Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-28T21:21:36.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Hydrodynamic-driven morphogenesis of karst draperies: spatio-temporal analysis of the two-dimensional impulse response

Published online by Cambridge University Press:  22 January 2021

Pier Giuseppe Ledda Open the ORCID record for Pier Giuseppe Ledda [Opens in a new window] ,
Gioele Balestra Open the ORCID record for Gioele Balestra [Opens in a new window] ,
Gaétan Lerisson ,
Benoit Scheid Open the ORCID record for Benoit Scheid [Opens in a new window] ,
Matthieu Wyart  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Pier Giuseppe Ledda*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Gioele Balestra
Affiliation:
iPrint Institute, HEIA-FR, HES-SO University of Applied Sciences and Arts Western Switzerland, FribourgCH-1700, Switzerland
Gaétan Lerisson
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Benoit Scheid
Affiliation:
TIPs, Université Libre de Bruxelles, Avenue F.D. Roosevelt 50, 1050Bruxelles, Belgium
Matthieu Wyart
Affiliation:
Physics of Complex Systems Laboratory, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
François Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
*
†Email address for correspondence: pier.ledda@epfl.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the role of hydrodynamic instabilities in the morphogenesis of some typical karst draperies structures encountered in limestone caves. The problem is tackled using the long wave approximation for the fluid film that flows under an inclined substrate, in the presence of substrate variations that grow according to a deposition law. We numerically study the linear and nonlinear evolution of a localized initial perturbation both in the fluid film and on the substrate, i.e. the Green function. A novel approach for the spatio-temporal analysis of two-dimensional signals resulting from linear simulations is introduced, based on the concepts of the Riesz transform and the monogenic signal, the multidimensional complex continuation of a real signal. This method allows for a deeper understanding of the pattern formation. The linear evolution of an initial localized perturbation in the presence of deposition is studied. The deposition linearly selects substrate structures aligned along the streamwise direction, as the spatio-temporal response is advected away. Furthermore, the growth of the initial defect produces a quasi-steady region also characterized by streamwise structures both on the substrate and the fluid film, which is in good agreement with the Green function for a steady defect on the substrate, in the absence of deposition.

JFM classification

Interfacial Flows (free surface): Thin films Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A53
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Arratia, C., Mowlavi, S. & Gallaire, F. 2018 Absolute/convective secondary instabilities and the role of confinement in free shear layers. Phys. Rev. Fluids 3 (5), 053901.CrossRefGoogle Scholar
Babchin, A.J., Frenkel, A.L., Levich, B.G. & Sivashinsky, G.I. 1983 Nonlinear saturation of Rayleigh–Taylor instability in thin films. Phys. Fluids 26 (11), 3159–3161.CrossRefGoogle Scholar
Barlow, N.S., Helenbrook, B.T. & Weinstein, S.J. 2017 Algorithm for spatio-temporal analysis of the signalling problem. IMA J. Appl. Maths 82 (1), 1–32.CrossRefGoogle Scholar
Bers, A. 1975 Linear waves and instabilities. In Physique des Plasmas (ed. C. DeWitt & J. Peyraud), pp. 117–215. Gordon & Breach.Google Scholar
Bertagni, M.B. & Camporeale, C. 2017 Nonlinear and subharmonic stability analysis in film-driven morphological patterns. Phys. Rev. E 96 (5), 053115.CrossRefGoogle ScholarPubMed
Brancher, P. & Chomaz, J.-M. 1997 Absolute and convective secondary instabilities in spatially periodic shear flows. Phys. Rev. Lett. 78 (4), 658.CrossRefGoogle Scholar
Brevdo, L. 1991 Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows. Z. Angew. Math. Phys. 42 (6), 911–942.CrossRefGoogle Scholar
Briggs, R.J. 1964 Electron stream interaction with plasmas. In Handbook of Plasma Physics. MIT Press.CrossRefGoogle Scholar
Brun, P.-T., Damiano, A., Rieu, P., Balestra, G. & Gallaire, F. 2015 Rayleigh–Taylor instability under an inclined plane. Phys. Fluids 27 (8), 084107.CrossRefGoogle Scholar
Buhmann, D. & Dreybrodt, W. 1985 The kinetics of calcite dissolution and precipitation in geologically relevant situations of karst areas: 1. Open system. Chem. Geol. 48 (1–4), 189–211.CrossRefGoogle Scholar
Bulow, T. & Sommer, G. 2001 Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case. IEEE Trans. Signal Process. 49 (11), 2844–2852.CrossRefGoogle Scholar
Camporeale, C. 2015 Hydrodynamically locked morphogenesis in karst and ice flutings. J. Fluid Mech. 778, 89–119.CrossRefGoogle Scholar
Camporeale, C. & Ridolfi, L. 2012 Hydrodynamic-driven stability analysis of morphological patterns on stalactites and implications for cave paleoflow reconstructions. Phys. Rev. Lett. 108, 238501.CrossRefGoogle ScholarPubMed
Carriere, P. & Monkewitz, P.A. 1999 Convective versus absolute instability in mixed Rayleigh–bénard– poiseuille convection. J. Fluid Mech. 384, 243–262.CrossRefGoogle Scholar
Charogiannis, A., Denner, F., van Wachem, B.G.M., Kalliadasis, S., Scheid, B. & Markides, C.N. 2018 Experimental investigations of liquid falling films flowing under an inclined planar substrate. Phys. Rev. Fluids 3 (11), 114002.CrossRefGoogle Scholar
Cohen, C., Berhanu, M., Derr, J. & Courrech du Pont, S. 2016 Erosion patterns on dissolving and melting bodies. Phys. Rev. Fluids 1, 050508.CrossRefGoogle Scholar
D'Alessio, S.J.D., Pascal, J.P., Jasmine, H.A. & Ogden, K.A. 2010 Film flow over heated wavy inclined surfaces. J. Fluid Mech. 665, 418–456.CrossRefGoogle Scholar
Decré, M.M.J. & Baret, J.-C. 2003 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166.CrossRefGoogle Scholar
Delbende, I. & Chomaz, J.-M. 1998 Nonlinear convective/absolute instabilities in parallel two-dimensional wakes. Phys. Fluids 10 (11), 2724–2736.CrossRefGoogle Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.CrossRefGoogle Scholar
Felsberg, M. & Sommer, G. 2001 The monogenic signal. IEEE Trans. Signal Process. 49 (12), 3136– 3144.CrossRefGoogle Scholar
Fermigier, M., Limat, L., Wesfreid, J.E., Boudinet, P. & Quilliet, C. 1992 Two-dimensional patterns in Rayleigh–Taylor instability of a thin layer. J. Fluid Mech. 236, 349–383.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J.-M. 2003 Mode selection in swirling jet experiments: a linear stability analysis. J. Fluid Mech. 494, 223–253.CrossRefGoogle Scholar
Hahn, S.L. 2003 Complex signals with single-orthant spectra as boundary distributions of multidimensional analytic functions. Bull. Pol. Ac.: Tech. 2 (2), 155–161.Google Scholar
Hayes, M., O'Brien, S.B.G. & Lammers, J.H. 2000 Green's function for steady flow over a small two-dimensional topography. Phys. Fluids 12 (11), 2845–2858.CrossRefGoogle Scholar
Heining, C. & Aksel, N. 2009 Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21 (8), 083605.CrossRefGoogle Scholar
Hill, C.A., Forti, P. & Shaw, T.R. 1997 Cave Minerals of the World, vol. 238. National Speleological Society.Google Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.CrossRefGoogle Scholar
Juniper, M.P. 2007 The full impulse response of two-dimensional jet/wake flows and implications for confinement. J. Fluid Mech. 590, 163–185.CrossRefGoogle Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G.M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12 (8), 1889–1898.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.G. & García, M. 2011 Falling Liquid Films, vol. 176. Springer Science & Business Media.Google Scholar
Kofman, N., Rohlfs, W., Gallaire, F., Scheid, B. & Ruyer-Quil, C. 2018 Prediction of two-dimensional dripping onset of a liquid film under an inclined plane. Intl J. Multiphase Flow 104, 286–293.CrossRefGoogle Scholar
Ledda, P.G., Lerisson, G., Balestra, G. & Gallaire, F. 2020 Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets. J. Fluid Mech. 904, A23.CrossRefGoogle Scholar
Lerisson, G. 2017 Stabilité d'une onde de gravité interne, analyse locale, globale et croissance transitoire. PhD thesis, thèse de doctorat dirigée par Chomaz, Jean-Marc et Ortiz Clerc, Sabine Mécanique des fluides Paris Saclay 2017.Google Scholar
Lerisson, G., Ledda, P.G., Balestra, G. & Gallaire, F. 2019 Dripping down the rivulet. Phys. Rev. Fluids 4, 100504.CrossRefGoogle Scholar
Lerisson, G., Ledda, P.G., Balestra, G. & Gallaire, F. 2020 Instability of a thin viscous film flowing under an inclined substrate: steady patterns. J. Fluid Mech. 898, A6.CrossRefGoogle Scholar
Lister, J.R., Rallison, J.M. & Rees, S.J. 2010 The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling. J. Fluid Mech. 647, 239–264.CrossRefGoogle Scholar
Marthelot, J., Strong, E.F., Reis, P.M. & Brun, P.-T. 2018 Designing soft materials with interfacial instabilities in liquid films. Nat. Commun. 9 (1), 4477.CrossRefGoogle ScholarPubMed
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. A 466 (2115), 659–694.CrossRefGoogle Scholar
Melville, W.K. 1983 Wave modulation and breakdown. J. Fluid Mech. 128, 489–506.CrossRefGoogle Scholar
Mowlavi, S., Arratia, C. & Gallaire, F. 2016 Spatio-temporal stability of the kármán vortex street and the effect of confinement. J. Fluid Mech. 795, 187–209.CrossRefGoogle Scholar
Rayleigh, Lord 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. s1-14 (1), 170–177.CrossRefGoogle Scholar
Scheid, B., Kofman, N. & Rohlfs, W. 2016 Critical inclination for absolute/convective instability transition in inverted falling films. Phys. Fluids 28 (4), 044107.CrossRefGoogle Scholar
Short, M.B., Baygents, J.C., Beck, J.W., Stone, D.A., Toomey, R.S. & Goldstein, R.E. 2005 a Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. Phys. Rev. Lett. 94, 018501.CrossRefGoogle ScholarPubMed
Short, M.B, Baygents, J.C & Goldstein, R.E. 2005 b Stalactite growth as a free-boundary problem. Phys. Fluids 17 (8), 083101.CrossRefGoogle Scholar
Stein, E.M. & Weiss, G. 2016 Introduction to Fourier Analysis on Euclidean Spaces, vol. 32. Princeton University Press.Google Scholar
Taylor, G.I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Trefethen, L.N. & Bau III, D. 1997 Numerical Linear Algebra, vol. 50. SIAM.CrossRefGoogle Scholar
Tseluiko, D, Blyth, M.G. & Papageorgiou, D.T. 2013 Stability of film flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729, 638–671.CrossRefGoogle Scholar
Unser, M., Sage, D. & Van De Ville, D. 2009 Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform. IEEE Trans. Image Process. 18 (11), 2402–2418.CrossRefGoogle ScholarPubMed
Van Saarloos, W. 2003 Front propagation into unstable states. Phys. Rep. 386 (2–6), 29–222.CrossRefGoogle Scholar
Vesipa, R., Camporeale, C. & Ridolfi, L. 2015 Thin-film-induced morphological instabilities over calcite surfaces. Proc. R. Soc. A 471 (2176), 20150031.CrossRefGoogle ScholarPubMed
Weinstein, S.J. & Ruschak, K.J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Wilson, S.D.R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209–221.CrossRefGoogle Scholar