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Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids

Published online by Cambridge University Press:  26 April 2006

Hamda Ben Hadid  and
Bernard Roux
Hamda Ben Hadid
Affiliation:
Laboratoire de Mécanique des Fluides & d'Acoustique, URA 263, Ecole Centrale de Lyon, BP 163, 69131 Ecully Cedex, France
Bernard Roux
Affiliation:
Institut de Mécanique de Marseille, UM 34, 1, rue Honnorat 13003, Marseille, France
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Abstract

The influence of thermocapillary forces on buoyancy-driven convection is numerically simulated for shallow open cavities with differentially heated endwalls and filled with low-Prandtl-number fluid. Calculations are carried out by solving two-dimensional Navier-Stokes equations coupled to the energy equation, for three aspects ratios A = (length/height) = 4, 12.5 and 25, and several values of the Grashof number (up to 6 × 104) and Reynolds number (|Re| ≤ 1.67 × 104). Thermocapillarity can have a quite significant effect on the stability of a primarily buoyancy-driven flow. The result of the combination of the two basic mechanisms (thermo-capillarity) and buoyancy) depends on whether their effects are additive (positive Re) or opposing (negative Re); counter-acting mechanisms yield more complex flow patterns. The critical Grashof number Grc for the onset of the unsteady regime is found to decrease substantially within a small range of negative Re, and to increase for positive Re (and also for large negative Re). For Gr = 4 × 104, A = 4 and small negative Reynolds numbers, −2.4 × 103 ≤ Re < 0, mono-periodic and bi- or quasi-periodic regimes are shown to exist successively, followed by a reverse transition. The development of the instabilities from an initial steady-state regime has been investigated by varying Re for Gr = 1.5 × 104 (below Grc at Re = 0); the onset of buoyant instabilities is enhanced in a narrow range of Re only (-1200 < Re < -200). It is also noteworthy that for small enough Grashof numbers (e.g. Gr = 3 × 103), a steady-state solution prevails over the whole range of Reynolds numbers investigated. This means that a critical Grashof number exists below which the effect of the thermocapillary forces is no longer destabilizing.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 1 - 36
Copyright
© 1992 Cambridge University Press

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References

Azouni, M. A. 1981 Time-dependent natural convection in crystal growth systems. PhysicoChem. Hydrodyn. 2, 295–309.Google Scholar
Ben Hadid, H. 1989 Etude numeArique des mouvements convectifs au sein des fluides de faible nombre de Prandtl. TheGse de Doctorat d'Etat, University of Aix-Marseille II, France.
Ben Hadid, H., Laure, P. & Roux, B. 1989 Thermocapillarity effect on the stability of buoyancy-driven flows in shallow cavities. PhysicoChem. Hydrodyn. 11, 625–644.Google Scholar
Ben Hadid, H. & Roux, B. 1989 Buoyancy- and thermocapillary-driven flow in a shallow open cavity: unsteady flow regimes. J. Cryst. Growth 97, 217–225.Google Scholar
Ben Hadid, H. & Roux, B. 1990a Buoyancy-driven oscillatory flows in shallow cavities filled with low-Prandtl-number flows. In Numerical Simulation of Oscillatory Convection in Low-Pr Fluids (ed. B. Roux), pp. 25–34. Vieweg.
Ben Hadid, H. & Roux, B. 1990b Thermocapillary convection in long horizontal layers of low-Prandtl-number melts subject to horizontal temperature gradient. J. Fluid Mech. 221, 77–103.Google Scholar
Ben Hadid, H., Roux, B., Laure, P., Tison, P., Camel, D. & Favier, J. J. 1988 Surface tension-driven flows in horizontal liquid metal layers. Adv. Space Res. 8, 293–304.Google Scholar
Bergman, T. L. & Keller, J. R. 1988 Combined buoyancy surface-tension flow in liquid metals. Numer. Heat Transfer 13, 49–63.Google Scholar
Bergman, T. L. & Ramadhyani, S. 1986 Combined buoyancy- and thermocapillary-driven convection in open square cavities. Numer. Heat Transfer 9, 441–451.Google Scholar
Birikh, R. V. 1966 Thermocapillary convection in horizontal layer of liquid. J. Appl. Mech. Tech. Phys. 7, 43–49.Google Scholar
Bradley, M. C. & Homsy, G. M. 1989 Combined buoyancy-thermocapillary flow in a cavity. J. Fluid Mech. 207, 121–132.Google Scholar
Camel, D. & Favier, J. J. 1988 Transport processes during directional solidification and crystal growth: scaling and experimental study. In Physico-Chemical-Hydrodynamics Interfacial Phenomena (ed. M. G. Velarde), NATO ASI Series, B: vol. 174, pp. 595–618.
Camel, D., Tison, P. & Favier, J. J. 1985 Marangoni flow regimes in liquid metals. Acta Astronautica 13, 723–726.Google Scholar
Carruthers, J. R. 1977a Crystal growth in low gravity environment. J. Cryst. Growth 42, 379–385.Google Scholar
Carruthers, J. R. 1977b Thermal convection instabilities relevant to crystal growth from liquids. In Preparation and Properties of Solid State Materials, Vol. 3 (ed. W. R. Wilcox & R. A. Lefever). Marcel Dekker.
Chun, Ch. H 1980 Experiments on steady and oscillatory temperature distribution in a floating zone due to the Marangoni convection. Acta Astronautica 7, 479–488.Google Scholar
Cuvelier, C. & Driessen, J. M. 1986 Thermocapillary free boundaries in crystal growth. J. Fluid Mech. 169, 1–26.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19, 403–435.Google Scholar
Favier, J. J., Rouzaud, A. & Comera J. 1986 Influence of various hydrodynamic regimes in melts on solidification interface. Rev. Phys. Appl. 22, 195–200.Google Scholar
Gershuni, G. Z., Zhukhovitsky, E. M. & Nepomniashchy, A. A. 1989 Stability of Convective Flows.Moscow:Nauka.
Hirsh, R. 1975 Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90–109.Google Scholar
Hurle, D. T. J. 1967 Thermo-hydrodynamic oscillation in liquid metals: the cause of impurities striations in melt-grown crystals. J. Phys. Chem. Solids Suppl. no. 1, 659–669.Google Scholar
Hurle, D. T. J. 1983 Convective transport in melt growth systems. J. Cryst. Growth 65, 124–132.Google Scholar
Hurle, D. T. J., Jakeman, E. & Johnson, C. P. 1974 Convective temperature oscillations in molten Gallium. J. Fluid Mech. 64, 565–576.Google Scholar
Kamotani, Y., Ostrach, S. & Lowry, S. 1982 An experimental study of heat induced surface-tension driven flow. In xMaterials Processing in the Reduced Gravity Environment of Space (ed. G. E. Rindone), pp. 161–171. North-Holland.
Kirdyashkin, A. G. 1984 Thermogravitational and thermocapillary flows in a horizontal liquid layer under the conditions of a horizontal temperature gradient. Intl J. Heat Mass Transfer 27, 1205–1218.Google Scholar
Lamprecht, R., Schwabe, D. & Scharmann, A. 1990 Thermocapillary and buoyancy convection in an open cavity under normal and reduced gravity. J. Fluid Mech. (submitted).Google Scholar
Langlois W. E. 1985 Buoyancy-driven flows in crystal-growth melts. Ann. Rev. Fluid Mech. 17, 191–215.Google Scholar
Laure, P. & Roux, B. 1987 SyntheGse des reAsultats obtenus par l'eAtude de stabiliteA des mouvements de convection dans une caviteA horizontale de grande extension. C.R. Acad. Sci. Paris II, 305, 1137–1143.Google Scholar
Laure, P. & Roux, B. 1989 Linear and non-linear analyses of the Hadley circulation. J. Cryst. Growth 97, 226–234.Google Scholar
Levich, V. G. & Krylov, V. S. 1969 Surface tension-driven phenomena. Ann. Rev. Fluid Mech. 1, 293–316.Google Scholar
Metzger, J. & Schwabe, D. 1988 Coupled buoyant and thermocapillary convection. PhysicoChem. Hydrodyn. 10, 263–282.Google Scholar
MuUller, G. 1988 Convection and inhomogeneities in crystal growth from the melt. In Crystals, Properties, and Applications, vol. 12. Springer.
Myznikov, V. B. 1981 On the stability of steady advective motion in a horizontal layer with a free boundary relative to space disturbances. In Convective Flow (ed. E. M. Zukhovitskii), vol. 76, pp. 76–83. Perm State Pedagogical Institute.
Ostrach, S. 1982 Low-gravity fluid flows. Ann. Rev. Fluid Mech. 14, 313–345.Google Scholar
Ostrach, S. & Pradhan, A. 1978 Surface-tension induced convection at reduced gravity. AIAA J. 16, 419–424.Google Scholar
Pimputkar, S. M. & Ostrach, S. 1981 Convective effects in crystal grown from melts. J. Cryst. Growth 55, 614–646.Google Scholar
Polezhaev, V. I. 1984 Hydrodynamics of Heat and Mass Transfer During Crystal Growth. Springer.
Pulicani, J. P. 1989 Application des meAthodes spectrales aG l'eAtude d'eAcoulements de convection. Ph.D. thesis, University of Nice, France.
Randriamampianina, A., Crispo Del Arco, E., Fontaine, J. P. & Bontoux, P. 1990 Spectral method for two-dimensional time-dependent Pr0 convection. In Numerical Simulation of Oscillatory Convection in Low-Pr Fluids (ed. B. Roux), pp. 224–255. Vieweg.
Roux, B. (ed.) 1990 GAMM Workshop: Numerical Simulation of Oscillatory Convection in Low-Pr Fluids. Notes on Numerical Fluid Mechanics, vol. 27. Vieweg.
Roux, B., Ben Hadid, H. & Laure, P. 1989 Hydrodynamical regimes in metallic melts subjects to a horizontal temperature gradient. Eur. J. Mech. B 8, 375–396.Google Scholar
Roux, B., Bontoux, P., Loc, T. P. & Daube, O. 1979 Optimisation of Hermitian Methods for N.S. Equations in Vorticity and Stream Function Formulation. Lecture Notes in Mathematics, vol. 771, pp. 450–468. Springer.
Rosenberger, F. 1979 Fundamentals of Crystal Growth. Springer.
Schwabe, D. 1988 Surface-tension-driven flow in crystal growth melts. In Crystals, vol. 11, pp. 75–112. Springer.
Schwabe, D. & Scharmann, A. 1981 The magnitude of thermocapillary convection in large melt volumes. Adv. Space. Res. 1, 13–16.Google Scholar
Smith, M. K. & Davis, S. H. 1983 Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119–144.Google Scholar
Strani, M., Piva, R. & Graziani, G. 1983 Thermocapillary convection in a rectangular cavity: asymptotic theory and numerical simulation. J. Fluid Mech. 130, 347–376.Google Scholar
Thompson, J. F., Thames, F. C. & Mastin, C. W. 1974 Automatic numerical generation of a body-fitted curvilinear system. J. Comput. Phys. 15, 299–319.Google Scholar
Utech, H. P., Brower, S. W. & Early, J. G. 1967 Thermal convection and crystal growth in horizontal boats: flow pattern, velocity measurements, and solute distribution. In Crystal Growth (ed. Peiser), pp. 201–205. Pergamon.
Villers, D. 1989 Couplage entre les convections capillaire et thermogravitationnelle. Ph.D. thesis, University of Mons, Belgium.
Villers, D. & Platten, J. K. 1985 Marangoni convection in systems presenting a minimum in surface tension. PhysicoChem. Hydrodyn. 6, 435–451.Google Scholar
Villers, D. & Platten, J. K. 1987a Separation of Marangoni convection from gravitational convection in earth experiments. PhysicoChem. Hydrodyn. 8, 173–183.Google Scholar
Villers, D. & Platten, J. K. 1987b Thermocapillary convection when surface tension increases with temperature: comparison between numerical simulations and experimental results by LDV. In Numerical Methods in Laminar and Turbulent Flow (ed. C. Taylor), vol. 5, pp. 1268–1279. Pineridge.
Villers, D. & Platten, J. K. 1990 Influence of thermocapillarity on the oscillatory convection in low-Pr fluids. In Numerical Simulation of Oscillatory Convection in low-Pr Fluids (ed. B. Roux), pp. 108–115. Vieweg.
Wilke, H. & LoUser, W. 1983 Numerical calculation of Marangoni convection in a rectangular open boat. Crystl Res. Technol. 6, 825–833.Google Scholar
Winters, K. H. 1988 Oscillatory convection in liquid metals in a horizontal temperature gradient. Intl J. Numer. Methods Engng 25, 401–414.Google Scholar
Winters, K. H. 1990 A bifurcation analysis of oscillatory convection in liquid metals. In Numerical Simulation of Oscillatory Convection in low-Pr Fluids (ed. B. Roux), pp. 319–326. Vieweg.
Winters, K. H., Cliffe, K. A. & Jackson, C. P. 1987 The prediction of instabilities using bifurcation theory. Harwell Rep. HL86/1147. (Published in Transient and Coupled Systems (ed. R. W. Lewis, E. Hinton, P. Bettes & B. A. Schrefer). Wiley.)
Yih, C. S. 1968 Fluid motion induced by surface-tension variation. Phys. Fluids 3, 477–480.Google Scholar
Zebib, A., Homsy, G. M. & Meiburg, E. 1985 High Marangoni number convection in a square cavity. Phys. Fluids 28, 3467–3476.Google Scholar