are consistent with, and extend the observations in the literature regarding the solenoidal and dilatational components of velocity in compressible turbulent wall-bounded flows.

physics.flu-dyn↗

arXiv2024

Estimating Dynamic Flow Features in Groups of Tracked Objects

Interpreting motion captured in image sequences is crucial for a wide range of computer vision applications. Typical estimation approaches include optical flow (OF), which approximates the apparent motion instantaneously in a scene, and multiple object tracking (MOT), which tracks the motion of subjects over time. Often, the motion of objects in a scene is governed by some underlying dynamical system which could be inferred by analyzing the motion of groups of objects. Standard motion analyses, however, are not designed to intuit flow dynamics from trajectory data, making such measurements difficult in practice. The goal of this work is to extend gradient-based dynamical systems analyses to real-world applications characterized by complex, feature-rich image sequences with imperfect tracers. The tracer trajectories are tracked using deep vision networks and gradients are approximated using Lagrangian gradient regression (LGR), a tool designed to estimate spatial gradients from sparse data. From gradients, dynamical features such as regions of coherent rotation and transport barriers are identified. The proposed approach is affordably implemented and enables advanced studies including the motion analysis of two distinct object classes in a single image sequence. Two examples of the method are presented on data sets for which standard gradient-based analyses do not apply.

cs.CV↗

arXiv2023

Lagrangian Gradient Regression for the Detection of Coherent Structures from Sparse Trajectory Data

Lagrangian Coherent Structures (LCS) are flow features which are defined to objectively characterize complex fluid behavior over a finite time regardless of the orientation of the observer. Fluidic applications of LCS include geophysical, aerodynamic, biological, and bio-inspired flows -- among others -- and can be generalized to broader classes of dynamical systems. One of the prevailing paradigms for identifying LCS involves examining continuum-mechanical properties of the underlying flow. Such methods, including finite-time Lyapunov exponent (FTLE) and Lagrangian-averaged vorticity deviation (LAVD) analyses, provide consistent and physically-meaningful results but require expensive computations on a dense array of numerically integrated trajectories. Faster, more robust, sparse methods, on the other hand, are typically non-deterministic and require a-priori intuition of flow field behavior for interpretation. If LCS are to be used in the decision-making protocol of future autonomous technologies, a bridge between dense and sparse approaches must be made. This work begins to address this goal through the development of Lagrangian gradient regression (LGR), which enables the computation of deformation gradients and velocity gradients from sparse data via regression. Using LGR, velocity gradients, flow map Jacobians (thereby FTLE), and rotational Lagrangian metrics like the LAVD are accurately computed from sparse trajectory data which is orders of magnitude less dense than traditional approaches require. Moreover, there is no need to compute velocity or numerically differentiate at any point when using LGR. Therefore, it is a purely Lagrangian technique.

physics.flu-dyn↗

arXiv2023

Towards real-time reconstruction of velocity fluctuations in turbulent channel flow

We develop a framework for efficient streaming reconstructions of turbulent velocity fluctuations from limited sensor measurements with the goal of enabling real-time applications. The reconstruction process is simplified by computing linear estimators using flow statistics from an initial training period and evaluating their performance during a subsequent testing period with data obtained from direct numerical simulation. We address cases where (i) no, (ii) limited, and (iii) full-field training data are available using estimators based on (i) resolvent modes, (ii) resolvent-based estimation, and (iii) spectral proper orthogonal decomposition modes. During training, we introduce blockwise inversion to accurately and efficiently compute the resolvent operator in an interpretable manner. During testing, we enable efficient streaming reconstructions by using a temporal sliding discrete Fourier transform to recursively update Fourier coefficients using incoming measurements. We use this framework to reconstruct with minimal time delay the turbulent velocity fluctuations in a minimal channel at ${\rm Re}_τ\approx 186$ from sparse planar measurements. We evaluate reconstruction accuracy in the context of the extent of data required and thereby identify potential use cases for each estimator. The reconstructions capture large portions of the dynamics from relatively few measurement planes when the linear estimators are computed with sufficient fidelity. We also evaluate the efficiency of our reconstructions and show that the present framework has the potential to help enable real-time reconstructions of turbulent velocity fluctuations in an analogous experimental setting.

physics.flu-dyn↗

arXiv2023

Spatio-temporal characterization of non-linear forcing in turbulence

The quadratic convection term in the incompressible Navier-Stokes equations is considered as a non-linear forcing to the linear operator, and it is studied in the Fourier domain through the analysis of interactions between triadically compatible wavenumber-frequency triplets. Interaction coefficients are proposed to quantify the contribution to the forcing by each pair of triplets and are computed using data from direct numerical simulations of a turbulent channel at $Re_τ \approx 550$. The coefficients show the importance of interactions involving streamwise large scales. The regions of non-linear interactions permitted under the quasi-linear (QL) and generalized quasi-linear (GQL) assumptions are shown to be significant contributors to the forcing and increasing the number of GQL-large scales is shown to monotonically increase the total forcing captured, providing a possible reason for the success of QL and GQL simulations. The tools presented are expected to be useful for improving modeling of the nonlinearity, especially in QL, GQL, and resolvent analyses, and understanding the amplitude modulation mechanism relating large-scale fluctuations to the modulation of near-wall structures.

physics.flu-dyn↗

arXiv2023

The transformative potential of machine learning for experiments in fluid mechanics

The field of machine learning has rapidly advanced the state of the art in many fields of science and engineering, including experimental fluid dynamics, which is one of the original big-data disciplines. This perspective will highlight several aspects of experimental fluid mechanics that stand to benefit from progress advances in machine learning, including: 1) augmenting the fidelity and quality of measurement techniques, 2) improving experimental design and surrogate digital-twin models and 3) enabling real-time estimation and control. In each case, we discuss recent success stories and ongoing challenges, along with caveats and limitations, and outline the potential for new avenues of ML-augmented and ML-enabled experimental fluid mechanics.

physics.flu-dyn↗

arXiv2023

From resolvent to Gramians: extracting forcing and response modes for control

During the last decade, forcing and response modes produced by resolvent analysis have demonstrated great potential to guide sensor and actuator placement and design in flow control applications. However, resolvent modes are frequency-dependent, which, although responsible for their success in identifying scale interactions in turbulence, complicates their use for control purposes. In this work, we seek orthogonal bases of forcing and response modes that are the most responsive and receptive, respectively, across all frequencies. We show that these frequency-independent bases of \emph{representative} resolvent modes are given by the eigenvectors of the observability and controllability Gramians of the system considering full state inputs and outputs. We present several numerical examples where we leverage these bases by building orthogonal or interpolatory projectors onto the dominant forcing and response subspaces. Gramian-based forcing modes are used to identify dynamically relevant disturbances, to place point sensors to measure disturbances, and to design actuators for feedforward control in the subcritical linearized Ginzburg--Landau equation. Gramian-based response modes are used to identify coherent structures and for point sensor placement aiming at state reconstruction in the turbulent flow in a minimal channel at $\mathrm{Re}_τ=185$. The approach does not require data snapshots and relies only on knowledge of the steady or mean flow.

physics.flu-dyn↗

arXiv2021

Physics-informed dynamic mode decomposition (piDMD)

In this work, we demonstrate how physical principles -- such as symmetries, invariances, and conservation laws -- can be integrated into the dynamic mode decomposition (DMD). DMD is a widely-used data analysis technique that extracts low-rank modal structures and dynamics from high-dimensional measurements. However, DMD frequently produces models that are sensitive to noise, fail to generalize outside the training data, and violate basic physical laws. Our physics-informed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles -- conservation, self-adjointness, localization, causality, and shift-invariance -- and derive several closed-form solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of challenging problems in the physical sciences, including energy-preserving fluid flow, travelling-wave systems, the Schrödinger equation, solute advection-diffusion, a system with causal dynamics, and three-dimensional transitional channel flow. In each case, piDMD significantly outperforms standard DMD in metrics such as spectral identification, state prediction, and estimation of optimal forcings and responses.

math.DS↗

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

Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. In contrast, sparse identification of nonlinear dynamics (SINDy) learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the underlying dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to accurately recover the linear model contribution with this approach, disambiguating the effects of the implicitly defined nonlinear terms, resulting in a DMD-like model that is robust to strongly nonlinear dynamics. We demonstrate our approach on data from a wide range of nonlinear ordinary and partial differential equations that arise in the physical sciences. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control, and discovery of governing laws.

physics.flu-dyn↗

arXiv2021

Closing the loop: nonlinear Taylor vortex flow through the lens of resolvent analysis

We present an optimization-based method to efficiently calculate accurate nonlinear models of Taylor vortex flow. We use the resolvent formulation of McKeon & Sharma (2010) to model these Taylor vortex solutions by treating the nonlinearity not as an inherent part of the governing equations but rather as a triadic constraint which must be satisfied by the model solution. We exploit the low rank linear dynamics of the system to calculate an efficient basis for our solution, the coefficients of which are then calculated through an optimization problem where the cost function to be minimized is the triadic consistency of the solution with itself as well as with the input mean flow. Our approach constitutes, what is to the best of our knowledge, the first fully nonlinear and self-sustaining, resolvent-based model described in the literature. We compare our results to direct numerical simulation of Taylor Couette flow at up to five times the critical Reynolds number, and show that our model accurately captures the structure of the flow. Additionally, we find that as the Reynolds number increases the flow undergoes a fundamental transition from a classical weakly nonlinear regime, where the forcing cascade is strictly down scale, to a fully nonlinear regime characterized by the emergence of an inverse (up scale) forcing cascade. Triadic contributions from the inverse and traditional cascade destructively interfere implying that the accurate modeling of a certain Fourier mode requires knowledge of its immediate harmonic and sub-harmonic. We show analytically that this finding is a direct consequence of the structure of the quadratic nonlinearity of the governing equations formulated in Fourier space. Finally, we show that using our model solution as an initial condition to a higher Reynolds number DNS significantly reduces the time to convergence.

physics.flu-dyn↗

arXiv2020

Nonlinear mechanism of the self-sustaining process in the buffer and logarithmic layer of wall-bounded flows

The nonlinear mechanism in the self-sustaining process (SSP) of wall-bounded turbulence is investigated. Resolvent analysis is used to identify the principal forcing mode which produces the maximum amplification of the velocities in numerical simulations of the minimal channel for the buffer layer and a modified logarithmic (log) layer. The wavenumbers targeted in this study are those of the fundamental mode that is infinitely long in the streamwise direction and once periodic in the spanwise direction. The identified mode is then projected out from the nonlinear term of the Navier-Stokes equations at each time step from the simulation of the corresponding minimal channel. The results show that the removal of the principal forcing mode of the fundamental wavenumber can inhibit turbulence in both the buffer and log layer, with the effect being greater in the buffer layer. Removing other modes instead of the principal mode of the fundamental wavenumber only marginally affects the flow. Closer inspection of the dyadic interactions in the nonlinear term shows that contributions toward the principal forcing mode come from a limited set of wavenumber interactions. Using conditional averaging, the flow structures that are responsible for generating the nonlinear interaction to self-sustain turbulence are identified as spanwise rolls interacting with oblique streaks. This method, based on the equations of motion, validates the similarities in the SSP of the buffer and log layer, and characterises the underlying quadratic interactions in the SSP of the minimal channel.

physics.flu-dyn↗

arXiv2020

Data-driven resolvent analysis

Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input--output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement, and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: $1)$ dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and $2)$ a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg--Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

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