Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-13T22:35:36.904Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

Published online by Cambridge University Press:  30 January 2013

Andrew J. Wells ,
J. S. Wettlaufer  and
Steven A. Orszag
Andrew 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 Department of Physics, Yale University, New Haven, CT 06520-8109, USA Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
Steven A. Orszag
Affiliation:
Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

JFM classification

Convection: Convection in porous media Phase change: Phase change Geophysical and Geological Flows: Sea ice
Type
Papers
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , pp. 203 - 227
Copyright
©2013 Cambridge University Press

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

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
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A. & Sorensen, D. 1999 LAPACK Users’ Guide, 3rd edn. SIAM.Google Scholar
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
Brandon, M. A., Cottier, F. R. & Nilsen, F. 2010 Sea ice and oceanography. In Sea Ice (ed. Thomas, D. N. & Dieckmann, G. S.), pp. 79–112. Wiley-Blackwell.Google Scholar
Briggs, W. L., Henson, V. E. & McCormick, S. F. 2000 A Multigrid Tutorial. SIAM.Google Scholar
Chung, C. & Worster, M. 2002 Steady-state chimneys in a mushy layer. J. Fluid Mech. 455, 387–411.Google Scholar
Comiso, J. C. 2010 Variability and trends of the global sea ice cover. In Sea Ice (ed. Thomas, D. N. & Dieckmann, G. S.), pp. 205–246. Wiley-Blackwell.Google Scholar
Copley, S., Giamei, A., Johnson, S. & Hornbecker, M. 1970 The origin of freckles in unidirectionally solidified castings. Metall. Mater. Trans. B 1 (8), 2193–2204.Google Scholar
Emms, P. W. & Fowler, A. C. 1994 Compositional convection in the solidification of binary alloys. J. Fluid Mech. 262, 111–139.Google Scholar
Feltham, D. L., Untersteiner, N., Wettlaufer, J. S. & Worster, M. G. 2006 Sea ice is a mushy layer. Geophys. Res. Lett. 33 (14).CrossRefGoogle Scholar
Guba, P. & Worster, M. G. 2010 Interactions between steady and oscillatory convection in mushy layers. J. Fluid Mech. 645, 411–434.Google Scholar
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
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
Le Bars, M. & Worster, M. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.Google Scholar
Peppin, S. S. L., Huppert, H. E. & Worster, M. G. 2008 Steady-state solidification of aqueous ammonium chloride. J. Fluid Mech. 599, 465–476.CrossRefGoogle Scholar
Petrich, C. & Eicken, H. 2010 Growth, structure and properties of sea ice. In Sea Ice (ed. Thomas, D. N. & Dieckmann, G. S.), pp. 23–78. Wiley-Blackwell.Google Scholar
Rees Jones, D. W. & Worster, M. G. 2013 Fluxes through steady chimneys in a mushy layer during binary alloy solidification. J. Fluid Mech. 714, 127–151.Google Scholar
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
Schulze, T. & Worster, M. 2005 A time-dependent formulation of the mushy-zone free-boundary problem. J. Fluid Mech. 541, 193–202.CrossRefGoogle Scholar
Solomon, T. & 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.Google Scholar
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
Wells, A. J., Wettlaufer, J. S. & Orszag, S. A. 2011 Brine fluxes from growing sea ice. Geophys. Res. Lett. 38, L04501.CrossRefGoogle Scholar
Wettlaufer, J., Worster, M. & Huppert, H. 1997 Natural convection during solidification of an alloy from above with application to the evolution of sea ice. J. Fluid Mech. 344, 291–316.Google Scholar
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
Worster, M. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29, 91–122.Google Scholar
Worster, M. G. 1991 Natural convection in a mushy layer. J. Fluid Mech. 224, 335–359.CrossRefGoogle Scholar
Worster, M. G. 2000 Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.). pp. 393–446. Cambridge University Press.Google Scholar
Zhong, J.-Q., Fragoso, A. T., Wells, A. J. & Wettlaufer, J. S. 2012 Finite-sample-size effects on convection in mushy layers. J. Fluid Mech. 704, 89–108.Google Scholar