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The subject of natural convection heat transfer is motivated by a wide range of applications in engineering technology. The hemispherical cavity is a part of basic geometries although it is not widely studied. The effect of inclinaison on natural convection fluid motions in the gap between two eccentric hemispheres is numerically studied. The inner hemisphere is subjected to a heat flux of a constant density and the outer one is maintened isothermal. The walls separating the two hemispheres are thermally adiabatic. Equations are formulated with vorticity and stream-functions variables. It is also assumed the fluid incompressible and obeys the approximation of Boussinesq. These equations are written by using bispherical coordinates system and solved by using a finite difference method. The results show the topology of flow is strongly dependent on the inclinaison because the flow can change from a unicellular regime to a multicellular regime by varying the inclination from 0 to π. By increasing the Rayleigh number (10
^{3}<
Ra<10
^{7}), the flow intensifies. T
he results are shown in terms of streamlines and isotherms during their transient evolution.

Natural convection heat transfer in enclosures especially from objects limited by spheres surfaces is of considerable interest in many engineering areas. The spheres or hemispheres can be used as constructing and insulated surfaces The rectangular or square cavity has been extensively treated [

A schematic depiction of the problem is shown in

We considered the fluid Newtonian and the flow incompressible and bidimensional. Viscous dissipation, compressibility effects an thermal radiation are neglected. We further applied the Boussinesq approximation. An adapted conformal tranformation is used in order to reduce a curvilinear enclosure into a rectangular field. A bispherical coordinate system is chosen. The dimensionnal transformation from bispherical ( η , θ ) of Cartesian coordinate ( x , y ) [

{ x = a sin θ cosh η − cos θ y = a sinh η cosh η − cos θ (1)

where a is the parameter of torus pole.

The inner copula is materialized by line of coordinate h = h i and the outer hemisphere by h = h e .

On the basis of these assumptions, the equations governing the problem in a dimensionless vorticity-stream function are written in bispherical coordinate. To transform these dimensional equations into a set of dimensionless equations, we used the references parameters as follow:

a , a 2 α and q a λ which represent respectively the length, time and temperature

gradient. Heat and momentum equations in the dimensionless form as given respectively as:

∂ 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 ) (2)

∂ 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 φ ∂ η T − G 1 φ ∂ θ T ) (3)

where Ψ and Ω are defined by:

U = 1 K H ∂ θ Ψ , V = − 1 K H ∂ η Ψ and Ω → ′ = ∇ → ∧ V → ′ .

With

{ g 1 = g 1 ( η , θ ) = − 1 h ∂ θ k = 1 − cos θ cosh η cosh η − cos θ g 2 = g 2 ( η , θ ) = 1 h ∂ η k = − sin θ sinh η cosh η − cos θ (4)

{ G 1 φ = G 1 ( η , θ , φ ) = g 2 cos φ + g 1 sin φ G 2 φ = G 2 ( η , θ , φ ) = g 1 cos φ + g 2 sin φ (5)

We defined the stream-function which intresically verifies incompressibility condition. His particularty is he has the dimension of a volume flow.

Ψ = K ϕ (6)

With ϕ is the stream-function which has a surface flow.

The dimensionless equation of stream-function is:

Ω = 1 K 2 H ( G 2 φ ∂ η Ψ − G 1 φ ∂ θ Ψ ) − 1 K H 2 ( ∂ η 2 Ψ + ∂ θ 2 Ψ ) (7)

These equations are subject to the following boundary conditions and initial:

· For t = 0 :

Ω = Ψ = T = U η = V θ = 0 (8)

· On the inner hemisphere ( η = η i ):

U = V = Ψ = 0 (9)

Ω = − 1 K H ∂ η 2 Ψ (10)

∂ η T = H i (11)

· On the outer hemisphere ( η = η e ):

U = V = Ψ = 0 (12)

Ω = − 1 K H ∂ η 2 Ψ (13)

· On the vertical walls ( θ = 0 ) and ( θ = π )

U = V = Ψ = 0 (14)

∂ θ T = 0 (15)

The local and average Nusselt number are defined respectively by:

· For the inner hemisphere

N u i = 1 T i , m (16)

N u i ¯ = 1 S ∬ N u i d S (17)

· For the inner hemisphere

N u e = 1 H e T i , m ∂ η T ( η = η e ) (18)

N u e ¯ = 1 S ∬ N u e d S (19)

Our physical domain is complex. the bispheric coordinate system we used allowed us to translate our curved walls into a parallelepiped grid. The implicit method alterning direction (A.D.I.) [

∑ i , j ( | w i , j m + 1 − w i , j m w i , j m + 1 | ) ≤ 10 − 5 (20)

where m denotes the iteration and w stands for Ψ , Ω , T .

Computation ConditionsPreliminary tests have been done on the influence on the mess and time step (see ^{−}^{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 | 3.2827 | 3.4581 | 3.8719 |

Time computing (min) | 30 | 87 | 390 |

δ ¯ ( % ) | 15.22 | 10.69 | 0 |

Meshgrid | ||||||||
---|---|---|---|---|---|---|---|---|

21 × 21 | 21 × 41 | 41 × 41 | 41 × 51 | 41 × 81 | 51 × 51 | 51 × 81 | 81 × 81 | |

Nu | 3.1203 | 3.2509 | 3.2273 | 3.2220 | 3.2130 | 3.2173 | 3.2061 | 3.1985 |

Time computing | 14 | 15 | 45 | 55 | 88 | 85 | 133 | 334 |

δ ¯ ( % ) | 2.44 | 1.64 | 0.9 | 0.73 | 0.45 | 0.59 | 0.24 | 0 |

With δ ¯ = | N u ref − N u present study | N u ref .

Figures 2-5 present the evolution of the streamlines and isotherms over time respectively for R a = 10 3 , R a = 10 5 , R a = 10 6 and R a = 10 7 . For 10^{3}, at the first moments, the streamlines the flow is constitued by two cells that turns in opposite directions. The secondary cell disappears in favor of the main cell. The isotherms similar to eccentric circles marrying the internal wall at the first moments and are then slightly deformed at the beginning of time for R a = 10 3 due to the phenomena of conduction as we can see it on ^{3}, the flow is dominated by the conduction, when R a = 10 5 or 10^{6}, the convection predominates and when Ra reached the value 10^{7}, we notice the formation of multi-cell flow.

Figures 6-9 depicted the evolution of streamlines and isotherms for various inclinaison angles ( φ = 0 , π / 3 , π / 2 , 2 π / 3 and π). When the Rayleigh number is fixed at 10^{5} and the eccentricity at −0.5. We notice that streamlines for configurations φ = 0 and φ = π (

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

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

Present results | 2.125 | 3.0651 | 4.982 | 7.6874 | 11.671 |

[ | 2.062 | 3.062 | 4.977 | 7.7720 | 12.109 |

δ ¯ ( % ) | 3.54 | 0.10 | 0.10 | 0.42 | 3.62 |

[ | 2.098 | 3.15 | 5.034 | 7.794 | 12.109 |

δ ¯ ( % ) | 1.76 | 2.70 | 1.03 | 1.37 | 3.62 |

they occupy almost the entire hemispherical cavity. For the value of angle φ = π / 3 (

negative eccentricity −0.5 and for a positive eccentricity +0.5. The streamlines show the monocellular structure appears rather with the negative eccentricity.

Likewise, the isotherms are much more deformed and more fill the cavity for this structure. When the eccentricity increases, we note a decrease in heat exchanges.

The numerical simulation of transient natural convection between two inclined hemispheres whose the inner hemisphere is subjected to a heat flux of constant density while the outer one is maintened isothermal with time is conducted. The bispherical coordinate and finite difference methods are used.

It emerges from the results presented that when the Rayleigh number is equal to 10^{3}, heat transfers are dominated by conduction whatever the inclination and eccentricity. The convective transfers intensify as the Rayleigh number increases and there is a strongly destabilized regime for a Rayleigh number equal to 10^{7}. The simulations have also shown that the topology of the flow is strongly dependent on the inclinaison because the flow can change from a monocellular regime to a bicellular regime by varying the inclinaison. We remark also when the eccentricity increase, we note a decrease in heat exchanges. It would also be interesting to replace the Newtonian fluid with a complex fluid, for example, a fluid whose viscosity depends on the temperature.

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

Koita, M.N., Sow, M.L., Dramé, O., Mbow, B., Mbow, C. and Sarr, J. (2021) Numerical Simulation of Natural Air Convection in Inclined Eccentric Hemispheres Enclosure. Open Journal of Applied Sciences, 11, 722-735. https://doi.org/10.4236/ojapps.2021.116053

Grecs symbols

α Thermal diffusivity, m^{2}·s^{−1}

β Thermal expansion coefficient, K^{−1}

η , θ , ζ Bispheric coordinate, m

λ Thermal conductivity, W·m^{−1}·K^{−1}

μ Kinematic viscosity, kg·m^{−1}·s^{−1}

Ω Dimensionless vorticity

Ψ Dimensionless volume stream function

ρ Density, kg·m^{−3}

φ Angle of inclinaison, rad

Latin letters

a Characteristic length a = R e − R i , m

e Dimensionless eccentricity

g Gravitational acceleration, m·s^{−2}

G 1 , G 2 Coefficients

H, K Dimensionless parameters

Nu Nusselt number

Pr Prandlt number

Ra Rayleigh number

T Dimensionless temperature T = λ q a ( T ′ − T 0 )

t Dimensionless time

U, V Dimensionless velocity components

x , y , z Cartesian coordinates, m