Turbulent Rayleigh-Bénard convection of cold water near its maximum density in a vertical cylindrical container

doi:10.1016/j.ijheatmasstransfer.2017.09.021

International Journal of Heat and Mass Transfer

Volume 116, January 2018, Pages 185-193

https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.021 Get rights and content

Highlights

•
Turbulent R-B convection of cold water with the maximum density is investigated.
•
Different flow mechanisms are certified at different density inversion parameters.
•
Soft and hard turbulence states are observed in penetrative R-B convection.
•
A heat transfer correlation for the Nu_ave with Ra and Θ_m is proposed.

Abstract

In order to understand the effects of density inversion parameter and Rayleigh number on the fluid motion and heat transfer characteristics of penetrative Rayleigh-Bénard convection in a cylindrical container, a series of large eddy simulations were performed. The working fluid is cold water and Prandtl number is 11.57. Rayleigh number up to 10¹¹ is considered and the density inversion parameter ranges from 0 to 0.7. The results indicate that the effect of cold plumes on the larger scale circulation gradually diminishes and finally disappears with the increase of density inversion parameter. Furthermore, an increase of the temperature within the bulk region and a decrease of the penetration depth are also certified with the increase of density inversion parameter. With the increase of Rayleigh number, soft and hard turbulent states are successively observed. In the soft turbulence state, the evolution of Nusselt number shows a weak fluctuation, the corresponding histogram has a Gaussian distribution, and the penetration depth increases sharply with the increase of Rayleigh number. In the hard turbulence state, the evolution of Nusselt number presents a much stronger fluctuation than that in the soft turbulence state, the corresponding histogram fits more exponential distribution, and the variation of the penetration depth with Rayleigh number is small. A single power law between the heat transfer efficiency (Nusselt number) and Rayleigh number that covers soft and hard turbulence states is developed for each density inversion parameter. It is found that the average Nusselt number decreases linearly with the increase of density inversion parameter.

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 T_c and T_h (T_c < T_h) 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], $ρ (T) = ρ_{m} (1 - γ | T - T_{m} |^{q}),$ where γ = 9.297173 × 10⁻⁶ (°C)^−q, q = 1.894 816, ρ_m = 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.

References (39)

Y. Ganot et al.
Impact of thermal convection on CO₂ flux across the earth–atmosphere boundary in high-permeability soils
Agric. For. Meteorol.
(2014)
S.K. Choi et al.
Computation of a turbulent Rayleigh-Bénard convection with the elliptic-blending second-moment closure
Int. Commun. Heat Mass Transf.
(2008)
H. Shibata
Heat flux in Rayleigh-Bénard convection
Physica A
(2005)
M. Mastiani et al.
Density maximum effects on mixed convection in a square lid-driven enclosure filled with Cu-water nanofluids
Adv. Powder Technol.
(2017)
B. Gebhart et al.
A new density relation for pure and saline water
Deep Sea Res.
(1977)
D. Kizildag et al.
Large eddy and direct numerical simulations of a turbulent water-filled differentially heated cavity of aspect ratio 5
Int. J. Heat Mass Transf.
(2014)
M. Lappa
Some considerations about the symmetry and evolution of chaotic Rayleigh-Bénard convection: the flywheel mechanism and the “wind” of turbulence
CR Mécanique
(2011)
R.J.A.M. Stevens et al.
Heat transport and flow structure in rotating Rayleigh-Bénard convection
Eur. J. Mech. B-Fluids
(2013)
M. Corcione et al.
Penetrative convection of water in cavities cooled from below
Comput. Fluids
(2015)
G. Ahlers et al.
Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection
Rev. Mod. Phys.
(2009)

F. Chilla et al.

New perspectives in turbulent Rayleigh-Bénard convection

Eur. Phys. J. E

(2012)

D. Lohse et al.

Small-scale properties of turbulent Rayleigh-Bénard convection

Annu. Rev. Fluid Mech.

(2010)

C. Guervilly et al.

Generation of magnetic fields by large-scale vortices in rotating convection

Phys. Rev. E

(2015)

Y.P. Hu et al.

Rayleigh-Bénard convection of cold water near its density maximum in a cubical cavity

Phys. Fluids

(2015)

F. Heslot et al.

Transitions to turbulence in helium gas

Phys. Rev. A

(1987)

B. Castaing et al.

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

J. Fluid Mech.

(1989)

S.H. Peng et al.

Large-eddy simulation and deduced scaling analysis of Rayleigh-Bénard convection up to Ra=10⁹

J. Turbul.

(2006)

X. Riedinger et al.

Heat transport regimes in an inclined channel

Phys. Fluids

(2013)

X.L. Qiu et al.

Onset of coherent oscillations in turbulent Rayleigh-Bénard convection

Phys. Rev. Lett.

(2001)

Cited by (11)

Rotating thermal flow of water near its density inversion point in a vertical annulus: flow regime transition and heat transfer enhancement
2022, International Journal of Heat and Mass Transfer
Citation Excerpt :
For the convection heat transfer of cold water in vertical cylindrical containers, Li et al. [33] performed numerical studies to explore the flow state multiplicity in the Rayleigh-Bénard convection. Huang et al. [34] conducted large eddy simulations for the Rayleigh-Bénard convection to investigate the effects of the density inversion parameter and Rayleigh number on the flow feature, the penetration depth and the heat transfer correlation in both soft and hard turbulent states. The natural convection of cold water has also been studied in horizontal annuli.
Three-dimensional simulations are performed to identify the flow regimes of the rotating thermal flow of cold water near its density inversion point within a vertical annulus and evaluate the commensurate heat transfer abilities. The development of flow regimes and the variation in heat transfer behaviors in the special case with the density inversion parameter $Θ_{m} = 0.5$ are explored over a wide range of parameters, with the Rayleigh number $10^{4} \leq R a \leq 10^{6}$ and the Reynolds number $R e \leq 150$ . The convection flow is radially separated into two layers by the density-maximum isosurface, with the inner and outer layers dominated by a positive and negative radial density gradient, respectively. Three axisymmetric and four non-axisymmetric basic flow regimes with different generation mechanisms are defined to describe the double-layer convection flow. Eleven flow regimes are identified, each of which consists of one, two or three basic flow regimes. The spatio-temporal features of the flow regimes depend on $R a$ and $R e$ . The mixed convection parameter is demonstrated to be capable of qualitatively measuring the relative strength of the flow between the inner and outer layers in certain ranges of $R a$ . The mechanisms of both the flow regime transitions and the mode transitions between steady and oscillatory states are explored in terms of the combined action of density inversion effect, centrifugal effect and buoyancy effect. The heat transfer behaviors depend on the spatio-temporal features of flow regimes. The mechanism of heat transfer enhancement and that for the non-monotonic variation of heat transfer ability are revealed. The limitation of the axisymmetric assumption is quantified.
Three-dimensional numerical simulation of gas-liquid interfacial mass transfer with Rayleigh convection using hybrid LBM-FDM and its mass transfer coefficient model
2019, Chemical Engineering Science
Citation Excerpt :
Theoretical analysis about Rayleigh convection focused on the critical Rayleigh number (Ra) and onset time of Rayleigh convection (Tan et al., 2010, Sun, 2012, Fontana et al., 2015, 2016). The numerical simulation is another powerful method to study Rayleigh convection which provides information for exploring the mass transfer intensification mechanism caused by Rayleigh convection (Chang and Alexander, 2006; Farajzadeh et al., 2007; Chen et al., 2012, Rongy et al., 2012; Fu et al., 2013; Chevalier et al., 2015; Jia et al., 2018; Li et al., 2018). As for the models of mass transfer coefficient, there are classical two film model (Whitman, 1923), penetration model (Higbie, 1935), surface renewal model (Danckwerts, 1981), and other recently developed models such as small eddy model and surface divergence model, which is only based on velocity field (Fortescue and Pearson, 1967; Lamont and Scott, 1970; Luk and Lee, 1986; Banerjee et al., 2004).
Rayleigh convection has a significant effect on gas-liquid interfacial mass transfer. However, its intensified mechanism has not been clarified so far. In the present work, three-dimensional hybrid Lattice Boltzmann Method-Finite Difference Method (LBM-FDM) was employed to simulate the Rayleigh convection emerging in the CO₂-ethanol absorption process. A gas phase random disturbance model was applied to represent the disturbances on the gas-liquid interface which effectively induces Rayleigh convection. Then the front view images obtained by three-dimensional simulation were compared to the Schlieren observations for validation. Finally, a theoretical mass transfer coefficient model based on dissipation rate of concentration variance was proposed, by analyzing the overhead view of concentration contour and dissipation rate of concentration variance contour. Compared to the small eddy model and surface divergence model, the proposed model shows its strength in determining molecular diffusion stage and convective mass transfer stage of Rayleigh convection process, as well as eliminating the phase difference in model prediction.
Numerical simulation of evaporation with Rayleigh convection using LBM-FDM hybrid method and proposal of the interfacial mass transfer coefficient model
2019, International Journal of Heat and Mass Transfer
Citation Excerpt :
Previous research reported in the literature mainly concentrates on theoretical analysis about Rayleigh convection [8–11], e.g. critical Rayleigh number (Ra), onset time prediction of Rayleigh convection. Moreover, macroscale numerical simulation has been used to obtain more details about velocity field and concentration field, especially at the vicinity of the interface [12–20]. However, it is difficult to clarify the mass transfer mechanism of Rayleigh convection only from the experimental images and the measurement of intensified mass transfer rate.
The Rayleigh convection could be induced by solute evaporation in dual solvents, such as acetone-ethyl acetate and isopropanol-water system. The onset of Rayleigh convection and its evolutionary characteristics were simulated using LBM-FDM hybrid method with liquid phase random disturbance model. And the simulation results were in reasonable agreement with experimental measurements in four aspects: Rayleigh convection onset time, concentration field acquired by Schlieren and velocity field acquired by PIV, average mass transfer coefficient, and statistical average velocity distribution along the gravitational direction. It proves that the LBM-FDM hybrid method is effective in capturing the key features in this mesoscale phenomenon. Then three mass transfer model including small eddy model, surface divergence model and the new model based on theoretical parameters concentration variance, its dissipation rate and Reynolds mass flux were proposed to predict the interfacial mass transfer coefficient, and the new developed model had advantages over other two models, in the onset time of Rayleigh convection, the peak of mass transfer coefficient and tackling the non-synchronism between velocity and concentration field in fully developed convective stage. The new developed model shows its strength in determining the mass transfer coefficient in a theoretical way.
Turbulent Rayleigh-Bénard convection in a cubical container filled with cold water near its maximum density
2018, International Journal of Heat and Mass Transfer
Citation Excerpt :
The diffusion terms and the convection terms in the momentum and energy equations are discretized by the second-order central difference scheme and the second-order upwind scheme, respectively. In our previous works [33,34], LES was applied to resolve the Rayleigh-Bénard convection of cold water in a cylindrical container and a rectangular container. It was found that the results on the heat transfer and flow structures have a good agreement with those from the references.
In order to understand the effects of Rayleigh number and the density inversion parameter on the flow structures, the turbulent characteristics and the heat transfer ability, a series of large eddy simulations on turbulent Rayleigh-Bénard convection in a cubical container with cold water near its maximum density were carried out. Results indicate that the large scale circulation (LSC) moves along the diagonal plane at small and medium density inversion parameters. However, at a large density inversion parameter, the LSC disappears. With the increase of the density inversion parameter, there are an increase of temperature in the bulk region and a decrease of penetration depth. Furthermore, with the increase of Rayleigh number, the deformation of isothermal surfaces becomes intense until the plumes are observed. At the same time, the scale of vortices reduces and the fluctuation of Nusselt number amplifies with the increase of Rayleigh number. Based on the numerical results, the dependence of Nusselt number on Rayleigh number and the density inversion parameter is achieved within a satisfactory uncertainty.
Influence of Particle Parameters on the Melting/Migration of Silicon Particles and Turbulent Heat Transfer During Ccz Monocrystalline Silicon Growth
2024, SSRN
Effects of Variable Properties on the Convective Flow of Water near Its Density Extremum in an Inclined Enclosure with Entropy Generation
2022, Mathematics

View all citing articles on Scopus

View full text

Turbulent Rayleigh-Bénard convection of cold water near its maximum density in a vertical cylindrical container

Highlights

Abstract

Introduction

Section snippets

Physical and mathematical model

Effect of density inversion parameter

Conclusions

Acknowledgments

Conflict of interest

Agric. For. Meteorol.

Int. Commun. Heat Mass Transf.

Physica A

Adv. Powder Technol.

Deep Sea Res.

Int. J. Heat Mass Transf.

CR Mécanique

Eur. J. Mech. B-Fluids

Comput. Fluids

Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

Rev. Mod. Phys.

New perspectives in turbulent Rayleigh-Bénard convection

Eur. Phys. J. E

Small-scale properties of turbulent Rayleigh-Bénard convection

Annu. Rev. Fluid Mech.

Generation of magnetic fields by large-scale vortices in rotating convection

Phys. Rev. E

Rayleigh-Bénard convection of cold water near its density maximum in a cubical cavity

Phys. Fluids

Transitions to turbulence in helium gas

Phys. Rev. A

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

J. Fluid Mech.

Large-eddy simulation and deduced scaling analysis of Rayleigh-Bénard convection up to Ra=109

J. Turbul.

Heat transport regimes in an inclined channel

Phys. Fluids

Onset of coherent oscillations in turbulent Rayleigh-Bénard convection

Phys. Rev. Lett.

Large-eddy simulation and deduced scaling analysis of Rayleigh-Bénard convection up to Ra=10⁹