Cellular flow in a partially filled rotating drum: regular and chaotic advection

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here, we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous) nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. To prove these results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to highly compressible flow, governed by the Euler–Poisson equations, a case of cosmological relevance. The recurrence relations associated with the Lagrangian formulation of these incompressible and compressible problems are so closely related that the proofs of time-analyticity are basically identical.

Quasi-steady linked vortices with chaotic streamlines

This paper describes the motion and the flow geometry of two or more linked ring vortices in an otherwise quiescent, ideal fluid. The vortices are thin tubes of near-circular shape which lie on the surface of an immaterial torus of small aspect ratio. Since the vortices are assumed to be identical and evenly distributed on any meridional section of the torus, the flow evolution depends only on the number of vortices ($n$) and the torus aspect ratio (${r}_{1} / {r}_{0} $, where ${r}_{0} $ is the centreline radius and ${r}_{1} $ is the cross-section radius). Numerical simulations based on the Biot–Savart law showed that a small number of vortices ($n= 2, 3$) coiled on a thin torus (${r}_{1} / {r}_{0} \leq 0. 16$) progressed along and rotated around the symmetry axis of the torus in an almost uniform manner while each vortex approximately preserved its shape. In the comoving frame the velocity field always possesses two stagnation points. The transverse intersection, along $2n$ streamlines, of the stream tube emanating from the front stagnation point and the stream tube ending at the rear stagnation point creates a three-dimensional chaotic tangle. It was found that the volume of the chaotic region increases with increasing torus aspect ratio and decreasing number of vortices.