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Effects of Prandtl number and a new instability mode in a plane thermal plume

Published online by Cambridge University Press:  14 November 2007

R. LAKKARAJU  and
MEHEBOOB ALAM
R. LAKKARAJU
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur PO, Bangalore 560064, India
MEHEBOOB ALAM*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur PO, Bangalore 560064, India
*
†Author to whom correspondence should be addressed: meheboob@jncasr.ac.in
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Abstract

The effect of Prandtl number on the linear stability of a plane thermal plume is analysed under quasi-parallel approximation. At large Prandtl numbers (Pr > 100), we found that there is an additional unstable loop whose size increases with increasing Pr. The origin of this new instability mode is shown to be tied to the coupling of the momentum and thermal perturbation equations. Analyses of the perturbation kinetic energy and thermal energy suggest that the buoyancy force is the main source of perturbation energy at high Prandtl numbers that drives this instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 592 , 10 December 2007 , pp. 221 - 231
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Batchelor, G. K. 1954 Heat convection and buoyancy effects in fluids. Q. J. R. Met. Soc. 80, 339–359.Google Scholar
Bridges, T. J. & Morris, P. J. 1984 Differential eigenvalue problems in which the parameter appears nonlinearly. J. Comput. Phys. 55, 437–460.CrossRefGoogle Scholar
Fujii, T. 1963 Theory of the steady laminar natural convection above a horizontal line heat source and a point heat source. Intl J. Heat Mass Transfer 6, 597–606.Google Scholar
Gebhart, B., Pera, L. & Schorr, A. 1970 Steady laminar natural convection plumes above a horizontal line heat source. Intl J. Heat Mass Transfer 13, 161–171.CrossRefGoogle Scholar
Gebhart, B., Jaluria, J., Mahajan, R. L. & Sammakia, B. 1988 Buoyancy Induced Flows and Transport. Hemisphere.CrossRefGoogle Scholar
Gill, A. E. & Davey, A. 1969 Instabilities of a buoyancy-driven system. J. Fluid Mech. 35, 775–798.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Hieber, C. A. & Nash, E. J. 1975 Natural convection above a line heat source: higher order effects and stability. Intl J. Heat Mass Transfer 18, 1473–1479.CrossRefGoogle Scholar
Kaminski, E. & Jaupart, C. 2003 Laminar starting plumes in high-Prandtl-number fluids. J. Fluid Mech. 478, 287–298.Google Scholar
Lilly, K. L. 1966 On the instability of Ekman boundary flow. J. Atmos. Sci. 21, 481–494.2.0.CO;2>CrossRefGoogle Scholar
Lister, J. R. 1987 Long-wavelength instability of a line plume. J. Fluid Mech. 175, 571–591.CrossRefGoogle Scholar
Majumder, C. A. H., Yuen, D. A. & Vincent, A. P. 2004 Four dynamical regimes for a starting plume model. Phys. Fluids 16, 1516–1531.Google Scholar
Malik, M. R. 1990 Numerical methods for boundary layer stability. J. Comput. Phys. 86, 376–413.CrossRefGoogle Scholar
Nachtsheim, P. R. 1963 Stability of free convection boundary layers. NASA TN D-2089.Google Scholar
Pera, L. & Gebhart, B. 1971 On the stability of laminar plumes: some numerical solutions and experiments. Intl J. Heat Mass Transfer 14, 975–984.Google Scholar
Riley, N. 1974 Free convection from a horizontal line source of heat. Z. Angew. Math. Phys. 25, 817–828.Google Scholar
Wakitani, S. 1985 Non-parallel flow instability of a two dimensional buoyant plume. J. Fluid Mech. 159, 241–258.Google Scholar
Wang, X. 2004 Infinite Prandtl number limit of Rayleigh–Bénard convection. Commun. Pure Appl. Maths. 57, 1265–1282.Google Scholar
Worster, M. G. 1985 Asymptotic analysis of a high Prandtl number axisymmetric plume. Stud. Appl. Maths. 159, 241–258.Google Scholar