arXiv subjects

Cedric Beaume

Publications and source records attributed to Cedric Beaume.

8 recordsLinked to original sources

arXiv2026

Pattern Formation with Two Length Scales: Spatiotemporal Chaos

Three-wave interactions (or resonant triads) are the lowest-order nonlinear interaction in pattern formation and arise between waves with different orientations when the sum of two wavevectors equals a third one. When a pattern has only one length scale, stripe patterns are possible but three-wave interactions are responsible for the prevalence of hexagons close to onset. In problems with two length scales, there is a much wider range of possible three-wave interactions, leading to more complex structures such as superhexagons, stars, quasipatterns and even spatiotemporal chaos (STC). We investigate the role that nonlinear wave interactions play in the formation of STC in a model partial differential equation (PDE) in the case that the length scale ratio is $\sqrt{7}$, relevant to superlattice patterns in the Faraday wave experiment. The simpler aspects of the dynamics can be represented by a system of ordinary differential equations (ODEs) derived from the PDE using weakly nonlinear theory. We analyze the equilibrium patterns in these ODEs and evaluate their stability, comparing the results with direct numerical simulations of the model PDE. The ODEs predict parameter regimes where there are no stable simple equilibria, which is where we typically find complex behavior in the PDE. We have conducted a careful study of the transition from simple patterns to patterns that include modes beyond the finite-dimensional subspace imposed in the reduction to the ODEs, to time-dependent competition between different triads, ending up with fully developed STC. For our choice of length scale ratio, we show that four-wave interactions also play an important role. Our analysis is relevant to any pattern-forming system with three-wave interactions involving two length scales, such as the Faraday wave experiment, coupled reaction-diffusion systems, and pattern formation in dryland vegetation.

nlin.PS↗

arXiv2025

Spatially localised doubly diffusive convection in an axisymmetric spherical shell

Doubly diffusive convection describes the fluid motion driven by the competition of temperature and salinity gradients diffusing at different rates. While the convective motions driven by these gradients usually occupy the entire domain, parameter regions exist where the convection is spatially localised. Although well-studied in planar geometries, spatially localised doubly diffusive convection has never been investigated in a spherical shell, a geometry of relevance to astrophysics. In this paper, numerical simulation is used to compute spatially localised solutions of doubly diffusive convection in an axisymmetric spherical shell. Several families of spatially localised solutions, named using variants of the word convecton, are found and their bifurcation diagram computed. The various convectons are distinguished by their symmetry and by whether they are localised at the poles or at the equator. We find that because the convection rolls that develop in the spherical shell are not straight but curve around the inner sphere, their strength varies with latitude, and the system is spatially modulated. As a result spatially periodic states can no longer form and, much like in a planar system with non-standard boundary conditions, localised states are forced to arise via imperfect bifurcations. While the direct relevance is to doubly diffusive convection, parallels drawn with the Swift Hohenberg equation suggest a wide applicability to other pattern forming systems in similar geometries.

physics.flu-dyn↗

arXiv2021

Can neural networks predict dynamics they have never seen?

Neural networks have proven to be remarkably successful for a wide range of complicated tasks, from image recognition and object detection to speech recognition and machine translation. One of their successes is the skill in prediction of future dynamics given a suitable training set of data. Previous studies have shown how Echo State Networks (ESNs), a subset of Recurrent Neural Networks, can successfully predict even chaotic systems for times longer than the Lyapunov time. This study shows that, remarkably, ESNs can successfully predict dynamical behavior that is qualitatively different from any behavior contained in the training set. Evidence is provided for a fluid dynamics problem where the flow can transition between laminar (ordered) and turbulent (disordered) regimes. Despite being trained on the turbulent regime only, ESNs are found to predict laminar behavior. Moreover, the statistics of turbulent-to-laminar and laminar-to-turbulent transitions are also predicted successfully, and the utility of ESNs in acting as an early-warning system for transition is discussed. These results are expected to be widely applicable to data-driven modelling of temporal behaviour in a range of physical, climate, biological, ecological and finance models characterized by the presence of tipping points and sudden transitions between several competing states.

cs.LG↗

arXiv2021

Optimizing the control of transition to turbulence using a Bayesian method

The nonlinear robustness of laminar plane Couette flow is considered under the action of in-phase spanwise wall oscillations by computing properties of the edge of chaos, i.e., the boundary of its basin of attraction. Three measures are used to quantify the chosen control strategy on laminar-to-turbulent transition: the kinetic energy of edge states (local attractors on the edge of chaos), the form of the minimal seed (least energetic perturbation on the edge of chaos), and the laminarization probability (the probability that a random perturbation from the laminar flow of given kinetic energy will laminarize). A novel Bayesian approach is introduced to enable the accurate computation of the laminarization probability at a fraction of the cost of previous methods. While the edge state and the minimal seed provide useful information about the dynamics of transition to turbulence, neither measure is particularly useful to judge the effectiveness of the control strategy since they are not representative of the global geometry of the edge. In contrast, the laminarization probability provides global information about the edge and can be used to evaluate the control effectiveness by computing a laminarization score (the expected laminarization probability) and the associated expected dissipation rate of the controlled flow. These two quantities allow for the determination of optimal control parameter values subject to desired constraints. The results discussed in the paper are expected to be applied to a wide range of transitional flows and control strategies aimed at suppressing or triggering transition to turbulence.

physics.flu-dyn↗

arXiv2020

A probabilistic protocol for the assessment of transition and control

Transition to turbulence dramatically alters the properties of fluid flows. In most canonical shear flows, the laminar flow is linearly stable and a finite-amplitude perturbation is necessary to trigger transition. Controlling transition to turbulence is achieved via the broadening or narrowing of the basin of attraction of the laminar flow. In this paper, a novel methodology to assess the robustness of the laminar flow and the efficiency of control strategies is introduced. It relies on the statistical sampling of the phase space neighborhood around the laminar flow in order to assess the transition probability of perturbations as a function of their energy. This approach is applied to a canonical flow (plane Couette flow) and provides invaluable insight: in the presence of the chosen control, transition is significantly suppressed whereas plausible scalar indicators of the nonlinear stability of the flow, such as the edge state energy, do not provide conclusive predictions. The methodology presented here in the context of transition to turbulence is applicable to any nonlinear system displaying finite-amplitude instability.

physics.flu-dyn↗

arXiv2019

Dynamics of spatially localized states in transitional plane Couette flow

Unsteady spatially localized states such as puffs, slugs or spots play an important role in transition to turbulence. In plane Couette flow, steady versions of these states are found on two intertwined solution branches describing homoclinic snaking (Schneider $\textit{et al.}$ 2010a). These branches can be used to generate a number of spatially localized initial conditions whose transition can be investigated. From the low Reynolds numbers where homoclinic snaking is first observed ($Re < 175$) to transitional ones ($Re \approx 325$), these spatially localized states traverse various regimes where their relaminarisation time and dynamics are affected by the dynamical structure of phase space. These regimes are reported and characterised in this paper for a $4π$ periodic domain in the streamwise direction as a function of the two remaining variables: the Reynolds number and the width of the localized pattern. Close to the snaking, localized states are attracted by spatially localized periodic orbits before relaminarising. At larger values of the Reynolds number, the flow enters a chaotic transient of variable duration before relaminarising. Very long chaotic transients ($t > 10^4$) can be observed without difficulty for relatively low values of the Reynolds number ($Re \approx 250$).

physics.flu-dyn↗

arXiv2014

Reduced description of exact coherent states in parallel shear flows

Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using equations obtained from a large Reynolds-number asymptotic reduction of the Navier-Stokes equations. The reduced equations employ a decomposition into streamwise-averaged (mean) and streamwise-varying (fluctuation) components and are characterized by an effective order one Reynolds number in the mean equations along with a formally higher-order diffusive regularization of the fluctuation equations. A robust numerical algorithm for computing exact coherent states is introduced. Numerical continuation of the lower branch states to lower Reynolds numbers reveals the presence of a saddle-node; the saddle-node allows access to upper branch states that, like the lower branch states, appear to be self-consistently described by the reduced equations. Both lower and upper branch states are characterized in detail.

physics.flu-dyn↗

arXiv2014

Exact coherent structures in a reduced model of parallel shear flow

A reduced description of shear flows consistent with the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow [J. Wang et al., Phys. Rev. Lett. 98, 204501 (2007)] is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found for Waleffe flow [F. Waleffe, Phys. Fluids 9, 883--900 (1997)]. The lower branch solution is characterized by fluctuations that vary slowly along the critical layer while the upper branch solutions display a bimodal structure and are more strongly focused on the critical layer. The reduced model provides a rational framework for investigations of subcritical spatiotemporal patterns in parallel shear flows.

physics.flu-dyn↗