We explore the transition to chaos in a prototypical hydrodynamic oscillator, namely a globally unstable low-density jet subjected to external time-periodic forcing. As the forcing strengthens at an off-resonant frequency, we find that the jet exhibits a sequence of nonlinear states: period-1 limit cycle $\rightarrow $ quasiperiodicity $\rightarrow$ intermittency $\rightarrow$ low-dimensional chaos. We show that the intermittency obeys type-II Pomeau–Manneville dynamics by analysing the first return map and the scaling properties of the quasiperiodic lifetimes between successive chaotic epochs. By providing experimental evidence of the type-II intermittency route to chaos in a globally unstable jet, this study reinforces the idea that strange attractors emerge via universal mechanisms in open self-excited flows, facilitating the development of instability control strategies based on chaos theory.

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes the method proposed in Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), in that the base flow now arbitrarily depends on time, and the operator exponential formalism for the evolution of the perturbation is not used. Applied to the two-dimensional Lamb–Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain in the weakly nonlinear regime. In particular, the minimum amplitude of the linear optimal initial perturbation required for the amplitude equation to lose a solution, interpreted as the flow experiencing a bypass (subcritical) transition, is found to decay as a power law with the Reynolds number. Although with a different exponent, this is recovered in direct numerical simulations, showing a transition towards a tripolar state. The simplicity of the amplitude equation and the link made with the sensitivity formula permits a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.

Quasistatic magnetoconvection of a fluid with low Prandtl number (${\textit {Pr}}=0.025$) with a vertical magnetic field is considered in a unit-aspect-ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number ${\textit {Ha}}$, the onset of convection is known to result from a sidewall instability giving rise to the wall-mode regime. Here, we carry out three-dimensional direct numerical simulations of unprecedented length to map out the parameter space at ${\textit {Ha}} = 200, 500, 1000$, varying the Rayleigh number (${\textit {Ra}}$) over the range $6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$. We track the development of stable equilibria produced by this primary instability, identifying bifurcations leading to limit cycles and eventually to chaotic dynamics. At ${\textit {Ha}}=200$, the steady wall-mode solution undergoes a symmetry-breaking bifurcation producing a state that features a coexistence between wall modes and a large-scale roll in the centre of the domain, which persists to higher ${\textit {Ra}}$. However, under a stronger magnetic field at ${\textit {Ha}}=1000$, the steady wall-mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in ${\textit {Ra}}$, the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of ${\textit {Ra}}$, and chaotic dynamics primarily arise from a mixture of chaotic wall-mode dynamics and arrays of cellular structures.

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in a globally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for peak multifractality produce a complex network whose node degree distribution obeys an inverse power-law scaling with an exponent of $2 < \gamma < 3$, indicating scale-free topology and (iii) the Hurst exponent and the global clustering coefficient can serve as early warning indicators of global instability under specific operating and forcing conditions. By characterising the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics commonly observed in turbulent flows can also be observed in laminar flows under certain stochastic forcing conditions.

The whistling induced by a low-Mach turbulent flow through a deep axisymmetric cavity in a duct is investigated theoretically and experimentally. The experiments include acoustic measurements and stereoscopic particle image velocimetry (PIV). The paper focuses on the effect of a mean swirl on the dynamics of the azimuthal aeroacoustic modes. The mean swirl in the cavity has two origins: one component is imposed by a controlled tangential air injection upstream of the cavity, and the other component spontaneously arises under the action of the self-sustained azimuthal aeroacoustic mode, as explained in the companion paper, Part 1 (Faure-Beaulieu, Xiong, Pedergnana & Noiray, J. Fluid Mech., vol. 971, 2023, A21). Experiments show that the dynamics of the aeroacoustic wave is influenced by the imposed swirl. In particular, the spinning wave propagating against the swirl is promoted. To explain this, a linear perturbation analysis is performed around an incompressible mean swirling flow obtained from RANS simulations. It reveals that the dominant shear layer modes of azimuthal order 1 and −1 involved in the whistling phenomenon are helical modes winding respectively with and against the swirl, and spinning respectively in counterswirl and co-swirl directions. The counterswirl hydrodynamic mode is the least damped of the two, which is in agreement with the experimental observations. Finally, a low-order model based on the wave equation is derived. With only a few parameters, it fully reproduces the experimental observations for a wide range of imposed swirl intensity in the duct flow, and it allows us to disentangle the mechanisms responsible for this complex aeroacoustic instability.

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.

Low-order thermoacoustic models are qualitatively correct, but typically, they are quantitatively inaccurate. We propose a time-domain bias-aware method to make qualitatively low-order models quantitatively (more) accurate. First, we develop a Bayesian ensemble data assimilation method for a low-order model to self-adapt and self-correct any time that reference data become available. Second, we apply the methodology to infer the thermoacoustic states and heat-release parameters on the fly without storing data (real time). We perform twin experiments using synthetic acoustic pressure measurements to analyse the performance of data assimilation in all nonlinear thermoacoustic regimes, from limit cycles to chaos, and interpret the results physically. Third, we propose practical rules for thermoacoustic data assimilation. An increase, reject, inflate strategy is proposed to deal with the rich nonlinear behaviour; and physical time scales for assimilation are proposed in non-chaotic regimes (with the Nyquist–Shannon criterion) and in chaotic regimes (with the Lyapunov time). Fourth, we perform data assimilation using data from a higher-fidelity model. We introduce an echo state network to estimate in real time the forecast bias, which is the model error of the low-fidelity model. We show that: (i) the correct acoustic pressure, parameters, and model bias can be inferred accurately; (ii) the learning is robust as it can tackle large uncertainties in the observations (up to 50 % of the mean values); (iii) the uncertainty of the prediction and parameters is naturally part of the output; and (iv) both the time-accurate solution and statistics can be inferred successfully. Data assimilation opens up new possibility for real-time prediction of thermoacoustics by combining physical knowledge and experimental data synergistically.

We propose a theoretical approach to derive amplitude equations governing the weakly nonlinear evolution of non-normal dynamical systems, when they experience transient growth or respond to harmonic forcing. This approach reconciles the non-modal nature of these growth mechanisms and the need for a centre manifold to project the leading-order dynamics. Under the hypothesis of strong non-normality, we take advantage of the fact that small operator perturbations suffice to make the inverse resolvent and the inverse propagator singular, which we encompass in a multiple-scale asymptotic expansion. The methodology is outlined for a generic nonlinear dynamical system, and four application cases highlight common non-normal mechanisms in hydrodynamics: the streamwise convective non-normal amplification in the flow past a backward-facing step, and the Orr and lift-up mechanisms in the plane Poiseuille flow.

A set of several exact coherent states in plane Couette flow is computed under spanwise wall oscillation control, with a range of wall oscillation amplitudes and periods $({A_w}, T)$. It is found that the wall oscillation generally stabilises the upper branch of the equilibrium solutions and achieves the corresponding drag reduction, while it influences modestly the lower branch. The stabilisation effect is found to increase with the oscillation amplitude with an optimal time period around ${T^{+}} \approx 100$. The exact coherent states reproduce some key dynamical behaviours of streaks observed in previous studies, while exhibiting the rich coherent structure dynamics that cannot be extracted from a phase average of turbulent states. Visualisation of state portraits shows that the size of the state space supporting turbulent solution is reduced by the spanwise wall oscillation, and the upper-branch equilibrium solutions become less repelling, with many of their unstable manifolds being stabilised. This change of the state space dynamics leads to a significant reduction in lifetime of turbulence. Finally, the main stabilisation mechanism of the exact coherent states is found to be the suppression of the lift-up effect of streaks, explaining why previous linear analyses have been so successful for turbulence stabilisation modelling and the resulting drag reduction.

This article deals with the symmetry breaking of azimuthal thermoacoustic modes in annular combustors. Using a nominally symmetric annular combustor, we present experimental evidence of a predicted spontaneous reflectional symmetry breaking, and also an unexpected explicit rotational symmetry breaking in the neighbourhood of the Hopf bifurcation which separates linearly stable azimuthal thermoacoustic modes from self-oscillating modes. We derive and solve a multidimensional Fokker–Planck equation to unravel a unified picture of the phase space topology. We demonstrate that symmetric probability density functions of the thermoacoustic state vector are elusive, because the effect of asymmetries, even imperceptible ones, is magnified close to the bifurcation. This conclusion implies that the thermoacoustic oscillations of azimuthal modes in real combustors will systematically exhibit a statistically dominant orientation of the mode in the vicinity of the Hopf bifurcation.

Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier–Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iteration-based shooting methods. While computationally slow, our methods converge from very inaccurate initial guesses.

We take a complex systems approach to investigating experimentally the collective dynamics of a network of four self-excited thermoacoustic oscillators coupled in a ring. Using synchronization metrics, we find a wide variety of emergent multi-scale behaviour, such as (i) a transition from intermittent frequency locking on a $\mathbb {T}^{3}$ quasiperiodic attractor to a breathing chimera, (ii) a two-cluster state of anti-phase synchronization on a periodic limit cycle, and (iii) a weak anti-phase chimera. We then compute the cross-transitivity from recurrence networks to identify the dominant direction of the coupling between the heat-release-rate ($q^{\prime }_{\mathbb {X}}$) and pressure ($p^{\prime }_{\mathbb {X}}$) fluctuations in each individual oscillator, as well as that between the pressure ($p^{\prime }_{\mathbb {X}}$ and $p^{\prime }_{\mathbb {Y}}$) fluctuations in each pair of coupled oscillators. We find that networks of non-identical oscillators exhibit circumferentially biased $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ coupling, leading to mode localization, whereas networks of identical oscillators exhibit globally symmetric $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ coupling. In both types of networks, we find that the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ coupling can be symmetric or asymmetric, but that the asymmetry is always such that $q^{\prime }_{\mathbb {X}}$ exerts a greater influence on $p^{\prime }_{\mathbb {X}}$ than vice versa. Finally, we show through a cluster analysis that the $p^{\prime }_{\mathbb {X}}$–$p^{\prime }_{\mathbb {Y}}$ interactions play a more critical role than the $p^{\prime }_{\mathbb {X}}$–$q^{\prime }_{\mathbb {X}}$ interactions in defining the collective dynamics of the system. As well as providing new insight into the interplay between the $p^\prime_{\mathbb{X}}\text{--}p^\prime_{\mathbb{Y}}$ and $p^\prime_{\mathbb{X}}\text{--}q^\prime_{\mathbb{X}}$ coupling, this study shows that even a small network of four ring-coupled thermoacoustic oscillators can exhibit a wide variety of collective dynamics. In particular, we present the first evidence of chimera states in a minimal network of coupled thermoacoustic oscillators, paving the way for the application of oscillation quenching strategies based on chimera control.

The 2021 Nobel Prize in Physics recognizes advances in the understanding of complex systems, and underscores that ‘complex’ does not mean ‘imponderable’.

We explore experimentally whether axisymmetry breaking can be exploited for the open-loop control of a prototypical hydrodynamic oscillator, namely a low-density inertial jet exhibiting global self-excited axisymmetric oscillations. We find that when forced transversely or axially at a low amplitude, the jet always transitions first from a period-1 limit cycle to $\mathbb {T}^2$ quasiperiodicity via a Neimark–Sacker bifurcation. However, we find that the subsequent transition, from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in, depends on the spatial symmetry of the applied perturbations: axial forcing induces a saddle-node bifurcation at small detuning but an inverse Neimark–Sacker bifurcation at large detuning, whereas transverse forcing always induces an inverse Neimark–Sacker bifurcation irrespective of the detuning. Crucially, we find that only transverse forcing can enable both asynchronous and synchronous quenching of the natural mode to occur without resonant or non-resonant amplification of the forced mode, resulting in substantially lower values of the overall response amplitude across all detuning values. From this, we conclude that breaking the jet axisymmetry via transverse forcing is a more effective control strategy than preserving the jet axisymmetry via axial forcing. Finally, we show that the observed synchronization phenomena can be modelled qualitatively with just two forced coupled Van der Pol oscillators. The success of such a simple low-dimensional model in capturing the complex synchronization dynamics of a multi-modal hydrodynamic system opens up new opportunities for axisymmetry breaking to be exploited for the open-loop control of other globally unstable flows.

The study concerns a slender, heavy body moving with streamwise velocity in a boundary layer. Modelling assumptions on body size reduce the governing equations for the body motion to a pair of nonlinear integro-differential equations (IDEs) which displays a wide range of distinguished behaviours, including eventual collision with the wall (‘crash’), escape to infinity (‘fly away’) and repeatedly travelling far from the wall and back again without ever colliding or escaping (‘bouncing’). The paper gives a survey of the variety of behaviour, as well as asymptotic analysis and insight into each category of fluid/body interaction and the conditions under which crash, fly away and bouncing occur.

In jet engines and gas turbines, the annular shape of the combustion chamber allows the appearance of self-oscillating azimuthal thermoacoustic modes. We report experimental evidence of a new type of modal dynamics characterised by periodic switching of the spinning direction and develop a theoretical model that fully reproduces this phenomenon and explains the underlying mechanisms. It is shown that tiny asymmetries of the geometry, the mean temperature field, the thermoacoustic response of the flames or the acoustic impedance of the walls, present in any real systems, can induce these heteroclinic orbits. The model also explains experimental observations showing a statistically dominant spinning direction despite the absence of swirling flow, or pairs of preferred nodal line directions.

Mass-spring models are essential for the description of sloshing resonances in engineering. By experimentally measuring the liquid's centre of mass in a horizontally oscillated rectangular tank, we show that low-amplitude sloshing obeys the Duffing equation. A bending of the response curve in analogy to a softening spring is observed, with growing hysteresis as the driving amplitude increases. At large amplitudes, complex wave patterns emerge (including wave breaking and run up at the tank walls), competition between flow states is observed and the dynamics departs progressively from Duffing. We also provide a quantitative comparison of wave shapes and response curves to the predictions of a multimodal model based on potential flow theory (Faltinsen & Timokha, Sloshing. Cambridge University Press, 2009) and show that it systematically overestimates the sloshing amplitudes and the hysteresis. We find that the phase lag between the liquid's centre of mass and the forcing is the key predictor of the nonlinear response maxima. The phase lag reflects precisely the onset of deviations from Duffing dynamics and – most importantly – at resonance the sloshing motion always lags the driving by $90^{\circ }$ (independently of the wave pattern). This confirms the theoretical $90^{\circ }$-phase-lag criterion (Cenedese & Haller, Proc. R. Soc. A, vol. 476, no. 2234, 2020, 20190494).

This work describes the application of a technique that extracts branched manifolds from time series to study numerically generated fluid particle behaviour in the wake past a cylinder performing a rotary oscillation at low Reynolds numbers, and compares it with the results obtained for a paradigmatic analytical model of Lagrangian motion: the driven double gyre. The approach does not require prior knowledge of the underlying equations defining the dataset. The time series taken as input corresponds to the evolution of a position coordinate of an individual fluid particle. A delay embedding is used to reconstruct the dynamics in phase space, and a cell complex is built to characterize the topology of the embedding. Fluid particles are said to belong to the same topological class when the Betti numbers, orientability chains and weak boundaries of the associated cell complexes coincide. Topological colouring consists of labelling or ‘colouring’ advected particles with the topological class obtained in their finite-time analyses. The results suggest that topological colouring can be used to distinguish between regions of the flow where trajectories exhibit different finite-time dynamics.