Dan Lucas

Spatiotemporal dynamics in two-dimensional 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.

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 [G. J. Chandler and R. R. Kerswell, “Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow,” J. Fluid Mech. 722, 554–595 (2013)] and shown to give improved predictions for the dissipation and energy pdfs of the chaos via periodic orbit theory. Analysis of the recurrent flows shows that the energy is largely trapped in the smallest wavenumbers through a combination of the inverse cascade process and a feature of the advective nonlinearity in 2D. Over the extended torus at low forcing amplitudes, some extracted states mimic the statistics of the spatially localised chaos present surprisingly well recalling the findings of Kawahara and Kida [“Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst,” J. Fluid Mech. 449, 291 (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.

A family of helically symmetric vortex equilibria

We present a family of steadily rotating equilibrium states consisting of helically symmetric vortices in an incompressible inviscid irrotational unbounded fluid. These vortices are described by contours bounding regions of uniform axial vorticity. Helical symmetry implies material conservation of axial vorticity (in the absolute frame of reference) when the flow field parallel to vortex lines is proportional to (1+ϵ 2 r 2 ) −1/2 , where ϵ is the pitch and r is the distance from the axis. This material conservation property enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterized by their mean radius and centroid position. In the case of a single vortex, parameter space cannot be fully filled by our numerical approach. We conjecture multiply connected contours will characterize equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. Stability properties are investigated numerically using a helical CASL algorithm.

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.

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

Sustaining processes from recurrent flows in body-forced turbulence

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Research data supporting "Layer formation in horizontally forced stratified turbulence: connecting exact coherent structures to linear instabilities".

imposed in the third, vertical direction in a fluid with kinematic viscosity

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

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

. 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

Layer formation and relaminarisation in plane Couette flow with spanwise stratification.

mathscr{E}= chi/(chi+mathcal{D})