Turbulent Rayleigh-Bénard convection of cold water near its maximum density in a vertical cylindrical container
Introduction
Rayleigh-Bénard convection is a classical system of thermal convection, in which the fluid is heated from the bottom and cooled on the top by horizontal uniform boundary temperatures or heat fluxes [1], [2], [3]. Rayleigh-Bénard convection plays a crucial role in nature and many engineering fields, such as the convection in the atmosphere [4], in the earth [5], in the electronic devices and in the solar collectors and so on [6].
As early as 1987, Heslot et al. [7] and Castaing et al. [8] in the University of Chicago experimentally studied Rayleigh-Bénard convection of helium at low temperature. They observed a soft turbulence state (2.5 × 105 < Ra < 4 × 107) and a hard turbulence state (4 × 107 < Ra < 6 × 1012). Then, many researchers paid their attention to soft and hard turbulence states in Rayleigh-Bénard convection by numerical and experimental methods. Peng et al. [9] analyzed the scaling relationship in an open-ended domain by Large Eddy Simulation (LES), where the Rayleigh (Ra) number varied from 6.3 × 105 to 109. They obtained a single relationship for the variation of Nusselt (Nu) number with Rayleigh number where the exponent is 0.286. Choi and Kim [10] numerically studied the scale of Rayleigh number with Nusselt number in the soft turbulence state (2 × 106 < Ra < 4 × 107) and the hard turbulence state (108 < Ra < 109) with the elliptic-blending second-moment closure. They predicted that Nusselt number follows two different correlations in soft and hard turbulence states. Shibata [11] calculated the heat conductivity in turbulent Rayleigh-Bénard convection. They found that the heat conductivity diverges stronger in the hard turbulence state than that in the soft turbulence state. Riedinger et al. [12] measured the heat flux in a slightly titled channel. They found that different flow regimes develop from a soft turbulence state to a hard turbulence state depending on the increase of the angle and the applied power. Qiu and Tong [13], [14] experimentally studied the coherent events in an aspect-ratio-one cylindrical cell. A sharp transition from a random chaotic state to a correlated turbulence state is found when Rayleigh number exceeds 5 × 107, which offers new perceptions on the soft and hard turbulence states.
The above numerical simulation studies on Rayleigh-Bénard convection adopted the Oberbeck-Boussinesq approximation [15], [16], where the density of the fluid is assumed to be linear function of temperature. However, the density has the maximum value near 4 °C for the cold water, in which the Oberbeck-Boussinesq approximation is no longer applicable. This maximum density phenomenon changes the flow dynamics and heat transfer characteristics significantly. If the maximum density point is located between the bottom and top walls, the fluid in the convection cell could be divided into an unstable layer and a stratified stable layer. Convection occurs in the lower unstable fluid layer, the fluid motion penetrates to the upper stable layer. Therefore, it is called as penetrative convection. Veronis [17] employed the perturbation method to investigate how far the fluid motion penetrated into the stable layer and put forward the term “penetrative convection”. Kuznetsova and Sibgatullin [18] described the evolution from conductive state to chaos in penetrative convection with the increase of Rayleigh number. A series of flow bifurcations were determined. Large and Andereck [19] carried out some experimental investigations on penetrative Rayleigh-Bénard convection in water near its maximum density point. The results showed that symmetric flow structures are formed if the whole layer is unstable. However, the symmetry will be destroyed by the existence of the stable layer. Mastiani et al. [20] and Hu et al. [21] investigated heat transfer characteristics of penetrative Rayleigh-Bénard convection. They found that heat transfer rate in penetrative Rayleigh-Bénard convection is lower than that in classical Rayleigh-Bénard convection.
The above existing literatures on penetrative convection are primarily concerned with laminar flow. Little attention has been paid to turbulent penetrative convection. Furthermore, there are a few investigations on turbulent penetrative convection in Rayleigh-Bénard system. In the present paper, we reported a series of the simulation results on turbulent Rayleigh-Bénard convection of cold water near its maximum density value by LES.
Section snippets
Physical and mathematical model
We consider a cylindrical container with diameter D and height H. It is filled with cold water near its maximum density. The aspect ratio of the cylindrical container is D/H = 1. Constant temperatures Tc and Th (Tc < Th) are applied at the top and bottom walls, respectively. The sidewall is adiabatic. No-slip and impermeable conditions are imposed on all walls.
The density of cold water near 4 °C is described as follows [22],where γ = 9.297173 × 10−6 (°C)−q, q = 1.894 816, ρm = 999.972 kg/m
Effect of density inversion parameter
The flow behavior of Rayleigh-Bénard convection of cold water near its maximum density is rich, varied and complicated. This section commits to study the flow dynamics at different density inversion parameters when Rayleigh number is fixed at Ra = 109.
Up to now, the formulation mechanism of large scale circulation (LSC) in Rayleigh-Bénard convection of common fluids has been widely studied [27], [29]. Xi et al. [30] reported an experimental result on Rayleigh-Bénard convection from the steady
Conclusions
Large Eddy Simulation with WALE sub-gird model was performed for Rayleigh-Bénard convection of cold water with the effects of the maximum density. Based on the simulation results, the following conclusions can be drawn.
- (1)
At small density inversion parameter, the LSC is driven by hot and cold plumes. With the increase of density inversion parameter, the act of cold plumes on the LSC gradually diminishes until actually disappears. The temperature in the bulk region for all density inversion
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant No. 51376199).
Conflict of interest
No.
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