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

Area of Science:

• Fluid Dynamics
• Computational Physics
• Nonlinear Dynamics

Background:

• Lid-driven cavities are fundamental models for studying fluid flow phenomena.
• Understanding the transition to chaos and the role of invariant tori is crucial in fluid mechanics.

Purpose of the Study:

• To numerically investigate steady three-dimensional flows in lid-driven cavities.
• To analyze critical points, limit cycles, and Kolmogorov-Arnold-Moser (KAM) tori.
• To determine the dependence of KAM tori on the Reynolds number.

Main Methods:

• Utilized a high-order spectral-element solver for incompressible Navier-Stokes equations.
• Focused on analyzing critical points, limit cycles, and KAM tori in the flow field.
• Examined flow behavior in both finite-length cuboidal and infinite spanwise square cross-section cavities.

Main Results:

• At low Reynolds numbers, chaotic streamlines form thin layers near walls, while KAM tori shrink as Reynolds number increases.
• KAM tori undergo resonances and eventually vanish, with chaotic layers widening.
• In Taylor-Görtler vortices, KAM tori initially shrink near walls and then grow in the central region.

Conclusions:

• The study provides accurate data on the location of closed streamlines and KAM tori, extending close to the moving lid.
• Flow dynamics in lid-driven cavities exhibit complex transitions involving chaotic streamlines and invariant tori.
• The behavior of KAM tori is highly dependent on the Reynolds number and cavity geometry.