Quasiperiodic routes to chaos in confined two-dimensional differential convection

Abstract

The complete cascade of bifurcations from steady to chaotic convection, as the Rayleigh number is varied, is considered numerically inside an air-filled differentially heated cavity. The system is assumed to be two-dimensional and is invariant under a generalized reflection about the center of the cavity. In the neighborhood of several codimension-two points, two main routes emerge, characterized by different symmetries of the first oscillatory eigenstate. Along these two competing routes, different sequences of bifurcations and symmetry breakings lead from the steady base flow to the hyperchaotic regime. Several families of two- and three-frequency tori have been identified via the computation of the leading Lyapunov exponents. Modal structures extracted from time series reveal the occurrence of slow internal oscillations in the center of the cavity and faster wall modes confined to vertical boundary layers. Further quasiperiodicity windows have been detected on each route. The different regimes eventually disappear in a boundary crisis in favor of a single, globally symmetric, hyperchaotic regime.

Onset of chaos in helical vortex breakdown at low Reynolds number

“…Then a quasiperiodic regime with three incommensurable frequencies is identified at a Reynolds value of Re = 205. Sensitivity to the initial condition suggests that the present attractor is a stable torus T 3 , similar to that in [46]. The interpretation of this regime as a stable torus T 3 is not in contradiction with the Ruelle-Takens-Newhouse route to chaos since the latter does not exclude such nonchaotic states [47].…”

“…This scenario is a typical Ruelle-Takens-Newhouse route to chaos, which has already appeared in several 064701-13 hydrodynamic examples (see, e.g., [18,20,25,44,45]). In particular, Oteski et al [46] reported this route to chaos including a stable three-frequency torus T 3 in a confined two-dimensional differential convection case. In the Ruelle-Takens-Newhouse scenario, the onset of the transition from a torus T 3 to a strange attractor is not predictable [47].…”

Periodic orbits in tall laterally heated rectangular cavities

“…A complete description of the transition to turbulence is given in Ref. [10] for air-filled 2D containers of Γ = 2, under a dynamical systems point of view. Two main routes that gradually break the symmetries of the solutions as sets were found.…”

Non-Hyperbolic Transport in Confined Oscillatory Convection

“…We focus in this article on a model convection flow inside a two-dimensional cavity filled with air, heated on one lateral wall and cooled on the opposite wall, while the top and bottom walls are considered adiabatic. The cavity has a geometrical as-pect ratio height/width of two, and the flow is known to become oscillatory beyond a given value of the Rayleigh number Ra = Ra c = 1.5865 × 10 8 , where Ra is proportional to the temperature difference between the two walls 8 . Chaotic advection of non-diffusive tracers in the oscillatory regime has been considered numerically by Oteski et al 5 .…”