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

Abstract

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.

Stokesian motion of a spherical particle near a right corner made by tangentially moving walls

“…This resembles a localised three-dimensional version of a two-dimensional Moffatt eddy (Moffatt 1964) and is, in fact, a three-dimensional saddle focus, similar to what is found in a three-dimensional lid-driven cavity flow (see e.g. figure 6 c in Romanò, Türkbay & Kuhlmann (2020 b )). Obviously, the small counter-rotating vortex in figure 4( d ) does not arise in the case of a sphere in its fixed point when the wall moves towards the edge (figure 4 a – c ), because the velocity a fixed sphere rotating with the same angular velocity induces at the periphery of the corner eddy is much smaller than the velocity induced by the moving wall at the same location.…”

Attractors for the motion of a finite-size particle in a two-sided lid-driven cavity

“…However, the size, location, and structure of the KAM tori are different due to the curvature of the moving walls. For the present system with curved moving walls we have reconstructed the KAM tori at Re = 400 in the same Romanò, Türkbay & Kuhlmann (2020). Owing to the point symmetry of the flow in a single cell, all KAM tori arise as two point-symmetric sets of tori.…”

“…For the present system with curved moving walls we have reconstructed the KAM tori at in the same way as in Romanò et al. (2017), Romanò, Hajisharifi & Kuhlmann (2017) and Romanò, Türkbay & Kuhlmann (2020). Owing to the point symmetry of the flow in a single cell, all KAM tori arise as two point-symmetric sets of tori.…”

Experimental and numerical investigation of the flow in a cylindrical cavity with an unsteady rotating lid

“…The Froude number and capillary number extinct at the limits of increasing Reynold's numbers. The chaotic mixing in lid-driven cavities has been analysed by [18,19]. They found that in general terms, at lower Reynolds number the chaotic mixing can be more important than higher Reynolds numbers.…”