Complex dynamics in double-diffusive convection

Abstract

The dynamics of a small Prandtl number binary mixture in a laterally heated cavity is studied numerically. By combining temporal integration, steady state solving and linear stability analysis of the full PDE equations, we have been able to locate and characterize a codimension-three degenerate Takens-Bogdanov point whose unfolding describes the dynamics of the system for a certain range of Rayleigh numbers and separation ratios near S=-1.Comment: 8 pages, 5 figure

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

“…This algorithm has been used previously by some of the authors to study binary mixtures. [18][19][20] The method employs a pressure boundary condition, which in conjunction with stiffly stable schemes prevents propagation and accumulation of time differencing errors. 21 On Chebyshev collocation points, the Helmholtz and Poisson equations resulting from the time splitting are solved efficiently by using a complete diagonalization of operators in both directions.…”

Transitions between symmetric and nonsymmetric regimes in binary-mixture convection

“…This conjunction of codimension-two bifurcations has been found in many problems of very different nature, from systems biology [22] to lasers [23]. In [3] it was found that this conjunction and all the low-Rayleigh number dynamics could be described as a part of the unfolding of a degenerate (codimension-three) Takens-Bogdanov point, the saddle case found by Dumortier et al [24].…”

“…Luckily, as in Ref. [3], the topology of the bifurcation lines could be rationalized as associations of codimension-two bifurcations that could be linked to higher codimension phenomena, which in fact resulted to be a guide to find and organize them. We think that such connections constitute an important result of this study.…”

“…To explore the genericity of the results of [2] and to bridge the gap with the pure fluid case, a study of the successive transformations of the global connection that gives birth to the aforementioned low-frequency nonsymmetric periodic solution was undertaken in [3]. It was found that this global connection changed in nature from a saddle-loop to a SNIC and then again to a saddle-loop, finally to become a Hopf bifurcation in a Takens-Bogdanov (TB) point.…”

Continuation of travelling-wave solutions of the Navier–Stokes equations

“…We have used this method in a second-order time-integration scheme to analyse two-dimensional oscillatory binary-uid convection, both in large aspect ratio containers heated from below [16] and in laterally heated cavities [17]. In this last problem, a ÿrst-order time-stepping formulation to carry out Newton's method, as described above, was used to calculate the steady solutions [18].…”