Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T21:20:45.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamically locked morphogenesis in karst and ice flutings

Published online by Cambridge University Press:  30 July 2015

C. Camporeale
C. Camporeale*
Affiliation:
Department of Environment, Land and Infrastructure Engineering, Politecnico di Torino, Corso Duca Abruzzi 24, 10129 Turin, Italy
*
†Email address for correspondence: carlo.camporeale@polito.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Two of the most widespread and fascinating patterns observed on cave walls and icefalls – karst and ice flutings – are demonstrated to share the same morphogenesis, whose core is a water film-induced locking mechanism. Creeping flow-based parallel and non-parallel stability analyses are developed through a numerical and analytical approach. These instabilities are shown to develop at inverted overhung conditions. A sharp transition between fluting and ripple-like patterns is presented. The non-parallel problem is solved with the use of Papkovich–Neuber solutions in order to obtain a finite wavelength selection close to the critical conditions. The method and results can be extended to similar problems where the temporal evolution of the interface is linearly related to the film depth.

JFM classification

Phase change: Morphological instability Phase change: Solidification/melting Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 89 - 119
Copyright
© 2015 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benney, B. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 150–155.CrossRefGoogle Scholar
Brevdo, L., Laure, P., Dias, F. & Bridges, T. J. 1999 Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71.CrossRefGoogle Scholar
Camporeale, C., Canuto, C. & Ridolfi, L. 2012 A spectral approach for the stability analysis of turbulent open-channel flows over granular beds. Theor. Comput. Fluid Dyn. 26 (1–4), 51–80.CrossRefGoogle Scholar
Camporeale, C. & Ridolfi, L. 2012a Hydrodynamic-driven stability analysis of morphological patterns on stalactites and implications for cave paleoflow reconstructions. Phys. Rev. Lett. 108, 238501.CrossRefGoogle ScholarPubMed
Camporeale, C. & Ridolfi, L. 2012b Ice ripple formation at large Reynolds numbers. J. Fluid Mech. 694, 225–251.CrossRefGoogle Scholar
Chan, P. Y. & Goldenfeld, N. 2007 Steady states and linear stability analysis of precipitation pattern formation at geothermal hot springs. Phys. Rev. E 76, 046104.CrossRefGoogle ScholarPubMed
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Charru, F. 2001 Hydrodynamic Instabilities. Cambridge University Press.Google Scholar
Chen, A. S.-H.2014 Experiments on the growth and form of icicles. PhD thesis, University of Toronto.Google Scholar
Chen, A. S.-H. & Morris, S. W. 2013 On the origin and evolution of icicle ripples. New J. Phys. 15 (103012), 248–252.CrossRefGoogle Scholar
Cheng, M. & Chang, H.-C. 1992 Subharmonic instabilities of finite amplitude monochromatic waves. Phys. Fluids 4, 505–523.CrossRefGoogle Scholar
Chu, K. J. & Dukler, A. E. 1974 Statistical characteristics of thin, wavy films: part 2. Studies of substrate and its wave structure. AIChE J. 20 (4), 695–706.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1196.CrossRefGoogle Scholar
DeBruin, G. J. 1974 Stability of a layer of liquid flowing down an inclined plane. J. Engng Maths 8 (3), 259–270.CrossRefGoogle Scholar
Dressler, R. F. 1978 New nonlinear shallow-flow equations with curvature. J. Hydraul Res. 16 (3), 205–222.CrossRefGoogle Scholar
Dreybrodt, W. 1988 Processes in Karst Systems. Springer.CrossRefGoogle Scholar
Dreybrodt, W. & Buhmann, D. 1991 A mass-transfer model for dissolution and precipitation of calcite from solutions in turbulent motion. Chem. Geol. 90 (1–2), 107–122.CrossRefGoogle Scholar
Fairchild, I. J., Smith, C. L., Baker, A., Fuller, L., Spotl, C., Mattey, D., McDermott, F. & E.I.M.P 2006 Modification and preservation of environmental signals in speleothems. Earth-Sci. Rev. 75, 105–153.CrossRefGoogle Scholar
Floryan, J. M., Davis, S. H. & Kelly, R. E. 1987 Instabilities of a liquid-film flowing down a slightly inclined plane. Phys. Fluids 30 (4), 983–989.CrossRefGoogle Scholar
Hammer, O., Dysthe, D. K., Lelu, B., Lund, H., Meakin, P. & Jamtveit, B. 2008 Calcite precipitation instability under laminar, open-channel flow. Geochim. Cosmochim. Acta 72 (20), 5009–5021.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Godreche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Hutter, K. 1983 Theoretical Glaciology. Kluwer.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, Applied Mathematical Sciences, vol 176, p. 440. Springer,; ISBN: 978-1-84882-366-2.CrossRefGoogle Scholar
Kapitsa, P. L. & Kapitsa, S. P. 1949 Wave flow of thin liquid layers. Zh. Eksp. Teor. Fiz. 19, 105–120.Google Scholar
Kaufmann, G. 2003 Stalagmite growth and palaeo-climate: the numerical perspective. Earth Planet. Sci. Lett. 214 (1–2), 251–266.CrossRefGoogle Scholar
Kaufmann, G. & Dreybrodt, W. 2007 Calcite dissolution kinetics in the system $\text{CaCO}_{3}{-}\text{H}_{2}\text{O}{-}\text{CO}_{2}$ at high undersaturation. Geochim. Cosmochim. Acta 71 (6), 1398–1410.CrossRefGoogle Scholar
Lin, T.-S., Kondic, L. & Filippov, A. 2012 Thin films lowing down inverted substrates: three-dimensional flow. Phys. Fluids 24 (022105), 1–17.CrossRefGoogle Scholar
Luo, H. & Pozrikidis, C. 2006 Effect of inertia on film flow over oblique and three-dimensional corrugations. Phys. Fluids 18, 078107.Google Scholar
Martin-Perez, A., Martin-Garcia, R. & Alonso-Zarza, A. M. 2012 Diagenesis of a drapery speleothem from castanar cave: from dissolution to dolomitization. Int. J. Speleo. 41 (2), 251–266.CrossRefGoogle Scholar
McDermott, F., Mattey, D. P. & Hawkesworth, C. 2001 Centennial-scale holocene climate variability revealed by a high-resolution speleothem delta O-18 record from SW Ireland. Science 294 (5545), 1328–1331.CrossRefGoogle Scholar
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. Lond. A 466 (2115), 659–694.Google Scholar
Nusselt, W. 1916 The surface condensation of water vapour. Z. Verein. Deutsch. Ing. 60, 541–546.Google Scholar
Pierson, F. W. & Whitaker, S. 1977 Some theoretical and experimental observations of the wave structure of falling liquid films. Ind. Engng Chem. Fundam. 16, 401–408.CrossRefGoogle Scholar
Pradas, M., Tseluiko, D. & Kalliadasis, S. 2011 Rigorous coherent-structure theory for falling liquid films: viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23 (4), 044104.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 357–369.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.CrossRefGoogle Scholar
Shen, J. 1994 Efficient spectral-Galerkin methods I. Direct solvers for the second and fourth order equations using Legendre polynomials. SIAM J. Sci. Comput. 15 (6), 1489–1505.CrossRefGoogle Scholar
Short, M. B., Baygents, J. C., Beck, J. W., Stone, D. A., Toomey, R. S. & Goldstein, R. E. 2005a Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. Phys. Rev. Lett. 94 (1), 018501.CrossRefGoogle ScholarPubMed
Short, M. B., Baygents, J. C. & Goldstein, R. E. 2005b Stalactite growth as a free-boundary problem. Phys. Fluids 17 (8), 083101.CrossRefGoogle Scholar
Sivashinsky, G. I. & Michelson, D. M. 1980 On irregular wavy flow of a liquid-film down a vertical plane. Progr. Theoret. Phys. 63 (6), 2112–2114.CrossRefGoogle Scholar
Tran-Cong, T. & Blake, J. R. 1982 General solutions of the Stokes’ flow equations. J. Math. Anal. Applics. 90, 72–84.CrossRefGoogle Scholar
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
Vesipa, R., Camporeale, C. & Ridolfi, L. 2015 Thin-film induced morphological instabilities over calcite surfaces. Proc. R. Soc. Lond. A 471, 20150031.Google ScholarPubMed
Wang, C. Y. 1981 Liquid film flowing slowly down a wavy incline. AIChE J. 27 (2), 207–212.CrossRefGoogle Scholar
Wierschem, A., Bontozoglou, V., Heining, C., Uecker, H. & Aksel, N. 2008 Linear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 34, 580–589.CrossRefGoogle Scholar
Wooding, R. A. 1991 Growth of natural dams by deposition from steady supersaturated shallow flow. J. Geophys. Res. 96 (B1), 667–682.CrossRefGoogle Scholar
Yih, C. S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12, 434–435.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar