Coarse Graining the State Space of a Turbulent Flow Using Periodic Orbits

Gökhan, Yalnız,; Hof, Björn; Budanur, Nazmi Burak

doi:10.1103/physrevlett.126.244502

Cited by 30 publications

(26 citation statements)

References 31 publications

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“…(Cvitanovic et al. 2005), as it has been recently done by Yalnız & Budanur (2020); Yalnız, Hof & Budanur (2021) using algebraic topology techniques.…”

Section: Discussionmentioning

confidence: 98%

Unveiling the competitive role of global modes in the pattern formation of rotating sphere flows

et al. 2022

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The wake flow past a streamwise rotating sphere is a canonical model of numerous applications, such as particle-driven flows, sport aerodynamics and freely rising or falling bodies, where the changes in particles’ paths are related to the destabilization of complex flow regimes and associated force distributions. Herein, we examine the spatio-temporal pattern formation, previously investigated by Lorite-Díez & Jiménez-González (J. Fluid Mech., vol. 896, 2020, A18) and Pier (J. Fluids Struct., vol. 41, 2013, pp. 43–50), from a dynamical system perspective. A systematic study of the mode competition between rotating waves, which arise from the linearly unstable modes of the steady-state, exhibits their connection to previously observed helical patterns present within the wake. The organizing centre of the dynamics turns out to be a triple Hopf bifurcation associated with three non-axisymmetric, oscillating modes with respective azimuthal wavenumbers $m=-1,-1$ and $-2$ . The unfolding of the normal form unveils the nonlinear interaction between the rotating waves to engender more complex states. It reveals that for low values of the rotation rate, the flow field exhibits a similar transition to the flow past the static sphere, but accompanied by a rapid variation of the frequencies of the flow with respect to the rotation. The transition from the single helix pattern to the double helix structure within the wake displays several regions with hysteric behaviour. Eventually, the interaction between single and double helix structures within the wake lead towards temporal chaos, which here is attributed to the Ruelle–Takens–Newhouse route. The onset of chaos is detected by the identification of an invariant state of the normal form constituted by three incommensurate frequencies. The evolution of the chaotic attractor is determined using of time-stepping simulations, which were also performed to confirm the existence of bi-stability and to assess the fidelity of the computations performed with the normal form.

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“…(Cvitanovic et al. 2005), as it has been recently done by Yalnız & Budanur (2020); Yalnız, Hof & Budanur (2021) using algebraic topology techniques.…”

Section: Discussionmentioning

confidence: 98%

Unveiling the competitive role of global modes in the pattern formation of rotating sphere flows

et al. 2022

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“…Suri et al (2020) found seven numerically converged periodic orbits and showed that these closely matched the dynamics in an experimental configuration of Kolmogorov flow. Recently, Yalníz, Hof & Budanur (2021) used an existing Newton-hookstep method to find 18 periodic orbits in three-dimensional Kolmogorov flow, which was sufficient to reduce the dynamics of the system to a remarkably simple Markov chain model. We also concentrate on the case of 2-D Kolmogorov flow, for the reasons listed above, as well as for comparison with this previous work.…”

Section: Introductionmentioning

confidence: 99%

Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

Parker

Schneider

2022

J. Fluid Mech.

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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.

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“…Hence, it appears that controlling UPOs in spatio-temporal systems whose dynamics are not completely described by finite-dimensional dynamics require nontrivial extensions to this method that will be the subject of a follow-up investigation. Finally, there are a number of compelling applications that may benefit from this control approach, including space mission design [16][17][18] and fluid flow control [6][7][8][9][10][11][12][13][14][15]. These topics are the subject of ongoing work, although SINDy has recently been applied to several fluid flows with promising results [68][69][70].…”

Section: Discussionmentioning

confidence: 99%

Data-Driven Stabilization of Periodic Orbits

Bramburger,

Kutz,

Brunton

2020

Preprint

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Periodic orbits are among the simplest non-equilibrium solutions to dynamical systems, and they play a significant role in our modern understanding of the rich structures observed in many systems. For example, it is known that embedded within any chaotic attractor are infinitely many unstable periodic orbits (UPOs) and so a chaotic trajectory can be thought of as 'jumping' from one UPO to another in a seemingly unpredictable manner. A number of studies have sought to exploit the existence of these UPOs to control a chaotic system. These methods rely on introducing small, precise parameter manipulations each time the trajectory crosses a transverse section to the flow. Typically these methods suffer from the fact that they require a precise description of the Poincaré mapping for the flow, which is a difficult task since there is no systematic way of producing such a mapping associated to a given system. Here we employ recent model discovery methods for producing accurate and parsimonious parameterdependent Poincaré mappings to stabilize UPOs in nonlinear dynamical systems. Specifically, we use the sparse identification of nonlinear dynamics (SINDy) method to frame model discovery as a sparse regression problem, which can be implemented in a computationally efficient manner. This approach provides an explicit Poincaré mapping that faithfully describes the dynamics of the flow in the Poincaré section and can be used to identify UPOs. For each UPO, we then determine the parameter manipulations that stabilize this orbit. The utility of these methods are demonstrated on a variety of differential equations, including the Rössler system in a chaotic parameter regime.

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Coarse Graining the State Space of a Turbulent Flow Using Periodic Orbits

Cited by 30 publications

References 31 publications

Unveiling the competitive role of global modes in the pattern formation of rotating sphere flows

Unveiling the competitive role of global modes in the pattern formation of rotating sphere flows

Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

Data-Driven Stabilization of Periodic Orbits

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