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Variational determination of minimal absorbing zones in incompressible shear flows
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DOI：10.1017/jfm.2026.11777.png)
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Abstract
En 中文
Absorbing sets are a central organising concept in the long-time analysis of dissipative partial differential equations; especially those arising in fluid mechanics. Roughly; an absorbing set is a bounded region in a suitable function space that eventually contains every trajectory that starts from any bounded set of initial data. Once the dynamics enter this region; they remain controlled in norm; and this ultimate boundedness becomes the gateway to more refined statements: existence of global attractors; finite-dimensional long-time behaviour. A variational framework; based on the background method; is developed to determine an ‘absorbing ball’ in the state space of incompressible shear flows described by the Navier–Stokes equations. This region is defined by the Reynolds–Orr energy identity and is guaranteed to contain all long-term dynamics; including chaotic and non-chaotic attractors. We employ a gradient-based optimisation to find a background flow corresponding to minimal absorbing radius for plane Poiseuille and Couette flows over a range of Reynolds numbers. The optimised background profiles are compared with the turbulent mean flow and they exhibit significantly steeper near-wall gradients than turbulent mean profiles obtained from direct numerical simulations and do not reproduce the universal law of the wall. Despite this quantitative discrepancy; the methodology provides rigorous; provable bounds on the region of state space accessible to the flow dynamics. This offers a novel and promising foundation for improving the global stability limit of the laminar state.
Keywords:
instability
transition to turbulence
variational methods
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3.9
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2.0W
Citations:
9.4W
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