Bifurcation and nonlinear evolution of convection in binary fluid mixtures with weak Soret effect

Abstract

Rayleigh-Bénard (RB) convection in binary fluid mixtures, which shows rich and interesting pattern formation behavior, is a paradigm for understanding instabilities, bifurcations, self-organization with complex spatiotemporal behavior and turbulence, with many applications in atmospheric and environmental physics, astrophysics, and process technology. In this paper, by using a high-order compact finite difference method to solve the full hydrodynamic field equations, we study numerically the RB convection in binary fluid mixtures such as ethanol-water with a very weak Soret effect (separation ratio <inline-formula><tex-math id="M2">\begin{document}$\psi=-0.02$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191836_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191836_M2.png"/></alternatives></inline-formula>) in a rectangular container heated uniformly from below. The direct numerical simulations are conducted in the rectangular container with aspect ratio of <inline-formula><tex-math id="M3">\begin{document}$\varGamma=12$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191836_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191836_M3.png"/></alternatives></inline-formula> and with four no-slip and impermeable boundaries, isothermal horizontal and perfectly insulated vertical boundaries. The bifurcation and the origin and evolution of pattern in RB convection for the considered physical parameters are studied, and the bifurcation diagram is presented. By performing two-dimensional simulations, we observe three stable states of Blinking state, localized traveling wave and stationary overturning convection (SOC) state, and discuss the transitions between them. The results show that there is a hysteresis in the transition from the Blinking state to the localized traveling wave state for the considered separation ratio, and the evolution of the oscillation frequency, convection amplitude and Nusselt number are discontinuous. Near the lower bound of the Rayleigh number range where the Blinking state exists, a asymmetric initial disturbance is the inducement for the formation of the Blinking state. Inside the range, its inducing effect is weakened, and the oscillatory instability becomes the main reason. It is further confirmed that reflections of lateral walls are responsible for the survival of the stable Blinking state. With the increase of the Rayleigh number, the critical SOC state undergoes multiple bifurcations and forms multiple SOC states with different wave numbers, and then transitions to a chaotic state. There are no stable undulation traveling wave states at both ends of the critical SOC branch.