Quasi-periodicity and chaos in a differentially heated cavity

Abstract

Convective flows of a small Prandtl number fluid contained in a two-dimensional vertical cavity subject to a lateral thermal gradient perpendicular to a constant gravity, are studied numerically. The chosen geometry and the values of the material parameters are relevant to semiconductor crystal growth experiments in the horizontal configuration of the Bridgman method. For increasing Rayleigh numbers we find a transition from a steady flow to periodic solutions through a supercritical Hopf bifurcation that maintains the centro-symmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf Bifurcation, the periodic solution loses stability in a subcritical Neimark-Sacker bifurcation, which gives rise to a branch of quasiperiodic states. In this branch, several intervals of frequency locking have been identified. Inside the resonance horns the stable limit cycles lose and gain stability via some typical scenarios in the bifurcation of periodic solutions. From a complicated bifurcation diagram of the stable limit cycle of the 1:10 resonance horn, a branch of chaotic solutions is obtained.

Bifurcations and chaos in single-roll natural convection with low Prandtl number

“…They are of the type 1 / q: namely 1:12, 1:11, and 1:10. Inside these resonance horns the stable limit cycles lose and gain stability via some typical scenarios of bifurcations of periodic solutions, 31 similar, for instance, to those found in Refs. 16 and 30.…”

“…The associated quasiperiodic and periodic solutions have already been described in great detail in Ref. 31. We will describe here the main lines of the dynamics on this branch of solutions.…”

Natural convection in a horizontal cylinder with axial rotation

“…We expect that the complex bifurcation scenarios we have identified for moderate values of the Rayleigh number and slow rotation rates lead to complex spatiotemporal dynamics, as it occurred in other analogous two-dimensional laterally heated systems, with similar aspect ratios and comparable values of the Prandtl number, that we had analyzed in the past [8][9][10][11]. However, a complete study of this emerging nonlinear dynamics would merit further study and is beyond the scope of this paper.…”

“…For this reason, many studies have focused on the study of the oscillatory threshold in low-Prandtl-number fluids in different geometrical configurations [2][3][4][5][6][7]. The system is also interesting from a fundamental fluid dynamics point of view, since it exhibits a rich nonlinear behavior that leads to complex spatiotemporal dynamics [8][9][10][11]. A bounded cylinder can be used to represent realistically the melt zone in the horizontal Bridgman crystal growth process.…”

Secondary flows in a laterally heated horizontal cylinder

“…2 The system is also interesting from a fundamental fluid dynamics point of view, since it exhibits a rich nonlinear behavior that leads to complex spatiotemporal dynamics. 3 Some simplified geometries have been considered in the theoretical study of natural convection induced by a horizontal temperature gradient. In one of such simplifications a channel is considered; the container is supposed to be unbounded in one horizontal direction, and a constant horizontal temperature gradient is applied to the lateral boundaries.…”