Prandtl–Batchelor theorem for flows with quasiperiodic time dependence

Arbabi, Hassan; Mezić, Igor

doi:10.1017/jfm.2018.998

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…An important feature of these time-dependent flows is that at very high Reynolds numbers the vorticity approaches a uniform distribution in their rotating core. This is known as Prandtl-Batchelor theorem (Prandtl 1904;Batchelor 1956) and was recently extended to unsteady flows by Arbabi & Mezić (2019). In this paper, we show that this uniform distribution of vorticity leads to uniform distribution of Lagrangian time periods in the mean flow, and that leads to weaker mixing in the core, compared to areas adjacent to the walls and secondary vortices.…”

Section: Introductionmentioning

confidence: 53%

“…The classical version of this theory states that in regions of the flow with closed streamlines and small viscous forces, the vorticity will be constant (Prandtl 1904;Batchelor 1956). In a previous work, we extended this theory to stationary time-dependent flows (Arbabi & Mezić 2019). The unsteady version of theory holds for any recirculating structure that may move with the flow, however, in the lid-driven cavity flow the location of the central vortex is almost fixed in time, and therefore averaging in time preserves the uniform distribution of vorticity in the core.…”

Section: Slow Mixing In the Core And Prandtl-batchelor Theoremmentioning

confidence: 99%

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Spectral analysis of mixing in 2D high-Reynolds flows

Arbabi,

Mezic

2019

Preprint

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We use spectral analysis of Eulerian and Lagrangian dynamics to study the advective mixing in an incompressible 2D bounded cavity flow. A significant property of this flow at high Reynolds numbers is that mixing in its rotational core is slower than wall-adjacent areas and corner eddies. We explain this property by appealing to the Prandtl-Batchloer theorem for unsteady flows which predicts a flat distribution of tracer circulation periods in the core of mean flow -similar to rigid body rotation -at high Reynolds numbers. When this inviscid core is exposed to velocity fluctuations arising from bifurcation at high Reynolds, it shows more resilience toward resonance in Lagrangian motion and hence mixes more slowly compared to other areas.We also investigate how well the truncations of Koopman mode decomposition and proper orthogonal decomposition approximate the mixing process in the flow. For periodic and quasi-periodic flows, the mixing is accurately explained by models comprising of a few Koopman modes, while for flows with aperiodic time dependence, the number of modes required to represent the mixing is substantially larger.

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Section: Introductionmentioning

confidence: 53%

Section: Slow Mixing In the Core And Prandtl-batchelor Theoremmentioning

confidence: 99%

Spectral analysis of mixing in 2D high-Reynolds flows

Arbabi,

Mezic

2019

Preprint

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Lagrangian chaos in steady three-dimensional lid-driven cavity flow

Romanò

TÜRKBAY>

Kuhlmann

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Steady three-dimensional flows in lid-driven cavities are investigated numerically using a high-order spectral-element solver for the incompressible Navier–Stokes equations. The focus is placed on critical points in the flow field, critical limit cycles, their heteroclinic connections, and on the existence, shape, and dependence on the Reynolds number of Kolmogorov–Arnold–Moser (KAM) tori. In finite-length cuboidal cavities at small Reynolds numbers, a thin layer of chaotic streamlines covers all walls. As the Reynolds number is increased, the chaotic layer widens and the complementary KAM tori shrink, eventually undergoing resonances, until they vanish. Accurate data for the location of closed streamlines and of KAM tori are provided, both of which reach very close to the moving lid. For steady periodic Taylor–Görtler vortices in spanwise infinitely extended cavities with a square cross section, chaotic streamlines occupy a large part of the flow domain immediately after the onset of Taylor–Görtler vortices. As the Reynolds number increases, the remaining KAM tori vanish from the Taylor–Görtler vortices, while KAM tori grow in the central region further away from the solid walls.

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Prandtl–Batchelor Theory for Kolmogorov Flows

Kim

Okamoto

2020

J. Phys. Soc. Jpn.

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Prandtl–Batchelor theorem for flows with quasiperiodic time dependence

Cited by 3 publications

References 15 publications

Spectral analysis of mixing in 2D high-Reynolds flows

Spectral analysis of mixing in 2D high-Reynolds flows

Lagrangian chaos in steady three-dimensional lid-driven cavity flow

Prandtl–Batchelor Theory for Kolmogorov Flows

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