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Finite-sample-size effects on convection in mushy layers

Published online by Cambridge University Press:  02 July 2012

J.-Q. Zhong ,
A. T. Fragoso ,
A. J. Wells  and
J. S. Wettlaufer
J.-Q. Zhong
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA
A. T. Fragoso
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA Department of Physics, Yale University, New Haven, CT 06520, USA
A. J. Wells
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA
J. S. Wettlaufer*
Affiliation:
Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA Department of Physics, Yale University, New Haven, CT 06520, USA NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
*
†Email address for correspondence: john.wettlaufer@yale.edu
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Abstract

We report theoretical and experimental investigations of the flow instability responsible for mushy-layer convection with chimneys, drainage channels devoid of solid, during steady-state solidification of aqueous ammonium chloride. Under certain growth conditions a state of steady mushy-layer growth with no flow is unstable to the onset of convection, resulting in the formation of chimneys. We present regime diagrams to quantify the state of the flow as a function of the initial liquid concentration, the porous-medium Rayleigh number, and the sample width. For a given liquid concentration, increasing both the porous-medium Rayleigh number and the sample width drove a transition from a weakly convecting chimney free state to a state of mushy-layer convection with fully developed chimneys. Increasing the concentration ratio stabilized the system and suppressed the formation of chimneys. As the initial liquid concentration increased, onset of convection and formation of chimneys occurred at larger values of the porous-medium Rayleigh number, but the critical cell widths for chimney formation are far less sensitive to the liquid concentration. At the highest liquid concentration, the mushy-layer mode of convection did not occur in the experiment. The formation of multiple chimneys and the morphological transitions between these states are discussed. The experimental results are interpreted in terms of a previous theoretical analysis of finite amplitude convection with chimneys, with a single value of the mushy-layer permeability consistent with the liquid concentrations considered in this study.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media Phase change: Solidification/melting
Type
Papers
Information
Journal of Fluid Mechanics , Volume 704 , 10 August 2012 , pp. 89 - 108
Copyright
Copyright © Cambridge University Press 2012

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References

1. Adams, J. C. 1989 mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations. Appl. Math. Comput. 34, 113–146.Google Scholar
2. Amberg, G. & Homsy, G. M. 1993 Nonlinear analysis of buoyant convection in binary solidification with application to channel formation. J. Fluid Mech. 252, 79–98.CrossRefGoogle Scholar
3. Anderson, D. M. & Worster, M. G. 1995 Weakly nonlinear analysis of convection in mushy layers during the solidification of binary alloys. J. Fluid Mech. 302, 307–331.CrossRefGoogle Scholar
4. Aussillous, P., Sederman, A. J., Gladden, L. F., Huppert, H. E. & Worster, M. G. 2006 Magnetic resonance imaging of structure and convection in solidifying mushy layers. J. Fluid Mech. 552, 99–125.Google Scholar
5. Beckermann, C., Gu, J. & Boettinger, W. 2000 Development of a freckle predictor via Rayleigh number method for single-crystal nickel-base superalloy castings. Metall. Mater. Trans. A 31, 2545–2557.CrossRefGoogle Scholar
6. Bergman, M. I. & Fearn, D. R. 1994 Chimneys on the Earth’s inner–outer core boundary? Geophys. Res. Lett. 21 (6), 477–480.CrossRefGoogle Scholar
7. Briggs, W. L., Henson, V. E. & McCormick, S. F. 2000 A Multigrid Tutorial. SIAM.Google Scholar
8. Chen, C. F. & Chen, F. 1991 Experimental study of directional solidification of aqueous ammonium chloride solution. J. Fluid Mech. 227, 567–586.Google Scholar
9. Chen, F., Lu, J. W. & Yang, T. L. 1994 Convective instability in ammonium chloride solution directionally solidified from below. J. Fluid Mech. 276, 163–187.CrossRefGoogle Scholar
10. Chung, C. A. & Worster, M. G. 2002 Steady-state chimneys in a mushy layer. J. Fluid Mech. 455, 387–411.CrossRefGoogle Scholar
11. Copley, S. M., Giamei, A. F., Johnson, S. M. & Hornbecker, M. F. 1970 The origin of freckles in unidirectionally solidified castings. Metall. Trans. 1 (8), 2193–2204.CrossRefGoogle Scholar
12. Felicelli, S. D., Heinrich, J. C. & Poirier, D. R. 1998 Three-dimensional simulations of freckles in binary alloys. J. Cryst. Growth 191, 879–888.CrossRefGoogle Scholar
13. Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. 2006 Sea ice is a mushy layer. Geophys. Res. Lett. 33, L14501.CrossRefGoogle Scholar
14. Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.Google Scholar
15. Giamei, A. F. & Kear, B. H. 1970 On nature of freckles in nickel base superalloys. Metall. Trans. 1, 2185–2192.CrossRefGoogle Scholar
16. Guba, P. & Worster, M. G. 2006 Nonlinear oscillatory convection in mushy layers. J. Fluid Mech. 553, 419–443.Google Scholar
17. Guba, P. & Worster, M. G. 2010 Interactions between steady and oscillatory convection in mushy layers. J. Fluid Mech. 645, 411–434.Google Scholar
18. Guo, J. & Beckermann, C. 2003 Three-dimensional simulation of freckle formation during binary alloy solidification: effect of mesh spacing. Numer. Heat Transfer A 44 (6), 559–576.Google Scholar
19. Happel, J. & Brenner, H. 1986 Low Reynolds Number Hydrodynamics. M. Nijhoff.Google Scholar
20. Heinrich, J. C. & Poirier, D. R. 2004 Convection modelling in directional solidification. C. R. Méc. 332 (5–6), 429–445.CrossRefGoogle Scholar
21. Jain, J., Kumar, A. & Dutta, P. 2007 Role of plume convection and remelting on the mushy layer structure during directional solidification. J. Phys. D 40 (4), 1150–1160.CrossRefGoogle Scholar
22. Katz, R. F. & Worster, M. G. 2008 Simulation of directional solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell. J. Comput. Phys. 227 (23), 9823–9840.Google Scholar
23. Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. Rabinowitz, P. ), pp. 359–384. Academic.Google Scholar
24. Mullins, W. W. & Sekerka, R. F. 1964 Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35 (2), 444–451.Google Scholar
25. Neufeld, J. A. & Wettlaufer, J. S. 2008a An experimental study of shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 363–385.CrossRefGoogle Scholar
26. Neufeld, J. A. & Wettlaufer, J. S. 2008b Shear-enhanced convection in a mushy layer. J. Fluid Mech. 612, 339–361.Google Scholar
27. Oertling, A. B. & Watts, R. G. 2004 Growth of and brine drainage from NaCl– freezing: a simulation of young sea ice. J. Geophys. Res. 109 (C4), C04013.Google Scholar
28. Peppin, S. S. L., Aussillous, P., Huppert, H. E. & Worster, M. G. 2007 Steady-state mushy layers: experiments and theory. J. Fluid Mech. 570, 69–77.Google Scholar
29. Peppin, S. S. L., Huppert, H. E. & Worster, M. G. 2008 Steady-state solidification of aqueous ammonium chloride. J. Fluid Mech. 599, 465–476.Google Scholar
30. Roper, S. M., Davis, S. H. & Voorhees, P. 2007 Convection in a mushy zone forced by sidewall heat losses. Metall. Mater. Trans. A 38A, 1069–1079.Google Scholar
31. Roper, S. M., Davis, S. H. & Voorhees, P. 2011 Localisation of convection in mushy layers by weak background flow. J. Fluid Mech. 675, 518–528.CrossRefGoogle Scholar
32. Schulze, T. & Worster, M. 1998 A numerical investigation of steady convection in mushy layers during the directional solidification of binary alloys. J. Fluid Mech. 356, 199–220.Google Scholar
33. Schulze, T. & Worster, M. 2005 A time-dependent formulation of the mushy-zone free-boundary problem. J. Fluid Mech. 541, 193–202.Google Scholar
34. Solomon, T. H. & Hartley, R. 1998 Measurements of the temperature field of mushy and liquid regions during solidification of aqueous ammonium chloride. J. Fluid Mech. 358, 87–106.CrossRefGoogle Scholar
35. Tait, S. & Jaupart, C. 1992 Compositional convection in a reactive crystalline mush and melt differentiation. J. Geophys. Res. 97, 6735–6756.CrossRefGoogle Scholar
36. US National Research Council, 2003 International Critical Tables of Numerical Data, Physics, Chemistry, and Technology.Google Scholar
37. Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2010 Maximum potential energy transport: a variational principle for solidification problems. Phys. Rev. Lett. 105, 254502.Google Scholar
38. Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2011 Brine fluxes from growing sea ice. Geophys. Res. Lett. 38, L04501.Google Scholar
39. Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. 1997a Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316.CrossRefGoogle Scholar
40. Wettlaufer, J. S., Worster, M. G. & Huppert, H. E. 1997b The phase evolution of young sea ice. Geophys. Res. Lett. 24 (10), 1251–1254.Google Scholar
41. Whiteoak, S. H., Huppert, H. E. & Worster, M. G. 2008 Conditions for defect-free solidification of aqueous ammonium chloride in a quasi two-dimensional directional solidification facility. J. Cryst. Growth 310 (15), 3545–3551.Google Scholar
42. Worster, M. G. 1992 Instabilities of the liquid and mushy regions during solidification of alloys. J. Fluid Mech. 237, 649–669.CrossRefGoogle Scholar
43. Worster, M. G. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29, 91–122.Google Scholar