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Natural convection heat transfer in open or closed cavities takes place in different engineering areas. The hemispherical cavity is a part of basic geometries although it is not widely studied. The present paper reports the numerical study of natural convection in a closed hemispherical annulus delimited by two vertically eccentric hemispheres filled with Newtonian fluid (air in this case with
*Pr* = 0.7) is conducted. The inner hemisphere is heated by a heat flux of constant density and the outer one is maintained isothermal. Based on the Boussinesq assumptions, the governing equations are numerically studied using unsteady natural convection formulated with vorticity and stream-function variables. These equations are written by using bispherical coordinates system and solved by using a finite difference method. The effect of the control parameters such as the Rayleigh number (
10^{3} ≤ *Ra* ≤ 10^{6}) or the eccentricity (
*e* = ±0.2, ±0.5, 0) in the dynamic and thermal behaviours of the fluid is investigated.

For several decades, heat transfer by natural convection has been the subject of much research and also offers a diverse field of application as electronics, nuclear industry, building or solar energy. Many theorical and experimental studies on rectangular [^{3}. The average Nusselt increases with the Prandtl number. Numerical studies have been done in other geometry cavities. Mack and Hardee [

Consider as depicted in _{i} and R_{e} centered respectively on θ i and θ e . The eccentricity e' is defined as the algebraic distance separating the two centers of hemispheres.

Initially, the enclosure is at a uniform temperature. From this time, a constant wall heat flux (q') is suddenly applied on the inner hemisphere while the outer is maintained isothermal ( T ′ c t e ). The walls separating the two hemispheres ( θ = 0 , θ = π ) are insulated. The two hemispheres being heated differently, a transient convection developpes inside the enclosed space.

We assume that the physical properties of the fluid are constant unless this density in terms of gravity, which in a first approximation of Boussinesq hypothesis, varies linearly with temperature. Viscous dissipation, thermal radiation and compressibility effects are neglected. We considered the fluid Newtonian and the flow incompressible and bidimensionnal. In order to reduce the curvilinear enclosure into a rectangular field a curvilinear coordinate system is used. A conformal transformation [

{ x = a sin θ ch η − cos θ y = a sh η ch η − cos θ (1)

The walls located on the vertical axis are given by θ = 0 and θ = π . The inner

hemisphere is materialized by line of coordinate ( h = h i ) and the outer hemisphere ( h = h e ).

Under these assumptions, the equations governing the problem in a dimensionless vorticity-stream function are the equation of the momentum and heats equations respectively:

∂ ∂ t ( Ω K ) + 1 H [ U − 3 P r g 2 K ] ∂ η ( Ω K ) + 1 H [ V + 3 P r g 1 K ] ∂ θ ( Ω K ) = P r H 2 [ ∂ η 2 ( Ω K ) + ∂ θ 2 ( Ω K ) ] + R a P r K H ( G 2 ∂ η T − G 1 ∂ θ T ) (2)

∂ t T + 1 H [ U − g 2 K ] ∂ η T + 1 H [ V + g 1 K ] ∂ θ T = 1 H 2 [ ∂ η 2 T + ∂ θ 2 T ] (3)

With:

{ x = a sin θ ch η − cos θ y = a sh η ch η − cos θ (4)

{ U = 1 K H ∂ Ψ ∂ θ V = − 1 K H ∂ Ψ ∂ η (5)

{ g 1 = 1 − cos θ ch η − cos θ y = − sin θ sh η ch η − cos θ (6)

We define the stream-function which intrinsically verifies incompressibility condition and which has the dimension of a volume flow.

Ψ = K Φ (7)

is the stream function which has a dimension of a surface flow.

The equation of stream function is given as:

Ω = 1 K 2 H [ g 2 ∂ η Φ − g 1 ∂ θ Φ ] − 1 K H 2 [ ∂ η 2 Φ − ∂ θ 2 Φ ] (8)

The above equations are complemented by the following initial and the boundary conditions:

For t = 0:

Ω = Ψ = T = U = V = 0 (9)

And for t > 0:

· On the inner hemisphere: η = η i

U = V = Ψ = 0 (10)

Ω = − 1 K H ∂ η 2 Ψ (11)

∂ η T = H i = ch η i sh 2 η i (12)

· On the outer hemisphere: η = η e

U = V = Ψ = 0 (13)

Ω = − 1 K H ∂ η 2 Ψ (14)

· On the vertical walls θ = 0 and θ = π :

U = V = Ψ = 0 (15)

∂ η T = 0 (16)

Ω = − 1 K H ∂ θ 2 Ψ (17)

The heat energy transmitted by the active walls is characterized by the Nusselt number. The local and the average Nusselt numbers relative to the inner and the outer hemisphere are as follows:

· For the inner hemisphere:

N u i = 1 T i , m (18)

N u i ¯ = 1 S i ∫ N u i d S i (19)

· For the outer hemisphere:

N u i = 1 H e T i , m ∂ η T (20)

N u e ¯ = 1 S e ∫ N u e d S e (21)

The implicit method alternating direction (A.D.I) [

At each instant, we assume that variable converges to its numerical value if the following stop criterion is satisfied:

| W n + 1 − W n | max | W n + 1 | max ≤ 10 − 5 (22)

where n denotes the iteration step and W stands for

Tests have been done on the influence of the mesh and on the time step. ^{−5} for the time which constitutes a good compromise between a precision and an acceptable computation time.

Time steps | |||
---|---|---|---|

10^{−4} | 10^{−5} | 10^{−6} | |

Nu | 4.7751 | 4.9524 | 5.0033 |

Difference (%) | 4.56 | 1.02 | 0 |

Time computing (min) | 24 | 95 | 471 |

Mesh grid | ||||||||
---|---|---|---|---|---|---|---|---|

21 * 21 | 21 * 41 | 41 * 41 | 41 * 51 | 41 * 81 | 51 * 51 | 51 * 81 | 81 * 81 | |

Nu | 5.0217 | 5.0617 | 4.9614 | 4.9647 | 4.9524 | 4.9670 | 4.9624 | 4.9618 |

Difference (%) | 1.82 | 2.63 | 0.60 | 0.67 | 0.71 | 0.42 | 0.38 | 0 |

Time computing (min) | 7 | 12 | 37 | 45 | 70 | 75 | 145 | 232 |

Ra | |||||
---|---|---|---|---|---|

10^{3} | 10^{4} | 10^{5} | 10^{6} | 10^{7} | |

Nu (our results) | 2.125 | 3.0651 | 4.982 | 7.6874 | 11.671 |

Nu (Tazi et al., 1997) | 2.098 | 3.15 | 5.034 | 7.794 | 12.109 |

Difference (%) | 1.76 | 2.70 | 1.03 | 1.37 | 3.62 |

Nu (Sow et al., 2007) | 2.062 | 3.062 | 4.977 | 7.7720 | 12.109 |

Difference (%) | 3.54 | 0.10 | 0.10 | 0.42 | 3.62 |

was less than 3.7% for all cases presented.

Figures 2(a)-(c) present side by side for each value of eccentricity the time evolution of the isotherms and the streamlines. The eccentricity varies from −0.5 to +0.5 and the Rayleigh number and the Prandtl number are equal respectively to 10^{6} and 0.7.

The shape of isothermal lines initially matches the internal wall during the first moment. However the isotherms are increasingly deformed over time for all the studies values of eccentricities. In general, the fluid heats up as it rises along

the internal wall and then goes down again along the external wall. When the modified Rayleigh number is high, isotherms are strongly deformed due to the motion. The maximum value of the stream function Ψ max increase with the eccentricity. This increasing is more important with negative value of eccentricity. We noted the fluid vortex center upwards when eccentricity increases.

Figures 3(a)-(c) show respectively the evolution of the Nusselt number, the stream function and the temperature of the heated wall.

^{3} is more important than that obtained in the case of two spheres [

T = λ ( T ′ − T ′ 0 ) q D (23)

q = v 2 λ P r q β D 4 R a (24)

The dimensionless temperature is inversely proportional to the heat flux density applied on the internal hemisphere and thus to the Rayleigh number. Indeed, when the heat flux density increases, the convection also increases. We notice the

value of the dimensionless temperature is greater than that obtained with Sow [

The time evolution of the minimum of stream function presented in

This is explaining by the fact at the beginning, the convection is predominant and the motion of fluid is important and the phenomena convection subside when the steady state is reached.

Figures 4(a)-(c) show respectively the evolution of the average Nusselt number, the minimum of stream function and the dimensionless temperature of the heated wall

The numerical study of unsteady natural convection between two eccentric hemispheres was investigated. The inner hemisphere is subjected to a heat flux constant while the outer one is maintained isotherm. The bispherical coordinates and the ADI and SOR methods based on finite difference method are used. The vorticity and stream-function variables are assumed.

The values obtained for the isotherms agree well with those of Sow [

The study of the effects of eccentricity shows that the center of rotation moves toward the top of enclosure when the eccentricity increases. This increase is more important with negative value of eccentricity.

The average Nusselt number increases with the modified Rayleigh number. However, the dimensionless temperature decreases when the modified Rayleigh number increases regardless of the value of eccentricity and the minimum of stream function decreases as the eccentricity increases.

The authors declare no conflicts of interest regarding the publication of this paper.

Koita, M.N., Sow, M.L., Thiam, O.N., Traoré, V.B., Mbow, C. and Sarr, J. (2021) Unsteady Natural Convection between Two Eccentric Hemispheres. Open Journal of Applied Sciences, 11, 177-189. https://doi.org/10.4236/ojapps.2021.112012

a, parameter of torus pole, m

e, eccentricity e = O i O e a

g, gravity intensity, m∙s^{−2}

g 1 and g 2 coefficients g 1 = g 1 ( η , θ ) = 1 − cos θ ch η ch η − cos θ , g 2 = g 2 ( η , θ ) = − sin θ sh η ch η − cos θ

H and K, metrics coefficient dimensionless H = 1 ch η − cos θ , K = sin θ ch η − cos θ

N u e , Nusselt number for the outer hemisphere, N u i = 1 T i , m

N u e , Nusselt number for the inner hemisphere, N u i = 1 H e T i , m ∂ η T

O i and O e , respectively center of inner and outer hemisphere

Pr, Prandtl number P r = ν α

q, Heat flux density W∙m^{−2}

R i and R e , respectively radius of inner and outer hemisphere

Ra, Rayleigh number, R a = g β a 4 q ν λ α

t, dimensionless time, t = α a 2 t ′

t ′ , dimension time, s

T, dimensionless temperature, T = λ q a ( T ′ − T ′ 0 )

U and V, dimensionless velocity components in the transformed planes

x and y, cartesian coordinates, m

α , thermal diffusivity, m^{2}∙s^{−1}

β , thermal expansion coefficient, K^{−1}

Δ t , time step, s

Δ T , difference of temperatures between the two hemispheres, K

η and θ , bispherical coordinates m

λ , thermal conductivity, W∙K^{−1}∙m^{−1}

ν , kinematical viscosity, m^{2}∙s^{−1}

Ψ , dimensionless stream-function, Ψ = K α a Ψ ′

Ψ ′ , dimension stream-function

Ω , dimensionless vorticity, Ω = a α 2 Ω ′

Ω , dimension vorticity