arXiv subjects

Dan Lucas

Publications and source records attributed to Dan Lucas.

13 recordsLinked to original sources

arXiv2024

Dynamically relevant recurrent flows obtained via a nonlinear recurrence function from two-dimensional turbulence

This paper demonstrates the efficient extraction of unstable recurrent flows from two-dimensional turbulence by using nonlinear triads to diagnose recurrence in direct numerical simulations. Nearly recurrent episodes are identified from simulations and then converged using a standard Newton- GMRES-hookstep method, however with much greater diversity than previous studies which performed this 'recurrent flow analysis'. Unstable periodic and relative periodic orbits are able to be identified which span larger values of dissipation rate, i.e. corresponding to extreme bursting events. The triad variables are found to provide a more natural way to weight the greater variety of spatial modes active in such orbits than a standard Euclidian norm of complex Fourier amplitudes. Moreover the triad variables build in a reduction of the continuous symmetry of the system which avoids the need to search over translations when obtaining relative periodic orbits. Armed with these orbits we investigate optimal weightings when reconstructing the statistics of turbulence and suggest that, in fact, a simple heuristic weighting based on the solution instability provides a very good prediction, provided enough dynamically relevant orbits are included in the expansion.

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arXiv2024

Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback

We demonstrate the first successful non-invasive stabilisation of nonlinear travelling waves in a straight cylindrical pipe using time-delayed feedback control (TDF) working in various symmetry subspaces. By using an approximate linear stability analysis and by analysing the frequency domain effect of the control using transfer functions, we find that solutions with well separated unstable eigenfrequencies can have narrow windows of stabilising time-delays. To mitigate this issue we employ a "multiple time-delayed feedback" (MTDF) approach, where several control terms are included to attenuate a broad range of unstable eigenfrequencies. We implement a gradient descent method to dynamically adjust the gain functions in order to reduce the need for tuning a high dimensional parameter space. This results in a novel control method where the properties of the target state are not needed in advance and speculative guesses can result in robust stabilisation. This enables travelling waves to be stabilised from generic turbulent states and unknown travelling waves to be obtained in highly symmetric subspaces.

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arXiv2020

Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback

Time-delayed feedback control, attributed to Pyragas (1992 Physics Letters 170(6) 421-428), is a method known to stabilise periodic orbits in low dimensional chaotic dynamical systems. A system of the form $\dot{\mathbf{x}}(t)=f(\mathbf{x})$ has an additional term $G(\mathbf{x}(t)-\mathbf{x}(t-T))$ introduced where $G$ is some `gain matrix' and $T$ a time delay. The form of the delay term is such that it will vanish for any orbit of period $T,$ therefore making it also an orbit of the uncontrolled system. This non-invasive feature makes the method attractive for stabilising exact coherent structures in fluid turbulence. Here we begin by validating the method for the basic flow in Kolmogorov flow; a two-dimensional incompressible Navier-Stokes flow with a sinusoidal body force. The linear predictions for stabilisation are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a known travelling wave solution is able to be stabilised up to relatively high Reynolds number. We discover that the famous `odd-number' limitation of this time-delayed feedback method can be overcome in the fluid problem by using the symmetries of the system. This leads to the discovery of 8 additional exact coherent structures which can be stabilised with this approach. This means that certain unstable exact coherent structures can be obtained by simply time-stepping a modified set of equations, thus circumventing the usual convergence algorithms.

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arXiv2019

Layer formation and relaminarisation in plane Couette flow with spanwise stratification

Recent research has shed light on the role of coherent structures in forming layers when stably stratified turbulence is forced with horizontal shear (Lucas, Caulfield & Kerswell, J. Fluid Mech., vol. 832, 2017, pp. 409-437). Here we extend our previous work to investigate the effect of rigid boundaries on the dynamics by studying stably-stratified plane Couette flow with gravity oriented in the spanwise direction. We observe near-wall layering and associated new mean flows in the form of large scale spanwise-flattened streamwise rolls. The layers exhibit the expected buoyancy scaling $l_z\sim U/N$ where $U$ is a typical horizontal velocity scale and $N$ the buoyancy frequency. We associate the new coherent structures with a stratified modification of the well-known large scale secondary flow in plane Couette and find that the possibility of the transition to sustained turbulence is controlled by the relative size of this buoyancy scale to the spanwise spacing of the streaks. We also investigate the influence on the transition to turbulence of the newly discovered linear instability in this system (Facchini et. al. 2018 arXiv:1711.11312).

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arXiv2017

Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities

We consider turbulence in a stratified 'Kolmogorov' flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratifica- tion is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connection between linear theory predicting instabilities from infinitesimal perturbations to the robust finite amplitude nonlinear layered state observed in the turbulence. We investigate how the observed vertical length scales are related to the primary linear instabilities and compare to previously considered examples of shear instability leading to layer formation in other horizontally sheared flows.

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arXiv2017

Irreversible mixing by unstable periodic orbits in buoyancy dominated stratified turbulence

We consider turbulence driven by a large-scale horizontal shear in Kolmogorov flow (i.e. with sinusoidal body forcing) and a background linear stable stratification with buoyancy frequency $N_B^2$ imposed in the third, vertical direction in a fluid with kinematic viscosity $ν$. This flow is known to be organised into layers by nonlinear unstable steady states, which incline the background shear in the vertical and can be demonstrated to be the finite-amplitude saturation of a sequence of instabilities, originally from the laminar state. Here, we investigate the next order of motions in this system, i.e. the time-dependent mechanisms by which the density field is irreversibly mixed. This investigation is achieved using 'recurrent flow analysis'. We identify (unstable) periodic orbits, which are embedded in the turbulent attractor, and use these orbits as proxies for the chaotic flow. We find that the time average of an appropriate measure of the 'mixing efficiency' of the flow $\mathscr{E}= χ/(χ+\mathcal{D})$ ($\mathcal{D}$ is the volume-averaged kinetic energy dissipation rate and $χ$ is the volume-averaged density variance dissipation rate) varies non-monotonically with the time-averaged buoyancy Reynolds numbers $\overline{Re}_B= \overline{\mathcal{D}}/(νN_B^2)$, and is bounded above by $1/6$, consistently with the classical model of Osborn (1980). There are qualitatively different physical properties between the unstable orbits that have lower irreversible mixing efficiency at low $\overline{Re}_B \sim O(1)$ and those with nearly optimal $\mathscr{E} \lesssim 1/6$ at intermediate $\overline{Re}_B \sim 10$. The weaker orbits, inevitably embedded in more strongly stratified flow, are characterised by straining or 'scouring' motions, while the more efficient orbits have clear overturning dynamics in more weakly stratified, and apparently shear-unstable flow.

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arXiv2016

Sustaining processes from recurrent flows in body-forced turbulence

By extracting unstable invariant solutions directly from body-forced three-dimensional turbulence, we study the dynamical processes at play when the forcing is large scale and either unidirectional in the momentum or the vorticity equations. In the former case, the dynamical processes familiar from recent work on linearly-stable shear flows - variously called the Self-Sustaining Process (Waleffe 1997) or Vortex-Wave Interaction (Hall & Smith 1991; Hall & Sherwin 2010) - are important even when the base flow is linearly unstable. In the latter case, where the forcing drives Taylor-Green vortices, a number of mechanisms are observed from the various types of periodic orbits isolated. In particular, two different transient growth mechanisms are discussed to explain the more complex states found.

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arXiv2016

Precession resonance in water waves

We describe the theory and present numerical evidence for a new type of nonlinear resonant interaction between gravity waves on the surface of deep water. The resonance constitutes a generalisation of the usual 'exact' resonance as we show that exchanges of energy between the waves can be enhanced when the interaction is three-wave rather than four and the linear frequency mismatch, or detuning, is non-zero i.e. $ω_1\pmω_2\pmω_3 \neq0.$ This is possible because the resonance condition is now a match between the so-called 'precession frequency' of a given $\textit{triad interaction}$ and an existent nonlinear frequency in the system. In the limit of weak nonlinearity this precession frequency is simply due to the linear 'drift' of the triad phase; therefore, it tends toward the detuning. This means precession resonance of this type can occur at finite amplitudes, with nonlinear corrections contributing to the resonance. We report energy transfer efficiencies of up to 40%, depending on the model options. To the authors' knowledge this represents the first new type of nonlinear resonance in surface gravity waves since the seminal work of Benjamin & Feir (1967).

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arXiv2015

Atypical late-time singular regimes accurately diagnosed in stagnation-point-type solutions of 3D Euler flows

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t < T^* - 10^{-9000}$, where $T^*$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating point double precision ($\approx 10^{-16}$) creates a `machine-epsilon' barrier. Due to this limitation on the \emph{original} independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^*- t \approx 10^{-140}.$

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arXiv2015

Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

Motivated by recent success in the dynamical systems approach to transitional flow, we study the efficiency and effectiveness of extracting simple invariant sets (recurrent flows) directly from chaotic/turbulent flows and the potential of these sets for providing predictions of certain statistics of the flow. Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a sinusoidal body force) is studied both over a square [0, 2π]2 torus and a rectangular torus extended in the forcing direction. In the former case, an order of magnitude more recurrent flows are found than previously (Chandler & Kerswell 2013) and shown to give improved predictions for the dissipation and energy pdfs of the chaos via periodic orbit theory. Over the extended torus at low forcing amplitudes, some extracted states mimick the statistics of the spatially-localised chaos present surprisingly well recalling the striking finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At higher forcing amplitudes, however, success is limited highlighting the increased dimensionality of the chaos and the need for larger data sets. Algorithmic developments to improve the extraction procedure are discussed.

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arXiv2014

Symmetry-plane model of 3D Euler flows and mapping to regular systems to improve blowup assessment using numerical and analytical solutions

Motivated by the work on stagnation-point type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (1999) and the subsequent demonstration of finite-time blowup by Constantin (2006) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields. The value of the model's parameter determines if there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model's parameter. We use a representative parameter value, for which the solution blows up in finite-time, as a benchmark for the systematic study of errors in numerical solutions. We compare numerical integrations of our "original" model with a "mapped" version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations (Bustamante 2011). We show that the mapped system's numerical solution increases accuracy in estimates of supremum norms and singularity time while entailing only a small additional computational cost. We study the Fourier spectrum of the model's numerical solution and find that the analyticity-strip width (a measure of the solution's analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet 2012) between the analyticity-strip method and the Beale-Kato-Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions of the 3D Euler fluid equations.

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arXiv2014

Spatiotemporal dynamics in 2D Kolmogorov flow over large domains

Kolmogorov flow in two dimensions - the two-dimensional Navier-Stokes equations with a sinusoidal body force - is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimicks the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn-Hilliard-type equation and as a result coarsening dynamics are observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially-localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) PDEs based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions - a kink and antikink - which connect two steady one-dimensional flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

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arXiv2014

Robust energy transfer mechanism via precession resonance in nonlinear turbulent wave systems

A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We introduce the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronised triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

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