arXiv subjects

Rich R. Kerswell

Publications and source records attributed to Rich R. Kerswell.

At least 19 recordsLinked to original sources

arXiv2026

Oscillatory shear flows and a new 2D self-sustaining process

A new self-sustaining process (SSP) is identified in an oscillating spatially-homogeneous shear flow using a nonlinear Kelvin mode model. This SSP consists of energetic spanwise-invariant 1D `sheets' and weaker spanwise-dependent 2D fluctuations. It is instantaneously 2D, with a time-dependent invariant direction in the flow-cross-shear plane, and has a varicose symmetry. The sheets oscillate in time and are linearly unstable generating fluctuations which then nonlinearly self-interact to reinforce the sheets. The linear instability of the sheet is shown to be due to a lift-up mechanism acting in tandem with `push-forward', which arises due to the `streamwise' shear of the sheets. As the Reynolds number $Re\rightarrow\infty$, both asymptotics and numerics show SSP lower branches with fluctuation velocity $|\boldsymbol{u}'|\sim Re^{-1/2}$. Implications are discussed for oscillatory wall-bounded flows.

physics.flu-dyn↗

arXiv2026

Non-normal weakly nonlinear analysis: asymptotic consistency and non-universality

Non-normality can induce large transient growth in linearly stable systems. Determining whether this growth triggers a transition in the underlying nonlinear system, however, requires understanding the interaction between non-normality and nonlinearity. Here, we develop a weakly nonlinear theory for linearly-stable, non-normal systems subject to harmonic forcing, enabling a systematic analysis of this interaction. Following Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), we define a formal small parameter $\varepsilon$ as the reciprocal of the system's maximum linear amplification. However, we ensure asymptotic consistency by providing a framework that naturally adapts to the underlying structure of the system. The approach is applied to a harmonically forced channel flow and to a two-dimensional model mimicking the structure of the Orr-Sommerfeld-Squire equations. Unlike classical weakly nonlinear analysis near bifurcation points, the resulting amplitude equations are non-universal. In fact, a single linear mode amplified by the non-normality can nonlinearly excite a multi-modal and multi-frequency response at leading-order, which is system- or even regime-specific. Nevertheless, the method yields asymptotically consistent amplitude equations that capture this complexity provided a limit in which $\varepsilon\rightarrow0$ can be identified. As the forcing amplitude increases, the reduced equations capture stable nonlinear states emerging from the laminar flow, their subsequent bifurcations, and their eventual collision with the boundary of their basin of attraction. Thus, the amplitude equations can capture subcritical transitions driven by forcing and varied initial conditions and enable the identification of critical parameters beyond which no stable weakly nonlinear state exists.

physics.flu-dyn↗

arXiv2026

Localised Arrowheads: The building blocks of elastic turbulence in rectilinear, sheared polymer flows

Pressure-driven flow of a dilute polymer solution has been numerically observed to possess a form of elastic turbulence which is organised around the interactions of localised versions of 2-dimensional `arrowhead' travelling waves (Page et al. Phys. Rev. Lett. 125, 154501, 2020). As a step to confirming this theoretically, we identify spanwise-localised arrowhead travelling waves by tracking a symmetry-breaking bifurcation of the known spanwise-invariant (2D) arrowhead to a spanwise-periodic state and then discovering a secondary modulational instability to a spanwise-localised travelling wave. Spanwise-symmetric and asymmetric localised arrowheads exist with the latter having a phase speed slightly inclined to the streamwise direction. Computations capture the flow randomly switching between spanwise local and global arrowhead states in a streamwise-restricted domain, suggesting they form the building blocks of the chaos. Splitting events are also seen, in which a single localised state spawns multiple arrowheads. However, both cross-shear and spanwise velocities are small suggesting that this elastic turbulence will not be a good mixer.

physics.flu-dyn↗

arXiv2026

Elastic turbulence in highly entangled polymers and wormlike micelles

We show theoretically that an initially homogeneous planar Couette flow of a concentrated polymeric fluid is linearly unstable to the growth of two-dimensional (2D) perturbations, within two widely used constitutive models: the Johnson-Segalman model and the Rolie-Poly model. We perform direct nonlinear simulations of both models in 2D to show that this instability leads to a state of elastic turbulence comprising several narrow shear bands that dynamically coalesce, split and interact. Importantly, we show that this 2D instability arises not only in fluids that have a non-monotonic constitutive curve, and therefore show shear banding in 1D calculations, but also in shear thinning fluids with a monotonic constitutive curve, for which an initially homogeneous base state is stable in 1D. For the former category, the high shear branch of the constitutive curve is unstable to 2D instability in both models, so that the high shear band may be turbulent. In the Rolie-Poly model, the low shear branch is also likewise unstable. Our work provides the first simulation evidence for elastic turbulence in highly entangled polymeric fluids. It also potentially explains rheo-chaotic states seen experimentally in shear banding wormlike micelles. We additionally demonstrate elastic turbulence within both models in the planar Poiseuille geometry.

cond-mat.soft↗

arXiv2025

Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics

We present a deep learning emulator for stochastic and chaotic spatio-temporal systems, explicitly conditioned on the parameter values of the underlying partial differential equations (PDEs). Our approach involves pre-training the model on a single parameter domain, followed by fine-tuning on a smaller, yet diverse dataset, enabling generalisation across a broad range of parameter values. By incorporating local attention mechanisms, the network is capable of handling varying domain sizes and resolutions. This enables computationally efficient pre-training on smaller domains while requiring only a small additional dataset to learn how to generalise to larger domain sizes. We demonstrate the model's capabilities on the chaotic Kuramoto-Sivashinsky equation and stochastically-forced beta-plane turbulence, showcasing its ability to capture phenomena at interpolated parameter values. The emulator provides significant computational speed-ups over conventional numerical integration, facilitating efficient exploration of parameter space, while a probabilistic variant of the emulator provides uncertainty quantification, allowing for the statistical study of rare events.

cs.LG↗

arXiv2025

Linear instability in planar viscoelastic Taylor-Couette flow with and without explicit polymer diffusion

Elastic turbulence has been found in computations of planar viscoelastic Taylor-Couette flow using the Oldroyd-B model, apparently generated by a linear instability (van Buel et al. Europhys. Lett., 124, 14001, 2018). We demonstrate that no such linear instability exists in the governing equations used unless some diffusion is added to the polymer conformation tensor equation, as might occur through a diffusive numerical scheme. With this addition, the polymer diffusive instability (PDI) (Beneitez et al. (Phys. Rev. Fluids, 8, L101901, 2023)) exists and leads to chaotic flows resembling those found by van Buel et al. (2018). We show how finite volume or finite-difference discretisations of the governing equations can naturally introduce diffusive errors near boundaries which are sufficient to trigger PDI. This suggests that PDI could well be important in numerical solutions of wall-bounded viscoelastic flows modelled using Oldroyd-B and FENE-P even with no polymer stress diffusion explicitly included.

physics.flu-dyn↗

arXiv2025

Nonlinear optimals and their role in sustaining turbulence in channel flow

We investigate the energy transfer from the mean profile to velocity fluctuations in channel flow by calculating nonlinear optimal disturbances,i.e. the initial condition of a given finite energy that achieves the highest possible energy growth during a given fixed time horizon. It is found that for a large range of time horizons and initial disturbance energies, the nonlinear optimal exhibits streak spacing and amplitude consistent with DNS at least at Re_tau = 180, which suggests that they isolate the relevant physical mechanisms that sustain turbulence. Moreover, the time horizon necessary for a nonlinear disturbance to outperform a linear optimal is consistent with previous DNS-based estimates using eddy turnover time, which offers a new perspective on how some turbulent time scales are determined.

physics.flu-dyn↗

arXiv2025

Deep Learning of the Evolution Operator Enables Forecasting of Out-of-Training Dynamics in Chaotic Systems

We demonstrate that a deep learning emulator for chaotic systems can forecast phenomena absent from training data. Using the Kuramoto-Sivashinsky and beta-plane turbulence models, we evaluate the emulator through scenarios probing the fundamental phenomena of both systems: forecasting spontaneous relaminarisation, capturing initialisation of arbitrary chaotic states, zero-shot prediction of dynamics with parameter values outside of the training range, and characterisation of dynamical statistics from artificially restricted training datasets. Our results show that deep learning emulators can uncover emergent behaviours and rare events in complex systems by learning underlying mathematical rules, rather than merely mimicking observed patterns.

cs.LG↗

arXiv2025

The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

cs.LG↗

arXiv2024

Predicting burst events in a forced 2D flow: A wavelet-based analysis

Predicting and perhaps mitigating against rare, extreme events in fluid flows is an important challenge. Due to the time-localised nature of these events, Fourier-based methods prove inefficient in capturing them. Instead, this paper uses wavelet-based methods to understand the underlying patterns in a forced flow over a 2-torus which has intermittent high-energy burst events interrupting an ambient low energy 'quiet' flow. Two wavelet-based methods are examined to predict burst events: (1) a wavelet proper orthogonal decomposition (WPOD) based method which uncovers and utilises the key flow patterns seen in the quiet regions and the bursting episodes; and (2) a wavelet resolvent analysis (WRA) based method that relies on the forcing structures which amplify the underlying flow patterns. These methods are compared to a straightforward energy tracking approach which acts as a benchmark. Both the wavelet-based approaches succeed in producing better predictions than a simple energy criterion, i.e. earlier prediction times and/or fewer false positives and the WRA-based technique always performs better than WPOD. However, the improvement of WRA over WPOD is not as substantial as anticipated. We conjecture that this is because the mechanism for the bursts in the flow studied is found to be largely modal, associated with the unstable eigenfunction of the Navier-Stokes operator linearized around the mean flow. The WRA approach should deliver much better improvement over the WPOD approach for generically non-modal bursting mechanisms where there is a lag between the imposed forcing and the final response pattern.

physics.flu-dyn↗

arXiv2024

Early turbulence in viscoelastic flow past a periodic cylinder array

Early turbulence in periodic cylinder arrays is of particular interest in many practical applications to enhance mixing and material/heat exchange. In this study, we reveal a new early transition pathway to a chaotic wavy state and drag enhancement with the addition of polymers. Using 2D direct numerical simulations with sufficiently small polymer diffusion ($ε=10^{-5}$), we show that viscoelastic flow past periodic cylinder arrays become unstable at a Reynolds number $Re\approx 10$, significantly lower than the Newtonian counterpart of $Re\approx 150-200$. The chaotic wavy state which ensues exhibits sheets and `arrowhead' polymer conformation structures, consistent with the saturated centre-mode instability observed in wall-bounded parallel flows (Page et al., Phys. Rev. Lett., 125, 154501, 2020). Analysis of the kinetic energy budget reveals the purely elastic origin of the chaos. However, inertial forces, in conjunction with elastic forces, can reshape the base state, affecting the formation of an invariant polymer sheet. This sheet facilitates the stretching and recoiling of polymers, which in turn induces flow fluctuations and maintains the chaos. Exploring various maximum polymer extensions $b$, and polymer concentrations $β$ highlights the role of elastic forces in stretching the upstream separation zone while suppressing the downstream separation zone, resulting in drag enhancement at finite $Re$. Surprisingly, these modifications to the base state can suppress the invariant polymer sheet under large elastic forces (large $b$ or small $β$), thereby achieving a stable polymer-modified laminar state.

physics.flu-dyn↗

arXiv2024

Transition route to elastic and elasto-inertial turbulence in polymer channel flows

Viscoelastic shear flows support additional chaotic states beyond simple Newtonian turbulence. In vanishing Reynolds number flows, the nonlinearity in the polymer evolution equation alone can sustain inertialess 'elastic' turbulence (ET) while 'elasto-inertial' turbulence (EIT) appears to rely on an interplay between elasticity and finite-$Re$ effects. Despite their distinct phenomenology and industrial significance, transition routes and possible connections between these states are unknown. We identify here a common Ruelle-Takens transition scenario for both of these chaotic regimes in two-dimensional direct numerical simulations of FENE-P fluids in a straight channel. The primary bifurcation is caused by a recently-discovered 'polymer diffusive instability' associated with small but non-vanishing polymer stress diffusion which generates a finite-amplitude, small-scale travelling wave localised at the wall. This is found to be unstable to a large-scale secondary instability which grows to modify the whole flow before itself breaking down in a third bifurcation to either ET or EIT. The secondary large-scale instability waves resemble 'centre' and 'wall' modes respectively - instabilities which have been conjectured to play a role in viscoelastic chaotic dynamics but were previously only thought to exist far from relevant areas of the parameter space.

physics.flu-dyn↗

arXiv2024

Threshold transient growth as a criterion for turbulent mean profiles

Lozano-Duran et al (J. Fluid Mech., 914, A8, 2021) have recently identified the ability of streamwise-averaged turbulent streak fields $U(y,z,t)\widehat{\mathbf{x}}$ in minimal channels to produce short-term transient growth as the key linear mechanism needed to sustain turbulence at $Re_τ=180$. Here, in an attempt to extend this result to larger domains and higher $Re_τ$, we model this streak transient growth as a two-stage linear process by first selecting the dominant streak structure expected to emerge over the eddy turnover time on the turbulent mean profile $U(y)\widehat{\mathbf{x}}$, and then examining the secondary growth on this (frozen) streak field $U(y,z)\widehat{\mathbf{x}}$. Choosing the mean streak amplitude and eddy turnover time consistent with simulations captures the growth thresholds found by Lozano-Duran et al. (2021) for sustained turbulence. In a larger domain at $Re_τ=180$, the most energetic near-wall streaks observed in simulations are close to the predicted optimal streaks. This most energetic streak spacing, approaches the optimal streak at $Re_τ=550$ where the secondary growth possible on each also comes together. A key prediction from the model is that the threshold transient growth required to sustain turbulence decreases with increasing $Re_τ$. More fundamentally, the work of Lozano-Duran et al. (2021) and our results suggest a subtle but significant revision of Malkus's (J. Fluid Mech.}, 521, 1, 1956) classic hypothesis concerning realisable turbulent mean profiles. The key property for a realisable turbulent mean profile could be the ability to generate sufficient short-term transient growth rather than dependence on its (long-term) linear stability characteristics which was Malkus's original idea.

physics.flu-dyn↗

arXiv2024

Inertial enhancement of the polymer diffusive instability

Beneitez et al. (Phys. Rev. Fluids, 8, L101901, 2023) have recently discovered a new linear "polymer diffusive instability" (PDI) in inertialess rectilinear viscoelastic shear flow using the FENE-P model when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette (PCF) and plane Poiseuille (PPF) flows under varying Weissenberg number $W$, polymer stress diffusivity $\varepsilon$, solvent-to-total viscosity ratio $β$, and Reynolds number $Re$, considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with $Re$. For instance, as $Re$ increases with $β$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$, it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$. The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabiliser, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.

physics.flu-dyn↗

arXiv2023

Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets

We present a novel probabilistic deep learning approach, the 'Stochastic Latent Transformer' (SLT), designed for the efficient reduced-order modelling of stochastic partial differential equations. Stochastically driven flow models are pertinent to a diverse range of natural phenomena, including jets on giant planets, ocean circulation, and the variability of midlatitude weather. However, much of the recent progress in deep learning has predominantly focused on deterministic systems. The SLT comprises a stochastically-forced transformer paired with a translation-equivariant autoencoder, trained towards the Continuous Ranked Probability Score. We showcase its effectiveness by applying it to a well-researched zonal jet system, where the interaction between stochastically forced eddies and the zonal mean flow results in a rich low-frequency variability. The SLT accurately reproduces system dynamics across various integration periods, validated through quantitative diagnostics that include spectral properties and the rate of transitions between distinct states. The SLT achieves a five-order-of-magnitude speedup in emulating the zonally-averaged flow compared to direct numerical simulations. This acceleration facilitates the cost-effective generation of large ensembles, enabling the exploration of statistical questions concerning the probabilities of spontaneous transition events.

cs.LG↗

arXiv2023

Recurrent flow patterns as a basis for turbulence: predicting statistics from structures

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space transiently visiting the neighbourhoods of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303, 1948). The hope has always been to turn this appealing picture into a predictive framework where the statistics of the flow follows from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: (1) paucity of known solutions and (2) the lack of a rational theory for predicting the required weights. Here we describe a method to substantially solve these problems, and thereby provide the first compelling evidence that the PDFs of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our new method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimising a trajectory-dependent loss function. We use this approach to find hundreds of new solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. To our knowledge, this is the first time the PDFs of a spatio-temporally-chaotic system have been successfully reproduced with a set of simple invariant states, and provides a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

physics.flu-dyn↗

arXiv2023

Physics-informed neural network to augment experimental data: an application to stratified flows

We develop a physics-informed neural network (PINN) to significantly augment state-of-the-art experimental data and apply it to stratified flows. The PINN is a fully-connected deep neural network fed with time-resolved, three-component velocity fields and density fields measured simultaneously in three dimensions at $Re = O(10^3)$ in a stratified inclined duct experiment. The PINN enforces incompressibility, the governing equations for momentum and buoyancy, and the boundary conditions by automatic differentiation. The physics-constrained, augmented data are output at an increased spatio-temporal resolution and demonstrate five key results: (i) the elimination of measurement noise; (ii) the correction of distortion caused by the scanning measurement technique; (iii) the identification of weak but dynamically important three-dimensional vortices; (iv) the revision of turbulent energy budgets and mixing efficiency; and (v) the prediction of the latent pressure field and its role in the observed Holmboe wave dynamics. These results mark a significant step forward in furthering the reach of experiments, especially in the context of turbulence, where accurately computing three-dimensional gradients and resolving small scales remain enduring challenges.

physics.flu-dyn↗

arXiv2023

Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders

Convolutional autoencoders are used to deconstruct the changing dynamics of two-dimensional Kolmogorov flow as $Re$ is increased from weakly chaotic flow at $Re=40$ to a chaotic state dominated by a domain-filling vortex pair at $Re=400$. The highly accurate embeddings allow us to visualise the evolving structure of state space and are interpretable using `latent Fourier analysis' (Page {\em et. al.}, \emph{Phys. Rev. Fluids} \textbf{6}, 2021). Individual latent Fourier modes decode into vortical structures with a streamwise lengthscale controlled by the latent wavenumber, $l$, with only a small number $l \lesssim 8$ required to accurately represent the flow. Latent Fourier projections reveal a detached class of bursting events at $Re=40$ which merge with the low-dissipation dynamics as $Re$ is increased to $100$. We use doubly- ($l=2$) or triply- ($l=3$) periodic latent Fourier modes to generate guesses for UPOs (unstable periodic orbits) associated with high-dissipation events. While the doubly-periodic UPOs are representative of the high-dissipation dynamics at $Re=40$, the same class of UPOs move away from the attractor at $Re=100$ -- where the associated bursting events typically involve larger-scale ($l=1$) structure too. At $Re=400$ an entirely different embedding structure is formed within the network in which no distinct representations of small-scale vortices are observed; instead the network embeds all snapshots based around a large-scale template for the condensate. We use latent Fourier projections to find an associated `large-scale' UPO which we believe to be a finite-$Re$ continuation of a solution to the Euler equations.

physics.flu-dyn↗