Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Abstract

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in RayleighBénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading order Lyapunov exponent and we quantify the details of their response to the dynamics of defects. The leading order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

Three-dimensional coherent structures of electrokinetic instability

“…The flow becomes a combination of unsteady hexagons, quadrangles, and triangles. In the present investigation, the advent of spatiotemporal chaos is found by visual inspection; in a future extension of this work, we intend to detect it by a calculation of the maximal Lyapunov exponent, as was done in [20].…”

“…The two-dimensional electroconvective rolls generated by surfaces bearing a charge varying in space are considred in [19]. The electrokinetic instability is reminiscent of Rayleigh-Bénard convection [20], but from both the physical and mathematical points of view, it is much more complicated. The Reynolds number in the electrokinetic instability is very small and, hence, the dissipation is very large and the nonlinear terms in the Navier-Stokes system are negligibly small.…”

Route to hyperchaos in Rayleigh-Bénard convection

“…In [19], a bifurcation diagram is presented where a series of steady states representing different patterns is numerically obtained as a function of Ra in a small cylindrical domain. Also in cylindrical domains, hyperchaotic states were found in studies of spiral defect chaos using simulations of three-dimensional Rayleigh-Bénard convection, where the spectrum of Lyapunov exponents was used to quantify extensivity in spatiotemporal chaos [20,21].…”

Gyrotactic bioconvection at pycnoclines

“…The advantage of parallel computing is that it provides us with the capability of simulating bioconvection in large-aspect-ratio chambers with realistic boundary conditions. This code has been used and verified in several numerical simulations of Rayleigh-Bénard convection discussed in the literature (Paul et al 2001(Paul et al , 2003Karimi & Paul 2012). Also, it has been employed in various simulations of bioconvection phenomenon (Karimi 2012;Karimi & Paul 2013) and its outcomes have been compared with the results of Ghorai & Hill (2007) in terms of the characteristics of the flow and the cell concentration.…”