Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-25T11:32:39.613Z Has data issue: false hasContentIssue false

Dissolution instability and roughening transition

Published online by Cambridge University Press:  26 October 2017

Philippe Claudin*
Physique et Mécanique des Milieux Hétérogènes, UMR 7636 ESPCI – CNRS – Univ. Paris-Diderot – Univ. P. M. Curie, 10 rue Vauquelin, 75005 Paris, France
Orencio Durán
Department of Ocean Engineering, Texas A & M University, College Station, TX 77843-3136, USA
Bruno Andreotti
Laboratoire de Physique Statistique, UMR 8550 Ecole Normale Supérieure – CNRS – Univ. Paris-Diderot – Univ. P. M. Curie, 24 rue Lhomond, 75005 Paris, France
Email address for correspondence:


We theoretically investigate the pattern formation observed when a fluid flows over a solid substrate that can dissolve or melt. We use a turbulent mixing description that includes the effect of the bed roughness. We show that the dissolution instability at the origin of the pattern is associated with an anomaly at the transition from a laminar to a turbulent hydrodynamic response with respect to the bed elevation. This anomaly, and therefore the instability, disappears when the bed becomes hydrodynamically rough, above a threshold roughness-based Reynolds number. This suggests a possible mechanism for the selection of the pattern amplitude. The most unstable wavelength scales as observed in nature on the thickness of the viscous sublayer, with a multiplicative factor that depends on the dimensionless parameters of the problem.

© 2017 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.)


Anderson, C. H., Behrens, C. J., Floyd, G. A. & Vining, M. R. 1998 Crater firn caves of Mount St. Helens, Washington. J. Cave Karst Studies 60, 4450.Google Scholar
Ashton, G. D. & Kennedy, J. F. 1972 Ripples on the underside of river ice covers. J. Hydraul. Div. ASCE 98, 16031624.Google Scholar
Blumberg, P. N. & Curl, R. L. 1974 Experimental and theoretical studies of dissolution roughness. J. Fluid Mech. 65, 735751.Google Scholar
Camporeale, C. & Ridolfi, L. 2012 Ice ripple formation at large Reynolds numbers. J. Fluid Mech. 694, 225251.Google Scholar
Carey, K. L. 1966 Observed configuration and computed roughness of the underside of river ice, St Croix River, Wisconsin. US Geol. Survey Prof. Paper 550, B192B198.Google Scholar
Charru, F., Andreotti, B. & Claudin, P. 2013 Sand ripples and dunes. Annu. Rev. Fluid Mech. 45, 469493.Google Scholar
Chen, A. S.-H. & Morris, S. W. 2013 On the origin and evolution of icicle ripples. New J. Phys. 15, 103012.Google Scholar
Claudin, F. & Ernstson, K.2004 Regmaglypts on clasts from the Puerto Mínguez ejecta, Azuara multiple impact event (Spain). From Scholar
Cohen, C., Berhanu, M., Derr, J. & Courrech du Pont, S. 2016 Erosion patterns on dissolving and melting bodies. Phys. Rev. Fluids 1, 050508.Google Scholar
Curl, R. L. 1974 Deducing flow velocity in cave conduits from scallops. NSS Bull. 36, 15.Google Scholar
Eames, I. W., Marr, N. J. & Sabir, H. 1997 The evaporation coefficient of water: a review. Intl J. Heat Mass Transfer 40, 29632973.Google Scholar
Falter, J. L., Atkinson, M. J. & Coimbra, C. F. M. 2005 Effects of surface roughness and oscillatory flow on the dissolution of plaster forms. Limnol. Oceanogr. 50, 246254.Google Scholar
Flack, K. A. & Schultz, M. P. 2010 Review of hydraulic roughness scales in the fully rough regime. Trans. ASME J. Fluids Engng 132, 041203.Google Scholar
Fourrière, A., Claudin, P. & Andreotti, B. 2010 Bedforms in a turbulent stream: formation of ripples by primary linear instability and of dunes by nonlinear pattern coarsening. J. Fluid Mech. 649, 287328.Google Scholar
Frederick, K. A. & Hanratty, T. J. 1988 Velocity measurements for a turbulent non-separated flow over solid waves. Exp. Fluids 6, 477486.Google Scholar
Gilpin, R. R. 1981 Ice formation in a pipe containing flows in the transition and the turbulent regimes. Trans. ASME J. Heat Transfer 103, 363368.Google Scholar
Gilpin, R. R., Hirata, T. & 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, 619640.Google Scholar
Goldenfeld, N., Chan, P. Y. & Veysey, J. 2006 Dynamics of precipitation pattern formation at geothermal hot springs. Phys. Rev. Lett. 96, 254501.Google Scholar
Gualtieri, C., Angeloudis, A., Bombardelli, F., Jha, S. & Stoesser, T. 2017 On the values for the turbulent Schmidt number in environmental flows. Fluids 2, 17.Google Scholar
Hanratty, T. J. 1981 Stability of surfaces that are dissolving or being formed by convective diffusion. Annu. Rev. Fluid Mech. 13, 231252.Google Scholar
Herzfeld, U. C., Mayer, H., Caine, N., Losleben, M. & Erbrecht, T. 2003 Morphogenesis of typical winter and summer snow surface patterns in a continental alpine environment. Hydro. Proces. 17, 619649.CrossRefGoogle Scholar
Huang, J. M., Moore, M. N. J. & Ristroph, L. 2015 Shape dynamics and scaling laws for a body dissolving in fluid flow. J. Fluid Mech. 765, R3.Google Scholar
Kiver, E. P. & Mumma, M. D. 1971 Summit firn caves, Mount Rainier, Washington. Science 173, 320322.Google Scholar
Leighly, J. 1948 Cuspate surfaces of melting ice and firn. Geograph. Rev. 38, 301306.Google Scholar
Lin, T. C. & Qun, P. 1986 On the formation of regmaglypts on meteorites. Fluid Dyn. Res. 1, 191199.Google Scholar
Lister, D. H., Gauthier, P., Goszczynski, J. & Slade, J. 1998 The accelerated corrosion of CANDU primary piping. In Proceedings Japan Atomic Indus. Forum Int. Conf. on Water Chemistry in Nuclear Power Plants, 442.Google Scholar
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. Lond. A 466, 659.Google Scholar
Nakouzi, E., Goldstein, R. E. & Steinbock, O. 2014 Do dissolving objects converge to a universal shape? Langmuir 31, 4145.Google Scholar
Ogawa, N. & Furukawa, Y. 2002 Surface instability of icicles. Phys. Rev. E 66, 041202.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.Google Scholar
Rhodes, J. J., Armstrong, R. L. & Warren, S. G. 1987 Mode of formation of ablation hollows controlled by dirt content of snow. J. Glaciol. 33, 135139.Google Scholar
Richardson, W. E. & Harper, R. D. M. 1957 Ablation polygons on snow – further observations and theories. J. Glaciol. 3, 2527.Google Scholar
Schultz, M. P. & Flack, K. A. 2009 Turbulent boundary layers on a systematically varied rough wall. Phys. Fluids 21, 015104.Google Scholar
Sharp, R. P. 1947 The Wolf Creek Glaciers, St. Elias Range, Yukon Territory. Geograph. Rev. 37, 2652.Google Scholar
Thomas, R. M. 1979 Size of scallops and ripples by flowing water. Nature 277, 281283.Google Scholar
Ueno, K. & Farzaneh, M. 2011 Linear stability analysis of ice growth under supercooled water film driven by a laminar airflow. Phys. Fluids 23, 042103.Google Scholar
Villen, B., Zheng, Y. & Lister, D. H. 2001 The scalloping phenomenon and its significance in flow-assisted corrosion. In Proceedings 26th Annual CNS–CNA Student Conf., Toronto.Google Scholar
Villen, B., Zheng, Y. & Lister, D. H. 2005 Surface dissolution and the development of scallops. Chem. Engng Commun. 192, 125136.Google Scholar
Zilker, D. P., Cook, G. W. & Hanratty, T. J. 1977 Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 1. Non-separated flows. J. Fluid Mech. 82, 2951.Google Scholar