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Cooling of a Pr<1 fluid in a rectangular container

Published online by Cambridge University Press:  15 February 2007

WENXIAN LIN ,
S. W. ARMFIELD  and
JOHN C. PATTERSON
WENXIAN LIN*
Affiliation:
School of Engineering, James Cook University, Townsville, QLD 4811, Australia Solar Energy Research Institute, Yunnan Normal University, Kunming 650092, China
S. W. ARMFIELD
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia
JOHN C. PATTERSON
Affiliation:
School of Engineering, James Cook University, Townsville, QLD 4811, Australia
*
†Author to whom correspondence should be addressed: [email protected].
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Abstract

The flow behaviour associated with the cooling of an initially quiescent isothermal Newtonian fluid with Prandtl number Pr less than unity in a rectangular container by unsteady natural convection with an imposed lower temperature on vertical sidewalls is investigated by scaling analysis and direct numerical simulation. The flow is dominated by two distinct stages of development. i.e the boundary-layer development stage adjacent to the sidewall and the subsequent cooling-down stage. The first stage can be further divided into a start-up stage, transitional stage, and steady-state stage. The parameters characterizing the flow behaviour are the boundary-layer thickness, the maximum vertical velocity within the boundary layer, the time for the boundary layer to reach the steady state, the Nusselt number across the sidewall at the boundary-layer development stage, the time for the fluid in the container to be fully cooled down, and the average fluid temperature over the whole volume of the container.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 574 , 10 March 2007 , pp. 85 - 108
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Armfield, S. W. 1991 Finite-difference solutions of the Navier–Stokes equations on staggered and non-staggered grids. Computers Fluids 20, 1–17.CrossRefGoogle Scholar
Armfield, S. W. & Debler, W. 1993 Purging of density stabilized basins. Intl J. Heat Mass Transfer 36, 519–530.CrossRefGoogle Scholar
Armfield, S. W. & Patterson, J. C. 1991 Direct simulation of wave interactions in unsteady natural convection in a cavity. Intl J. Heat Mass Transfer 34, 929–940.CrossRefGoogle Scholar
Armfield, S. W. & Patterson, J. C. 1992 Wave properties of natural-convection boundary layers. J. Fluid Mech. 239, 195–211.CrossRefGoogle Scholar
Armfield, S. W. & Street, R. L. 1999 The fractional-step method for the Navier–Stokes equations on staggered grids: the accuracy of three variations. J. Comput. Phys. 153, 660–665.CrossRefGoogle Scholar
Catton, I. 1978 Natural convection in enclosures. In Proc. 6th Intl Heat Transfer Conf., pp. 13–43.CrossRefGoogle Scholar
Gebhart, B., Jaluria, Y., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy-Induced Flows and Transport. Hemisphere.CrossRefGoogle Scholar
Goldstein, R. J. & Briggs, D. G. 1964 Transient free convection about vertical plates and circular cylinders. Trans. ASME: J. Heat Transfer 86, 490–500.CrossRefGoogle Scholar
Goldstein, R. J., Eckert, E. R. G., Ibele, W. E. et al. 2003 Heat transfer – a review of 2001 literature. Intl J. Heat Mass Transfer 46 1887–1992.CrossRefGoogle Scholar
Hyun, J. M. 1984 Transient process of thermally stratifying an initially homogeneous fluid in an enclosure. Intl J. Heat Mass Transfer 27, 1936–1938.CrossRefGoogle Scholar
Hyun, J. M. 1985 Effect of the Prandtl number on heatup of a stratified fluid in a enclosure. Trans. ASME: J. Heat Transfer 107, 982–984.CrossRefGoogle Scholar
Hyun, J. M. 1994 Unsteady buoyant convection in an enclosure. Adv. Heat Transfer 24, 277–320.CrossRefGoogle Scholar
Incropera, F. P. & DeWitt, D. P. 1996 Fundamentals of Heat and Mass Transfer, 4th Edn. John Wiley & Sons.Google Scholar
Javam, A., Imberger, J. & Armfield, S. W. 1999 Numerical study of internal wave reflection from sloping boundaries. J. Fluid Mech. 396, 183–201.CrossRefGoogle Scholar
Kwak, H. S., Kuwahara, K. & Hyun, J. M. 1998 Convective cool-down of a contained fluid through its maximum density temperature. Intl J. Heat Mass Transfer 41, 323–333.CrossRefGoogle Scholar
Leonard, B. P. 1979 A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comp. Meth. Appl. Mech. Engng 19, 59–98.CrossRefGoogle Scholar
Lin, W. 2000 Scaling analysis and direct simulation of unsteady weak fountains and natural convection flow. PhD Thesis, The University of Sydney, Sydney.Google Scholar
Lin, W. & Armfield, S. W. 1999 Direct simulation of natural convection cooling in a vertical circular cylinder. Intl J. Heat Mass Transfer 42, 4117–4130.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2000 Direct simulation of weak axisymmetric fountains in a homogeneous fluid. J. Fluid Mech. 403, 67–88.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2001 Natural convection cooling of rectangular and cylindrical containers. Intl J. Heat Fluid Flow 22, 72–81.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2002 Weak fountains in a stratified fluid. Phys. Rev. E 66 066308.Google Scholar
Lin, W. & Armfield, S. W. 2003 The Reynolds and Prandtl number dependence of weak fountains. Computational Mech. 31, 379–389.CrossRefGoogle Scholar
Lin, W. & Armfield, S. W. 2004 Long-term behaviour of cooling fluid in a rectangular container. Phys. Rev. E 69, 056315.Google Scholar
Nicolette, V. F. & Yang, K. T. 1985 Transient cooling by natural convection in a two-dimensional square enclosure. Intl J. Heat Mass Transfer 28, 1721–1732.CrossRefGoogle Scholar
Ostrach, S. 1953 An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. NACA Rep. 1111.Google Scholar
Ostrach, S. 1982 Natural convection heat transfer in cavities and cells. In Proc. 7th Intl Heat Transfer Conf., pp. 365–379.Google Scholar
Otis, D. R. & Roessler, J. 1987 Development of stratification in a cylindrical enclosure. Intl J. Heat Mass Transfer 30, 1633–1636.CrossRefGoogle Scholar
Patterson, J. C. 1984. Unsteady natural convection in a cavity with internal heating and cooling. J. Fluid Mech. 140, 135–151.CrossRefGoogle Scholar
Patterson, J. C. & Armfield, S. W. 1990 Transient features of natural convection in a cavity. J. Fluid Mech. 219, 469–497.CrossRefGoogle Scholar
Patterson, J. C. & Imberger, J. 1980. Unsteady natural convection in a rectangular cavity. J. Fluid Mech. 100, 65–86.CrossRefGoogle Scholar
Schladow, S. G., Patterson, J. C. & Street, R. L. 1989 Transient flow in a side-heated cavity at high Rayleigh number: a numerical study. J. Fluid Mech. 200, 121–148.CrossRefGoogle Scholar
Tritton, D. J. 1995 Physical Fluid Dynamics, 2nd Edn. Oxford University Press.Google Scholar