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Research Papers: Natural and Mixed Convection

Convective Motion and Heat Transfer in a Slowly Rotating Fluid Quasi-Sphere With Uniform Heat Source and Axial Gravity

[+] Author and Article Information
Gerardo Anguiano-Orozco

Facultad de Ciencias,
UAEM, Av. Instituto Literario 100,
50000 Toluca, Mexico;
Depto. de Física,
Instituto Nacional
de Investigaciones Nucleares,
Apdo. Postal 18-1027,
Mexico D.F.
e-mail: gerardo.anguiano@gmail.com

Ruben Avila

Departamento de Termofluidos,
Facultad de Ingeniería,
Universidad Nacional Autónoma de México,
C.P. 04510,
Mexico D.F.
e-mail: ravila@unam.mx

Syed Shoaib Raza

Pakistan Institute of Engineering
and Applied Sciences (PIEAS),
P.O. Nilore,
Islamabad 45650, Pakistan
e-mail: fac246@pieas.edu.pk;
ssraza@msn.com

1In Memoriam, 2010.

2Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received November 23, 2010; final manuscript received November 29, 2012; published online March 20, 2013. Assoc. Editor: Darrell W. Pepper.

J. Heat Transfer 135(4), 042501 (Mar 20, 2013) (10 pages) Paper No: HT-10-1549; doi: 10.1115/1.4023126 History: Received November 23, 2010; Revised November 29, 2012

The laminar natural convection of a rotating fluid quasi-sphere in the presence of an axial gravity field and uniform heat source is presented. The influence of the Rayleigh number Ra and the Taylor number Ta on the flow pattern and heat transfer rate from the fluid quasi-sphere is discussed. The governing nonsteady, three-dimensional Navier–Stokes equations for an incompressible fluid, formulated in a Cartesian coordinate system, have been numerically solved by using the h/p spectral element method. It is shown that for a given Ta number, as the Ra number is increased, the heat transfer on the northern hemisphere is enhanced whereas the average Nusselt number on the southern hemisphere is reduced. On the other hand for a given Ra number, as the Ta number is increased, the heat transfer is a function of the convective motion intensity. It has been found that for low and high Ra numbers the heat transfer rate slightly depends on the rotation rate. However at intermediate Ra numbers, the net effect of an increased rotation rate is a reduction of the heat transfer through the wall, hence an increase of the maximum temperature of the fluid sphere is observed. We show that the net effect of the Coriolis force is to damp the convective motion and to allow a redistribution of the vorticity field.

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References

Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford, UK.
Roberts, P. H., 1965, “On the Thermal Instability of a Highly Rotating Fluid Sphere,” Astrophysical J., 141, pp. 240–250. [CrossRef]
Roberts, P. H., 1968, “On the Thermal Instability of a Rotating-Fluid Sphere Containing Heat Sources,” Philos. Trans. R. Soc. London, Ser. A, 263, pp. 93–117. [CrossRef]
Evonuk, M., and Glatzmaier, G. A., 2007, “The Effects of Rotation Rate on Deep Convection in Giant Planets With Small Solid Cores,” Planet. Space Sci., 55, pp. 407–412. [CrossRef]
Anguiano-Orozco, G., and Avila, R., 2009, “Vortex Ring Formation Within a Spherical Container With Natural Convection,” Comput. Model. Simul. Eng., 49, pp. 217–254.
Sharma, A. K., and Balaji, C., 2004, “A Numerical Study of Natural Convection From a Localized Heat Source in the Lower Plenum of a Fast Breeder Reactor Under Failed Conditions,” Int. J. Heat Mass Transfer, 40, pp. 853–858. [CrossRef]
Dinh, T. N., Nourgaliev, R. R., and Sehgal, B. R., 1997, “On Heat Transfer Characteristics of Real and Stimulant Melt Pool Experiments,” Nucl. Eng. Des., 169, pp. 151–164. [CrossRef]
Dinh, T. N., and Nourgaliev, R. R., 1997, “Turbulence Modelling for Large Volumetrically Heated Liquid Pools,” Nucl. Eng. Des., 169, pp. 131–150. [CrossRef]
Whitley, H. G., III, and Vachon, R. I., 1972, “Transient Laminar Free Convection in Closed Spherical Containers,” ASME J. Heat Transfer, 94(4), pp. 360–366. [CrossRef]
Chow, M. Y., and Akins, R. G., 1975, “Pseudosteady-State Natural Convection Inside Spheres,” ASME J. Heat Transfer, 97(1), pp. 54–59. [CrossRef]
Hutchins, J., and Marschall, E., 1989, “Pseudosteady-State Natural Convection Heat Transfer Inside Spheres,” Int. J. Heat Mass Transfer, 32, pp. 2047–2053. [CrossRef]
Zhang, Y., Khodadadi, J., and Shen, F., 1999, “Pseudosteady-State Natural Convection Inside Spherical Containers Partially Filled With a Porous Medium,” Int. J. Heat Mass Transfer, 42, pp. 2327–2336. [CrossRef]
Ladeinde, F., and Torrance, K. E., 1991, “Convection in a Rotating, Horizontal Cylinder With Radial and Normal Gravity Forces,” J. Fluid Mech., 228, pp. 361–385. [CrossRef]
Shew, W. L., and Lathrop, D. P., 2005, “Liquid Sodium Model of Geophysical Core Convection,” Phys. Earth Planet. Interiors, 153, pp. 136–149. [CrossRef]
Asfia, F. J., and Dhir, V. K., 1996, “An Experimental Study of Natural Convection in a Volumetrically Heated Spherical Pool Bounded on Top With a Rigid Wall,” Nucl. Eng. Des., 163, pp. 333–348. [CrossRef]
Dallman, R. J., and Douglass, R. W., 1980, “Convection in a Rotating Spherical Annulus With a Uniform Axial Gravitational Field,” Int. J. Heat Mass Transfer, 23, pp. 1303–1312. [CrossRef]
Heinrich, J. C., and Pepper, D. W., 1989, “Flow Visualization of Natural Convection in a Differentially Heated Sphere,” Flow Visualization, B.Khalighi, M. J.Braun, and C. J.Freitas, eds., ASME FED, New York, Vol. 85, pp. 135–141.
Pepper, D. W., and Heinrich, J. C., 1993, “Transient Natural Convection Within a Sphere Using a 3-D Finite Element Method,” Finite Elements in Fluids, K.Morgan, E.Oñate, J.Periaux, J.Peraire, and O. C.Zienkiewicz, eds., Pineridge Press, UK, pp. 369–378.
Greenspan, H. P., 1968, The Theory of Rotating Fluids, Cambridge University Press, New York.
King, E. M., Stellmach, S., Noir, J., Hansen, U., and Aurnou, J. M., 2009, “Boundary Layer Control of Rotating Convection Systems,” Nature, 457, pp. 301–304. [CrossRef] [PubMed]
Patera, A. T., 1984, “A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion,” J. Comput. Phys., 54, pp. 468–488. [CrossRef]
Rønquist, E. M., 1988, “Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier-Stokes Equations,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
Karniadakis, G. E., and Sherwin, S. J., 1999, Spectral/hp Element Methods for CFD, Oxford University Press, New York.
Avila, R., and Solorio, F. J., 2009, “Numerical Solution of 2D Natural Convection in a Concentric Annulus With Solid-Liquid Phase Change,” Comput. Model. Eng. Sci., 44, pp. 177–202.
Cabello-Gonzalez, A., and Avila, R., 2010, “Numerical Simulation of the Convective Flow Patterns Within a Rotating Concentric Annulus With Radial Gravity,” 63rd Annual Meeting of the APS Division of Fluid Dynamics, Long Beach, CA, Nov. 21–23.
Osorio, A., Avila, R., and Cervantes, J., 2004, “On the Natural Convection of Water Near Its Density Inversion in an Inclined Square Cavity,” Int. J. Heat Mass Transfer, 47, pp. 4491–4495. [CrossRef]
Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, Clarendon Press, Oxford, UK.

Figures

Grahic Jump Location
Fig. 1

Natural convection and heat transfer in a rotating fluid sphere with uniform heat source and axial gravity. The origin of the Cartesian coordinate system is located at the center of the sphere. The dimensional constant angular velocity, axial gravity and uniform heat source are defined by Ω = Ω3i3, g = g3i3, and Qv, respectively.

Grahic Jump Location
Fig. 2

Spectral element method mesh with 256 nonregular hexahedra macroelements with polynomial expansion of order 6. The origin of the Cartesian coordinate system is located at the center of the quasi-sphere. Left panel: discretization of the bounding spherical surface. Right panel: an interior view of the mesh.

Grahic Jump Location
Fig. 3

SEM GLL points distribution in the quasi-sphere. Left panel: x2 - x3 meridional plane. Right panel: GLL points on the bounding spherical surface at three positions of the x1 axis (directed out of the page), (i) external circle at x1 = 0 (meridional plane), (ii) middle circle at x1 = 0.7, and (iii) internal circle at x1 = 0.91. Circles: SEM-GLL points. Continuous line: perfect sphere with radius 0.985. Order of the polynomial equal to 6.

Grahic Jump Location
Fig. 4

Nonsteady heat conduction within a quasi-sphere with homogeneous and constant heat source. Dimensionless temperature distribution along the radial direction. Left panel: polynomial interpolation p of order 6. Right panel: polynomial interpolation p of order 8. Continuous line: analytical solution for a perfect sphere, see Eqs. (6) and (7). Circles: SEM method results. At t = 1, the steady state condition is reached. Dimensional values: ΔT = Qv R2/6k = 161.7 K, Qv = 1000 W/m3, R = 0.985 m, k = 1 W/m-K, α = 1 m2/s.

Grahic Jump Location
Fig. 5

Natural convection in a fluid sphere for different Ra numbers without rotation Ta = 0. Meridional fields of vorticity (left column), pressure (middle column), and temperature (right column). (a): Ra = 10, (b): Ra = 1 × 103, (c): Ra = 1 × 104, (d): Ra = 1 × 105, (e): Ra = 5 × 105, (f): Ra = 1 × 106, and (g): Ra = 1 × 107.

Grahic Jump Location
Fig. 6

Dimensionless maximum temperature Tmax in the fluid sphere (left panel) and average Nusselt number Nu¯ (right panel) as functions of the Taylor number Ta and the Rayleigh number Ra. The convection coefficient h, used to obtain Nu¯, has been calculated from the energy balance (see Eqs. (12) and (13)). (i) ○ Ra = 10, (ii) * Ra = 1 × 103, (iii) □ Ra = 1 × 104, (iv) Δ Ra = 1 × 105, (v) ⋆ Ra = 1 × 106, and (vi) ⋄ Ra = 1 × 107.

Grahic Jump Location
Fig. 7

Average Nusselt number as a function of the Ra and Ta numbers. The convection coefficient h, used to obtain Nu¯, has been calculated from the energy balance (see Eqs. (12) and (13)). (i) ○ Ta = 0, (ii) □ Ta = 1600, (iii) Δ Ta = 6400, and (iv) ⋆ Ta = 14,400.

Grahic Jump Location
Fig. 8

Meridional convective vorticity fields with rotation. First column (left): Ta = 0, second column: Ta = 1600, third column: Ta = 6400, and fourth column (right): Ta = 14,400. (a): Ra = 10, (b): Ra = 1 × 103, (c): Ra = 1 × 104, (d): Ra = 1 × 105, (e): Ra = 1 × 106, and (f): Ra = 1 × 107.

Grahic Jump Location
Fig. 9

Meridional convective temperature fields with rotation. Same caption as Fig. 8.

Grahic Jump Location
Fig. 10

Dimensionless meridional (x2, x3) vorticity (i1 component) in a fluid (quasi sphere) for Ra = 10. Left panel: Ta = 1600. Middle panel: Ta = 6400. Right panel: Ta = 14,400. Bold line represents the average vorticity equal to zero, and a measure of the thickness of the Ekman boundary layer δE.

Grahic Jump Location
Fig. 11

Dimensionless Ekman boundary layer thickness δE, and thermal boundary layer thickness δT in terms of the Ra number. Left panel: Ta = 6400, δE ≈ 0.136 (continuous line). Right panel: Ta = 14,400, δE ≈ 0.113 (continuous line). ○: δT at the north pole region, □: δT at the equatorial region.

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