Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T03:30:20.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Destabilization of free convection by weak rotation

Published online by Cambridge University Press:  21 September 2011

A. Yu. Gelfgat
A. Yu. Gelfgat*
Affiliation:
School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, 69978, Tel-Aviv, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This study offers an explanation of the recently observed effect of destabilization of free convective flows by weak rotation. After studying several models where flows are driven by the simultaneous action of convection and rotation, it is concluded that destabilization is observed in cases where the centrifugal force acts against the main convective circulation. At relatively low Prandtl numbers, this counter-action can split the main vortex into two counter-rotating vortices, where the interaction leads to instability. At larger Prandtl numbers, the counter-action of the centrifugal force steepens an unstable thermal stratification, which triggers the Rayleigh–Bénard instability mechanism. Both cases can be enhanced by advection of azimuthal velocity disturbances towards the axis, where they grow and excite perturbations of the radial velocity. The effect was studied by considering a combined convective and rotating flow in a cylinder with a rotating lid and a parabolic temperature profile at the sidewall. Next, explanations of the destabilization effect for rotating-magnetic-field-driven flow and melt flow in a Czochralski crystal growth model were derived.

JFM classification

Convection: Buoyancy-driven instability Instability: Parametric instability Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 377 - 412
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Akamatsu, M., Kakimoto, K. & Ozoe, H. 1997 Numerical computation for the secondary convection in a Czochralski crystal growing system with a rotating crucible and static crystal rod. J. Mater. Process. Manuf. Sci. 5, 329–348.Google Scholar
2. Albensoeder, S. & Kuhlmann, H. C. 2002 Three-dimensional instability of two counter-rotating vortices in a rectangular cavity driven by parallel wall motion. Eur. J. Mech. B/Fluids 21, 307–316.CrossRefGoogle Scholar
3. Ali, M. E. & McFadden, G. B. 2005 Linear stability of cylindrical Couette flow in the convection regime. Phys. Fluids 17, 054112.CrossRefGoogle Scholar
4. Ali, M. & Weidman, P. D. 1990 On the stability of circular Couette flow with radial heating. J. Fluid Mech. 220, 53–84.CrossRefGoogle Scholar
5. Banerjee, J. & Muralidhar, K. 2006 Simulation of transport processes during Czochralski growth of YAG crystals. J. Cryst. Growth 286, 350–364.CrossRefGoogle Scholar
6. Bodenshatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
7. Boronska, K. & Tuckerman, L. S. 2010 Extreme multiplicity in cylindrical Rayleigh–Bénard convection. I. Time dependence and oscillations. Phys. Rev. E 81, 036320.CrossRefGoogle ScholarPubMed
8. Brummel, N., Hart, J. E. & Lopez, J. M. 2000 On the flow induced by centrifugal buoyancy in a differentially-heated rotating cylinder. Theor. Comput. Fluid Dyn. 14, 39–54.CrossRefGoogle Scholar
9. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
10. Chen, S. & Li, M. 2007 Flow instability of molten GaAs in the Czochralski configuration. J. Mater. Sci. Technol. 23, 395–401.Google Scholar
11. Crnogorac, N., Wilke, H., Cliffe, K. A., Gelfgat, A. Yu. & Kit, E. 2008 Numerical modelling of instability and supercritical oscillatory states in a Czochralski model system of oxide melts. Cryst. Res. Technol. 43, 606–615.CrossRefGoogle Scholar
12. Fernando, H. J. S. & Smith, D. C. IV 2001 Vortex structures in geophysical convection. Eur. J. Mech. B/Fluids 20, 437–470.CrossRefGoogle Scholar
13. Gelfgat, A. Yu. 2002 Three-dimensionality of trajectories of experimental tracers in a steady axisymmetric swirling flow: effect of density mismatch. Theor. Comput. Fluid Dyn. 16, 29–41.CrossRefGoogle Scholar
14. Gelfgat, A. Yu. 2007 Three-dimensional instability of axisymmetric flows: solution of benchmark problems by a low-order finite volume method. Intl J. Numer. Meth. Fluids 54, 269–294.CrossRefGoogle Scholar
15. Gelfgat, A. Yu. 2008 Numerical study of three-dimensional instabilities of Czochralski melt flow driven by buoyancy convection, thermocapillarity and rotation. In Studies of Flow Instabilities in Bulk Crystal Growth (ed. Gelfgat, A. ), Transworld Research Network, 2007 , pp. 57–82.Google Scholar
16. Gelfgat, A. Yu., Bar-Yoseph, P. Z. & Solan, A. 1996 Stability of confined swirling flow with and without vortex breakdown. J. Fluid Mech. 311, 1–36.CrossRefGoogle Scholar
17. Gelfgat, A. Yu., Bar-Yoseph, P. Z. & Solan, A. 2000 Axisymmetry breaking instabilities of natural convection in a vertical Bridgman growth configurations. J. Cryst. Growth 220, 316–325.CrossRefGoogle Scholar
18. Gelfgat, A. Yu., Bar-Yoseph, P. Z. & Solan, A. 2001 Three-dimensional instability of axisymmetric flow in a rotating lid–cylinder enclosure. J. Fluid Mech. 438, 363–377.CrossRefGoogle Scholar
19. Gelfgat, A. Yu., Bar-Yoseph, P. Z., Solan, A. & Kowalewski, T. 1999 An axisymmetry-breaking instability in axially symmetric natural convection. Intl J. Transport Phenom. 1, 173–190.Google Scholar
20. Gemeny, L. E., Martin Witkowski, L. & Walker, J. S. 2007 Buoyant instability in a laterally heated vertical cylinder. Intl J. Heat Mass Transfer 50, 1010–1017.CrossRefGoogle Scholar
21. Goldstein, H. F., Knobloch, E., Mercader, I. & Net, M. 1993 Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers. J. Fluid Mech. 248, 583–604.CrossRefGoogle Scholar
22. Gorbachev, L. P., Nikitin, N. V. & Ustinov, A. L. 1974 Magnetohydrodynamic rotation of an electrically conductive liquid in a cylindrical vessel of finite dimensions. Magn. Gidrodin. 10, 406–414.Google Scholar
23. Grants, I. & Gerbeth, G. 2001 Stability of axially symmetric flow driven by a rotating magnetic field in a cylindrical cavity. J. Fluid Mech. 431, 407–426.CrossRefGoogle Scholar
24. Grants, I. & Gerbeth, G. 2002 Linear three-dimensional instability of a magnetically driven rotating flow. J. Fluid Mech. 463, 229–239.CrossRefGoogle Scholar
25. Hintz, P., Schwabe, D. & Wilke, H. 2001 Convection in Czochralski crucible – Part I: non-rotating crystal. J. Cryst. Growth 222, 343–355.CrossRefGoogle Scholar
26. Kakimoto, K., Eguchi, M., Watanabe, H. & Hibiya, T. 1990 Flow instability of molten silicon in the Czochralski configuration. J. Cryst. Growth 102, 16–20.CrossRefGoogle Scholar
27. Kishida, Y., Tanaka, M. & Esaka, H. 1993 Appearance of a baroclinic wave in Czochralski silicon melt. J. Cryst. Growth 130, 75–84.CrossRefGoogle Scholar
28. Kloosterziel, R. C. & Carnavale, G. F. 2003 Closed-form linear stability conditions for rotating Rayleigh–Bénard convection with rigid stress-free upper and lower boundaries. J. Fluid Mech. 480, 25–42.CrossRefGoogle Scholar
29. Knobloch, E. 1998 Rotating convection: recent developments. Intl J. Engng Sci. 36, 1421–1450.CrossRefGoogle Scholar
30. Koschmieder, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.Google Scholar
31. Lewis, G. M. 2010 Mixed-mode solutions in an air-filled differentially heated rotating annulus. Physica D 239, 1843–1854.CrossRefGoogle Scholar
32. Lopez, J. M. & Marques, F. 2009 Centrifugal effects in rotating convection: nonlinear dynamics. J. Fluid Mech. 628, 269–297.CrossRefGoogle Scholar
33. Lucas, P. G. J., Pfitenhauer, J. M. & Donelly, J. 1983 Stability and heat transfer of rotating cryogens. Part 1. Influence of rotation on the onset of convection in liquid 4He. J. Fluid Mech. 129, 251–264.CrossRefGoogle Scholar
34. Müller, G. 2007 Review: the Czochralski method – where we are 90 years after Jan Czochralski’s invention. Cryst. Res. Technol. 42, 1150–1161.CrossRefGoogle Scholar
35. Munakata, T. & Tanasawa, I. 1990 Onset of oscillatory flow in CZ growth melt. J. Cryst. Growth 106, 566–576.CrossRefGoogle Scholar
36. Nienhüser, Ch. & Kuhlmann, H. C. 2002 Stability of thermocapillary flows in non-cylindrical liquid bridges. J. Fluid Mech. 458, 35–73.CrossRefGoogle Scholar
37. Ozoe, H., Toh, K. & Inoue, T. 1991 Transition mechanism of flow modes in Czochralski convection. J. Cryst. Growth 110, 472–480.CrossRefGoogle Scholar
38. Rubio, A., Lopez, J. M. & Marques, F. 2010 Onset of Kupper–Lortz-like dynamics in finite rotating thermal convection. J. Fluid Mech. 644, 337–357.CrossRefGoogle Scholar
39. Schwabe, D., Sumathi, R. R. & Wilke, H. 2004 An experimental and numerical effort to simulate the interface deflection of YAG. J. Cryst. Growth 265, 440–452.CrossRefGoogle Scholar
40. Seidl, A., McCord, G., Müller, G. & Leister, H.-J. 1994 Experimental observation and numerical simulation of wave patterns in a Czochralski silicon melt. J. Cryst. Growth 137, 326–334.CrossRefGoogle Scholar
41. Sørensen, J. N., Naumov, I. & Mikkelsen, R. 2006 Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity. Exp. Fluids 41, 425–440.CrossRefGoogle Scholar
42. Sung, H. J., Jung, Y. J. & Ozoe, H. 1995 Prediction of transient oscillating flow in Czochralski convection. Intl J. Heat Mass Transfer 38, 1627–1636.CrossRefGoogle Scholar
43. Suzuki, T. 2004 Temperature fluctuations in a LiNbO3 melt during crystal growth. J. Cryst. Growth 270, 511–516.CrossRefGoogle Scholar
44. Szmyd, J. S., Jaszczur, M. & Ozoe, H. 2002 Numerical calculation of spoke pattern in Bridgman top seeding convection. Numer. Heat Transfer A 41, 685–695.CrossRefGoogle Scholar
45. Teitel, M., Schwabe, D. & Gelfgat, A. Yu. 2008 Experimental and computational study of flow instabilities in a model of Czochralski growth. J. Cryst. Growth 310, 1343–1348.CrossRefGoogle Scholar
46. Zeng, Z., Chen, J., Mizuseki, H., Shimamura, K., Fukuda, T. & Kawazoe, Y. 2003 Three-dimensional oscillatory convection of LiCaAlF6 melts in Czochralski crystal growth. J. Cryst. Growth 252, 538–549.CrossRefGoogle Scholar

Gelfgat supplementary movie

Movie 1. An example of cold thermals instability observed in experiments of Teitel et al. (2008) (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 8.1 MB

Gelfgat supplementary movie

Movie 1. An example of cold thermals instability observed in experiments of Teitel et al. (2008) (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 9.5 MB

Gelfgat supplementary movie

Movie 2. Cold thermals instability by superposition of base flow and perturbation (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 6.9 MB

Gelfgat supplementary movie

Movie 2. Cold thermals instability by superposition of base flow and perturbation (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 8.2 MB

Gelfgat supplementary movie

Movie 3. Cold thermals instability by solution of full non-linear problem (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 7.8 MB

Gelfgat supplementary movie

Movie 3. Cold thermals instability by solution of full non-linear problem (for Fig. 5).

Download Gelfgat supplementary movie(Video)
Video 14.8 MB

Gelfgat supplementary movie

Movie 4. An example of the rotating (oscillatory) jet instability (for Fig. 12).

Download Gelfgat supplementary movie(Video)
Video 1.3 MB

Gelfgat supplementary movie

Movie 4. An example of the rotating (oscillatory) jet instability (for Fig. 12).

Download Gelfgat supplementary movie(Video)
Video 2 MB

Gelfgat supplementary movie

Movie 5. Isotherm oscillations for the cold thermals instability (for Fig. 13).

Download Gelfgat supplementary movie(Video)
Video 1.8 MB

Gelfgat supplementary movie

Movie 5. Isotherm oscillations for the cold thermals instability (for Fig. 13).

Download Gelfgat supplementary movie(Video)
Video 4.7 MB

Gelfgat supplementary movie

Movie 6. Isotherm oscillations for the cold thermals instability in Czochralski model flow without crystal rotation (for Fig. 17).

Download Gelfgat supplementary movie(Video)
Video 4.8 MB

Gelfgat supplementary movie

Movie 6. Isotherm oscillations for the cold thermals instability in Czochralski model flow without crystal rotation (for Fig. 17).

Download Gelfgat supplementary movie(Video)
Video 6.5 MB

Gelfgat supplementary movie

Movie 7. Isotherm oscillations for the cold thermals instability in Czochralski model flow with crystal rotation (for Fig. 18).

Download Gelfgat supplementary movie(Video)
Video 4.5 MB

Gelfgat supplementary movie

Movie 7. Isotherm oscillations for the cold thermals instability in Czochralski model flow with crystal rotation (for Fig. 18).

Download Gelfgat supplementary movie(Video)
Video 8 MB