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Natural convection of water near its density maximum between horizontal cylinders
International Journal of Heat and Mass Transfer 54 (2011) 2550–2559
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Natural convection of water near its density maximum between horizontal cylinders You-Rong Li a,b,⇑, Xiao-Feng Yuan a, Chun-Mei Wu a, Yu-Peng Hu a a b
College of Power Engineering, Chongqing University, Chongqing 400044, China Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Ministry of Education, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 21 September 2010 Received in revised form 25 January 2011 Accepted 26 January 2011 Available online 23 February 2011 Keywords: Natural convection Heat transfer Density inversion Horizontal annulus Flow pattern
a b s t r a c t In order to understand the characteristics of natural convection of cold water near its density maximum between horizontal cylinders, a series of unsteady two-dimensional numerical simulations were conducted by using finite volume method. The radius ratio of horizontal cylinders ranged from 1.2 to 2.0, density inversion parameter from 0 to 1, and the vertical eccentricity from 0 to 1.0 for eccentric annulus. The results show that the flow pattern mainly depends on the density inversion parameter and Rayleigh number. The formation of small cell at the top or bottom of annulus corresponds to the Rayleigh–Bénard instability within the converse density gradient layer. The width of annulus has slightly influence on the flow structure. However, the number of Bénard cells decreases with the increase of the radius ratio. For the oscillatory flow at a large Rayleigh number, the vertical converse density gradient in the top of annulus or the horizontal density gradient in the middle of annulus plays an important role for the formation of oscillatory flow when the density inversion parameter is in a small or moderate range. But the vertical density gradient in the bottom of annulus and the horizontal density gradient in the middle of annulus work together for oscillatory flow when the density inversion parameter is high. Average Nusselt number on the inner wall increases with the increase of Rayleigh number and radius ratio. However, there exists the minimum value of average Nusselt number at a moderate density inversion parameter. The flow pattern in eccentric annulus has the characteristics of coupling flows in the narrow-gap at the bottom with in the large-gap at the top of annulus. With the increase of the eccentricity, heat transfer is enhanced and the average Nusselt number increases slightly. Based on the simulation results, the new heat transfer correlation has been proposed according to the multiple linear regression technique. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection in a fluid-filled cavity is a very interesting topic for many researchers due to its wide range of applications in science, engineering and technology. Generally, fluid density changes as a linear function of temperature. But, for some fluids, the density–temperature relation exhibits an extremum. For example, the mass density of pure water reaches its maximum value at about 4 °C at the sea-level atmospheric pressure. Above this temperature, the mass density of water decreases with the increase of temperature in a manner similar to other fluids. However, when the temperature is below 4 °C, the trend is reversed, i.e. the mass density of water increases with the increase of its temperature. This is so-called density inversion phenomenon and it results in complicated flow patterns and heat transfer characteristics for the natural convection of water near its density maximum. During the past decades, a lot of experimental observations and numerical simulations have been reported for the natural convec⇑ Corresponding author at: College of Power Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 6511 2284; fax: +86 23 6510 2473. E-mail address: [email protected] (Y.-R. Li). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.02.006
tion of water near its density maximum inside a rectangle cavity with heated side walls and insulated horizontal surfaces. The representative works on steady-state [1–10] and transient [11–20] natural convection have all revealed that the density inversion phenomenon had great effect on the flow pattern, temperature field and heat transfer characteristic. Meanwhile, influences of internal heat generation [21] and magnetic field [22] on natural convection of water near its density maximum in a rectangular cavity were also investigated. Moreover, Kandaswamy et al. [23,24] performed a numerical simulation for the transient natural convective heat transfer of water near its density maximum in a square cavity with partially active vertical walls. The results indicated that the heat transfer was enhanced when the heating area located in the middle of the vertical wall. On the other hand, Kalabin et al. [25] and Zubkov et al. [26,27] studied the natural convection of water near its density maximum in a square cavity with adiabatic vertical walls and isothermal horizontal walls. Four different kinds of steady convection and at least three kinds of unsteady oscillatory convection were found under various Grashof numbers and initial conditions, and it has been concluded that only one flow pattern was asymmetric and all the other flow patterns were symmetric about the central vertical axis among the unsteady convections.
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Nomenclature d e g Nu P Pr q r R R⁄ Ra T V
distance between the center of inner cylinder and outer cylinder eccentricity, d/(ro ri) acceleration of gravity, m/s2 Nusselt number, Nu ¼ Ri lnðR Þ@ H @R R¼Ri
dimensionless pressure Prandtl number, m/a exponent in density–temperature Eq. (1) radius, m dimensionless radial coordinate radius ratio, ro/ri Rayleigh number based on the gap width, Ra = gc(Ti To)q(ro ri)3/(ma) temperatures, °C dimensionless velocity
Greek symbols a thermal diffusivity, m2/s
In the design of the energy storage systems, we often encounter the natural convective heat transfer of water near its density maximum between horizontal cylinders with uniform temperature boundary condition on the inner and outer surfaces, respectively. However, up to now, only a few investigations about this heat transfer problem have been reported. Seki et al. [28], Nguyen et al. [29], Vasseur et al. [30] and Raghavarao and Sanyasiraju [31] investigated steady natural convective heat transfer of water near its density maximum between two horizontal concentric cylinders with constant surface temperatures for different Rayleigh (Ra) number, the radius ratio and the density inversion parameter. It was found that the flow pattern was greatly influenced by the density inversion parameter of water, and concluded that the effect of density inversion was unexpectedly large and the average Nusselt (Nu) number was different from that of Boussinesq liquids whose density decreased linearly with the increase of temperature. Ho and Lin [32] and Raghavarao and Sanyasiraju [33] performed a numerical simulation on steady-state natural convection of cold water near its density maximum within an eccentric horizontal annulus. Results indicated that the flow patterns and heat transfer characteristics were strongly influenced by the combined effect induced by the density inversion of water and the position of the inner cylinder of the annulus. A minimum on heat transfer appeared with the inversion parameter between 0.4 and 0.5 depending primarily on the position of the inner cylinder. For natural convection between horizontal cylinders, the fluid is subjected not only to the horizontal temperature gradient in the middle region, but also to the vertical temperature gradient in the top and bottom regions, which results in more complicated flow patterns. Therefore, the aim of this paper is to reveal these complicated flow patterns and heat transfer characteristics of natural convection of water near its density maximum from steady-state to oscillatory flow between horizontal cylinders with different radius ratio, density inversion parameter and eccentricity.
c l m h
H
q s w
coefficient in density–temperature Eq. (1), (°C)q dynamic viscous, kg/(m s) kinematic viscosity, m2/s azimuthal coordinate dimensionless temperature, H = (T To)/(Ti To) density, kg/m3 dimensionless time dimensionless stream function
Subscripts ave average i inner wall o outer wall m density inversion point max the maximum R radial p periodical h azimuthal
respectively. Their centers locate at O and O0 , respectively. The eccentricity e is measured by the non-dimensional distance between the center of inner and outer cylinders. The following assumptions are introduced in this model: (1) the velocity is low and the flow is in laminar region; (2) all physical properties are independent of temperature except for the density in buoyancy force term; (3) viscous dissipation is neglected. Moreover, the nonlinear density–temperature relation of water, which is proposed by Gebhart and Mollendorf [34], is adopted in the following form:
qðTÞ ¼ qm ð1 cjT T m jq Þ;
ð1Þ 3
where the maximum density qm = 999.972 kg/m at the temperature Tm = 4.029325 °C, q = 1.894816 and c = 9.297173 106 (°C)q. The thermophysical properties of cold water at Tm are l = 1.567 103 kg/(m s), k = 0.562045 W/(m K), a = 1.344 107 m2/s, and Pr = 11.67. Based on the above assumptions, the non-dimensional governing equations of continuity, momentum and energy are given by introducing (ro ri), (ro ri)2/m, m/(ro ri) and ml/(ro ri)2 as the characteristic length, time, velocity and pressure, respectively.
1 @ðV R RÞ @V h ¼ 0; þ R@h R @R @V R @V R @V R V 2h þ VR þ Vh @s @R R@h R @P 1 @ @V R @2V R 2 @V h ¼ þ R þ 2 2 2 @R R @R @R R @h R @h V R Ra 2 jH Hm jq sin h; Pr R
2. Physical and mathematical models The physical model of the problem and the coordinate system are shown in Fig. 1. The fluid is contained between two infinite horizontal cylinders with the inner radius ri and outer radius ro. The inner and outer walls of the horizontal annulus are isothermally held at constant uniform temperatures Ti and To (Ti > To),
Fig. 1. Physical model and coordinate system.
ð2Þ
ð3Þ
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@V h @V h @V h V R V h þ VR þ Vh þ @s @R R@h R @P 1 @ @V h @2V h 2 @V R þ 2 2þ 2 ¼ þ R R@h R @R @R R @h R @h V h Ra 2 jH Hm jq cos h; Pr R
Table 2 Comparison of average Nusselt numbers for different Rayleigh number in a square enclosure. Ra
ð4Þ
" # @H @H V h @H 1 1 @ @H @2H þ VR R þ ¼ þ 2 2 : @s @R R @h Pr R @R @R R @h
ð5Þ
The corresponding initial condition and boundary conditions on the two cylinder walls are as follows:
s ¼ 0; V R ¼ V h ¼ 0; H ¼ 0;
ð6a—cÞ
V R ¼ V h ¼ 0;
ð7a—cÞ
R ¼ Ri ;
R ¼ e sin h þ
H ¼ 1;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2o ðe cos hÞ2 ;
V R ¼ V h ¼ 0;
H ¼ 0:
ð8a—cÞ
The density inversion parameter in Eqs. (3) and (4) is given as:
Hm ¼ ðT m T o Þ=ðT i T o Þ:
ð9Þ
The local and average Nusselt numbers at the inner wall are defined as
@ H Nu ¼ Ri lnðR Þ ; @R R¼Ri Nuave ¼
Ri 2psp
lnðR Þ
ð10Þ
Z s0 þsp Z 2p @ H dh ds: @R R¼Ri s0 0
ð11Þ
The velocity field is displayed in terms of the non-dimensional stream function w, which is defined as
VR ¼
@w ; R@h
Vh ¼
@w : @R
ð12Þ
3. Numerical method and validation check The governing equations (2)–(5) are discretized by the finite volume method. A central difference approximation is applied to the diffusion term and the QUICK scheme is used for the convective term. The SIMPLE algorithm is adopted to handle the pressure– velocity coupling. The dimensionless time step is chosen between 104 and 102. At each time step, the convergence is reached if the maximum relative error for velocity and temperature are less than 105. In order to evaluate the grid convergence, simulations with several different meshes were conducted. As shown in Table 1, the average Nusselt number of inner wall were presented with different grids at Ra = 5 104 and Hm = 0.7. From this table, it clearly shows that 50R 220h, 70R 160h and 70R 90h grids are dense enough for the accurate simulation at Ra = 5 104 and Hm = 0.7 for R⁄ = 1.2, 1.5 and 2.0, respectively. For other Ra number and den-
Table 1 Comparison of the average Nusselt number for various grids at Ra = 5 104 and Hm = 0.7. R⁄ = 1.2
Mesh (R h) Nuave
30 140 1.813
50 220 1.785
80 340 1.770
R⁄ = 1.5
Mesh (R h) Nuave
50 120 2.166
70 160 2.162
100 220 2.159
R⁄ = 2.0
Mesh (R h) Nuave
50 70 2.535
70 90 2.530
100 120 2.526
103 104 105 106
Nuave Ref. [4]
Present
1.001 1.076 2.080 4.090
0.989 1.065 2.079 4.124
sity inversion parameter, the grid convergence check is confirmed as well. In order to check the accuracy of the present numerical method, some numerical simulations have been carried out for the steady state natural convection in a square enclosure filled with water near its density maximum under the same conditions as Ref. [4]. The average Nusselt numbers are compared with those given by Nansteel et al. [4] at different Rayleigh numbers, as shown in Table 2. It could be found that the present results are very close to those in Ref. [4]. Further validation simulations have also been conducted for the problem of oscillatory convection of cold water in a tall vertical rectangular cavity and a vertical annulus considered by Ho and Tu [16,35]. The streamlines and isotherms are in a good agreement with those results in Refs. [16,35], and the maximum deviation of the average Nusselt number is about 5.6%. Therefore, all these results confirm that the present numerical method is accurate enough to study the natural convection of water near its density maximum. 4. Results and discussion 4.1. The narrow-gap horizontal concentric annulus For the narrow-gap horizontal concentric annulus, for example, R⁄ = 1.2, the steady-state natural convection of cold water is always symmetry about the vertical axis for any density inversion parameter. At Hm = 0.3, a crescent-shaped flow pattern appears in whole region of concentric annulus at a small Ra number, and the fluid flows upward along the inner wall and downward along the outer wall, as shown in Fig. 2(a). In this case, the isotherms are a set of concentric lines. With the increase of Rayleigh number, the strength of flow is enhanced and a small counter-rotating cell starts appearing near the top of annulus. Due to the change of flow pattern, the isotherms are firstly distorted near the top. With the further increase of Rayleigh number, more small cells are formed at the top region of annulus, as shown in Fig. 2(b) and (c). In Fig. 3, the local Nusselt number of inner wall is presented. From this figure, it is easy to find that a large variation of local Nusselt number appears when the small cells exist near the top region of annulus. Obviously, the Nusselt number increases if the fluid flows towards the inner wall and decreases if the fluid flows depart from the inner wall. As shown in Fig. 3, when Rayleigh number is 104, there exist two extremums of local Nusselt number at Hm = 0.3 and the maximum value is located at the top region of annulus. For the case of Hm = 0.5, a reverse bi-cellular flow structure is clearly observed, as shown in Fig. 2(d). In this case, the fluid flows upward along both inner wall and outer walls, and downward along the interface of the two adjacent cells. Due to the opposite flow direction, the two adjacent cells are suppressed each other, therefore, the strength of flow gets weakened at the same Rayleigh number and the maximum value of the stream function gets decreased as well. At the same time, the heat transfer ability is decreased due to the change of flow pattern. As shown in Fig. 3, the local Nusselt number is almost a constant close to 1 at Ra = 104, which means that conductive heat transfer is dominant.
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Fig. 2. Streamlines (left) and isotherms (right) at R⁄ = 1.2. Solid lines denote positive value of stream function and dotted lines denote negative value. dH = 0.125. (a) Hm = 0.3, Ra = 3 103, wmax = 13.23; (b) Hm = 0.3, Ra = 4.5 103, wmax = 20.05; (c) Hm = 0.3, Ra=104, wmax = 46.75; (d) Hm = 0.5, Ra = 2 104, wmax = 17.85; (e) Hm = 0.7, Ra=104, wmax = 48.70; (f) Hm = 0.7, Ra = 9 104, wmax = 230.35.
produced with the increase of Rayleigh number. Therefore, the additional Bénard cells appear firstly in these regions when the Rayleigh number exceeds a certain threshold value. Furthermore, in order to confirm that this flow pattern is the result of the Rayleigh–Bénard instability, the local Rayleigh number RaL is
(a)
Fig. 3. The distribution of local Nu number on the inner wall at R⁄ = 1.2 and Ra = 104. 1: Hm = 0.3; 2: Hm = 0.5; 3: Hm = 0.7.
(c) τ0
4
τ0+τp/5
τ0+2τp/5
τ0+3τp/5
τ0+4τp/5
top 3
Nu
For the case of Hm = 0.7, the crescent-shaped cell also appears in the whole region of concentric annulus for a small Ra number, however, the flow direction is opposite as compared with the case of Hm = 0.3, i.e., the fluid flows upward along the outer wall and downward along the inner wall. Furthermore, one or more small cells are formed near the bottom of annulus with the increase of Rayleigh number, as shown in Fig. 2(e) and (f). In this case, the distortion of the isotherms appears firstly at the bottom of annulus, where convective heat transfer begins to become dominant. Therefore, the maximum value of local Nusselt number appears at the bottom region for Hm = 0.7, as shown in Fig. 3. Due to nonlinear density correlation of cold water, the maximum density contour is located on the mid-plane of annulus at Hm = 0.5. Since the dense fluid near the mid-plane falls while the less dense fluid adjacent to the inner and outer walls rises, a reversal bi-cellular flow structure is formed. With the decrease or increase of density inversion parameter, the maximum density contour moves toward the outer or inner wall. The crescentshaped cell close to the outer or inner wall becomes stronger and larger. At the top and bottom of annulus for Hm = 0.3 and 0.7, respectively, there exists a converse density gradient layer. Within this converse density gradient layer, Rayleigh–Bénard instability is
(b)
2
1
(c) −π/2
0
π/2
θ
π
3π/2
Fig. 4. Snapshots of streamlines (a) and isotherms (b), and temporal evolution (c) of local Nu number on the inner wall during a period at R⁄ = 1.2, Hm = 0.3 and Ra = 2 104. dH = 0.125, wmax = 83.94.
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estimated, which is defined as RaL = gb(DT)qL3/(ma), where DT is temperature difference through the converse density gradient layer and L is the thickness of the converse density gradient layer. The simulation results show that the local Rayleigh numbers are about 1253.87 and 1201.51 for Hm = 0.3 and 0.7, respectively. It
(a)
(b)
(c) τ0
4
τ0+τp/5
τ0+2τp/5
τ0+3τp/5
τ0+4τp/5
top
Nu
3
right
left
2
1
(c) −π/2
0
π/2
θ
π
3π/2
Fig. 5. Snapshots of streamlines (a) and isotherms (b), and temporal evolution (c) of local Nu number on the inner wall during a period at R⁄ = 1.2, Hm = 0.5 and Ra = 105. dH = 0.125, wmax = 191.21. Arrows denote the moving direction of the cells.
shows that the local Rayleigh number would exceed the critical Rayleigh number value, i.e., Racri = 1100.65, which was obtained by the linear stability analysis [36] for the incipience of the Rayleigh–Bénard instability in an infinitely extended fluid layer with a free and a solid surface subjected to a constant vertical temperature gradient. Therefore, a conclusion could be made that this flow pattern is the result of the Rayleigh–Bénard instability within the converse density gradient layer. With the further increase of Rayleigh number, the steady-state flow starts to become unstable, i.e., oscillatory convection flow, and the symmetry of flow and temperature fields disappears. At Hm = 0.3, the flow instability starts firstly near the top of annulus, and then some small cells appear at the top of annulus for a high Ra number. As shown in Fig. 4, it is presented the snapshots of streamlines and isotherms, and temporal evolution of local Nusselt number on the inner wall during a period at R⁄ = 1.2 and Ra = 2 104. From this figure, it is obviously observed that there exist a set of Bénard cells with the different rotation direction near the top of annulus and two crescent-shaped cells located at the lower-left and lower-right sides of annulus, respectively. The centers of the Bénard cells are almost fixed, and the flow strength of two adjacent Bénard cells at the top of annulus is almost constant. However, the flow strength of the Bénard cells adjacent to the crescent-shaped cells shifts continuously, which results in the movement of the crescent-shaped cells. Therefore, it is indicated that the vertical converse density gradient at the top of annulus is responsible for this oscillatory flow at Hm = 0.3. In Fig. 4(c), it is clearly shows that the local Nusselt number on the inner wall varies with the shift of the Bénard cells adjacent to the crescent-shaped cells. It is found that the local Nusselt number varies with the center and strength of the cells, however, the average Nusselt number on the inner wall is independent of time. For the case of Hm = 0.5, the local Rayleigh number through the vertical converse density gradient layer at the bottom of annulus is greater than that at the top of annulus. Therefore, the Rayleigh– Bénard convection firstly occurs at the bottom, and then at the top of annulus with the increase of Rayleigh number. Moreover, the reversal double-circulation multicellular flow pattern was
Fig. 6. Streamlines (left) and isotherms (right) at R⁄ = 2.0. dH = 0.125. (a) Hm = 0.3, Ra = 103, wmax = 2.27; (b) Hm = 0.3, Ra = 5 104, wmax = 58.35; (c) Hm = 0.5, Ra = 5 105, wmax = 43.73; (d) Hm = 0.5, Ra = 106, wmax = 87.73; (e) Hm = 0.7, Ra = 5 104, wmax = 58.99; (f) Hm = 0.7, Ra = 5 105, wmax = 73.16.
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observed in the left and right sides of the vertical section of annulus at a high Rayleigh number. This pattern corresponds to the thermal instability based on the horizontal temperature gradient in the vertical region. The number of cells increases and the region occupied by the cells expands with the increase of Rayleigh number. The small cells, which split from the reversal two bi-cellular cells, originate from the bottom of annulus, then drift upward and grow up, finally decay at the top of annulus, as shown in Fig. 5. It is similar to natural convection of water near its density maximum inside a tall rectangular enclosure with differential heating at the vertical walls [16]. However, the cells near the top and bottom of annulus are almost changeless. It suggests that the horizontal density gradient plays the main role for this oscillatory flow at Hm = 0.5. From Fig. 5(c), it is found that the local Nusselt number varies with the movement of the cells in the left and right of the vertical section of annulus. However, it keeps a constant near the top and bottom of annulus. The average Nusselt number on the inner wall is also independent of time. When Hm = 0.7, the multiple like-rotating cells appear in the vertical section of annulus and the additional Bénard cells at the bottom of annulus extend upwards for a high Ra number. The movement of cells in the vertical section of annulus is the
Fig. 7. The distribution of local Nu number on the inner wall at R⁄ = 2.0 and Ra = 5 104. 1: Hm = 0.3; 2: Hm = 0.5; 3: Hm = 0.7
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same as that at Hm = 0.5, and the movement of cells at the bottom of annulus is similar with that at Hm = 0.3. Therefore, the vertical density gradient at the bottom of annulus and the horizontal density gradient in the middle of annulus work together to induce the oscillatory flow in this case. 4.2. The large-gap horizontal concentric annulus In the case of large-gap horizontal concentric annulus, the flow pattern of steady-state natural convection of cold water is also symmetry about the vertical axis for any density inversion parameter. At Hm = 0.3, the center of the crescent-shaped cells moves upwards and there appears another small crescent cell adjacent to the outer wall at the bottom of annulus with the increase of the Rayleigh number, as shown in 6(a) and (b). Moreover, the distortion of isotherms also appears firstly near the top of annulus, and then extends to the bottom of annulus. A large local temperature gradient is formed near the outer wall at the top of annulus and near the inner wall at the bottom of annulus, where the convective heat transfer is dominant. Therefore, the maximum value of local Nusselt number on the inner wall for Hm = 0.3 at Ra = 5 104 locates at the bottom of annulus, as shown in Fig. 7. When Hm = 0.5, a reversal bi-cellular flow structure is also formed at a very small Ra number. However, with the increase of Rayleigh number, the cell near the outer wall gets stronger and becomes more angular in shape. On the contrary, the cell near the inner wall is restrained by that near the outer wall, which results in the gradual disappearing of the co-rotating cell, as shown in Fig. 6(c) and (d). In this case, the maximum value of local Nusselt number on the inner wall locates at the top of annulus, where the fluid flows towards the inner wall. For the case of Hm = 0.7, the Rayleigh–Bénard convection also occurs at the bottom of annulus with a moderate Rayleigh number, as shown in Fig. 6(e). However, only one Bénard cell appears due to the large gap of horizontal concentric annulus. At a large Rayleigh number, for example Ra = 5 105, the crescent-shaped cell becomes more angular and the counter-rotating cell at the bottom
Fig. 8. Snapshots of streamlines (upper) and isotherms (down) at R⁄ = 2.0. dH = 0.125. (a) Hm = 0.3, Ra = 105, wmax = 80.26; (b) Hm = 0.3, Ra = 106, wmax = 215.36; (c) Hm = 0.7, Ra = 1.8 106, wmax = 123.05.
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of annulus vanishes, as shown in Fig. 6(e). In this case, the local Nusselt number is also dependent on the flow pattern. Because there exist the crescent-shaped cell and a Bénard cell near the bottom at Ra = 5 104, local Nusselt numbers on the inner wall have large values at the top and bottom of the annulus. The minimum value of local Nusselt number locates at interface of two cells, where the fluid flows depart from the inner wall, as shown in Fig. 7. When the Rayleigh number exceeds a certain critical value, the steady-state flow in a large-gap horizontal concentric annulus will transit to an oscillatory flow. For the case of Hm = 0.3, variations of the streamlines and isotherms are similar to the ones in a narrowgap annulus. The Rayleigh–Bénard convection also occurs at the top region and becomes unstable. However, the flow pattern is symmetrical even at Ra = 106. The crescent-shaped cells near the inner wall become more angular in shape, and several cells are observed in the crescent-shaped cells as a result of free-shear instability, which holds the self-sustained oscillation. Furthermore, the small crescent cells close to the outer wall extend upward (see Fig. 8(a) and (b)). When Hm = 0.7, several similar cells are observed in the interior of large crescent-shaped cells with the increase of Ra number, as shown in Fig. 8(c). For the large-gap horizontal concentric annulus, the instability mechanism of natural convection of cold water under different density inversion parameter is the same as one in a narrow-gap annulus.
4.3. The vertical eccentric annulus The effect of vertical eccentricity on natural convection of water near its density maximum in a horizontal annulus is similar for different radius ratio. Therefore, only the results at radius ratio R⁄ = 1.5 are given. The steady-state flow for Hm = 0.3 is similar with that in horizontal concentric annulus. When there is a vertical eccentricity, the convective space spreads and the critical Rayleigh number for the Bénard cells decreases with the increase of eccentricity. Fig. 9 gives the snapshots of streamlines and isotherms of oscillatory convections for different eccentricity at Hm = 0.3. It is found that the flow is also symmetrical and periodic oscillation appears at a moderate Rayleigh number, for example Ra = 3 104. In this case, with the increase of eccentricity, the number of the Bénard cells near the top of annulus increases. The location of the cells is almost fixed for the periodic oscillation. However, periodic expansion and shrinkage of the Bénard cells will drive the movement of crescent-shaped cells along the azimuthal direction, which results in a periodic variation of the local Nusselt number. When the Rayleigh number increases to Ra = 105, the oscillatory convection becomes asymmetry, as shown in Fig. 9(b), and the number of the Bénard cells near the top of annulus increases. At Hm = 0.5, the reversal bi-cellular flow structure is formed and the corresponding distortion of isotherms is consistent with
Fig. 9. Snapshots of streamlines (upper) and isotherms at R⁄ = 1.5 and Hm = 0.3. dH = 0.125.
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that in horizontal concentric annulus at a small Rayleigh number. With the increase of eccentricity, the flow is enhanced and the non-dimensional stream function increases. However, the stagnant region of the fluid flow at the top of annulus extends, as shown in Fig. 10(a). Furthermore, the distortion of crescent-shaped cells augments. Especially, the head of crescent-shaped cells become large. At e = 0.3, with the increase of Rayleigh number, the flow transits from the steady-state convection to the oscillatory convection. The co-rotating cells near the bottom of annulus and the reversal bi-cellular multiple like-rotating cells have only very small change comparing with these in horizontal concentric annulus. But the flow near the top of annulus becomes chaotic. The increase of eccentricity results in a large-gap at the top of annulus, where several chaotic secondary vortexes near the inner wall is formed at e = 0.5 and 0.7, as shown in Fig. 10(b). When Hm = 0.7, the flow region is reduced and the multicellular steady flow for narrow-gap annulus appears at the bottom of annulus with the increase of eccentricity. Furthermore, the large crescent-shaped cells become more angular at the top of annulus with the increase of Rayleigh number. The cells closed to the inner wall extend upwards. At a large Rayleigh number, the flow transits to the oscillatory multi-cellular convection in the narrow-gap bottom of annulus and the chaotic convection with several vortexes in the crescent-shaped cells at the vertical large-gap part of annulus.
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4.4. Correlation equation for heat transfer characteristics Fig. 11 shows the variation of average Nusselt number on the inner wall of the annulus with the Rayleigh number at different radius ratio and density inversion parameter from steady-state to oscillatory flow. Obviously, with the increase of Ra number, the flow is enhanced, therefore, the average Nusselt number gets increased. Furthermore, when the flow pattern transits from steady state to oscillatory state, the average Nusselt number at different Ra number locates near a straight line in bi-logarithm coordinate system at a fixed radius ratio and density inversion parameter. It suggests that variation law of average Nusselt number with the Rayleigh number is independent of the flow pattern transition. Therefore, the critical condition of the flow patterns transition is not presented in this paper. Around Hm = 0.5, the reversal bi-cellular flow structure results in that the flow near inner wall is suppressed by the flow cell near the outer wall, therefore, the average Nusselt number is always the least at the same Ra number. When the density inversion parameter increases or decreases from Hm = 0.5, the average Nusselt number all increases, as shown in Fig. 11. On the other hand, the width of the annulus has also important effect on the flow intensity and heat transfer characteristic. With the increase of the radius ratio, the flow space of natural convection expands and the flow is
Fig. 10. Snapshots of streamlines (upper) and isotherms at R⁄ = 1.5 and Hm = 0.5. dH = 0.125.
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4
(a)
(a) Nu ave
Nu ave
3
10
2
4
2
1 10 3
10 4
Ra
10 5
1 103
10 6
5 4
104
105
106
107
105
106
107
105
106
107
Ra
10
(b)
(b) Nu ave
Nu ave
3
2
4
2
1 103
104
105
106
107
1 103
Ra 7 6 5
Ra
10
(c)
(c) Nu ave
4
Nu ave
104
3
4
2
2 1 103
104
105
Ra
106
107
1 103
Fig. 11. The variation of average Nusselt number with Rayleigh number. (a) R⁄ = 1.2; (b) R⁄ = 1.5; (c) R⁄ = 2.0. 4: Hm = 0.3 (steady); N: Hm = 0.3 (oscillation); s: Hm = 0.5 (steady); d: Hm = 0.5 (oscillation); h: Hm = 0.7 (steady); j: Hm = 0.7 (oscillation).
enhanced, therefore, the average Nusselt number increases at the same Rayleigh number. Fig. 12 gives the variation of average Nusselt number on the inner wall with the Rayleigh number at the different eccentricity. Although the vertical eccentricity has the effect on the flow pattern, the variation of the average Nusselt number is small. At Hm = 0.3, the average Nusselt number increases slightly with the increase of the eccentricity at a fixed Rayleigh number. For Hm = 0.5 and 0.7, it also increases with the increase of the eccentricity at a small Rayleigh number. However, at a large Rayleigh number, because the multi-cellular flow for narrow-gap annulus appears, this variation become very complex with the increase of eccentricity. For eccentric annulus, the effect of density inversion parameter on average Nusselt number is in accordance with that in concentric annulus. Based on the simulation results, a multiple linear regression technique is carried out for these data, and a correlation equation for average Nusselt number on the inner wall is given as follows:
Nuave ¼ 0:122Ra0:261 R0:312 jHm 0:496j0:104 ð1 þ eÞ0:166 The application ranges of Eq. (13) are as follows:
ð13Þ
104
Ra
Fig. 12. The variation of average Nusselt number on the inner wall. (a) e = 0.3; (b) e = 0.5; (c) e = 0.7. 4: Hm = 0.3 (steady); N: Hm = 0.3 (oscillation); s: Hm = 0.5 (steady); d: Hm = 0.5 (oscillation); h: Hm = 0.7 (steady); j: Hm = 0.7 (oscillation).
1:2 6 R 6 2:0;
104 6 Ra 6 8:5 106 ;
0 < Hm < 1;
06e<1
The present simulation data falls within ±15% of Eq. (13) in terms of the average Nusselt number, and the average deviation of the average Nusselt number between the simulation data and the proposed correlation is about 4.8%. 5. Conclusions The natural convection of cold water near its density maximum between horizontal cylinders is numerically investigated by the finite volume method. The radius ratio of horizontal concentric cylinders ranged from 1.2 to 2.0 and density inversion parameter from 0 to 1. The following conclusions could be made: For steady-state flow, the flow and temperature fields are always symmetry about the vertical axis for any density inversion parameter. The flow pattern depends mainly on density inversion parameter and Rayleigh number. The formation of small cell at the top or bottom of annulus attribute to the Rayleigh–Bénard instability within the converse density gradient
Y.-R. Li et al. / International Journal of Heat and Mass Transfer 54 (2011) 2550–2559
layer. At a small density inversion parameter, Bénard cells appear near the top of annulus, and they also appear near the bottom of annulus for a large density inversion parameter. The width of annulus has slightly influence on the flow structure. However, the number of Bénard cells decreases with the increase of the radius ratio. When the flow transits to oscillatory state, the flow and temperature fields become asymmetrical. Furthermore, the vertical converse density gradient at the top of annulus and the horizontal density gradient in the middle of annulus play the main role for the formation of oscillatory flow at a small and moderate density inversion parameter, respectively. But the vertical density gradient at the bottom of annulus and the horizontal density gradient in the middle of annulus work together for oscillatory flow at a large density inversion parameter. For vertical eccentric annulus, the flow pattern has the characteristics of coupling the narrow-gap flow at the bottom with the large-gap flow at the top of annulus. The convective space at the top of annulus spreads and the flow is enhanced. The critical Rayleigh number for the Bénard cells at the top of annulus decreases with the increase of eccentricity. Furthermore, the number of the Bénard cells increases. However, when oscillatory convection happens, the mechanisms of flow instability under different density inversion are the same as that in horizontal concentric annulus. Average Nusselt number on the inner wall increases with the increase of Ra number and radius ratio. However, there exists the minimum value of average Nusselt number at about Hm = 0.5. With the increase of the eccentricity, heat transfer is enhanced and the average Nusselt number increases slightly. Based on the simulation results, the new heat transfer correlation has been proposed according to the multiple linear regression technique. The average deviation of the proposed average Nusselt number correlation is about 4.8%.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 50976129) and the Fundamental Research Funds for the Central Universities (Grant No. CDJXS10 14 22 24). References [1] A. Watson, The effect of the inversion temperature on the convection of water in an enclosed rectangular cavity, Quart. J. Mech. Appl. Math. 25 (1972) 423– 446. [2] V.S. Desai, R.E. Forbes, Free convection in water in the vicinity of maximum density, Environ. Geophys. Heat Transfer 93 (1971) 41–47. [3] M.W. Nansteel, K. Medjani, D.S. Lin, Natural convection of water near its density maximum in a rectangle enclosure: low Rayleigh number calculations, Phys. Fluids 30 (1986) 312–317. [4] D.S. Lin, M.W. Nansteel, Natural convection heat transfer in a square enclosure containing water near its density maximum, Int. J. Heat Mass Transfer 30 (1987) 2319–2329. [5] W. Tong, J.N. Koster, Natural convection of water in a rectangular cavity including density inversion, Int. J. Heat Fluid Flow 14 (1993) 366–375. [6] W. Tong, J.N. Koster, Coupling of natural convection flow across a vertical density inversion interface, Heat Mass Transfer 28 (1993) 471–479. [7] N. Seki, S. Fukusako, H. Inaba, Free convective heat transfer with density inversion in a confined rectangular vessel, Heat Mass Transfer 11 (1978) 145– 156. [8] H. Inaba, T. Fukuda, An experimental study of natural convection in an inclined rectangular cavity filled with water at its density extremum, ASME J. Heat Transfer 106 (1984) 109–115.
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[9] A. Osorio, R. Avila, J. Cervantes, On the natural convection of water near its density inversion in an inclined square cavity, Int. J. Heat Mass Transfer 47 (2004) 4491–4495. [10] S. Sivasankaran, C.J. Ho, Effect of temperature dependent properties on natural convection of water near its density maximum in enclosures, Numer. Heat Transfer, Part A 53 (2008) 507–523. [11] P. Vasseur, L. Robillard, Transient natural convection heat transfer in a mass of water cooled through 4 °C, Int. J. Heat Mass Transfer 23 (1980) 1195–1205. [12] L. Robillard, P. Vasseur, Transient natural convection heat transfer of water with maximum density effect and supercooling, ASME J. Heat Transfer 103 (1981) 528–534. [13] W. Tong, J.N. Koster, Density inversion effect on transient natural convection in a rectangular enclosure, Int. J. Heat Mass Transfer 37 (1994) 927–938. [14] S.L. Braga, R. Viskanta, Transient natural convection of water near its density extremum in a rectangular cavity, Int. J. Heat Mass Transfer 35 (1992) 861– 875. [15] T. Nishimura, Y. Hayashida, M. Mineoka, A. Wake, Oscillatory natural convection of water near the density extremum at high Rayleigh numbers, Int. J. Heat Mass Transfer 40 (1997) 3449–3465. [16] C.J. Ho, F.J. Tu, Numerical study on oscillatory convection of cold water in a tall vertical enclosure, Int. J. Numer. Methods Heat Fluid Flow 9 (1999) 487–508. [17] C.J. Ho, F.J. Tu, Visualization and prediction of natural convection of water near its density maximum in a tail rectangular enclosure at high Rayleigh numbers, ASME J. Heat Transfer 123 (2001) 84–95. [18] M. Zivic, M. Savar, S. Krizmanic, Effect of the asymmetric density–temperature relationship of water on natural convection in an enclosure, Trans. Famena 32 (2008) 49–56. [19] N.O. Moraga, S.A. Vega, Unsteady three-dimensional natural convection of water cooled inside a cubic enclosure, Numer. Heat Transfer, Part A 45 (2004) 825–839. [20] C.J. Ho, F.J. Tu, Transition to oscillatory natural convection of water near its density maximum in a tall enclosure – assessment of three-dimensional effects, Int. J. Numer. Methods Heat Fluid Flow 11 (2001) 626–641. [21] M.A. Hossain, D.A.S. Rees, Natural convection flow of water near its density maximum in a rectangular enclosure having isothermal walls with heat generation, Heat Mass Transfer 41 (2005) 367–374. [22] S. Sivasankaran, C.J. Ho, Effect of temperature dependent properties on MHD convection of water near its density maximum in a square cavity, Int. J. Therm. Sci. 47 (2008) 1184–1194. [23] P. Kandaswamy, S. Sivasankaran, N. Nithyadevi, Buoyancy-driven convection of water near its density maximum with partially active vertical walls, Int. J. Heat Mass Transfer 50 (2007) 942–948. [24] N. Nithyadevi, S. Sivasankaran, P. Kandaswamy, Buoyancy-driven convection of water near its density maximum with time periodic partially active vertical walls, Meccanica 42 (2007) 503–510. [25] E.V. Kalabin, E.M. Sviridov, P.T. Zubkov, Steady and unsteady water flows in square cavity with large Grashof numbers, Heat Transfer Sci. Technol. (2000) 217–222. [26] P.T. Zubkov, E.V. Kalabin, A.V. Yakovlev, Investigation of natural convection in a cubic cavity near 4 °C, Fluid Dynam. 37 (2002) 847–853. [27] P.T. Zubkov, E.V. Kalabin, Numerical Investigation of the natural convection of water in the neighborhood of the density inversion point for Grashof numbers up to 106, Fluid Dynam. 36 (2001) 944–951. [28] N. Seki, S. Fukusako, M. Nakaoka, Experimental study on natural convection heat transfer with density inversion of water between two horizontal concentric cylinders, ASME J. Heat Transfer 97 (1975) 556–561. [29] T.H. Nguyen, P. Vasseur, L. Robillard, Natural convection between horizontal concentric cylinders with density inversion of water for low Rayleigh numbers, Int. J. Heat Mass Transfer 25 (1982) 1559–1568. [30] P. Vasseur, L. Robillard, S.B. Chandra, Natural convection heat transfer of water within a horizontal cylindrical annulus with density inversion effects, ASME J. Heat Transfer 105 (1983) 117–123. [31] C.V. Raghavarao, Y.V.S.S. Sanyasiraju, Natural convection heat transfer of cold water between concentric cylinders. For high Rayleigh numbers – a numerical study, Int. J. Eng. Sci. 32 (1994) 1437–1450. [32] C.J. Ho, Y.H. Lin, Natural convection heat transfer of cold water within an eccentric horizontal cylindrical annulus, ASME J. Heat Transfer 110 (1988) 894–900. [33] C.V. Raghavarao, Y.V.S.S. Sanyasiraju, Natural convection heat transfer of cold water in an eccentric horizontal cylindrical annulus – a numerical study, Comput. Mech. 18 (1996) 464–470. [34] B. Gebhart, J.C. Mollendorf, A new density relation for pure and saline water, Deep Sea Res. 24 (1977) 831–848. [35] C.J. Ho, F.J. Tu, Transition to oscillatory natural convection of cold water in a vertical annulus, Int. J. Heat Mass Transfer 41 (1998) 1559–1572. [36] D.L. Zeng, Engineering Nonequilibrium Thermodynamics, Science China Press, Beijing, 1991.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Natural convection of water near its density maximum between horizontal cylinders You-Rong Li a,b,⇑, Xiao-Feng Yuan a, Chun-Mei Wu a, Yu-Peng Hu a a b
College of Power Engineering, Chongqing University, Chongqing 400044, China Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Ministry of Education, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 21 September 2010 Received in revised form 25 January 2011 Accepted 26 January 2011 Available online 23 February 2011 Keywords: Natural convection Heat transfer Density inversion Horizontal annulus Flow pattern
a b s t r a c t In order to understand the characteristics of natural convection of cold water near its density maximum between horizontal cylinders, a series of unsteady two-dimensional numerical simulations were conducted by using finite volume method. The radius ratio of horizontal cylinders ranged from 1.2 to 2.0, density inversion parameter from 0 to 1, and the vertical eccentricity from 0 to 1.0 for eccentric annulus. The results show that the flow pattern mainly depends on the density inversion parameter and Rayleigh number. The formation of small cell at the top or bottom of annulus corresponds to the Rayleigh–Bénard instability within the converse density gradient layer. The width of annulus has slightly influence on the flow structure. However, the number of Bénard cells decreases with the increase of the radius ratio. For the oscillatory flow at a large Rayleigh number, the vertical converse density gradient in the top of annulus or the horizontal density gradient in the middle of annulus plays an important role for the formation of oscillatory flow when the density inversion parameter is in a small or moderate range. But the vertical density gradient in the bottom of annulus and the horizontal density gradient in the middle of annulus work together for oscillatory flow when the density inversion parameter is high. Average Nusselt number on the inner wall increases with the increase of Rayleigh number and radius ratio. However, there exists the minimum value of average Nusselt number at a moderate density inversion parameter. The flow pattern in eccentric annulus has the characteristics of coupling flows in the narrow-gap at the bottom with in the large-gap at the top of annulus. With the increase of the eccentricity, heat transfer is enhanced and the average Nusselt number increases slightly. Based on the simulation results, the new heat transfer correlation has been proposed according to the multiple linear regression technique. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Natural convection in a fluid-filled cavity is a very interesting topic for many researchers due to its wide range of applications in science, engineering and technology. Generally, fluid density changes as a linear function of temperature. But, for some fluids, the density–temperature relation exhibits an extremum. For example, the mass density of pure water reaches its maximum value at about 4 °C at the sea-level atmospheric pressure. Above this temperature, the mass density of water decreases with the increase of temperature in a manner similar to other fluids. However, when the temperature is below 4 °C, the trend is reversed, i.e. the mass density of water increases with the increase of its temperature. This is so-called density inversion phenomenon and it results in complicated flow patterns and heat transfer characteristics for the natural convection of water near its density maximum. During the past decades, a lot of experimental observations and numerical simulations have been reported for the natural convec⇑ Corresponding author at: College of Power Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 6511 2284; fax: +86 23 6510 2473. E-mail address: [email protected] (Y.-R. Li). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.02.006
tion of water near its density maximum inside a rectangle cavity with heated side walls and insulated horizontal surfaces. The representative works on steady-state [1–10] and transient [11–20] natural convection have all revealed that the density inversion phenomenon had great effect on the flow pattern, temperature field and heat transfer characteristic. Meanwhile, influences of internal heat generation [21] and magnetic field [22] on natural convection of water near its density maximum in a rectangular cavity were also investigated. Moreover, Kandaswamy et al. [23,24] performed a numerical simulation for the transient natural convective heat transfer of water near its density maximum in a square cavity with partially active vertical walls. The results indicated that the heat transfer was enhanced when the heating area located in the middle of the vertical wall. On the other hand, Kalabin et al. [25] and Zubkov et al. [26,27] studied the natural convection of water near its density maximum in a square cavity with adiabatic vertical walls and isothermal horizontal walls. Four different kinds of steady convection and at least three kinds of unsteady oscillatory convection were found under various Grashof numbers and initial conditions, and it has been concluded that only one flow pattern was asymmetric and all the other flow patterns were symmetric about the central vertical axis among the unsteady convections.
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Nomenclature d e g Nu P Pr q r R R⁄ Ra T V
distance between the center of inner cylinder and outer cylinder eccentricity, d/(ro ri) acceleration of gravity, m/s2 Nusselt number, Nu ¼ Ri lnðR Þ@ H @R R¼Ri
dimensionless pressure Prandtl number, m/a exponent in density–temperature Eq. (1) radius, m dimensionless radial coordinate radius ratio, ro/ri Rayleigh number based on the gap width, Ra = gc(Ti To)q(ro ri)3/(ma) temperatures, °C dimensionless velocity
Greek symbols a thermal diffusivity, m2/s
In the design of the energy storage systems, we often encounter the natural convective heat transfer of water near its density maximum between horizontal cylinders with uniform temperature boundary condition on the inner and outer surfaces, respectively. However, up to now, only a few investigations about this heat transfer problem have been reported. Seki et al. [28], Nguyen et al. [29], Vasseur et al. [30] and Raghavarao and Sanyasiraju [31] investigated steady natural convective heat transfer of water near its density maximum between two horizontal concentric cylinders with constant surface temperatures for different Rayleigh (Ra) number, the radius ratio and the density inversion parameter. It was found that the flow pattern was greatly influenced by the density inversion parameter of water, and concluded that the effect of density inversion was unexpectedly large and the average Nusselt (Nu) number was different from that of Boussinesq liquids whose density decreased linearly with the increase of temperature. Ho and Lin [32] and Raghavarao and Sanyasiraju [33] performed a numerical simulation on steady-state natural convection of cold water near its density maximum within an eccentric horizontal annulus. Results indicated that the flow patterns and heat transfer characteristics were strongly influenced by the combined effect induced by the density inversion of water and the position of the inner cylinder of the annulus. A minimum on heat transfer appeared with the inversion parameter between 0.4 and 0.5 depending primarily on the position of the inner cylinder. For natural convection between horizontal cylinders, the fluid is subjected not only to the horizontal temperature gradient in the middle region, but also to the vertical temperature gradient in the top and bottom regions, which results in more complicated flow patterns. Therefore, the aim of this paper is to reveal these complicated flow patterns and heat transfer characteristics of natural convection of water near its density maximum from steady-state to oscillatory flow between horizontal cylinders with different radius ratio, density inversion parameter and eccentricity.
c l m h
H
q s w
coefficient in density–temperature Eq. (1), (°C)q dynamic viscous, kg/(m s) kinematic viscosity, m2/s azimuthal coordinate dimensionless temperature, H = (T To)/(Ti To) density, kg/m3 dimensionless time dimensionless stream function
Subscripts ave average i inner wall o outer wall m density inversion point max the maximum R radial p periodical h azimuthal
respectively. Their centers locate at O and O0 , respectively. The eccentricity e is measured by the non-dimensional distance between the center of inner and outer cylinders. The following assumptions are introduced in this model: (1) the velocity is low and the flow is in laminar region; (2) all physical properties are independent of temperature except for the density in buoyancy force term; (3) viscous dissipation is neglected. Moreover, the nonlinear density–temperature relation of water, which is proposed by Gebhart and Mollendorf [34], is adopted in the following form:
qðTÞ ¼ qm ð1 cjT T m jq Þ;
ð1Þ 3
where the maximum density qm = 999.972 kg/m at the temperature Tm = 4.029325 °C, q = 1.894816 and c = 9.297173 106 (°C)q. The thermophysical properties of cold water at Tm are l = 1.567 103 kg/(m s), k = 0.562045 W/(m K), a = 1.344 107 m2/s, and Pr = 11.67. Based on the above assumptions, the non-dimensional governing equations of continuity, momentum and energy are given by introducing (ro ri), (ro ri)2/m, m/(ro ri) and ml/(ro ri)2 as the characteristic length, time, velocity and pressure, respectively.
1 @ðV R RÞ @V h ¼ 0; þ R@h R @R @V R @V R @V R V 2h þ VR þ Vh @s @R R@h R @P 1 @ @V R @2V R 2 @V h ¼ þ R þ 2 2 2 @R R @R @R R @h R @h V R Ra 2 jH Hm jq sin h; Pr R
2. Physical and mathematical models The physical model of the problem and the coordinate system are shown in Fig. 1. The fluid is contained between two infinite horizontal cylinders with the inner radius ri and outer radius ro. The inner and outer walls of the horizontal annulus are isothermally held at constant uniform temperatures Ti and To (Ti > To),
Fig. 1. Physical model and coordinate system.
ð2Þ
ð3Þ
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@V h @V h @V h V R V h þ VR þ Vh þ @s @R R@h R @P 1 @ @V h @2V h 2 @V R þ 2 2þ 2 ¼ þ R R@h R @R @R R @h R @h V h Ra 2 jH Hm jq cos h; Pr R
Table 2 Comparison of average Nusselt numbers for different Rayleigh number in a square enclosure. Ra
ð4Þ
" # @H @H V h @H 1 1 @ @H @2H þ VR R þ ¼ þ 2 2 : @s @R R @h Pr R @R @R R @h
ð5Þ
The corresponding initial condition and boundary conditions on the two cylinder walls are as follows:
s ¼ 0; V R ¼ V h ¼ 0; H ¼ 0;
ð6a—cÞ
V R ¼ V h ¼ 0;
ð7a—cÞ
R ¼ Ri ;
R ¼ e sin h þ
H ¼ 1;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2o ðe cos hÞ2 ;
V R ¼ V h ¼ 0;
H ¼ 0:
ð8a—cÞ
The density inversion parameter in Eqs. (3) and (4) is given as:
Hm ¼ ðT m T o Þ=ðT i T o Þ:
ð9Þ
The local and average Nusselt numbers at the inner wall are defined as
@ H Nu ¼ Ri lnðR Þ ; @R R¼Ri Nuave ¼
Ri 2psp
lnðR Þ
ð10Þ
Z s0 þsp Z 2p @ H dh ds: @R R¼Ri s0 0
ð11Þ
The velocity field is displayed in terms of the non-dimensional stream function w, which is defined as
VR ¼
@w ; R@h
Vh ¼
@w : @R
ð12Þ
3. Numerical method and validation check The governing equations (2)–(5) are discretized by the finite volume method. A central difference approximation is applied to the diffusion term and the QUICK scheme is used for the convective term. The SIMPLE algorithm is adopted to handle the pressure– velocity coupling. The dimensionless time step is chosen between 104 and 102. At each time step, the convergence is reached if the maximum relative error for velocity and temperature are less than 105. In order to evaluate the grid convergence, simulations with several different meshes were conducted. As shown in Table 1, the average Nusselt number of inner wall were presented with different grids at Ra = 5 104 and Hm = 0.7. From this table, it clearly shows that 50R 220h, 70R 160h and 70R 90h grids are dense enough for the accurate simulation at Ra = 5 104 and Hm = 0.7 for R⁄ = 1.2, 1.5 and 2.0, respectively. For other Ra number and den-
Table 1 Comparison of the average Nusselt number for various grids at Ra = 5 104 and Hm = 0.7. R⁄ = 1.2
Mesh (R h) Nuave
30 140 1.813
50 220 1.785
80 340 1.770
R⁄ = 1.5
Mesh (R h) Nuave
50 120 2.166
70 160 2.162
100 220 2.159
R⁄ = 2.0
Mesh (R h) Nuave
50 70 2.535
70 90 2.530
100 120 2.526
103 104 105 106
Nuave Ref. [4]
Present
1.001 1.076 2.080 4.090
0.989 1.065 2.079 4.124
sity inversion parameter, the grid convergence check is confirmed as well. In order to check the accuracy of the present numerical method, some numerical simulations have been carried out for the steady state natural convection in a square enclosure filled with water near its density maximum under the same conditions as Ref. [4]. The average Nusselt numbers are compared with those given by Nansteel et al. [4] at different Rayleigh numbers, as shown in Table 2. It could be found that the present results are very close to those in Ref. [4]. Further validation simulations have also been conducted for the problem of oscillatory convection of cold water in a tall vertical rectangular cavity and a vertical annulus considered by Ho and Tu [16,35]. The streamlines and isotherms are in a good agreement with those results in Refs. [16,35], and the maximum deviation of the average Nusselt number is about 5.6%. Therefore, all these results confirm that the present numerical method is accurate enough to study the natural convection of water near its density maximum. 4. Results and discussion 4.1. The narrow-gap horizontal concentric annulus For the narrow-gap horizontal concentric annulus, for example, R⁄ = 1.2, the steady-state natural convection of cold water is always symmetry about the vertical axis for any density inversion parameter. At Hm = 0.3, a crescent-shaped flow pattern appears in whole region of concentric annulus at a small Ra number, and the fluid flows upward along the inner wall and downward along the outer wall, as shown in Fig. 2(a). In this case, the isotherms are a set of concentric lines. With the increase of Rayleigh number, the strength of flow is enhanced and a small counter-rotating cell starts appearing near the top of annulus. Due to the change of flow pattern, the isotherms are firstly distorted near the top. With the further increase of Rayleigh number, more small cells are formed at the top region of annulus, as shown in Fig. 2(b) and (c). In Fig. 3, the local Nusselt number of inner wall is presented. From this figure, it is easy to find that a large variation of local Nusselt number appears when the small cells exist near the top region of annulus. Obviously, the Nusselt number increases if the fluid flows towards the inner wall and decreases if the fluid flows depart from the inner wall. As shown in Fig. 3, when Rayleigh number is 104, there exist two extremums of local Nusselt number at Hm = 0.3 and the maximum value is located at the top region of annulus. For the case of Hm = 0.5, a reverse bi-cellular flow structure is clearly observed, as shown in Fig. 2(d). In this case, the fluid flows upward along both inner wall and outer walls, and downward along the interface of the two adjacent cells. Due to the opposite flow direction, the two adjacent cells are suppressed each other, therefore, the strength of flow gets weakened at the same Rayleigh number and the maximum value of the stream function gets decreased as well. At the same time, the heat transfer ability is decreased due to the change of flow pattern. As shown in Fig. 3, the local Nusselt number is almost a constant close to 1 at Ra = 104, which means that conductive heat transfer is dominant.
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Fig. 2. Streamlines (left) and isotherms (right) at R⁄ = 1.2. Solid lines denote positive value of stream function and dotted lines denote negative value. dH = 0.125. (a) Hm = 0.3, Ra = 3 103, wmax = 13.23; (b) Hm = 0.3, Ra = 4.5 103, wmax = 20.05; (c) Hm = 0.3, Ra=104, wmax = 46.75; (d) Hm = 0.5, Ra = 2 104, wmax = 17.85; (e) Hm = 0.7, Ra=104, wmax = 48.70; (f) Hm = 0.7, Ra = 9 104, wmax = 230.35.
produced with the increase of Rayleigh number. Therefore, the additional Bénard cells appear firstly in these regions when the Rayleigh number exceeds a certain threshold value. Furthermore, in order to confirm that this flow pattern is the result of the Rayleigh–Bénard instability, the local Rayleigh number RaL is
(a)
Fig. 3. The distribution of local Nu number on the inner wall at R⁄ = 1.2 and Ra = 104. 1: Hm = 0.3; 2: Hm = 0.5; 3: Hm = 0.7.
(c) τ0
4
τ0+τp/5
τ0+2τp/5
τ0+3τp/5
τ0+4τp/5
top 3
Nu
For the case of Hm = 0.7, the crescent-shaped cell also appears in the whole region of concentric annulus for a small Ra number, however, the flow direction is opposite as compared with the case of Hm = 0.3, i.e., the fluid flows upward along the outer wall and downward along the inner wall. Furthermore, one or more small cells are formed near the bottom of annulus with the increase of Rayleigh number, as shown in Fig. 2(e) and (f). In this case, the distortion of the isotherms appears firstly at the bottom of annulus, where convective heat transfer begins to become dominant. Therefore, the maximum value of local Nusselt number appears at the bottom region for Hm = 0.7, as shown in Fig. 3. Due to nonlinear density correlation of cold water, the maximum density contour is located on the mid-plane of annulus at Hm = 0.5. Since the dense fluid near the mid-plane falls while the less dense fluid adjacent to the inner and outer walls rises, a reversal bi-cellular flow structure is formed. With the decrease or increase of density inversion parameter, the maximum density contour moves toward the outer or inner wall. The crescentshaped cell close to the outer or inner wall becomes stronger and larger. At the top and bottom of annulus for Hm = 0.3 and 0.7, respectively, there exists a converse density gradient layer. Within this converse density gradient layer, Rayleigh–Bénard instability is
(b)
2
1
(c) −π/2
0
π/2
θ
π
3π/2
Fig. 4. Snapshots of streamlines (a) and isotherms (b), and temporal evolution (c) of local Nu number on the inner wall during a period at R⁄ = 1.2, Hm = 0.3 and Ra = 2 104. dH = 0.125, wmax = 83.94.
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estimated, which is defined as RaL = gb(DT)qL3/(ma), where DT is temperature difference through the converse density gradient layer and L is the thickness of the converse density gradient layer. The simulation results show that the local Rayleigh numbers are about 1253.87 and 1201.51 for Hm = 0.3 and 0.7, respectively. It
(a)
(b)
(c) τ0
4
τ0+τp/5
τ0+2τp/5
τ0+3τp/5
τ0+4τp/5
top
Nu
3
right
left
2
1
(c) −π/2
0
π/2
θ
π
3π/2
Fig. 5. Snapshots of streamlines (a) and isotherms (b), and temporal evolution (c) of local Nu number on the inner wall during a period at R⁄ = 1.2, Hm = 0.5 and Ra = 105. dH = 0.125, wmax = 191.21. Arrows denote the moving direction of the cells.
shows that the local Rayleigh number would exceed the critical Rayleigh number value, i.e., Racri = 1100.65, which was obtained by the linear stability analysis [36] for the incipience of the Rayleigh–Bénard instability in an infinitely extended fluid layer with a free and a solid surface subjected to a constant vertical temperature gradient. Therefore, a conclusion could be made that this flow pattern is the result of the Rayleigh–Bénard instability within the converse density gradient layer. With the further increase of Rayleigh number, the steady-state flow starts to become unstable, i.e., oscillatory convection flow, and the symmetry of flow and temperature fields disappears. At Hm = 0.3, the flow instability starts firstly near the top of annulus, and then some small cells appear at the top of annulus for a high Ra number. As shown in Fig. 4, it is presented the snapshots of streamlines and isotherms, and temporal evolution of local Nusselt number on the inner wall during a period at R⁄ = 1.2 and Ra = 2 104. From this figure, it is obviously observed that there exist a set of Bénard cells with the different rotation direction near the top of annulus and two crescent-shaped cells located at the lower-left and lower-right sides of annulus, respectively. The centers of the Bénard cells are almost fixed, and the flow strength of two adjacent Bénard cells at the top of annulus is almost constant. However, the flow strength of the Bénard cells adjacent to the crescent-shaped cells shifts continuously, which results in the movement of the crescent-shaped cells. Therefore, it is indicated that the vertical converse density gradient at the top of annulus is responsible for this oscillatory flow at Hm = 0.3. In Fig. 4(c), it is clearly shows that the local Nusselt number on the inner wall varies with the shift of the Bénard cells adjacent to the crescent-shaped cells. It is found that the local Nusselt number varies with the center and strength of the cells, however, the average Nusselt number on the inner wall is independent of time. For the case of Hm = 0.5, the local Rayleigh number through the vertical converse density gradient layer at the bottom of annulus is greater than that at the top of annulus. Therefore, the Rayleigh– Bénard convection firstly occurs at the bottom, and then at the top of annulus with the increase of Rayleigh number. Moreover, the reversal double-circulation multicellular flow pattern was
Fig. 6. Streamlines (left) and isotherms (right) at R⁄ = 2.0. dH = 0.125. (a) Hm = 0.3, Ra = 103, wmax = 2.27; (b) Hm = 0.3, Ra = 5 104, wmax = 58.35; (c) Hm = 0.5, Ra = 5 105, wmax = 43.73; (d) Hm = 0.5, Ra = 106, wmax = 87.73; (e) Hm = 0.7, Ra = 5 104, wmax = 58.99; (f) Hm = 0.7, Ra = 5 105, wmax = 73.16.
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observed in the left and right sides of the vertical section of annulus at a high Rayleigh number. This pattern corresponds to the thermal instability based on the horizontal temperature gradient in the vertical region. The number of cells increases and the region occupied by the cells expands with the increase of Rayleigh number. The small cells, which split from the reversal two bi-cellular cells, originate from the bottom of annulus, then drift upward and grow up, finally decay at the top of annulus, as shown in Fig. 5. It is similar to natural convection of water near its density maximum inside a tall rectangular enclosure with differential heating at the vertical walls [16]. However, the cells near the top and bottom of annulus are almost changeless. It suggests that the horizontal density gradient plays the main role for this oscillatory flow at Hm = 0.5. From Fig. 5(c), it is found that the local Nusselt number varies with the movement of the cells in the left and right of the vertical section of annulus. However, it keeps a constant near the top and bottom of annulus. The average Nusselt number on the inner wall is also independent of time. When Hm = 0.7, the multiple like-rotating cells appear in the vertical section of annulus and the additional Bénard cells at the bottom of annulus extend upwards for a high Ra number. The movement of cells in the vertical section of annulus is the
Fig. 7. The distribution of local Nu number on the inner wall at R⁄ = 2.0 and Ra = 5 104. 1: Hm = 0.3; 2: Hm = 0.5; 3: Hm = 0.7
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same as that at Hm = 0.5, and the movement of cells at the bottom of annulus is similar with that at Hm = 0.3. Therefore, the vertical density gradient at the bottom of annulus and the horizontal density gradient in the middle of annulus work together to induce the oscillatory flow in this case. 4.2. The large-gap horizontal concentric annulus In the case of large-gap horizontal concentric annulus, the flow pattern of steady-state natural convection of cold water is also symmetry about the vertical axis for any density inversion parameter. At Hm = 0.3, the center of the crescent-shaped cells moves upwards and there appears another small crescent cell adjacent to the outer wall at the bottom of annulus with the increase of the Rayleigh number, as shown in 6(a) and (b). Moreover, the distortion of isotherms also appears firstly near the top of annulus, and then extends to the bottom of annulus. A large local temperature gradient is formed near the outer wall at the top of annulus and near the inner wall at the bottom of annulus, where the convective heat transfer is dominant. Therefore, the maximum value of local Nusselt number on the inner wall for Hm = 0.3 at Ra = 5 104 locates at the bottom of annulus, as shown in Fig. 7. When Hm = 0.5, a reversal bi-cellular flow structure is also formed at a very small Ra number. However, with the increase of Rayleigh number, the cell near the outer wall gets stronger and becomes more angular in shape. On the contrary, the cell near the inner wall is restrained by that near the outer wall, which results in the gradual disappearing of the co-rotating cell, as shown in Fig. 6(c) and (d). In this case, the maximum value of local Nusselt number on the inner wall locates at the top of annulus, where the fluid flows towards the inner wall. For the case of Hm = 0.7, the Rayleigh–Bénard convection also occurs at the bottom of annulus with a moderate Rayleigh number, as shown in Fig. 6(e). However, only one Bénard cell appears due to the large gap of horizontal concentric annulus. At a large Rayleigh number, for example Ra = 5 105, the crescent-shaped cell becomes more angular and the counter-rotating cell at the bottom
Fig. 8. Snapshots of streamlines (upper) and isotherms (down) at R⁄ = 2.0. dH = 0.125. (a) Hm = 0.3, Ra = 105, wmax = 80.26; (b) Hm = 0.3, Ra = 106, wmax = 215.36; (c) Hm = 0.7, Ra = 1.8 106, wmax = 123.05.
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of annulus vanishes, as shown in Fig. 6(e). In this case, the local Nusselt number is also dependent on the flow pattern. Because there exist the crescent-shaped cell and a Bénard cell near the bottom at Ra = 5 104, local Nusselt numbers on the inner wall have large values at the top and bottom of the annulus. The minimum value of local Nusselt number locates at interface of two cells, where the fluid flows depart from the inner wall, as shown in Fig. 7. When the Rayleigh number exceeds a certain critical value, the steady-state flow in a large-gap horizontal concentric annulus will transit to an oscillatory flow. For the case of Hm = 0.3, variations of the streamlines and isotherms are similar to the ones in a narrowgap annulus. The Rayleigh–Bénard convection also occurs at the top region and becomes unstable. However, the flow pattern is symmetrical even at Ra = 106. The crescent-shaped cells near the inner wall become more angular in shape, and several cells are observed in the crescent-shaped cells as a result of free-shear instability, which holds the self-sustained oscillation. Furthermore, the small crescent cells close to the outer wall extend upward (see Fig. 8(a) and (b)). When Hm = 0.7, several similar cells are observed in the interior of large crescent-shaped cells with the increase of Ra number, as shown in Fig. 8(c). For the large-gap horizontal concentric annulus, the instability mechanism of natural convection of cold water under different density inversion parameter is the same as one in a narrow-gap annulus.
4.3. The vertical eccentric annulus The effect of vertical eccentricity on natural convection of water near its density maximum in a horizontal annulus is similar for different radius ratio. Therefore, only the results at radius ratio R⁄ = 1.5 are given. The steady-state flow for Hm = 0.3 is similar with that in horizontal concentric annulus. When there is a vertical eccentricity, the convective space spreads and the critical Rayleigh number for the Bénard cells decreases with the increase of eccentricity. Fig. 9 gives the snapshots of streamlines and isotherms of oscillatory convections for different eccentricity at Hm = 0.3. It is found that the flow is also symmetrical and periodic oscillation appears at a moderate Rayleigh number, for example Ra = 3 104. In this case, with the increase of eccentricity, the number of the Bénard cells near the top of annulus increases. The location of the cells is almost fixed for the periodic oscillation. However, periodic expansion and shrinkage of the Bénard cells will drive the movement of crescent-shaped cells along the azimuthal direction, which results in a periodic variation of the local Nusselt number. When the Rayleigh number increases to Ra = 105, the oscillatory convection becomes asymmetry, as shown in Fig. 9(b), and the number of the Bénard cells near the top of annulus increases. At Hm = 0.5, the reversal bi-cellular flow structure is formed and the corresponding distortion of isotherms is consistent with
Fig. 9. Snapshots of streamlines (upper) and isotherms at R⁄ = 1.5 and Hm = 0.3. dH = 0.125.
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that in horizontal concentric annulus at a small Rayleigh number. With the increase of eccentricity, the flow is enhanced and the non-dimensional stream function increases. However, the stagnant region of the fluid flow at the top of annulus extends, as shown in Fig. 10(a). Furthermore, the distortion of crescent-shaped cells augments. Especially, the head of crescent-shaped cells become large. At e = 0.3, with the increase of Rayleigh number, the flow transits from the steady-state convection to the oscillatory convection. The co-rotating cells near the bottom of annulus and the reversal bi-cellular multiple like-rotating cells have only very small change comparing with these in horizontal concentric annulus. But the flow near the top of annulus becomes chaotic. The increase of eccentricity results in a large-gap at the top of annulus, where several chaotic secondary vortexes near the inner wall is formed at e = 0.5 and 0.7, as shown in Fig. 10(b). When Hm = 0.7, the flow region is reduced and the multicellular steady flow for narrow-gap annulus appears at the bottom of annulus with the increase of eccentricity. Furthermore, the large crescent-shaped cells become more angular at the top of annulus with the increase of Rayleigh number. The cells closed to the inner wall extend upwards. At a large Rayleigh number, the flow transits to the oscillatory multi-cellular convection in the narrow-gap bottom of annulus and the chaotic convection with several vortexes in the crescent-shaped cells at the vertical large-gap part of annulus.
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4.4. Correlation equation for heat transfer characteristics Fig. 11 shows the variation of average Nusselt number on the inner wall of the annulus with the Rayleigh number at different radius ratio and density inversion parameter from steady-state to oscillatory flow. Obviously, with the increase of Ra number, the flow is enhanced, therefore, the average Nusselt number gets increased. Furthermore, when the flow pattern transits from steady state to oscillatory state, the average Nusselt number at different Ra number locates near a straight line in bi-logarithm coordinate system at a fixed radius ratio and density inversion parameter. It suggests that variation law of average Nusselt number with the Rayleigh number is independent of the flow pattern transition. Therefore, the critical condition of the flow patterns transition is not presented in this paper. Around Hm = 0.5, the reversal bi-cellular flow structure results in that the flow near inner wall is suppressed by the flow cell near the outer wall, therefore, the average Nusselt number is always the least at the same Ra number. When the density inversion parameter increases or decreases from Hm = 0.5, the average Nusselt number all increases, as shown in Fig. 11. On the other hand, the width of the annulus has also important effect on the flow intensity and heat transfer characteristic. With the increase of the radius ratio, the flow space of natural convection expands and the flow is
Fig. 10. Snapshots of streamlines (upper) and isotherms at R⁄ = 1.5 and Hm = 0.5. dH = 0.125.
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4
(a)
(a) Nu ave
Nu ave
3
10
2
4
2
1 10 3
10 4
Ra
10 5
1 103
10 6
5 4
104
105
106
107
105
106
107
105
106
107
Ra
10
(b)
(b) Nu ave
Nu ave
3
2
4
2
1 103
104
105
106
107
1 103
Ra 7 6 5
Ra
10
(c)
(c) Nu ave
4
Nu ave
104
3
4
2
2 1 103
104
105
Ra
106
107
1 103
Fig. 11. The variation of average Nusselt number with Rayleigh number. (a) R⁄ = 1.2; (b) R⁄ = 1.5; (c) R⁄ = 2.0. 4: Hm = 0.3 (steady); N: Hm = 0.3 (oscillation); s: Hm = 0.5 (steady); d: Hm = 0.5 (oscillation); h: Hm = 0.7 (steady); j: Hm = 0.7 (oscillation).
enhanced, therefore, the average Nusselt number increases at the same Rayleigh number. Fig. 12 gives the variation of average Nusselt number on the inner wall with the Rayleigh number at the different eccentricity. Although the vertical eccentricity has the effect on the flow pattern, the variation of the average Nusselt number is small. At Hm = 0.3, the average Nusselt number increases slightly with the increase of the eccentricity at a fixed Rayleigh number. For Hm = 0.5 and 0.7, it also increases with the increase of the eccentricity at a small Rayleigh number. However, at a large Rayleigh number, because the multi-cellular flow for narrow-gap annulus appears, this variation become very complex with the increase of eccentricity. For eccentric annulus, the effect of density inversion parameter on average Nusselt number is in accordance with that in concentric annulus. Based on the simulation results, a multiple linear regression technique is carried out for these data, and a correlation equation for average Nusselt number on the inner wall is given as follows:
Nuave ¼ 0:122Ra0:261 R0:312 jHm 0:496j0:104 ð1 þ eÞ0:166 The application ranges of Eq. (13) are as follows:
ð13Þ
104
Ra
Fig. 12. The variation of average Nusselt number on the inner wall. (a) e = 0.3; (b) e = 0.5; (c) e = 0.7. 4: Hm = 0.3 (steady); N: Hm = 0.3 (oscillation); s: Hm = 0.5 (steady); d: Hm = 0.5 (oscillation); h: Hm = 0.7 (steady); j: Hm = 0.7 (oscillation).
1:2 6 R 6 2:0;
104 6 Ra 6 8:5 106 ;
0 < Hm < 1;
06e<1
The present simulation data falls within ±15% of Eq. (13) in terms of the average Nusselt number, and the average deviation of the average Nusselt number between the simulation data and the proposed correlation is about 4.8%. 5. Conclusions The natural convection of cold water near its density maximum between horizontal cylinders is numerically investigated by the finite volume method. The radius ratio of horizontal concentric cylinders ranged from 1.2 to 2.0 and density inversion parameter from 0 to 1. The following conclusions could be made: For steady-state flow, the flow and temperature fields are always symmetry about the vertical axis for any density inversion parameter. The flow pattern depends mainly on density inversion parameter and Rayleigh number. The formation of small cell at the top or bottom of annulus attribute to the Rayleigh–Bénard instability within the converse density gradient
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layer. At a small density inversion parameter, Bénard cells appear near the top of annulus, and they also appear near the bottom of annulus for a large density inversion parameter. The width of annulus has slightly influence on the flow structure. However, the number of Bénard cells decreases with the increase of the radius ratio. When the flow transits to oscillatory state, the flow and temperature fields become asymmetrical. Furthermore, the vertical converse density gradient at the top of annulus and the horizontal density gradient in the middle of annulus play the main role for the formation of oscillatory flow at a small and moderate density inversion parameter, respectively. But the vertical density gradient at the bottom of annulus and the horizontal density gradient in the middle of annulus work together for oscillatory flow at a large density inversion parameter. For vertical eccentric annulus, the flow pattern has the characteristics of coupling the narrow-gap flow at the bottom with the large-gap flow at the top of annulus. The convective space at the top of annulus spreads and the flow is enhanced. The critical Rayleigh number for the Bénard cells at the top of annulus decreases with the increase of eccentricity. Furthermore, the number of the Bénard cells increases. However, when oscillatory convection happens, the mechanisms of flow instability under different density inversion are the same as that in horizontal concentric annulus. Average Nusselt number on the inner wall increases with the increase of Ra number and radius ratio. However, there exists the minimum value of average Nusselt number at about Hm = 0.5. With the increase of the eccentricity, heat transfer is enhanced and the average Nusselt number increases slightly. Based on the simulation results, the new heat transfer correlation has been proposed according to the multiple linear regression technique. The average deviation of the proposed average Nusselt number correlation is about 4.8%.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 50976129) and the Fundamental Research Funds for the Central Universities (Grant No. CDJXS10 14 22 24). References [1] A. Watson, The effect of the inversion temperature on the convection of water in an enclosed rectangular cavity, Quart. J. Mech. Appl. Math. 25 (1972) 423– 446. [2] V.S. Desai, R.E. Forbes, Free convection in water in the vicinity of maximum density, Environ. Geophys. Heat Transfer 93 (1971) 41–47. [3] M.W. Nansteel, K. Medjani, D.S. Lin, Natural convection of water near its density maximum in a rectangle enclosure: low Rayleigh number calculations, Phys. Fluids 30 (1986) 312–317. [4] D.S. Lin, M.W. Nansteel, Natural convection heat transfer in a square enclosure containing water near its density maximum, Int. J. Heat Mass Transfer 30 (1987) 2319–2329. [5] W. Tong, J.N. Koster, Natural convection of water in a rectangular cavity including density inversion, Int. J. Heat Fluid Flow 14 (1993) 366–375. [6] W. Tong, J.N. Koster, Coupling of natural convection flow across a vertical density inversion interface, Heat Mass Transfer 28 (1993) 471–479. [7] N. Seki, S. Fukusako, H. Inaba, Free convective heat transfer with density inversion in a confined rectangular vessel, Heat Mass Transfer 11 (1978) 145– 156. [8] H. Inaba, T. Fukuda, An experimental study of natural convection in an inclined rectangular cavity filled with water at its density extremum, ASME J. Heat Transfer 106 (1984) 109–115.
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