Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

Lucas, Dan; Kerswell, Rich R.

doi:10.1063/1.4917279

Cited by 49 publications

(66 citation statements)

References 28 publications

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“…They are currently found through Newton-GMRES-hook iterations (see, e.g., Kawahara et al (2006);Viswanath (2007); Willis et al (2013); Lucas & Kerswell (2015)) with the drawbacks discussed in the Introduction. Alternatives include the variational method of Lan & Cvitanović (2004) which shares the universally convergent property of our adjoint-based method.…”

Section: Discussionmentioning

confidence: 99%

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Farazmand

2016

J. Fluid Mech.

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We consider the incompressible Navier-Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and traveling wave solutions of the Navier-Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of 100% is observed, leading to the discovery of 21 new steady state and traveling wave solutions at Reynolds number Re = 40. Some of the new invariant solutions have spatially localized structures that were previously believed to only exist on domains with large aspect ratios. We show that one of the newly found steady state solutions underpins the temporal intermittencies, i.e., high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.

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Section: Discussionmentioning

confidence: 99%

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Farazmand

2016

J. Fluid Mech.

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“…As a consequence, properties of the ECS have something to say about how transition is triggered (Viswanath & Cvitanović 2009;Duguet et al 2010;Pringle et al 2012), the subsequent transitional process (Itano & Toh 2001;Skufca et al 2006;Schneider et al 2007;Mellibovsky et al 2009) and features of the turbulent state itself (e.g. Kawahara & Kida (2001); Hof et al (2004); Viswanath (2007); Gibson et al (2008) ;Chandler & Kerswell (2013) ;Willis et al (2013); Lucas & Kerswell (2015)). …”

Section: Introductionmentioning

confidence: 99%

Exact coherent structures in stably stratified plane Couette flow

Olvera

Kerswell

2017

J. Fluid Mech.

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. The existence of exact coherent structures in stably-stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds-Richardson number (ReRi b ) space for a fluid of unit Prandtl number (P r = 1) using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge-tracking -EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (2014) -and found to connect with 2-dimensional convective roll solutions when tracked to negative Ri b (the RayleighBenard problem with shear). Both these states and Nagata's (1990) original exact solution feel the presence of stable stratification when Ri b = O(Re −2 ) or equivalently when the Rayleigh number Ra := −Ri b Re 2 P r = O(1). This is confirmed via a stratified extension of the Vortex-Wave-Interaction (VWI) theory of Hall & Sherwin (2010). If the stratification is increased further, EQ7 is found to progressively spanwise and crossstream localise until a second regime is entered at Ri b = O(Re −2/3 ). This corresponds to a stratified version of the Boundary Reduced Equation (BRE) regime of Deguchi, Hall & Walton (2013). Increasing the stratification further, appears to lead to a third, ultimate regime where Ri b = O(1) in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar-turbulent boundary in the (Re-Ri b ) plane are briefly discussed.

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“…They called this solution a unimodal solution, and suggested that at high Reynolds numbers there is at least one unimodal solution, independent of the number of jets of the original parallel flow. Note that the Kolmogorov problem has been studied from several perspectives including transition to chaos (Platt, Sirovich & Fitzmaurice 1991), the orbital instability of chaotic flow (Inubushi et al 2012) and the extraction of invariant sets from chaotic attractors (Lucas & Kerswell 2015). We will compare our results on a sphere with those of Kolmogorov flows in a planar domain.…”

Section: Introductionmentioning

confidence: 89%

Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere

2015

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We study the bifurcation structure of zonal flows on a rotating sphere. The setting of our problem is similar to the Kolmogorov problem on a flat torus, where the vorticity forcing is given by a single eigenfunction of the Laplacian. First we prove the global stability of two-jet zonal flow for arbitrary Reynolds number and the rotation rate of the sphere. Then we study the bifurcation structure of steady solutions arising from three-jet zonal flow. In the non-rotating case, we find that two steady travelling-wave solutions bifurcate from a three-jet zonal flow via Hopf bifurcation. As the Reynolds number increases, steady-travelling solutions arise via pitchfork bifurcation from the steady-travelling solutions. On the other hand, in the rotating case, we find saddlenode bifurcations and closed-loop branches. We carry out time integration to study the properties of unsteady solutions at high Reynolds numbers. In the non-rotating case, the unsteady solution is chaotic and it wanders around the steady-travelling solutions bifurcating from three-jet zonal flow. We show that a linear combination of the steady and steady-travelling solutions gives a good approximation of the zonal-mean zonal flow of the unsteady solution, suggesting that the chaotic solution at high Reynolds numbers exists mostly within a relatively low-dimensional space spanned by the steady and steady-travelling solutions, which become unstable at low Reynolds numbers.

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Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow

Cited by 49 publications

References 28 publications

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

An adjoint-based approach for finding invariant solutions of Navier–Stokes equations

Exact coherent structures in stably stratified plane Couette flow

Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere

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