Heteroclinic path to spatially localized chaos in pipe flow

Budanur, Nazmi Burak; Hof, Björn

doi:10.1017/jfm.2017.516

Cited by 19 publications

(25 citation statements)

References 38 publications

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“…The nonlinear terms are evaluated following the 3/2-rule for dealiasing, leading to (768 × 144 × 64) grid points in the physical space. The adequacy of this resolution is confirmed in the previous studies [6,9], which examined similar parameter regimes using Openpipeflow with slightly lower resolutions.As in the ref. [7], the solutions that we consider are invariant under the reflection symmetry and azimuthal rotation by π; i.e.…”

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confidence: 65%

Geometry of transient chaos in streamwise-localized pipe flow turbulence

Budanur¹,

Dogra²,

Hof³

2019

Phys. Rev. Fluids

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In pipes and channels, the onset of turbulence is initially dominated by localized transients which eventually lead to a non-equilibrium phase transition to sustained turbulence. In the present study, we elucidate a state space structure that gives rise to transient chaos. Starting from the basin boundary separating laminar and turbulent flow, we identify transverse homoclinic orbits, presence of which necessitates chaos. The homoclinic tangle on the laminar-turbulent boundary verifies its fractal nature, a property that had been proposed in various earlier studies. By mapping the transversal intersections between the stable and unstable manifold of a periodic orbit, we identify the gateways that promote an escape from turbulence. 05.45.Jn, 47.27.ed In wall-bounded shear flows close to onset, turbulence can be observed as localized patches that coexist with the orderly laminar flow that is stable against infinitesimal disturbances. In the case of pipe flow, these localized structures are known as puffs and they have a simple phenomenology: After a chaotic time-evolution, a puff might suddenly decay and disappear or split and give rise to a new puff. As the governing parameter, the Reynolds number (Re), is increased, the mean puff lifetime before decaying increases while that before a splitting event decline. This phenomenology was recently exploited to conceptualize a spatiotemporal description of the transition [1, 2]: In an infinitely long pipe that is initially populated with many puffs, turbulence will be sustained if the rate of puff splitting exceeds that of decay. Consequently, a critical Re C is defined as the Re, at which both rates are equal.In addition to allowing for a clear definition of the critical Re C , the spatiotemporal dynamics also revealed universal properties of the transition to turbulence: critical exponents were determined in laboratory experiments for Couette flow [3] and in direct numerical simulations for Couette [3] and Waleffe flow [4] and these suggest that the transition falls into the universality class of directed percolation.Key to the above scenario is the existence of spatially discrete chaotic transients. While the transient chaotic nature of the puff dynamics is evident from observations, how such dynamics arise from the Navier-Stokes equations is not well understood. Even though localized asymptotic states [5] and invariant solutions [6,7] at the laminar-turbulent boundary were computed, dynamical routes for sudden puff decay across the stable manifold of such solutions were not identified. In this Letter, we demonstrate the existence of homoclinic orbits in the vicinity of a periodic solution, which itself is on the laminar-turbulent boundary. Based on this observation, we suggest a global picture of the state space, containing a Smale horseshoe, which gives rise to transient chaos with a fractal basin boundary and thus, infinitely many paths for puff decay.We simulate the incompressible fluid flow through a circular pipe of diameter D and length L = πD/0.1256637 ≈ 25...

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confidence: 65%

Geometry of transient chaos in streamwise-localized pipe flow turbulence

Budanur¹,

Dogra²,

Hof³

2019

Phys. Rev. Fluids

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“…This technical step was a straightforward extension of the first Fourier mode slice implementation of ref. [20]. Nevertheless, it was not implemented before and it successfully closes the continuous symmetry reduction problem for pipe flow.…”

Section: Discussionmentioning

confidence: 99%

“…where φ θ is the slice-fixing phase (18), i.e.ã = g θ (φ θ )â. For the derivation of these projection operators, we refer to the appendix of [20].…”

Section: Continuous Symmetry Reductionmentioning

confidence: 99%

Complexity of the laminar-turbulent boundary in pipe flow

Budanur

Hof

2018

Phys. Rev. Fluids

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Over the past decade, the edge of chaos has proven to be a fruitful starting point for investigations of shear flows when the laminar base flow is linearly stable. Numerous computational studies of shear flows demonstrated the existence of states that separate laminar and turbulent regions of the state space. In addition, some studies determined invariant solutions that reside on this edge. In this paper, we study the unstable manifold of one such solution with the aid of continuous symmetry-reduction, which we formulate here for the first time for the simultaneous quotiening of axial and azimuthal symmetries. Upon our investigation of the unstable manifold, we discover a previously unknown traveling wave solution on the laminar-turbulent boundary with a relatively complex structure. By means of low-dimensional projections, we visualize different dynamical paths that connect these solutions to the turbulence. Our numerical experiments demonstrate that the laminar-turbulent boundary exhibits qualitatively different regions whose properties are influenced by the nearby invariant solutions.

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“…For the last two decades, a large number of such solutions, in the form of stationary/travelling waves and periodic orbits, have been found (Nagata 1990;Waleffe 1998;Kawahara & Kida 2001;Waleffe 2003;Faisst & Eckhardt 2004;Wedin & Kerswell 2004;, and many others), and their understanding has played a central role in the recent advance in transition and turbulence at low Reynolds number. These solutions are often called 'exact coherent states', and they characterise the state-space skeleton of turbulence at low Reynolds numbers (Gibson et al 2008;Willis et al 2013Willis et al , 2016Budanur & Hof 2017).…”

Section: Introductionmentioning

confidence: 99%

Exact coherent states of attached eddies in channel flow

2019

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A new set of exact coherent states in the form of a travelling wave is reported in plane channel flow. They are continued over a range in $Re$ from approximately $2600$ up to $30\,000$, an order of magnitude higher than those discovered in the transitional regime. This particular type of exact coherent states is found to be gradually more localised in the near-wall region on increasing the Reynolds number. As larger spanwise sizes $L_{z}^{+}$ are considered, these exact coherent states appear via a saddle-node bifurcation with a spanwise size of $L_{z}^{+}\simeq 50$ and their phase speed is found to be $c^{+}\simeq 11$ at all the Reynolds numbers considered. Computation of the eigenspectra shows that the time scale of the exact coherent states is given by $h/U_{cl}$ in channel flow at all Reynolds numbers, and it becomes equivalent to the viscous inner time scale for the exact coherent states in the limit of $Re\rightarrow \infty$. The exact coherent states at several different spanwise sizes are further continued to a higher Reynolds number, $Re=55\,000$, using the eddy-viscosity approach (Hwang & Cossu, Phys. Rev. Lett., vol. 105, 2010, 044505). It is found that the continued exact coherent states at different sizes are self-similar at the given Reynolds number. These observations suggest that, on increasing Reynolds number, new sets of self-sustaining coherent structures are born in the near-wall region. Near this onset, these structures scale in inner units, forming the near-wall self-sustaining structures. With further increase of Reynolds number, the structures that emerged at lower Reynolds numbers subsequently evolve into the self-sustaining structures in the logarithmic region at different length scales, forming a hierarchy of self-similar coherent structures as hypothesised by Townsend (i.e. attached eddy hypothesis). Finally, the energetics of turbulent flow is discussed for a consistent extension of these dynamical systems notions to high Reynolds numbers.

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Heteroclinic path to spatially localized chaos in pipe flow

Cited by 19 publications

References 38 publications

Geometry of transient chaos in streamwise-localized pipe flow turbulence

Geometry of transient chaos in streamwise-localized pipe flow turbulence

Complexity of the laminar-turbulent boundary in pipe flow

Exact coherent states of attached eddies in channel flow

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