Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T11:19:07.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Ice ripple formation at large Reynolds numbers

Published online by Cambridge University Press:  02 February 2012

Carlo Camporeale  and
Luca Ridolfi
Carlo Camporeale*
Affiliation:
Department of Water Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
Luca Ridolfi
Affiliation:
Department of Water Engineering, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A free-surface-induced morphological instability is studied in the laminar regime at large Reynolds numbers () and on sub-horizontal walls (). We analytically and numerically develop the stability analysis of an inclined melting–freezing interface bounding a free-surface laminar flow. The complete solution of both the linearized flow field and the heat conservation equations allows the exact derivation of the upper and lower temperature gradients at the interface, as required by the Stefan condition, from which the dispersion relationship is obtained. The eigenstructure is obtained and discussed. Free-surface dynamics appears to be crucial for the triggering of upstream propagating ice ripples, which grow at the liquid–solid interface. The kinematic and the dynamic conditions play a key role in controlling the formation of the free-surface fluctuations; these latter induce a streamline distortion with an increment of the wall-normal velocities and a destabilizing phase shift in the net heat transfer to the interface. Three-dimensional effects appear to be crucial at high Reynolds numbers. The role of inertia forces, vorticity, and thermal boundary conditions are also discussed.

JFM classification

Waves/Free-surface Flows: Channel flow Phase change: Morphological instability Phase change: Solidification/melting
Type
Papers
Information
Journal of Fluid Mechanics , Volume 694 , 10 March 2012 , pp. 225 - 251
Copyright
Copyright © Cambridge University Press 2012

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. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions: With Formulas, Graphs and Mathematical Tables. Dover.Google Scholar
2. Ashton, G. D. & Kennedy, J. F. 1972 Ripples on underside of river ice covers. Proc. ASCE 98 (HY9), 1603–1624.Google Scholar
3. Batchelor, G. K. 2000 An Introduction to Fluid Mechanics. Cambridge University Press.Google Scholar
4. Bender, C. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
5. Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
6. Blumberg, P. N. & Curl, R. L. 1974 Experimental and theoretical studies of dissolution roughness. J. Fluid Mech. 65, 735–751.CrossRefGoogle Scholar
7. Bontozoglou, V. & Papapolymerou, G. 1997 Laminar flow down a wavy incline. Intl J. Multiphase Flow 23, 69–79.CrossRefGoogle Scholar
8. Camporeale, C., Canuto, C. & Ridolfi, L. 2011 A spectral approach for the stability analysis of turbulent open-channel flows over granular beds. Theor. Comput. Fluid Dyn. 26 (1), 51–80.CrossRefGoogle Scholar
9. Camporeale, C., Perona, P., Porporato, A. & Ridolfi, L. 2007 Hierarchy of models for meandering rivers and related morphodynamic processes. Rev. Geophys. 45, RG1001.CrossRefGoogle Scholar
10. Camporeale, C. & Ridolfi, L. 2011 Modal versus nonmodal linear stability analysis of river dunes. Phys. Fluids 23 (10), 104102.CrossRefGoogle Scholar
11. Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods. Fundamentals in Single Domains. Springer.CrossRefGoogle Scholar
12. Coleman, S. E. & Edling, B. 2000 Sand wavelets in laminar open-channel flows. J. Hydraul. Res. 38 (5), 331–338.CrossRefGoogle Scholar
13. Colombini, M. & Stocchino, A. 2005 Coupling or decoupling bed and flow dynamics: fast and slow sediment waves at high Froude numbers. Phys. Fluids 17 (3), 036602.CrossRefGoogle Scholar
14. Devauchelle, O., Malverti, L., Lajeunesse, E., Lagrée, P. Y., Josserand, C. & Nguyen Thu-Lam, K. D. 2010 Stability of bedforms in laminar flows with free surface: from bars to ripples. J. Fluid Mech. 642, 329–348.CrossRefGoogle Scholar
15. Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
16. Exner, F. M. 1925 Über Die Wechselwirkung Zwischen Wasser und Geschiebe in Flüssen, vol. 14, pp. 165–180. Akademie der Wissenschaften, Wien (in German).Google Scholar
17. Feltham, D. L. & Worster, M. G. 1999 Flow-induced morphological instability of a mushy layer. J. Fluid Mech. 391, 337–357.CrossRefGoogle Scholar
18. Ferguson, R. I. 1973 Sinuosity of supraglacial streams. Geol. Soc. Am. Bull. 84, 251–256.2.0.CO;2>CrossRefGoogle Scholar
19. Giannakis, D., Fischer, P. & Rosner, R. 2009 A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD. J. Comput. Phys. 228 (4), 1188–1233.CrossRefGoogle Scholar
20. Gilpin, R. R. 1981 Ice formation in a pipe containing flows in the transition and turbulent regimes. J. Heat Transfer 103, 363–368.CrossRefGoogle Scholar
21. Gilpin, T., Hirata, R. R. & Cheng, K. C. 1980 Wave formation and heat transfer at an ice–water interface in the presence of a turbulent flow. J. Fluid Mech. 99, 619–640.CrossRefGoogle Scholar
22. Godreche, C. & Manneville, P.  (Eds) 1998 Hydrodynamics and Nonlinear Instabilities. Cambridge University Press.CrossRefGoogle Scholar
23. Grosch, C. E. & Salwen, H. 1968 The stability of steady and time-dependent plane Poiseuille flow. J. Fluid Mech. 34, 177–194.CrossRefGoogle Scholar
24. Hutter, K. 1983 Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets. Springer.CrossRefGoogle Scholar
25. Kaser, G., Cogley, J. G., Dyurgerov, M. B., Meier, M. F. & Ohmura, A. 2006 Mass balance of glaciers and ice caps: consensus estimates for 1961–2004. Geophys. Res. Lett. 33 (19), L19501.CrossRefGoogle Scholar
26. Lock, G. S. H 1990 The Growth and Decay of Ice. Cambridge University Press.Google Scholar
27. Luo, H. & Pozrikidis, C. 2006 Effect of inertia on film flow over oblique and three-dimensional corrugations. Phys. Fluids 18, 078107.CrossRefGoogle Scholar
28. Neufeld, J. A., Goldstein, R. E. & Worster, M. G. 2010 On the mechanisms of icicle evolution. J. Fluid Mech. 647, 287–308.CrossRefGoogle Scholar
29. Ogawa, N. & Furukawa, Y. 2002 Surface instability of icicles. Phys. Rev. E 66 (4, part 1), 041202.CrossRefGoogle ScholarPubMed
30. Olsson, P. J. & Henningson, D. S. 1995 Optimal disturbance growth in water table flow. Stud. Appl. Maths 94, 183–210.CrossRefGoogle Scholar
31. Parker, G. 1975 Meandering of supraglacial melt streams. Water Resour. Res. 11 (4), 551–552.CrossRefGoogle Scholar
32. Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
33. Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences , vol. 142. Springer.CrossRefGoogle Scholar
34. Seminara, G. 2010 Fluvial sedimentary patterns. Annu. Rev. Fluid Mech. 42, 43–66.CrossRefGoogle Scholar
35. Shen, J. 1994 Efficient spectral-Galerkin methods. Part I. Direct solvers for the second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15 (6), 1489–1505.CrossRefGoogle Scholar
36. Short, M. B., Baygents, J. C. & Goldstein, R. E. 2006 A free-boundary theory for the shape of the ideal dripping icicle. Phys. Fluids 18 (8), 083101.CrossRefGoogle Scholar
37. Stefan, J. 1891 Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere. Ann. Phys. Chem. 42, 269–286.CrossRefGoogle Scholar
38. Thomas, R. R. 1979 Size of scallops and ripples formed by flowing water. Nature 277, 281–283.CrossRefGoogle Scholar
39. Thorsness, C. B. & Hanratty, T. J. 1979 Stability of dissolving or depositing surfaces. AIChE 25, 697–701.CrossRefGoogle Scholar
40. Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra. Princeton University Press.CrossRefGoogle Scholar
41. Ueno, K. 2003 Pattern formation in crystal growth under parabolic shear flow. Phys. Rev. E 68 (2, part 1), 021603.CrossRefGoogle ScholarPubMed
42. Ueno, K., Farzaneh, M., Yamaguchi, S. & Tsuji, H. 2010 Numerical and experimental verification of a theoretical model of ripple formation in ice growth under supercooled water film flow. Fluid Dyn. Res. 42 (2), 025508.CrossRefGoogle Scholar
43. Wang, C. Y. 1981 Liquid film flowing slowly down a wavy incline. AIChE J. 27 (2), 207–212.CrossRefGoogle Scholar
44. Wierschem, A. & Aksel, N. 2003 Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221–237.CrossRefGoogle Scholar
45. Worster, M. G. 1992 Instabilities of the liquid and mushy layers. J. Fluid Mech. 237, 649–669.CrossRefGoogle Scholar
46. Worster, M. G. 2000 Solidification of fluids. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Worster, M. G., Batchelor, G. K. & Moffat, H. K. ). Cambridge University Press (chapter 8).Google Scholar
47. Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar