Rayleigh-Bénard convection of a gas-vapor mixture with abnormal dependence of thermal expansion coefficient on temperature

Abstract

In order to reveal basic characteristics of Rayleigh-Bénard convection (RBC) of a gas mixture, which has abnormal dependence of thermal expansion coefficient on temperature, we presented a direct numerical simulation on RBC of a gas mixture of oxygen and cyclohexane near its maximum density. The influences of the density inversion parameter and the Rayleigh (Ra) number on the critical conditions of the flow onset and destabilization were analyzed. The flow evolution with Ra was exhibited. The results show the critical value of Rayleigh number of the flow onset is approximately proportional to the fourth power of the thickness of the upper stable sublayer. Regardless of density inversion parameter, the flow pattern at the flow appearance always has a multicellular structure. With increasing Ra, the multicellular flow structure would be transformed into various steady flow patterns, including parallel flow and pendant pattern. When the density inversion parameter is small, the unsteady flow cells exhibit a back and forth motion along the border. However, at a large density inversion parameter, the cells swing around the centers of the flow rolls. We also found that the local Nusselt (Nu) number on the bottom depends on the flow pattern. The average Nu increases gradually with increasing Ra under a small density inversion parameter. However, an abrupt rise of the average Nu appeared under a large density inversion parameter.

Introduction

Rayleigh-Bénard convection (RBC) is a flow driven by the buoyancy force in a fluid layer with a vertical temperature gradient. Therefore, the dependence of fluid density on temperature is critical for the onset of RBC and the evolution of the flow patterns with the Rayleigh (Ra) number. Up to now, many studies on RBC of the fluids with a linear variation of the density with temperature have been carried out [[1], [2], [3], [4], [5]]. However, commonly encountered fluids with a non-monotonic variation of the density with the temperature, like cold water [6] and the gas mixture of oxygen and cyclohexane [7], have not been studied yet.

For the fluid with a linear variation of the density with temperature, convection cells of RBC appear in the fluid layer when Rayleigh number exceeds the first critical value. The spatial variation of a convection structure is referred to as flow pattern that depends on the Prandtl number of the fluid, the enclosure geometry and the boundary conditions of the fluid layer. With the increasing Ra, the buoyancy force is enhanced. When Ra exceeds the second threshold value, the steady flow pattern will destabilize and bifurcate to three-dimensional (3D) unstable flows. Furthermore, the boundaries of the stability region of the flow patterns are determined by various destabilization mechanisms, which is known as the Busse balloon [1,8].

For the fluid with a non-monotonic variation density with regard to temperature, the buoyancy force will change its sign along the direction vertical to the fluid layer due to an imposed vertical temperature gradient, which results in a variety of convection modes. Zubkov et al. [9,10] numerically calculated RBC of water near the density maximum in a cubical cavity with isothermal horizontal walls and thermally insulated vertical walls. It was found that there are six stable flow patterns at different Grashof numbers. Furthermore, the ranges of Grashof number in which all the flow patterns coexist were determined. Large and Andereck [11] performed an experiment on RBC of cold water in a cavity. It was revealed that the flow first appears in the lower unstable layer, and then penetrates the upper stable region. At onset of the flow, the flow includes parallel and transverse rolls. When the unstable layer height is much smaller than the stable layer height, the flow only partially penetrate the upper stable region, within which weak counter-rotation flows are induced. With increasing Rayleigh number, the flow undergoes a secondary transition to a hexagonal cellular or a longitudinal roll state. Hu et al. [12,13] carried out a direct numerical simulation on RBC on cold water in various vertical cavities. Results show that the density inversion has an important effect on RBC. The critical Ra for the flow onset is larger than that of the common fluid, and it increases with increasing density inversion parameter Θ_m. The density inversion parameter is used to describe the location of the maximum density in the cavity and is defined as

$${\mathrm{\Theta }}_m=\frac{T_m-T_c}{T_h-T_c},$$

where T_h and T_c are the bottom and top temperatures of the liquid layer, respectively. T_m is the maximum density temperature, and T_h > T_m ≥ T_c.

Furthermore, Bekezhanova [14] analyzed the stability of the equilibrium state of a horizontal cold water layer with nonlinear temperature and pressure dependences of density. Lyubimov et al. [15] determined the influence of a given heat flux boundary condition on the convection onset in a cold water layer. Furthermore, Toppaladoddi and Wettlaufer [16] studied the turbulent penetrative RBC of the density extremum fluids. It was found that the penetration depth decreases with the density inversion parameter increasing. At the same time, Nusselt (Nu) number fluctuation amplifies with increasing Ra.

The density maximum phenomenon also appears in some gas-vapor mixtures at the critical temperature T_m, such as a gas mixture of oxygen and cyclohexane. Recently, Palymskly et al. [17,18] proposed a theoretical model of RBC of oxygen and cyclohexane mixture with the density maximum., and performed a linear stability analysis. However, for simplicity they used constant thermal expansion coefficients of β = −0.01583 and 0.0030537 at T ≤ T_m and T > T_m, respectively. Recently, based on the theoretical model proposed by Palymskly et al., Yu et al. [19] carried out two-dimensional numerical simulation on RBC of the oxygen and cyclohexane mixture. The results show that the strength of RBC is suppressed with the density inversion parameter increasing. However, two-dimensional simulation cannot show 3D flow structure and reveal the formation mechanism of the flow pattern.

In this work, a series of direct numerical simulations were performed to reveal the characteristics of RBC of the oxygen and cyclohexane mixture near the density maximum point. The thermal and flow fields at different conditions were exhibited. The flow evolution with Ra was emphasized.