Interfaces complexes : gouttes et bulles, milieux granulaires et

Transcription

Interfaces complexes : gouttes et bulles, milieux granulaires et
turbulence d’ondes
Christophe Josserand
Institut Jean Le Rond D’Alembert,
CNRS-Université Pierre et Marie Curie (Paris VI) UMR 7190,
Case 162, 4 place Jussieu, 75005 Paris, France.
13 mars 2008
ii
Préambule
Ce mémoire présente divers travaux de recherches portant sur des problèmes de dynamiques complexes d’interfaces, réalisés depuis au Laboratoire de Modélisation en Mécanique, maintenant Institut Jean Le Rond D’Alembert.
Depuis mon entrée au CNRS en octobre 1999 au Laboratoire de Modélisation en Mécanique, mon activité de recherche est en fait partagée entre quatre thématiques principales :
les écoulements interfaciaux, la modélisation de systèmes granulaires, quelques problèmes
liés aux écoulements turbulents (fusion de tourbillons, turbulence d’ondes) et les liquides
quantiques. Dans ce document, je m’attache cependant à décrire mes travaux liés aux
écoulements complexes d’interface, notamment des problèmes d’impacts de gouttes pour
lesquels la dynamique de l’interface, fortement déformée, est donc primordiale. En fin de
manuscrit, je présente très brièvement mes travaux effectués sur la modélisation d’écoulements granulaires (écoulements sec denses cisaillés et dynamique d’érosion notamment)
et un exemple de turbulence d’ondes avec les vibrations d’une plaque élastique.
Comme ce document en témoigne, mon intérêt se porte en général sur des problèmes
pour lesquels une approche combinée entre modélisation, calculs analytiques et approche
numérique est nécessaire. Il peut d’ailleurs être particulièrement pertinent d’utiliser la
simulation numérique certes pour interroger la modélisation mais aussi sous une forme
appelée parfois ”expériences numériques” où les résultats des simulations servent de base
au développement d’une approche analytique. L’aspect expérimental paraı̂t incontournable
pour ces écoulements, il n’est qu’à voir le nombre de publications récentes sur les impacts
de gouttes) et le développement de collaborations avec des équipes expérimentales a donc
été un souci constant pour moi durant ce travail.
Table des matières
I
II
Qui suis-je ?
3
Quelques problèmes d’interfaces.
11
1 Splash !
1.1 Historique et contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Formulation générale du problème . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Quelques ordres de grandeur . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Nombres sans dimension . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Mise en équation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Méthodes numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Calcul par volume de fluide (VOF) . . . . . . . . . . . . . . . . . . .
1.3.2 Suivi d’interface par marqueur . . . . . . . . . . . . . . . . . . . . .
1.3.3 Méthode intégrales de frontières . . . . . . . . . . . . . . . . . . . .
1.3.4 Autres approches : ”Level Set”, champ de phase, méthode particulaire
1.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Corolles, éclaboussures et étalements.
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Transition splash-étalement pour les impacts sur film liquide . . . . . . .
2.2.1 Théorie visqueuse du jet . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Impacts sur solide : splashes et obstacles, étalement et rétraction . . . . .
2.3.1 Déviation d’un film liquide par un obstacle. . . . . . . . . . . . . .
2.3.2 Impacts sur surfaces super-hydrophobes : étalement, rétraction et
rebond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Etalement et rétraction lors d’un impact sur surface hydrophobe .
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3 Quelques singularités d’interface
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3.1 Singularités et auto-similarités . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Conclusion-Perspectives
4.1 Ecoulements granulaires . . .
4.1.1 Un liquide granulaire ?
4.1.2 Méandres laminaires .
4.1.3 Perspectives . . . . . .
4.2 Turbulence d’ondes. . . . . .
4.3 Dynamiques d’interfaces . . .
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Table des matières
4.3.1
4.3.2
III
Scénario global d’impact . . . . . . . . . . . . . . . . . . . . . . . . . 70
Sélection d’épaisseurs caractéristiques . . . . . . . . . . . . . . . . . 71
Publications jointes
75
Première partie
Qui suis-je ?
5
CURRICULUM VITÆ
Christophe Josserand
Né le 10/07/69 à Sallanches, vie maritale, 2 enfants.
Chargé de Recherche au CNRS (CR1), Section 10
Institut Jean Le Rond D’Alembert
UMR 7190, CNRS-Paris VI Case 162, Université Pierre et Marie Curie,
4 Place Jussieu, 75252 Paris Cédex 05.
tél : 01 44 27 72 61 ; fax : 01 44 27 52 59
e-mail : [email protected].
http ://www.lmm.jussieu.fr/ josseran
Fonctions
– Octobre 1999-présent : Chargé de recherche à l’Institut Jean Le Rond D’Alembert (anciennement au Laboratoire de Modélisation en Mécanique), CNRS-Université
P. & M. Curie, UMR 7190, Paris. (CR1 depuis Octobre 2002).
– Septembre 2002-présent : Professeur chargé de cours à l’Ecole Polytechnique
Formation
– Octobre 1997-Septembre 1999 : Post-doc au James Franck Institute, Université
de Chicago sous la direction de Léo Kadanoff.
– 1993-1997 : Thèse de doctorat de l’université Pierre et Marie Curie (Paris
VI).
Sujet : ”Dynamiques de superfluide : nucléation de vortex et transition de phase du
premier ordre.”
Directeur de thèse : Yves POMEAU
– Septembre 1994-Août 1995 : Service militaire en tant que scientifique du contingent,à
l’Institut Non-Linéaire de Nice.
– 1994 : Agrégation de sciences physiques, option physique.
– 1993 : DEA de physique théorique de l’université Pierre et Marie Curie.
– 1990-1994 : Elève à l’Ecole Normale Supérieure (Paris).
Séjour de recherche à l’étranger
– Mai-Novembre 2007 : Séjour au Kavli Institute for Theoretical Research de
l’Université de Santa Barbara.
– Octobre 1997-Septembre 1999 : Post-doc au James Franck Institute, Université
de Chicago sous la direction de Léo Kadanoff.
– Février-Mars 1999 : Séjour au Center for non linear studies, Los Alamos,
New-Mexico.
– 1992 : Janvier-Juin : stage au département de Physique de l’Ohio State University, à Columbus.
Distinctions :
– Prix de Mme Claude Berthault 2005 de l’Académie des Sciences.
– Médaille de Bronze du CNRS 2007.
6
Encadrement
– Thèse de Laurent Duchemin 10/1999-10/2002 (30%). Position : Maı̂tre de Conférences à Marseille, Laboratoire IRPHE, depuis septembre 2005.
– Thèse d’Olivier Devauchelle 09/2004-09/2007 (50%). Actuellement ATER à l’Institut de Physique du Globe, Paris VII.
– Stage de recherche de DEA : 6 étudiants (50%).
– Stage de Master II et Projet Européen AlFa de Leonardo Gordillo, 10/2006-06/2007
(50%).
Enseignement
– Professeur chargé de cours à l’Ecole Polytechnique, Département de Mécanique (Mécanique des milieux continus, Mécanique des fluides).
– Cours de recherche au DEA de Mécanique (24h/an) : Milieux granulaires (3 années)
– Moniteur à l’Université de Versailles-Saint-Quentin (09/1995-09/1996).
– Agrégé-Préparateur, Ecole Normale Supérieure (09/1996-10/1997).
7
Liste de Publications
Toutes ces publications peuvent être trouvées en fichier pdf sur le site web : http ://www.lmm.jussieu.fr/ jo
seran
1. F. Hayot, C. Jayaprakash and C. Josserand, ”Long Wavelength properties of the
Kuramoto-Sivashinsky equation”, Physical Review E. 47, 911-915, (1993).
2. C. Josserand and Y. Pomeau, ”Generation of Vortices in a Model of Superfluid 4 He
by the Kadomtsev-Petviashvili Instability”, Europhys. Lett. 30 (1), pp. 43-48 (1995).
3. C. Josserand, Y. Pomeau and S. Rica, ”Cavitation versus Vortex Nucleation in a
Superfluid Model”, Phys. Rev. Lett. 75, 3150 (1995).
4. C. Josserand and S. Rica, ”Coalescence and Droplets in the Subcritical Nonlinear
Schrödinger Equation”, Phys. Rev. Lett., 78, 1215 (1997).
5. C. Josserand, ”Cavitation induced by explosion in a model of ideal fluid”, Phys. Rev.
E 60, 482 (1999).
6. C. Josserand, Y. Pomeau and S. Rica, ”Vortex Shedding in a Model of Superflow”,
Physica D 134, 111 (1999).
7. C. Josserand, ”A 2-D asymmetric exclusion model for granular flows”, Europhys.
Lett.48, 36 (1999).
8. R. Jordan and C. Josserand, ”Self-organization in nonlinear wave turbulence”, Phys.
Rev. E 61, 1527 (2000).
9. D. Blair, I. S. Aranson, G. W. Crabtree, V. Vinokur, L. S. Tsimring, and C. Josserand, ”Patterns in thin high-amplitude vibrated granular layers : Interfaces, hexagons, and superoscillons”, Phys. Rev. E 61, 5600 (2000).
10. C. Josserand and G. Debregeas, ”A self-similar model for shear flows in dense granular
materials”, Europhys. Lett. 52, 137 (2000).
11. C. Josserand, A. Tkachenko, D. Mueth and H. Jaeger, ”Memory effects in granular
materials”, Phys. Rev. Lett. 85, 3632 (2000).
12. R. Jordan and C. Josserand, ”Statistical equilibrium states for the nonlinear Schrödinger equation”, Mathematics and Computers in Simulation 1897, 1-15 (2000).
13. C. Josserand and Y. Pomeau, ”Non-linear aspects of the theory of Bose-Einstein
condensates”, Nonlinearity 14, R25-R62 (2001).
14. S. Villain-Guillot and C. Josserand, ”Non-linear growth of periodic interfaces”, Phys.
Rev. E 66, 036308 (2002).
15. L. Duchemin, S. Popinet, C. Josserand and S. Zaleski, ”Jet formation in bubbles
bursting at a free surface”, Phys. Fluids 14, 3000-3008 (2002).
16. Y. Renardy, S. Popinet, L. Duchemin, M. Renardy, S. Zaleski, C. Josserand, M.A.
Drumright-Clarke, D. Richard, C. Clanet and D. Quéré, ”Pyramidal and toroidal
water drops after impact on a solid surface”, J. Fluid Mech. 48, 69-83 (2003).
17. C. Josserand and S. Zaleski, ”Droplet splashing on a thin liquid film”, Phys. Fluids
15, 1650 (2003).
8
18. L. Duchemin, J. Eggers and C. Josserand, ”Inviscid coalescence of drops”, J. Fluid
Mech. 487, 167-178 (2003).
19. C. Josserand, P.-Y. Lagrée and D. Lhuillier, ”Stationary shear flows of dense granular
materials : a tentative continuum modelling”, EUR. PHYS. JOURNAL E 14 (2) :
127-135 (2004).
20. C. Josserand, ”Wave turbulence and Bose-Einstein condensates ”, C. R. PHYSIQUE
5 (1) : 77-90 (2004).
21. I. Delbende, T. Gomez, C. Josserand, C. Nore and M. Rossi, ”Different aspects of
fluid vortices”, C. R. Mécanique 332, 767-781 (2004).
22. C. Josserand, ”Stability of giant vortices in quantum liquids”, Chaos 14 (3), 875
(2004).
23. C. Josserand, L. Lemoyne, R. Troeger and S. Zaleski, ”Droplet impact on a dry
surface : triggering the splash with a small obstacle”, J. Fluid Mech. 524, 47-56
(2005).
24. L. Duchemin, C. Josserand and P. Clavin, ”Asymptotic behavior of the RayleighTaylor instability”, Phys. Rev. Lett. 94, 224501 (2005).
25. C. Josserand and Y. Pomeau, ”Vortices in condensate mixtures”, Phys. Rev. A 72,
023618 (2005).
26. C. Connaughton, C. Josserand, A. Picozzi, Y. Pomeau and S. Rica, ”Condensation
of classical nonlinear waves”, Phys. Rev. Lett. 95, 263901 (2005).
27. D. Bartolo, C. Josserand and D. Bonn, ”Retraction dynamics of aqueous drops upon
impact on nonwetting surfaces.” J. Fluid Mech. 545, 329-338 (2005).
28. C. Josserand, P.-Y. Lagrée and D. Lhuillier, ”Granular pressure and the thickness of
a layer jamming on a rough incline”, Europhys. Lett. 73, 363-369 (2006).
29. D. Bartolo, C. Josserand and D. Bonn, ”Singular jets and bubbles in drop impact”,
Phys. Rev. Lett. 96, 124501 (2006).
30. G. Düring, C. Josserand, and S. Rica, ”Weak Turbulence for a Vibrating Plate : Can
One Hear a Kolmogorov Spectrum ?”, Phys. Rev. Lett. 97, 025503 (2006)
31. C. Josserand, Y. Pomeau and S. Rica, ”Self-similar Singularities in the Kinetics of
Condensation”, J. Low Temp. Phys. 145, 231-265 (2006).
32. O. Devauchelle, C. Josserand and S. Zaleski, ”Forced dewetting on porous media.”,
J. Fluid Mech. 574, 343-364 (2007).
33. C. Josserand, Y. Pomeau and S. Rica, ”Coexistence of ordinary elasticity and superfluidity in a model of a defect-free supersolid”, Phys. Rev. Lett. 98, 195301 (2007).
34. C. Josserand, Y. Pomeau and S. Rica, ”Patterns and Supersolids”, Eur. Phys. J.
Special Topics 146, 47 (2007).
35. C. Josserand and M. Rossi, ”The merging of two co-rotating vortices : a numerical
study”, Eur. J. Mech./B Fluids 26, 779-794 (2007).
36. O. Devauchelle, C. Josserand, P.-Y. Lagrée and S. Zaleski, ”Morphodynamic modeling of erodible laminar channels”, Phys. Rev. E 76, 056318 (2007).
37. T.H. Vo Thi, J.-L. Rouet, P. Brault, J.-M. Bauchire, S. Cordier and C. Josserand,
”A continuous non-linear shadowing model of columnar growth”, J. Phys. D : Appl.
Phys. 41 (2), 022003 (2008).
9
38. N. Sepulveda, C. Josserand and S. Rica, ”Nonclassical rotational inertia fraction in
a one-dimensional model of a supersolid”, Phys. Rev. B 77, 054513 (2008).
actes de conférences :
– R. Jordan and C. Josserand : Coherent structures and statistical equilibrium states
in a model of dispersive wave turbulence, in Differential Equations and Nonlinear
Mechanics, par K. Vajravelu, Editions Kluwer Academic Publishers (Avril 2000).
– C. Josserand and S. Zaleski, actes du Congrès Français de Mécanique, Nancy (2001).
– R. Jordan and C. Josserand : Asymptotic behavior in a model of dispersive wave
turbulence, in Nonlinear PDE’s in condensed matter and reactive flows, H. Berestycki
and Y. Pomeau Eds, Kluwer Academic Publishers (2002).
– P. Carles, L. Duchemin, D. Gueyffier, C. Josserand and S. Zaleski : Droplet splashing
on a thin liquid film, actes de l’International Conference on Multiphase Flow, NewOrleans (2001).
– C. Josserand and S. Zaleski, actes de l’International Conference on Multiphase Flow,
Yokohama (2004).
– C. Josserand and S. Zaleski, actes de l’International Conference on Theoretical and
Applied Mechanics, Varsovie (2004).
– C. Josserand and S. Zaleski, actes du Congrès Français de Mécanique, Troyes (2005).
ouvrages collectifs
– M. Marengo, R. Scardovelli, C. Josserand and S. Zaleski, ”Isothermal drop-wall interactions : introduction to experimental and numerical studies” in The Navier-Stokes
equations : theory and numerical methods, lecture notes in pure and applied mathematics, ed. R. Salvi.
– C. Josserand, P.-Y. Lagrée and D. Lhuillier ”On different aspects of granular physics”
in Cours de l’Ecole de Peyresq, à paraı̂tre (2008).
10
Deuxième partie
Quelques problèmes d’interfaces.
Chapitre 1
Splash !
1.1
Historique et contexte
Les enfants jouant à s’éclabousser ou à jeter des cailloux dans les flaques d’eau imaginentils la complexités des phénomènes qu’ils provoquent ? Quels rôles jouent les gouttes de pluie
dans les échanges océan-atmosphère ou dans la dissémination de maladies végétales en agriculture ? Comment se répartissent les gouttes créées par atomisation d’un spray dans un
carburateur de moteur ? Ces exemples divers illustrent bien l’importance des situations
impliquant des dynamiques d’impact de gouttes. Pour bien décrire ces phénomènes, il faut
avoir une compréhension précise de la manière dont une goutte de liquide interagit avec
son environnement : autres gouttes, parois (chaudes ou tempérées), film liquide, surfaces
structurées, etc... La difficulté provient du fait qu’il y a une grande variabilité des résultats d’une expérience d’impact. allant du simple étalement au rebond en passant par la
formation de corolles, couronnes liquides et détachements de gouttelettes secondaires par
exemple. Il est en effet particulièrement fascinant d’observer comment la simple projection
d’une goutte de liquide sur une surface (sèche, humide ou recouverte d’un film liquide)
peut engendrer une grande variété de dynamiques qui restent encore mal comprises pour
l’essentiel. Pourtant, la dynamique de formation d’une corolle liquide lors de l’impact d’une
goutte sur un film liquide semble n’avoir plus de secret pour le (télé)-spectateur assidu :
de génériques de prévisions météo en publicités pour produits cosmétiques, nous sommes
habitués à voir tout naturellement l’impact d’une goutte se transformer en une magnifique
corolle de laquelle se détachent quelques gouttelettes. Ces apparentes familiarisations avec
es problèmes mettant en jeu des impacts entre gouttes, des films liquides, des bulles, sont
trompeuses et souvent ces illustrations sont plus le fruit de l’imagination de l’artiste que
d’une expérience ou un calcul particuliers. Ces images quasi-iconographiques pour le photographe amateur cachent en fait de nombreuses interrogations qui restent à élucider.
La physique des bulles, gouttes et plus généralement celle des interfaces est en fait très
liée à la notion de tension de surface. On peut donc faire remonter les premières études
sur ce type de problème à la fin du XVIIIeme siècle et aux travaux de Young, Laplace
et Benjamin Franklin notamment. Grâce au développement rapide de la photographie,
des clichés instantanés d’impacts de gouttes permirent de révéler la complexité de cette
dynamique dès la fin du XIXeme siècle. Les premières études s’appuyant sur l’observation de
photographies décomposant les différentes phases de l’impact remontent alors aux travaux
fondateurs pour le domaine de A.M. Worthington[111]. L’impact du livre qu’il publia
14
Splash !
en 1908[112] dépassa la communauté scientifique et fit découvrir à un plus large public
ces fascinantes images de splashes ! A sa suite, Edgerton, professeur au MIT, réalisa de
nombreux clichés d’impact grâce à des techniques de visualisation rapide toujours plus
performantes (ces nombreux travaux ont été réédités en DVD récemment par le MIT[39,
40]. Ces photographies rencontrèrent un large succès public grâce à leur diffusion dans
de nombreux magazines (National Geographic, revues de photographie...) et les fameuses
photographies de splashes de gouttes de lait sont devenus maintenant des ”classiques” de
toute revue de photographie.
Depuis Worthington, l’intérêt de la communauté scientifique pour ces problèmes d’impacts de gouttes ne s’est jamais dementie et on peut même remarquer une augmentation
notable des études scientifiques et technologiques sur ce type de sujet depuis plus de vingt
ans. Plusieurs raisons différentes s’ajoutent et se conjuguent pour expliquer cet intérêt :
l’impact de gouttes pose toujours des questions (et des problèmes) fondamentales pour
le mécanicien des fluides ; il représente un enjeu technologique et industriel réel et crucial, avec des implications nombreuses pour l’environnement ; finalement, l’impact d’une
goutte sur différents types de surface est devenu un des problèmes-test (benchmark) de
toute nouvelle méthode numérique multiphasique.
D’un point de vue ”académique”, on observe un nombre croissant d’études et de communications traitant d’impacts de gouttes sous différentes conditions. Outre les applications
nombreuses qui motivent indéniablement ces études, il me semble que l’extrême richesse
des comportements résultant d’expérience finalement relativement simple à réaliser explique également cet engouement. L’impact d’une goutte sur un film liquide mince peut
doner lieu à un étalement à faibles vitesses (ou grandes viscosités) où à la formation d’éclaboussures (phénomènes que nous appèlerons souvent ”splash” dans ce document , cédant
à cet anglicisme qui correspond justement à cette formation d’éclaboussures lors d’un impact) au plus grande vitesse. Cependant, si l’impact se fait dans un réservoir de liquide
profond, l’impact formera plutôt une cavité dont le collapse produira un jet de liquide fin
et la formation d’un anneau de vorticité se dirigeant vers le fond du réservoir. De même,
l’impact sur un subtrat sec peut conduire à des splashes intenses, à un étalement simple,
voire au rebond partiel ou total de la goutte ! On peut aussi remarquer que le simple impact d’une goutte est le siège d’un nombre important d’instabilités hydrodynamiques et
de comportements critiques : instabilités de Richtmyer-Meshkov au début de l’impact, de
Kelvin-Helmotz entre la goutte qui s’étale et le fluide au repos, celle de Rayleigh-Taylor
lorsque la corolle ralentit, instabilité de Rayleigh-Plateau lors du détachement des gouttes
secondaires, instabilité liée à l’effet Marangoni lorsque les écarts de températures sont
importants, transition transsonique etc... On peut donc imaginer une goutte en impact
comme un ”micro-laboratoire” d’étude des instabilités hydrodynamiques classiques ! Cela
explique aussi pourquoi il est difficile d’avoir un scénario global de la dynamique de l’impact qui permette de suivre avec précision les différentes phases de l’évolution de la goutte.
De même, étant donné cette diversité des dynamiques pouvant apparaı̂tre durant l’impact,
il est souvent malaisé d’obtenir (et donc de trouver dans la littérature) des règles simples
et robustes décrivant les résultats de l’impact. Par exemple, des questions aussi ”banales”
que ”va-ton observer un splash ou un étalement ?”, ”Combien de gouttes secondaires seront
produites par le splash ?”, ”Quelles tailles feront-elles ?” ou ”Quel sera le rayon maximal
d’extension de la goutte ?” peuvent engendrer une grande variété de réponses, suivant les
conditions d’expérience, les régimes dynamiques considérés ou les paramètres de l’impact !
1.2. Formulation générale du problème
15
Il est donc primordial de pouvoir extraire de ces résultats des lois d’impact les plus simples
possibles ainsi qu’une théorie et leur interprétation cohérentes.
Mieux comprendre les dynamiques mises en jeu lors de l’impact de gouttes est aussi
une nécessité pour de nombreuses applications industrielles. Pour l’industrie automobile, il
s’agit notamment de mieux maı̂triser et contrôler dans les carburateurs la taille des gouttes
formées lors d’impacts ou la quantité de liquide se déposant sur les parois. En effet, la zone
d’échange entre un liquide et un gaz se situe bien sûr à l’interface, c’est là qu’ont lieu les
réactions chimiques de la combustion notamment ; les déformations et transformations de
celle-ci lors d’un ou de plusieurs impacts peuvent donc influencer fortement ces échanges.
Le ruisselement d’eau sur les pares brises ainsi que l’évacuation des gouttes de pluie sont
des problématiques importantes apparues plus récemment. En aéronautique, là aussi la
combustion implique de bien comprendre les phénomènes d’impact : un exemple spectaculaire concerne l’aluminium présent comme catalyseur dans les boosters d’Ariane 5. Il
est projeté sous forme de gouttes d’alumine produites lors de la combustion contre les
parois internes du booster et forme in fine un volume liquide important retenu dans la
structure particulière du booster[51]. L’impact de gouttes entre en jeu dans de nombreux
autres contextes : impression jet d’encre et fabrication de micro-circuits par jet de matière,
sprays de refroidissement ou de revêtement de surface ou encapsulation en microfluidique
par exemples. L’impact de goutte est aussi un facteur important de dissémination de produits ou déchets toxiques et nucléaires lors de fuites dans les circuits de production et les
règles de sûreté industrielle peuvent avoir à tenir compte explicitement de ces risques.
Mais n’oublions pas que l’impact de gouttes est avant tout un ”évènement naturel”
dont le rôle en agriculture, dans les échanges thermiques ou chimiques en météorologie est
primordial. En agriculture, l’impact de gouttes peut-être bénéfique quand elle augmente
la dispersion des pollens mais cela peut aussi s’avérer dangereux pour les plantes lors
des pluies violentes ou par la dissémination des pesticides ou des maladies que l’impact
favorise[11]. L’impact des gouttes de pluies en mer représente aussi une contribution importante dans les échanges air-océan [20] : en effet, l’emprisonnement de bulles de gaz lors
de ces impacts favorise les échanges thermiques et gazeux ; de même, l’eclatement de ces
bulles d’air permet à de nombreux aérosols d’être volatilisés dans l’air [31].
Dans cette partie du manuscrit, je présente différents travaux portant sur des dynamiques
d’impact de gouttes. En général, il s’agit d’une étude expérimentale et/ou numérique sur
lesquels s’appuie une théorie ou une modélisation simple de l’impact.
1.2
Formulation générale du problème
On considère l’impact d’une goutte sur une surface plane recouverte d’un film liquide
d’épaisseur h (h = 0 correspondant ainsi à l’impact sur surface sèche), représenté sur la
figure (1.1).
On note ρL et ρG , µL et µG les densités et viscosités dynamiques dans le liquide et
le gaz respectivement, ainsi que σ la tension de surface liquide-gaz. La goutte que l’on
considérera en général sphérique de rayon R (diamètre D = 2R), est supposée tomber
en incidence normale depuis une hauteur initiale d et une vitesse verticale de chute U =
−U ez . A priori l’ensemble est soumis à la gravité g = −gez . Le film, d’épaisseur h, est
formé du même liquide que la goutte. On peut remarquer que le problème ainsi posé
contient déjà de nombreuses simplifications et qu’une formulation plus générale pourrait
16
Splash !
Fig. 1.1 – Représentation schématique de l’impact d’une goutte sur une surface plane.
considérer des impacts d’incidence non normale, un substrat solide non plan ou des liquides
distincts dans la goutte et le film par exemple. De même les échanges thermiques ou la
présence de substances dissoutes pourraient être modélisés, permettant une prise en compte
d’effets Marangoni notamment. Cependant, la situation simple présentée ici contient déjà
l’essentiel des variétés de comportement observés lors d’impacts de gouttes quelconques
et permet une approche systématique du problème avec un nombre limité de paramètres
variables. On supposera donc les densités et viscosités des fluides ainsi que la tension de
surface constants, ce qui revient notamment à considerer des fluides incompressibles. Le
gaz et le fluide sont des fluides distincts (on ne considère donc pas l’équilibre entre le
liquide et sa vapeur) et immiscibles. De même, nous garderons toujours à l’esprit que les
fluides que nous cherchons à étudier et modéliser sont ”réalistes” : air à des pressions de
l’ordre de la pression atmosphérique pour le gaz, solutions aqueuses, glycérol, alcools ou
carburants (diesel, hydrogène liquide par exemple) pour les liquides.
1.2.1
Quelques ordres de grandeur
Bien que les modèles et les calculs numériques s’appliquent à des impacts caractérisés par des nombres sans dimension, il est utile d’avoir à l’esprit quelques ordres de
grandeur, notamment des impacts réalisés en laboratoire. Les gouttes ont en général des
rayons variant de quelques microns à quelques millimt̀res. Les impacts se font à des vitesses correspondant à des chutes de gouttes réalisable en laboratoire (quelques mètres)
ou à des vitesses obtenues dans des conditions ”naturelles” (pluie par exemple, soit à
nouveau quelques mètres par seconde). Le temps durant lequel l’impact se produit corres-
17
pond à quelques temps caractéristiques τ = R/U (sauf effet particulier de la surface), soit
quelques millisecondes. Ceci explique l’utilisation en laboratoire de caméra rapide (plus
de 1000 images par seconde). On peut justifier l’hypothèse d’incompressibilité de l’écoulement puisque les nombres de Mach dans l’air et dans l’eau M = U/c (où U est la vitesse
caractéristique de l’impact et c la vitesse du son dans le fluide) seront de l’ordre de 0.01
pour l’air et 0.001 pour l’eau. Dans le cas de très grandes vitesses d’impact, il faut prendre
en compte la compressibilité du liquide notamment via la formation d’une onde choc se
propageant dans la goutte : des bulles de cavitation peuvent ainsi apparaı̂tre au sommet de
la goutte lorsque l’onde de choc de l’impact atteint le sommet de celle-ci et se réfléchit en
onde de détente (voir la revue de M. Lesser et J. Field sur le sujet [74]). Le gaz est lui aussi
considéré incompressible bien que très peu d’études (comparé à la littérature sur le sujet)
font varier les caractéristique du gaz. Il est en fait couramment admis, qu’étant donné
l’importante différence de densité entre le liquide et le gaz, on peut négliger l’influence du
gaz sur la dynamique en première approximation (cette approximation n’est en généralement pas invoqué dans le cadre de l’atomisation en revanche). Si on néglige complètement
l’influence du gaz (formellement, cela revient à prendre ρG = 0 et µG = 0) on se retrouve
dans le cadre de l’approximation des écoulements à surface libre, ce qui permet notamment des traitements analytiques et numériques simplifiés. L’importance du gaz (ou tout
au moins de sa compressibilité à faibles pressions) a été spectaculairement montrée récemment lors d’impacts de gouttes sur surface solide puisque diminuer la pression du gaz peut
supprimer la formation du splash[114] ! L’influence du gaz sur la dynamique de l’impact
reste donc en grande partie à étudier et sort des travaux présentés ici. Donc, par la suite,
l’influence du gaz et sa dynamique seront souvent négligés en première approximation.
Cependant, la dynamique du gaz sera résolue numériquement dans de nombreuses études
numériques, ce qui impliquera de prendre les paramètres physiques du gaz compatibles
avec les caractéristiques de la méthode numérique.
Dans ce document, nous considérerons que la viscosité des fluides et la tension de
surface liquide/gaz peuvent varier largement. Expérimentalement, alors qu’il est relativement aisé de faire varier la viscosité, la tension de surface pour les liquides couramment
utilisés change peu. Sous des conditions normales de pression et température (20◦ Celsius), l’eau a pour viscosité dynamique µeau = 10−3 P a · s, assez proche de l’ethanol (
µetha = 1.19 · 10−3 P a · s, l’ether est quatre fois moins visqueux (µethe ≈ 2.3 · 10−4 P a · s)
alors que le glycerol est plus de mille fois plus visqueux (µgly = 1.5 P a · s). On peut ainsi
avec des solutions aqueuses eau-glycerol faire varier continûment la viscosité du liquide
d’un facteur 1500 pour une faible évolution de la tension de surface. En effet, la tension
de surface de l’eau pure (eau/air) vaut σeau = 0.073 P a · m assez proche de celle du glycerol (σgly = 0.064 P a · m), celles l’ethanol et l’ether étant environ trois fois plus faibles
(σetha = 0.022 P a · m et σethe = 0.029 P a · m). Remarquons que parmi les liquides courants, seul le Mercure (d’usage délicat !) offre une tension de surface nettement différente
(σHg = 0.425 P a · m).
Finalement, bien que les expériences se fassent en général en présence de la gravité
terrestre (sauf quelques expériences réalisées en micro-gravité) qui accélère les gouttes
pendant leur chute, l’influence de la gravité pendant l’impact lui-même est en général
négligée. En effet, le nombre de Froude, qui caractérise le rôle de la gravité pendant
l’impact :
18
Splash !
Fr =
√
gD
U
se situe en général en dessous de 0.1. Ainsi, pour un impact développant une corolle et
des éclaboussures, la gravité aura certainement un rôle dans la dynamique longue de la
corolle, que nous n’étudierons pas particulièrement ici.
1.2.2
Nombres sans dimension
Plusieurs nombres sans dimension permettent en fait de caractériser l’impact étudié, au
delà des nombres de Mach et de Froude que nous omettrons en général dans nos études. Les
deux nombres caractérisants le poids relatif de l’inertie par rapport aux forces visqueuses
et capillaires pour le liquide sont le plus souvent utilisés ; il s’agit du nombre de Reynolds :
Re =
ρL U · R
µL
(1.1)
ρL U 2 · R
.
σ
(1.2)
et du nombre de Weber :
Re =
On peut quelquefois les remplacer par une combinaison de ses deux nombres, permettant de mieux mettre en évidence certains résultats. Lorsque l’on veut étudier l’influence
des paramètres de la goutte autres que sa vitesse d’impact, on utilisera le nombre de
Ohnesorge :
√
µL
We
Oh = √
=
.
Re
ρL · σ · R
(1.3)
D’autre part, le nombre capillaire permet de mesurer le rapport des forces visqueuses
et capillaires (et est indépendant du rayon de la goutte) :
Ca =
µL · U
We
=
.
σ
Re
(1.4)
A ces nombres couramment utilisés, il ne faut pas oublier d’ajouter les nombres caractérisant le rapport entre le liquide et le gaz :
Rρ =
ρG
µG
et Rµ =
ρL
µL
ainsi que les nombres liés à la géométrie ; le raport d’aspect :
Rasp =
h
R
et dans le cas où l’on considérerait des gouttes non sphériques, $ l’ellepticité de la
goutte lors de l’impact. Sauf mention contraire, nous ne ferons pas varier ces nombres au
sein de chaque problème étudié dans la suite.
1.2.3
19
Mise en équation
La dynamique de chaque fluide obéit à l’équation de Navier-Stokes (N-S par la suite) :
ρi
!
"
∂u
+ u · ∇(u) = −∇P + µi ∆u + ρi g
∂t
(1.5)
où l’indice i correspond soit à la phase liquide i = L, soit au gaz i = G. Le champ de
vitesse est décrit par le vecteur u(x, t) où x est le vecteur position et t le temps. La gravité
g a été prise en compte par souci d’une présentation générale. Nous l’omettrons par la
suite. Le champ de pression P (x, t) peut être interprété comme un paramètre de Lagrange
imposant l’incompressibilité de chaque fluide :
∇·u=0
(1.6)
A l’interface liquide-gaz (notée symboliquement S), la tension de surface impose la
condition de saut :
[(−P Id + 2µD)]S · n = σκn
(1.7)
Id et D sont les tenseurs d’ordre deux, identité et taux de déformation :
1
D = (t ∇u + ∇u),
2
la notation [ ]S indiquant la diférence entre les valeurs de chaque côté de l’interface
(orientée suivant sa normale n). κ est la courbure totale de la surface. Pour une goutte
sphérique de rayon R, au repos, la différence de presion entre le liquide et le gaz extérieur
correspond au résultat classique 2σ/R. D’autre part, la vitesse est continue à l’interface,
en présence de fluides visqueux :
[u]S = 0.
Finalement, on adoptera une écriture ”mixte” valable dans tout l’espace, l’équation de
Navier-Stokes diphasique :
ρ
!
"
∂u
+ u · ∇(u) = −∇P + µ∆u + σκδS n.
∂t
(1.8)
La densité ρ(x, t) et la viscosité µ(x, t) doivent maintenant être vues comme des champs
fonction de la position et du temps. Il s’agit en fait de fonctions constantes par domaine
a priori. Plus simplement, si on définit la fonction caractéristique χ(x, t), telle que :
χ = 0 dans le gaz,
χ = 1 dans le liquide,
on peut alors définir dans tout l’espace la densité et la viscosité :
ρ(x, t) = χ(x, t)ρL + (1 − χ(x, t))ρG , µ = χ(x, t)µL + (1 − χ(x, t))µG .
(1.9)
Ces définitions, qui peuvent sembler naturelle, cachent en fait une approximation dans
le calcul de la viscosité moyenne. En effet, cette formule, dans le cas d’un cisaillement
simple, se révèle exacte pour décrire un écoulement dont le taux de glissement serait
20
Splash !
constant. La définition d’une viscosité qui corresponde à un taux de cisaillement constant,
plus en accord avec les conditions à imposer à l’interface, donnerait :
1
χ(x, t) (1 − χ(x, t))
=
+
.
µ
µL
µG
Notons qu’à ce stade cette différence n’a pas de sens puisque ces formules sont équivalentes pour les valeurs discrète de χ (= (0, 1)). En revanche, cette distinction prend tout
son sens lorsqu’il faut définir la viscosité moyenne dans une maille traversée par l’interface.
Dans ce cas, les deux formules proposées ne se révèlent exactes que dans les limites différentes d’écoulements de glissement simple (normal ou tangent à l’interface). Il n’est donc
a priori pas possible de se donner une formule moyenne pour la viscosité et on prendra
dans la suite la convention (1.9).
L’évolution des champs de densité et de viscosité est donc liée à la dynamique de χ :
dχ
∂χ
∂χ
=
+ u · ∇χ =
+ ∇ · (χu) = 0
(1.10)
dt
∂t
∂t
où nous avons utilisé la propriété d’incompressibilité du champ de vitesse. Cette équation
traduit ainsi simplement la dynamique particulaire de la fonction caractéristique.
Cette écriture deux-fluides de l’équation de N-S que nous venons de déduire est en fait
celle utilisée pour détermier la plupart des schémas numériques.
1.3
Méthodes numériques
La résolution numérique de l’équation de Navier-Stokes diphasique reste un problème
complexe : aux difficultés habituelles rencontrées pour la résolution de l’équation de NS monophasique (avec ρ constant et uniforme) s’ajoute la prise en compte de l’interface
liquide-gaz. Cette interface, d’épaisseur nulle dans le modèle diphasique que nous utilisons, est un ensemble de courbes à lorsque le problème est bidimensionnel (2D, géométrie
plane ou cylindrique), ou un ensemble de surfaces dans le cas général (3D). La difficulté
mathématique est donc due à la nécessité de décrire une structure d’épaisseur nulle dans
un domaine de dimension supérieure (on peut formaliser cela en notant que l’interface est
de codimension 1 par rapport au domaine de calcul). Une première alternative se pose
alors si on veut développer un code de calcul diphasique performant : doit-on prendre un
maillage ad hoc, c’est à dire pris selon un maillage déterminé à l’avance (même si on pourra
se restreindre à des sous ensembles de ce maillage, variant au cours du temps, comme c’est
le cas pour un raffinement de maillage adaptatif) au sein duquel l’interface devra être soit
reconstruite (méthode Volume of Fluid, notée VOF[55, 72]), soit suivie par une collection
de marqueurs par exemple[108, 96] ? Doit-on au contraire utiliser un maillage asservi à
l’interface soit naturellement dans le cadre d’une méthode d’intégrale de frontière, soit en
utilisant un maillage orthogonale mobile propagé à partir de l’interface (voir [48, 79] par
exemple) ? Toutes ces méthodes ont des avantages et des inconvénients divers et le choix
de leur utilisation devrait dépendre idéalement du problème étudié, mais ce sont l’histoire
et la maı̂trise d’un code de la part de l’utilisateur et plus généralement du laboratoire
qui déterminent souvent la méthode utilisée. Les différentes méthodes utilisées dans le
cadre de ce manuscrit (VOF, marqueurs et intégrales de frontières) sont décrites ici, suivies d’une rapide discussion générale introduisant d’autres approches. La présentation des
méthodes numériques s’inspire donc largement des différentes publications du laboratoire
1.3. Méthodes numériques
21
décrivant ces schémas, et à la plupart desquelles l’auteur de ce manuscrit n’a pas ou a peu
contribué[72, 76, 52, 101, 96, 33, 34].
Pour les méthodes en maillage fixe, on se donnera en général (i.e. sauf mention contraire)
un maillage fixe et homogène, c’est à dire que les dimensions des mailles du calcul sont
constantes dans l’espace et le temps. On notera alors dx = dy = dz = dr = h suivant que
l’on sera à deux ou trois dimensions cartésiennes (x, y et éventuellement z) ou en trois
dimensions avec symétrie de révolution (coordonnées cylindriques r et z). La généralisation de ces méthodes à des maillages de taille variables (permettant ainsi un rafinement
de maillage dynamique) reste un problème difficile notamment pour le suivi de l’interface et le calcul des forces capillaires. A contrario, les méthodes à maillage mobile défini
par l’interface sont souvent adaptatives par construction. Pour la méthode VOF (et ces
nombreuses variantes), plusieurs méthodes de raffinement de maillage ont été réalisées
récemment seulement ; on peut citer en particulier ici la bibliothèque de calcul Gerris développée par Stéphane Popinet[94, 95]. Par simplicité on décrira les méthodes de calcul
dans le cas 2D, le passage à 3D, malgré de réelles difficultés et complications techniques,
ne demande pas de changement de technique. Volontairement, seules les aspects concernant la prise en compte des interfaces seront développés (rapidement) et plus de détails
pourront être trouvés dans les articles suivants [72, 76, 52, 101, 96].
1.3.1
Calcul par volume de fluide (VOF)
Les différentes quantités physique sont définies et évaluées sur la grille de maillage
suivant la méthode ”markers and cells” (MAC) qui consiste à évaluer les quantités scalaires
telles que la pression pij ou la fraction volumique cij au centre de chaque cellule de calcul
(indicée donc par (i, j) à deux dimensions et notée Ωij ) alors que les vecteurs et les flux
sont évaluées sur les faces de chaque cellule (voir figure 1.2).
Fig. 1.2 – Discrétisation suivant la méthode MAC. On montre ici la fraction volumique c,
la pression p et les flux de volume.
La fraction volumique dans chaque maille (appelée fonction de couleur) est définie via
la discrétisation de la fonction caractéristique χ :
22
Splash !
cij =
#
χ(r)dr
Ωij
Ainsi cij = 1 lorsque la maille ne contient que du liquide, cij = 0 s’il n’y a que du gaz
et 0 < cij < 1 si l’interface traverse la maille. Pour reconstruire l’interface à partir de la
donnée de la fonction de couleur uniquement, plusieurs options se présentent. Remarquons
tout d’abord que dans l’équation de N-S, la connaissance précise de l’interface (i.e. à une
échelle plus petite que la taille de la maille) n’est nécessaire que pour le calcul des termes
de tension de surface et éventuellement pour le terme de pression. En effet, il faut être
capable d’estimer le plus précisement possible la valeur moyenne de la courbure dans la
maille de calcul, ainsi que le vecteur normal associé. De même, la position du centre de
la maille (là où se calcule la pression) par rapport à l’interface peut aider à déterminer
plus précisément les gradients de pression numériques. Finalement, la connaissance de la
position de l’interface est très importante pour le calcul des termes d’advection de masse
(eq. (1.10)) et de quantité de mouvement (dans N-S).
La simplification supplémentaire faite en général dans le cadre des méthodes VOF
consiste à reconstruire l’interface sous forme de segment dans chaque maille traversée.
Cette approximation se justifie dans la mesure où il peut sembler illusoire de décrire
l’interface plus précisément avec la seule connaissance de la fraction volumique dans chaque
cellule. Dans un premier temps, les méthodes développées se sont attachées à reconstruire
l’interface sous la forme de segments parallèles à un des axes du maillage (plan a 3D),
comme indiqué sur la figure 1.3.
Fig. 1.3 – Reconstruction de l’interface (représentée en trait épais) par des segments
alignés avec les axes du maillage (méthode SLIC).
On peut imaginer une reconstruction plus fine en prenant des segments d’orientation
quelconque et en imposant la continuité de l’interface. On appelle généralement ces deux
métodes VOF/SLIC (simple linear interface calculation) et elles se révèlent toutes les deux
assez imprécises car elles ne tiennent finalement pas compte explicitement de l’orientation
réelle de lnterface (définie par sa normale n.
23
La technique la plus utilisée dans le cadre des méthodes VOF consiste en fait à estimer
le vecteur normal à l’interface, défini via le gradient de la fonction de couleur ∇c et
à construire l’interface sous forme de segments orthogonaux à cette normale en chaque
cellule contenant l’interface (voir figure 1.4). Le calcul des termes d’advection dans les
équations d’évolution se révèlent être bien plus précis dans ce cadre là. Notons que le
calcul du gradient de c devant se faire au centre de la cellule, cela nécessite un calcul
par différence finie sur neuf points à deux dimensions et donc vept-sept points à 3D. On
nomme les différentes méthodes s’inspirant de cette technique généralement VOF/PLIC
(piecewise linear interface calculation). Il s’agit de la méthode utilisée pour les codes VOF
dans ce manuscrit[76, 52].
Fig. 1.4 – Reconstruction de l’interface (représentée en trait épais) par des segments
orientés par la normale de la fraction volumique (méthode PLIC).
L’intégration temporelle du système d’équations (1.6,1.8,1.10) se fait en plusieurs
étapes successives. Tout d’abord l’équation (1.10) est résolue via la propagation de l’interface suivant le champ de vitesse. Cette propagation se fait suivant chaque direction
principale (x et y) successivement et permet de calculer le champ de couleur actualisé au
pas de temps suivant. L’équation de Navier-Stokes (1.8) et l’incompressibilité (1.6) sont
ensuite résolues en deux étapes : tout d’abord un champ de vitesse u∗ explicite est prédit
en ne prenant pas en compte la pression dans la dynamique. Le champ de vitesse ainsi
obtenu ne satisfait pas la condition d’incompressibilité et une équation de Poisson sur la
pression permet de déterminer la pression et le champ de vitesse actualisé :
!
"
1
∇
∇P = ∇ (u∗ · ∇(u∗ ))
ρ
1
u = u∗ − ∇Pdt
ρ
où dt est le pas de temps. Finalement, la résolution de l’équation de Poisson se fait grâce à
un shéma multi-grille[14] qui permet d’accélérer la dynamique de relaxation vers le champ
24
Splash !
de pression voulu. La figure (1.5) illustre l’impact d’une goutte sur un film liquide calculé
par la méthode VOF en géométrie cylindrique. Le champ de densité est montré pour
différents temps après le début de l’impact.
Fig. 1.5 – Impact d’une goutte de rayon 1 mm et de vitesse U0 = 3m · s−1 sur un film du
même liquide de 0.5 mm d’épaisseur calculé par laméthode VOF. Le liquide a la même
densité que l’eau ρ = 1000kg · m−3 , mais est dix fois plus visqueux µ = 0.01kg · m−1 · s−1 .
Le gaz est environ deux fois plus dense que l’air ρ = 2kg · m−3 et vingt fois plus visqueux
µ = 2 · 10−5 kg · m−1 · s−1 . La tension de surface est la même que pour l’interface air/eau :
σ = 0.07kg · s−2 . Les nombre de Weber et de Reynolds valent donc We = 129 et Re = 300.
Les profils de densité sont représentés aux temps t = 0.05, t = 0.25, t = 0.5 et t = 0.75
ms, soit t/τ = U0 t/R0 = 0.15, 0.75, 1.5 et 2.25, de gauche à droite et de haut en bas.
Dans les travaux liés à ce manuscrit, la méthode VOF a été utilisé pour étudier la
dynamique de splashes sur film liquide[70] et le contrôle de l’impact par une irrégularité
de surface[67].
1.3.2
Suivi d’interface par marqueur
La méthode des marqueurs consiste à imaginer des points matériels (marqueurs) attachés à l’interface. Cette méthode a été développé par Stéphane Popinet durant sa thèse au
LMM, sous la direction de Stéphane Zaleski[96]. Je décris ici succinctement les différentes
particularités de cette méthode. Leur dynamique est donc simplement donnée par leur
25
vitesse Lagrangienne, extrapolée à partir du champ de vitesse. L’interface est reconstruite
à l’aide de polynômes reliant les marqueurs entre eux et permettant ainsi de calculer simplement la résultante des effets capillaires sur chaque maille, comme le montre la figure
(1.6). En effet, la connaissance des points d’intersection de l’interface avec les faces de
la maille de calcul ainsi que le vecteur tangent à l’interface donne simplement la force
capillaire, proportionnelle à la différence des deux vecteurs tangents (tB − tA suivant la
figure (1.6)). La reconstruction polynomiale doit donc assurer que l’interface et les vecteurs
tangents soient continument différentiables afin d’obtenir un calcul cohérent des effets capillaires. Pour cela, des polynômes d’ordre au moins trois sont nécessaires et suffisants (des
polynômes d’ordre 3 sont utilisés dans les simulations numériques.
Fig. 1.6 – Illustration de la méthode des marqueurs : des marqueurs (non représentés
sur la figure) permettent une description continument différentiable de l’interface et des
vecteurs tangents à l’interface. Leur différence de chaque part d’une maille de calcul donne
la tension capillaire agissant sur la maille.
La connaissance très précise de la position ”réelle” de l’interface permet de connaı̂tre
la position du noeud de calcul par rapport à l’interface : il est ainsi possible d’avoir une
meilleure estimation de la contribution de la pression dans chaque maille en ”corrigeant”
cette contribution. En effet, il convient alors plutôt de pondérer la contribution de la pression sur une face de la maille traversée par l’interface par les valeurs de la pression aux
noeuds les plus proches dans la même phase fluide (pour plus de détails, voir [96]). Une
fois l’interface déterminée par la position des marqueurs et leurs polynômes associés, le
calcul du champ de couleur ci,j est immédiat par intégration et il est donc actualisée via
l’actualisation de la position des marqueurs. L’équation de Navier-Stokes est alors résolue
de manière similaire à la méthode VOF (la contribution de la tension de surface et de la
pression comme expliqué ci-dessus) en utilisant notamment une méthode multi-grille pour
assurer l’incompressibilité.
Dans les travaux présentés dans ce manuscrit, la version surface libre de cette méthode a
été utilisée. Cette version consiste formellement à supposer que le fluide extérieur est de
densité nul, ce qui donne une condition de cisaillement nul à l’interface (surface libre).
L’impact d’une goutte sur un film liquide est présenté figure (1.7). Les paramètres du
liquide sont les même que pour le cas VOF, mais le calcul est fait dans l’approximation
surface libre pour le gaz.
26
Splash !
4
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’toto.mov.0001’ u ($2)/51.2:($1)/51.2
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Fig. 1.7 – Même impact que dans le cas précédent (VOF) mais simulé par la méthode
marqueur avec surface libre. La forme de l’interface est représentés aux temps t = 0.05,
t = 0.25, t = 0.5 et t = 0.75 ms, soit t/τ = U0 t/R0 = 0.15, 0.75, 1.5 et 2.25, de gauche à
droite et de haut en bas.
La méthode marqueurs avec surface libre est liée aux travaux de collapse d’une bulle
sous l’action capillaire[35] et l’impact d’une goutte sur surface super-hydrophobe[100].
Dans ce cas pour lequel l’angle de contact du liquide sur la surface est très proche de 180◦ ,
on obtient un très bon accord qualitatif entre les expériences et les simulations numériques
(voir figure (1.8).
1.3.3
Méthode intégrales de frontières
Dans le cas d’un écoulement avec interface pour un fluide parfait (de viscosité nulle)
incompressible, on peut utiliser une propriété importante liée au potentiel de vitesse dont
le Laplacien est alors nul dans le volume liquide : la connaissance du potentiel vitesse sur
l’interface suffit à déterminer le champ de vitesse en tous point du volume liquide. Alors, on
peut remarquer que la dynamique de l’interface se fait simplement en calculant le champ
de vitesse à l’interface, de même que l’actualisation du potentiel vitesse se fait là aussi
sur l’interface seulement via l’équation de Bernoulli. Cette méthode où finalement tout le
calcul se ramène à des calculs (certes d’intégrales divergentes) sur l’interface seulement,
s’appelle la méthode par intégrales de frontières (Boundary Integral Methods ou BIM en
anglais). Son utilisation en simulation numérique en mécanique des fluides remonte au
papier fondateur de Baker, Meiron et Orszag en 1980[4]. Un avantage apparent de cette
méthode réside dans le fait que tous les calculs se font uniquement sur l’interface, indexée
par une série de marqueurs. Même si la détermination de la vitesse en chaque point de
l’interface nécessite le calcul d’une intégrale le long de l’interface, cette méthode est particulièrement rapide. Les travaux liés aux intégrales de frontière (toujours en collaboration
27
Fig. 1.8 – Comparaison entre expérience et simulation numérique pour l’impact d’une
goutte d’eau sur une surface hyper-hydrophobe, dont l’angle de contact est expérimentalement très proche de 180◦ , alors que le code marqueur prend précisément cet angle comme
conditions aux bords sur le solide. Notons, la forme elliptique initiale de la goutte due à
sa chute. On observe un très bon accord entre les profils de gouttes expérimentaux et ceux
obtenus par simulation numérique.
avec Laurent Duchemin) ont porté sur la fusion inviscide de deux gouttes[33] et le comportement asymptotique du pic formé par l’instabilité de Rayleigh-Taylor en surface libre[34].
Dans ces deux cas, une technique de raffinement de maillage adaptatif a été développé et
les détails de l’implémentation de la méthode BIM ainsi que du raffinement de maillage
sont décrits en détail dans ces deux publications. La figure (1.9) illustre la dynamique du
pic formé par l’instabilité de Rayleigh-Taylor dans ce cas.
1.3.4
Autres approches : ”Level Set”, champ de phase, méthode particulaire
Il reste essentiellement trois autres approches à décrire pour finir ce rapide panorama
rapide des différentes méthodes numériques permettant de prendre en compte l’interface
(i.e. sa cinématique et la dynamique du fluide environnant) : la méthode ”Level Set”, les
méthodes dites de champ de phase ou de second gradient et la méthode particulaire.
La méthode ”Level Set” consiste à définir une fonction distance F (r, t) de l’interface
telle que celle-ci soit donc définie par
F (r, t) = 0
28
Splash !
Fig. 1.9 – Développement de l’instabilité de Rayleigh-Taylor en fonction du temps. Le
liquide est en haut, la gravité orientée vers le bas. La condition initial est une sinusoı̈de
de longueur d’onde la largeur du domaine, d’amplitude très faible. La courbe de gauche
montre les différents profils de l’interface pour différents temps. La courbe de droite représente la vitesse verticale en fonction du temps pour plusieurs points répartis de manière
homgène sur l’interface. On discerne notamment la dynamique d’accélération des points
près du pic et la remontée à vitesse constante de la bulle.
à chaque instant. F est donc pris comme la distance du point considéré à l’interface. On
advecte l’interface via la cinématique de cette fonction F qui satisfait :
DF
=0
Dt
où l’opérateur D/Dt correspond à la dérivée particulaire. Une dynamique de relaxation
est ensuite introduite pour maintenir la norme du gradient de F égale à l’unité. Cette
méthode est très précise quant à la prise en compte des effets capillaires (on suit l’interface
plutôt que de la reconstruire) et elle permet aussi de traiter les changements topologiques
de l’interface (détachement, fusion) sans artifice particulier. En revanche, cette méthode
ne conserve pas la masse de manière satisfaisante et doit donc être couplée avec d’autres
techniques (par exemple particulaires) pour améliorer cette faiblesse. Pour trouver une
description plus détaillée de la méthode Level Set voir par exemple[92, 103, 91, 84].
La méthode dite de champ de phase ou de second gradient consiste à introduire un
paramètre d’ordre supplémentaire qui passe continûment d’une valeur fixe à une autre
au travers d’une interface. La dynamique de ce paramètre d’ordre est souvent déduite
d’une équation de conservation de type Cahn-Hilliard liée à un Hamiltonien du paramètre
d’ordre décrivant l’équilibre thermodynamique entre les phases. La tension de surface
apparaı̂t alors naturellement comme l’énergie liée à l’interface et les effets capillaires se
trouvent inclus dans la dynamique générale. Le système d’équations final est donc constitué d’un couplage entre ce paramètre d’ordre et le champ vitesse. Ce modèle prend ainsi
naturellement en compte les transitions de phase entre les fluides (évaporations, solidification par exemple). Cette modélisation, par sa prise en compte en compte intrinsèque de la
tension de surface, est très séduisante ; elle permet de plus une approche intéressante de la
29
dynamique de la ligne de contact dans le cas d’écoulements à trois phases[102]. Cependant,
cette description ”épaisse” de l’interface est particulièrement délicate à mettre en oeuvre
dans la limite (souhaitée pour les interfaces air-eau qui nous intéressent) où l’épaisseur de
l’interface doit être très faible. Pour un lecteur intéressé, voici quelques exemples d’utilisation de cette méthode pour les écoulements diphasiques[88, 58, 59]. La formation d’une
interface périodique entre deux fluides dans le cadre d’un de ces modèles de champs de
phase, l’équation de Cahn-Hilliard, a été l’objet d’une collaboration avec S. Villain-Guillot,
où nous avons obtenu une description analytique de la phase non-linéaire de la croissance
(voir la publication [109], ces travaux n’étant pas évoqués plus loin).
Finalement, on peut décrire la dynamique d’une interface (surface libre notamment)
par une méthode Lagrangienne de suivi de particule. L’interface est alors déterminée à partir de la position des différentes particules du liquide advectée par le champ de vitesse. Une
description détaillée de cette méthode particulaire appelée ”Smoothed Particles Hydrodynamics” ou SPH peut être trouvée dans les références suivantes par exemple[85, 86, 90].
1.3.5
Discussion
Les différentes méthodes évoquées ci-dessus présentent toutes des points forts indéniables mais aussi des handicaps importants dans la simulation d’écoulements diphasiques.
Le choix d’une méthode plutôt qu’une autre devrait ainsi être adapté au problème que
l’on cherche à résoudre. Malgré tout, on peut extraire quelques tendances et influences
récentes qui expliquent le succès grandissant de certaines de ces méthodes.
Ainsi, les méthodes de type VOF ont l’avantage d’être robustes et de conserver la masse
avec grande précision : en effet, la reconstruction d’interface n’exige aucune restriction
topologique sur les propriétés de l’interface. En revanche, la ”diffusion” intrinsèque de
l’interface sur les mailles voisines via le calcul des forces capillaires essentiellement, conduit
à la présence de courants parasites assez importants qui altèrent la précision de la méthode.
Pour les méthodes basées sur le suivi de marqueurs, les termes capillaires et la pression
sont pris en compte avec une précision accrue et bien contrôlée, mais alors les changements topologiques de l’interface nécessitent un traitement particulier. Remarquons que
bien qu’une telle prise en compte compliquerait fortement la méthode, elle permettrait
d’introduire les comportements auto-similaires en jeu lors de la coalescence ou la séparation fluides.
Le calcul par intégrale de frontière se rapproche d’une méthode marqueur par sa prise
en compte des termes capillaires et de la pression et par sa difficulté à modéliser les
changements topologiques. D’autre part, il concerne les écoulements inviscides et reste
donc principalement intéressant pour des études modèles. Finalement, notons que ces
deux dernières méthodes (marqueurs et BIM) sont assez complexes à mettre en oeuvre à
trois dimensions spatiales alors que cela se fait de manière plus systématique dans le cas
VOF, avec une implémentation en parallèle naturelle (voir par exemple [52]
La tendance actuelle des méthodes numériques pour interface se dirige, à mon avis,
vers deux directions prometteuses : d’une part, le raffinement de maillage adaptatif, qui
permet de traiter la dynamique proche de l’interface en résolution plus fine et si possible
alors, de réduire les erreurs numériques ; un couplage de méthode d’interface différentes
pour améliorer les méthodes de suivi ou de capture d’interfaces. Par exemple, un couplage
entre une méthode particulaire proche de l’interface avec une approche Level Set permet de
diminuer fortement la perte de masse de cette dernière méthode[46] ; de manière similaire,
30
Splash !
on peut envisager une méthode VOF pour la partie advection de l’interface et Level Set
ou marqueur pour la prise en compte des termes capillaires et de pression1 .
1
Communication privée avec F. Gibou
Chapitre 2
Corolles, éclaboussures et
étalements.
2.1
Introduction
L’impact d’une goutte sur un film liquide ou un substrat sec peut engendrer des résultats aussi divers que l’éjection de gouttelettes, la formation d’une corolle liquide, l’adhésion ou au contraire le rebond de la goutte sur substrat. Dans ce chapitre, nous décrivons quelques résultats obtenus lors de l’étude d’impacts de gouttes sur films liquides
minces ou sur substrats secs hydrophobes voire super-hydrophobes. La surface de ces
solides peut aussi être lisse ou présenter de fortes aspérités. L’angle de contact entre l’interface et le solide varie d’un solide à un autre et varie aussi en général en fonction de la
vitesse de l’interface. La prise en compte de la dynamique précise proche de cette ligne
triple reste encore incomplète, soit analytiquement (seuls des modèles empiriques décrivent
cette dynamique[23, 110], quelques approches en théorie de lubrification peuvent traiter
des problèmes stationnaires essentiellement[44]), soit numériquement (voir par exemple
l’implémentation dans la méthode VOF d’une loi empirique pour la dynamique de la ligne
triple[99]). Malgré des différences importantes entre ces impacts sur solides ou films liquides, il est intéressant d’observer quelques comportements similaires lors de ces impacts
effectués dans des conditions variées.
Ainsi, on a pu remarquer que l’essentiel de la dynamique de la goutte aux temps courts
était déterminée par l’inertie incidente de la goutte et la conservation de la masse. Ceci
apparaı̂t clairement dans le comportement du rayon d’étalement de la goutte en fonction
du temps. Ce rayon d’étalement R(t) est défini de manière naturelle à partir de la surface
d’étalement de la goutte sur le film ou le substrat. On entend par temps ”courts” ou
”longs”, des petits ou grands rapports entre le temps depuis le début de l’impact avec le
temps caractéristique de l’impact R0 /U0 . Des simulations numériques sur films minces et
solide super-hydrophobes (i.e. dont l’angle de contact est maintenu à 180◦ tout au long du
calcul) montrent que de rayon d’étalement se comporte comme le rayon de l’intersection
de la goutte incidente à vitesse constante U0 avec le film/substrat (voir figure (2.1)) :
R(t) ∼
$
2R0 U0 t
(2.1)
Ni la viscosité ni la tension de surface n’interviennent dans cette loi, même si la tension
de surface dans le cas d’impacts sur solides se révèle rapidement ralentir cette expansion,
32
Corolles, éclaboussures et étalements.
10
1.2*sqrt(2*x)
R(t)/R0
r/D
1
0.1
0.01
0.01
1
0.1
0.01
0.1
1
0.1
1
dimensionless time
10
100
Ut/D
Fig. 2.1 – Rayon d’expansion de la goutte en fonction du temps lors d’impacts sur a) films
liquides, b) surfaces super-hydrophobes. Pour des viscosité de liquide variées. Pour chaque
courbe, la droite représente la loi (2.1) avec un préfacteur multiplicatif 1.2.
jusqu’à provoquer la rétraction de la goutte. En revanche, dans le cas de l’impact sur films
liquides, cette loi se révèle être correcte bien au-delà de la validité de l’argument géométrique. Yarin et Weiss[115] ont donné un argument convaincant pour cette dynamique
aux temps longs à partir de équations inviscides des films minces. Le champ de vitesse
d’étalement obéit à une équation de Burgers, caractéristique de la formation de choc. En
interprétant la corolle formée éventuellement par l’impact comme la signature du choc
crée par la collision entre la goutte et le film initial, ils obtiennent une loi d’étalement
proportionnelle à la formule géométrique (2.1). Le raccord précis entre la loi géométrique
et la propagation cinématique de choc du film mince reste encore cependant à étudier.
De même, on a pu remarquer que le champ de pression dans la goutte aux temps courts
de l’impact ne dépendait pas de la viscosité ni de la tension de surface, sauf dans la région
proche de l’interface où un jet peut être éventuellement créé[70]. On peut observer ceci
sur la figure (2.2) qui montre le champ de pression dans la goutte au même instant pour
différentes viscosités. Ces champs de pression ont été obtenus par simulation numérique
par la méthode marqueur pour des gouttes de rayon R0 = 1 mm et de vitesse incidente
U0 = 2 m · s−1 . La tension de surface et la densité sont celles de l’eau. On peut retrouver
ce champ de pression du à l’inertie de l’impact par un argument de conservation de la
quantité de mouvement assez simple, qui s’inspire de la méthode de pression impulsive
(particulièrement bien décrites par Cooker et Peregrin[22]).
Cet argument part d’une hypothèse d’autosimilarité du champ de pression aux temps
courts qui consiste à estimer que la région de la goutte incidente où le champ de vitesse
est perturbé se comporte comme R(t). Les champs de pression obtenus à différents temps
après l’impact montrent clairement cette autosimilarité sur la figure (2.3).
Le volume de la goutte V (t) dont la vitesse verticale est passée de U0 à zero (si on considère qu’après l’impact l’essentiel de la vitesse est horizontal) se comporte donc comme :
2
V (t) ∼ πR(t)3
3
Et on obtient pour l’ordre de grandeur du champ de pression P (t) créé par l’impact
sur la surface d’intersection de rayon R(t) à la base de la goutte, en faisant le bilan de
quantité de mouvement verticale, la relation :
2.1. Introduction
33
πR(t)2 P (t) ∼ ρL U0
dV (t)
dt
Ce qui donne après simplifications :
P (t) ∼
ρL U02
%
2R0
U0 t
(2.2)
Cette loi d’échelle pour l’ordre de grandeur de la pression est en bon accord avec les
simulations numériques sur films liquides ou sur surface super-hydrophobes (voir la figure
(2.4)). On remarque notamment une chute importante de la pression lorsque t/τ ≥ 1, qui
correspond justement à la fin de la validité de l’argument de similarité proposé (puisque
toute l’inertie verticale de la goutte a été ”déviée” par le champ de pression).
On peut encore raffiner cette approche en considérant cette fois le même bilan de
quantité de mouvement entre les rayons r et r + dr, ce qui permet d’obtenir un champ de
réponse impulsive de la pression de la forme :
ρL U02 R0
P (t)
P (r, t) = $
= %
&
'2
R(t)2 − r2
r
2 1 − R(t)
(2.3)
dont le comportement quadratique en r pour r ( R(t) peut être observé sur la figure
(2.5).
A nouveau, on remarque que le champ de pression ne dépend que très faiblement de la
viscosité du liquide ou de la tension de surface. On peut alors s’interroger sur l’influence
de la viscosité et de la tension de surface sur la dynamique de l’impact. Il est pourtant
évident qu’elles jouent un rôle important notamment dans la formation et la structure des
nappes liquides éjectées lors de l’impact. Ces jets sont souvent à l’origine de la formation de
gouttelettes expulsées par l’impact, à des temps très courts (souvent appelé alors ”prompt
splash” avec des gouttelettes de tailles très reduites, de l’ordre du micromètre de diamètre),
et/ou plus tard lorsque la corolle éventuellement produite de déstabilise en gouttelettes
de rayons plus important (de l’ordre du dixième de millimètres). Lorsque l’impact est fait
à faibles vitesses (ou comme nous allons le voir, forte viscosité et/ou tension de surface
élevée) aucun jet n’est formé et la goutte s’étale, soit sur le film liquide, soit sur le substrat.
La transition splash/étalement ainsi définie se trouve de manière surprenante suivre une
loi expérimentale similaire pour les impacts sur films minces ou sur solides (voir figure
(2.6) et les articles expérimentaux[104, 115, 87]).
Ce critère de transition fait intervenir une combinaison entre l’inertie, la dissipation
visqueuse et la tension capillaire sous la forme :
We ·
√
Re = K
(2.4)
où K est un nombre sans dimension, appelé paramètre de splashes, qui dépend des
détails de la géométrie considérée, de manière plus ou moins connue. Ainsi pour l’impact
sur film liquide, ce nombre dépend en particulier du rapport d’aspect R0 /h entre le rayon
de la goutte et l’épaisseur du film. Pour les impacts sur surface solide, le nombre K est
dépendant en premier lieu de la rugosité de la surface solide, suivant une loi empirique
obtenue expérimentalement :
34
Dans les deux cas, très peu de choses sont connues sur l’influence du fluide extérieur
(l’air en général) sur le critère de splash. Il a été montré récemment que la pression extérieure lors de l’impact sur surface solide jouait un rôle important sur la transition entre
étalement et splashes[114] : on peut ainsi supprimer totalement le splash en diminuant la
pression de l’air suffisamment.
Avec Stéphane Zaleski, nous avons tâché d’éclairer le rôle de la viscosité dans cette
dynamique permettant d’expliquer la loi (2.4). En s’appuyant sur des simulations numériques avec des viscosités différentes, nous avons montré que l’épaisseur du jet produit
par l’impact était sélectionné par une longueur visqueuse. La loi (2.4) se retrouve alors
simplement à partir d’un argument de bilan de quantité de mouvement dans le jet. On
compare alors en fait la vitesse de ”recul” du jet due à la tension de surface (appelée vitesse
de Taylor-Culick[105, 25]) et la vitesse d’éjection de liquide obtenue par conservation de
la masse. Cette approche est décrite avec plus de détails dans le paragraphe 2.2 auquel est
joint la publication[70].
La formation d’un splash et/ou d’une corolle liquide peut-être provoquée lors de l’impact sur une surface solide par la présence d’irrégularités de la surface. Cet effet, qui se
manifeste en fait par la dépendance de la loi (2.4) suivant la rugosité de la surface peut
s’étudier de manière plus systématique en créant une irrégularité importante de la surface en un point. Avec Luis Lemoyne, Richard Troeger et Stéphane Zaleski, nous avons
étudié expérimentalement dans un premier temps, puis numériquement de manière simplifiée ensuite, la déviation du film obtenu lors de l’impact par la présence d’une marche de
faible épaisseur. Nous avons notamment cherché à comprendre l’angle de déviation de la
corolle en fonction des données du problème (voir le paragraphe 2.3.1 avec la publication
jointe[67]).
Le paragraphe 2.3.2 et la publication jointe[100] décrivent une collaboration notamment avec Yuriko Renardy dans laquelle on réalisait une comparaison qualitative détaillée
entre les expériences d’impacts sur surfaces super-hydrophobes et des simulations numériques par méthode marqueur imposant un angle de contact de 180◦ . Ce travail illustre à
mon avis à la fois le niveau de précision atteint par les méthodes numériques récemment
développées, qui sont capables de bien reproduire une dynamique déjà complexe de déformation d’interface et une utilisation prédictive du numérique qui permet d’avoir accès à
des quantités difficilement mesurables expérimentalement.
Finalement, le dernier paragraphe 2.3.3 de ce chapı̂tre correspond à un article de collaboration expérimentale avec Denis Bartolo et Daniel Bonn sur les différents régimes d’étalement et de rétraction d’une goutte lors de l’impact sur une surface hydrophobe (mais
non super-hydrophobe)[7]. On décrit notamment un régime de rétraction dominé par la
dynamique fortement hors d’équilibre de la ligne de contact mobile, via la relaxation de
l’angle de contact vers sa valeur d’équilibre. A contrario, cette étude illustre les limites
d’approches numériques ne prenant pas correctement en compte des effets localisés tels
que la dynamique de la ligne triple ; en effet, aucune étude numérique avec les méthodes
à notre disposition ne s’est révélée réellement pertinente dans le cadre de ce travail.
2.1. Introduction
35
Fig. 2.2 – Champ de pression obtenu à t = 10−4 (soit t/τ = 0.2) par la méthode marqueur
pour l’impact d’une goutte sur une surface super-hydrophobe à la vitesse de 2 m·s−1 . Les
paramètres physiques de la goutte sont proches de ceux de l’eau (ρL = 1000 kg·m−3 ,
σ = 0.05 kg·s−2 , pour des viscosité variant de celle de l’eau 0.001 à 0.005, 0.01 et 0.05
(kg·m−1 ·s−1 ) de gauche à droite et de haut en bas respectivement. On observe très peu de
variation du champ de pression pour des nombre de Reynolds variant ainsi de 40 à 2000.
36
Fig. 2.3 – Champ de pression pour l’impact précédent avec la viscosité 0.005 kg·m−1 ·s−1
à différents temps t/τ = 0.08, 0.12, 0.2 et 0.32, de gauche à droite et de haut en bas
respectivement. On observe ainsi que les variations importantes du champ de pression
sont essentiellement situées dans une zone de taille R(t) proche de la paroi. Ces différents
champs suggèrent une loi d’échelle autosimilaire pour le champ de pression.
2.1. Introduction
37
1e+01
10
1./sqrt(x)
1
1
P(t)/P0
Pressure
0.1
0.1
0.01
0.001
0.01
1e-04
1e-05
0.001
0.01
a)
0.1
t/!
0.1
1
dimensionless time
1
b)
Fig. 2.4 – Mesure de l’ordre de grandeur de la pression dans le cadre d’un calcul d’impact a) sur film mince (VOF) b) sur surface super-hydrophobe (marqueurs). La pression
est adimensionnée par la pression dynamique P0 = ρL U02 et le temps t/τ = U0 t/R0 . a)
représente la valeur de la pression maximale, obtenue à la base de la corolle formée pour
des différentes viscosités ; b) correspond à la valeur de la pression au centre de la zone
d’impact z = 0 et r = 0. Les courbes sont obtenues
pour différentes viscosités et tensions
$
de surface. Les droites indiquent la loi ∼ 1/ t/τ .
a)
b)
Fig. 2.5 – Profil radial du champ de pression en z = 0 en fonction à différents temps
dans le cas d’une simulation d’un impact de goutte sur surface super-hydrophobe par la
méthode Marker (W e = 80 et Re = 2000). a) Profil radial P (r, t)/P (0, t) en fonction de
r/R0 b) P (r, t)/P (0, t) − 1 en fonction de r/R(t) en log-log. La droite correspond au profil
parabolique valable aux faibles r suivant la formule (2.3).
38
a)
b)
Fig. 2.6 – Evaluation expérimentale de la transition entre splashes et étalement pour
des impacts de gouttes sur a) solide sec (figure empruntée à Mundo et al.[87], page 162)
et b) film liquide mince ((figure empruntée à Yarin et Weiss[115], page 149). Les lois
empiriques obtenues se ramènent à la formule (2.4) une fois l’écriture suivant les nombres
sans dimension appropriés effectuée.
2.2. Transition splash-étalement pour les impacts sur film liquide
2.2
2.2.1
39
Transition splash-étalement pour les impacts sur film
liquide
Théorie visqueuse du jet
Le rôle de la viscosité et de la tension de surface sur la formation d’un splash et donc
d’éclaboussures semble difficile à élucider à partir des variations de quantités telles que la
pression ou le rayon d’expansion de l’impact. Cependant, la formule (2.4) montre d’une
part que que ces paramètres sont primordiaux et qu’ils interagissent de manière complexe.
On peut aussi observer les changements des profils de densité pour des impacts qui diffèrent
par des variations de viscosité importantes (voir figure (2.7)).
Fig. 2.7 – Profils de densités obtenus avec une méthode VOF pour W e = 8000 et des
nombres de Reynolds Re = 1000, 100 et 40 de gauche à droite. Le rapport d’aspect entre
l’épaisseur du film (coloré en gris clair) et le diamètre de la goutte est 1/6. Les liquides
du film et de la goutte sont les mêmes et la différence de couleur est là pour illustrer la
provenance du liquide formant la corolle.
On peut alors observer que ces variations sont essentiellement concentrées dans la
corolle (si elle est formée) et notamment à la base de la corolle (ou plus généralement dans
la zone d’expansion de l’impact, i.e. proche de R(t)). Le calcul du champ de vorticité dans
l’écoulement (voir figure (2.8)) permet de voir qu’en effet, de la vorticité est précisément
créée dans cette zone, et d’autant plus forte que le nombre de Reynolds est élevé.
Pour expliquer la création de vorticité dans cette région, on peut invoquer la relation de
Kelvin qui indique qu’un écoulement le long d’une interface courbée produit de la vorticité.
Or la zone d’expansion de l’impact est précisément une région à forte courbure le long de
laquelle le liquide de la goutte glisse dans le liquide du film au repos. D’autre part, cette
zone est aussi le siège d’un pic de pression et donc d’un gradient de pression qui tend
à vouloir expulser le liquide (ce qui, en l’absence d’une surface libre ne créerait pas de
vorticité cependant). Ceci explique que la vorticité soit créée sous la forme d’un dipôle
correspondant à l’émission d’un jet de liquide (la figure (2.8) correspond effectivement à
des situations où un splash est observé). La vorticité est donc produite sur l’interface même
et est ensuite diffusée dans le liquide (mais aussi dans une moindre mesure dans le gaz où la
viscosité est en fait bien plus faible). On se trouve finalement en présence d’une dynamique
assez classique de développement d’une couche limite visqueuse à partir de la surface libre
courbée et en mouvement. Dès lors, on peut estimer que la zone de diffusion de la vorticité
√
doit évoluer à priori comme lν ∼ νt, ce qui n’est pas évident à observer sur la figure
40
Fig. 2.8 – Champs de vorticité pour W e = 8000 et des nombres de Reynolds Re = 1000
et 100 (de gauche à droite) obtenus par simulation numérique (même calculs que pour la
figure (2.7)).
présentant les champs de vorticité (2.8) à cause notamment du choix de la palette couleur.
Puisque, comme nous l’avons vu, l’écoulement hors de cette zone d’intersection semble très
peu dépendant de la viscosité, la quantité de vorticité créée n’en dépend que faiblement et
seule sa diffusion est influencée directement par la viscosité. On peut donc attendre que
le champ de vorticité ait une extension spatiale normale à l’interface proportionnelle à la
longueur visqueuse lν . Cela revient à avoir un pic de vorticité inversement proportionnel à
lν lorsque l’on fait varier la viscosité. On peut observer cette dépendance sur la figure (2.9)
où l’amplitude du dipôle est représentée en fonction du temps pour différents impacts où
seule la viscosité varie.
4000
1e+05
80000
3000
vorticity
vorticity
60000
2000
40000
1000
20000
0
0
0.5
1
time unit
1.5
0
0
1
time unit
Fig. 2.9 – Amplitude du dipôle de vorticité généré par l’impact en fonction du temps pour
viscosité variables telles que le nombre de Reynolds varie de 40 à 1000 (gauche). La courbe
de droite représente la même quantité divisée par la racine carrée du nombre de Reynolds
de chaque impact.
Lorsque l’on représente l’amplitude divisée par la racine carrée du Reynolds, on observe
2.2. Transition splash-étalement pour les impacts sur film liquide
41
que les courbes des différentes amplitudes se superposent raisonnablement, notamment aux
temps courts. Ceci suggère donc que l’épaisseur du jet selectionnée est bien déterminée par
la viscosité. On peut dès lors faire un bilan de masse au travers du jet d’épaisseur lν en
considérant que l’essentiel de la masse de liquide impactant le film est éjecté. On obtient
alors que cette vitesse d’éjection vj suit la loi :
%
Re
vj =
U0
2
qui est en bon accord qualitatif avec les mesures expérimentales de S.T. Thoroddsen[106].
On peut maintenant comparer cette vitesse d’éjection à la vitesse de retraction de TaylorCullick vret qu’on peut obtenir par un bilan de quantité de mouvement (stationnaire) à
travers le jet :
%
2σ
vret =
ρlν
La formation du jet est donc possible suivant ce modèle simplifié si la vitesse de rétraction
est inférieur à la vitesse d’éjection, ce qui conduit à la relation :
We ·
√
%
Re
U0 t
≥ 2.
R0
Finalement, les bilans mis en jeu dans ce calcul étant valables pour
comme critère pour la formation d’un splash :
√
W e Re ≥ K
U0 t
R0
≤ 1 on obtient
où K est une constante dépendant des autres paramètres du problème (qui peuvent
entrer en jeu notamment dans les préfacteurs des bilans de masse et de quantité de mouvement), en bon accord avec les lois expérimentales empiriques. Cette théorie qui introduit
une couche limite visqueuse lors de l’impact offre une explication cohérente de cette loi
expérimentale robuste mais appelle cependant quelques remarques supplémentaires :
-dans l’article de Yarin et Weiss[115], une longueur visqueuse avait été introduite dans leur
modèle et permettait de retrouver le critère de formation du splash. Cependant, dans le
cadre de ce modèle, cette longueur correspondait à l’épaisseur du film sur lequel l’impact
avait lieu et non à l’épaisseur du jet formé.
-une étude plus détailée de la couche limite et notamment le raccordement asymptotique
avec l’écoulement de fluide parfait dans la goutte a été développée par la suite dans [56].
-dans le modèle ci-dessus le point crucial est de considérer que la largeur du jet est déterminée par une longueur de couche limite visqueuse. Bien que les observations expérimentales
de [106] semble en accord avec une épaisseur dépendant du temps comme lν , la longueur
√
visqueuse du problème νR0 U0 aurait donné le même critère final. -le lien entre la formation de ce jet et la corolle ”classique” observée dans de nombreux impacts reste encore
mal compris. Il a été notamment proposé récemment de bien distinguer entre ce que l’on
appelle le ”prompt splash” qui serait l’éjection de liquide dans un jet fin, aux temps courts,
par le processus décrit plus haut et la formation plus tard d’une corolle (de laquelle se détachera souvent des gouttelettes secondaires) via la collision horizontale entre le film du à
l’étalement de la goutte avec le film liquide au repos[29]. Cette formation d’une corolle par
collision de jet a été présentée auparavant par Peregrine dans un article au titre évocateur
”The fascination of fluid mechanics”[93].
42
2.3
Impacts sur solide : splashes et obstacles, étalement et
rétraction
Nous avons évoqué ci-dessus deux mécanismes de formation d’un splash, qui ne sont
d’ailleurs pas indépendants et dont le lien reste en fait à éclairer : en bref, ce que l’on appelle
généralement les ”prompt splashes” induits par le champ de pression aux temps courts et
la déviation du film liquide créé par la goutte s’étalant au contact du film initial. On
peut observer la formation d’éclaboussures par d’autres mécanismes comme par exemple
celui, mal compris, mettant en jeu le rôle de l’air lors de l’impact sur surface sèche[114].
Le rôle de la rugosité et de la forme de la surface solide pour ces impacts sur surfaces
sèches est aussi crucial dans la formation de splashes. Il est intéressant de rappeler ici
que le critère de transition étalement-éclaboussure précédent (2.4) est aussi vrai pour des
impacts sur solides, le paramètre d’impact K étant alors fonction de la rugosité de la
surface notamment[104]. L’étude de l’influence d’une rugosité ”macroscopique” (fraction
de millimètre à comparer aux rugosités micrométriques usuelles) contrôlée peut donc se
révéler particulièrement instructive dans ce cadre (voir ci-dessous et la réf. [67]). L’impact
d’une goutte sur une surface sèche est donc sensible aux rugosités de la surface mais
aussi aux propriétés physico-chimiques de celle-ci, qui se manifestent notamment via une
dynamique complexe de la ligne de contact (ligne d’intersection entre les phases liquide,
solide et gazeuse). Loin de vouloir discuter en détail ici la dynamique de la ligne de contact
(on peut notamment voir l’article de revue de P.-G. De Gennes [28]), on peut remarquer
que celle-ci joue un rôle important lors de l’impact d’une goutte. Schématiquement, on
peut distinguer deux phases lors de l’impact : l’étalement (l’expansion) de la goutte sur la
surface et la rétraction partielle, jusqu’à atteindre un rayon d’équilibre obtenu par l’angle
de contact statique et le volume de liquide (cette rétraction peut même correspondre à
un étalement supplémentaire si le rayon maximal atteint lors de l’impact est inférieur au
rayon d’équilibre), ou la rétraction totale, i.e. allant jusqu’au détachement d’une partie ou
de la totalité du liquide. Le cas super-hydrophobe est étudié à l’aide simulation numérique
où l’on observe deux régimes d’impacts distincts lorsque le frottement de la surface est
négligé : soit le film d’étalement s’assèche (cela se fait alors au point d’impact) ce qui
devrait conduire au démouillage de la surface (transition que l’on ne peut pas modéliser
numériquement actuellement), soit la goutte rebondit après une déformation de l’interface
importante mais sans démouillage[100]. L’étude expérimentale de l’impact de gouttes sur
surface hydrophobe donne lieu à l’identification de quatre régimes, suivant quel balance
entre l’inertie, la capillarité et les effets visqueux domine l’impact et la rétraction[7].
2.3.1
Déviation d’un film liquide par un obstacle.
En collaboration avec Luis Lemoyne et Richard Troeger (Laboratoire de Mécanique
Physique, Saint-Cyr) et Stéphane Zaleski, nous avons proposé d’étudier l’impact d’une
goutte sur une surface lisse sur laquelle une marche d’épaisseur contrôlée est ajoutée.
L’épaisseur de la marche est grande comparée à la taille caractéristique de la rugosité de
la surface. On considère alors des paramètres d’impacts tels qu’en l’absence de l’obstacle
formé par la marche, aucune éclaboussure n’est observée. On observe alors expérimentalement et numériquement (la marche étant alors cylindrique) que cette marche dévie le film
mince d’étalement créé par l’impact de la goutte. Une corolle partielle se forme au delà de
cette marche conduisant à la formation d’éclaboussures et l’éjection de petites gouttelettes
2.3. Impacts sur solide : splashes et obstacles, étalement et rétraction
43
(voir figure (2.10).
Fig. 2.10 – Profils à différents instant d’une goutte impactant une surface lisse à laquelle
une marche a été ajoutée. On observe la formation d’une corolle à partir de la marche et
le détachement de gouttelettes de cette corolle.
L’angle formé par la corolle et l’horizontale obéit à une dynamique complexe comme
le montre la figure (2.11) : après une émission quasiment à angle droit, la corolle semble
atteindre ensuite un angle constant aux temps longs, qui dépend à la fois de l’épaisseur de
la marche et de la distance de la marche au point d’impact.
90
* 0.035mm
o 0.07mm
+ 0.2mm
80
Angle (°)
70
60
50
40
30
0
0.5
1
1.5
Time (t*U/D)
2
2.5
3
Fig. 2.11 – Angle de déviation de la corolle (mesuré aux temps longs) en fonction de la
distance r entre le point d’impact et la marche : (+) expériences, (*) simulation numériques
et la courbe pour le modèle.
On retrouve un comportement similaire en simulant numériquement cet impact en
géométrie cylindrique avec la méthode VOF. La marche est alors de facto cylindrique.
L’angle de déviation en fonction de la distance entre le point d’impact et la marche obtenu
dans les expérience et en simulation numérique est représenté sur la figure (2.12) les autres
paramètres de l’impact restant les mêmes. On observe que l’angle de déviation numérique
est inférieur à celui mesuré expérimentalement.
Un modèle simple basé sur la théorie classique d’impact d’un jet sur un obstacle retrouve la dépendance générale entre l’angle et la distance de l’impact : le modèle essaie
44
90
measurements
theory
numeric
sheet angle (degrees)
80
70
60
50
40
30
20
10
0
0.2
0.4
0.6
0.8
1
dimensionless obstacle distance dobs/D
1.2
Fig. 2.12 – Angle de la corolle en fonction du temps pour trois épaisseurs de marche
différentes. Les courbes correspondent à une interpolation des résultats expérimentaux
(représentés par les symboles).
d’estimer le champ de pression exercé par la marche sur un jet d’épaisseur h arrivant avec
une vitesse u sur une marche d’épaisseur e. Le bilan de quantité de mouvement horizontal
donnera alors l’angle de déviation. La difficulté réside dans l’estimation de la pression le
long de la marche, pression qui, si l’on peut négliger la viscosité, est de l’ordre de ρL u2 à
la base de la marche et proche de zero en haut de la marche (où on a une surface libre).
On obtient alors aux petits angles de déviation α une relation de proportionnalité entre e,
h et α :
e
= C $ α2
h
(2.5)
Une théorie potentielle existe qui donne cette loi avec la détermination de la constante
mais la viscosité semble jouer un rôle complexe puisque des simulations numériques
préliminaires donnent une relation en bon accord avec la loi (2.5) mais avec un préfacteur
C $ différent de celui prédit par la théorie potentielle (l’erreur est de l’ordre de 50%).
D’autre part, il reste à estimer la relation entre h(r) entre l’épaisseur du film liquide
impactant la marche à une distance r du point d’impact. Cette question de l’épaisseur des
jets ou films sélectionnée est récurrente dans les problèmes d’écoulement à surface libre
et notamment dans les impacts en général et reste encore à éclaircir. On peut remarquer
qu’un écoulement potentiel en l’absence de tension de surface ne donnerait pas d’autre
longueurs caractéristiques que celles liées à la géométrie de l’écoulement, ce qui conduit
souvent à des jets singuliers (par exemple dans le cas des calculs de jets en approximation de
pression impulsive[22] ou dans le cas général traité par Longuet-Higgins[77]). La sélection
de l’épaisseur du film fait donc intervenir de manière complexe la viscosité, l’inertie et
la tension de surface en général. Dans le cas de l’impact sur un film liquide nous avions
montré qu’une longueur visqueuse intervenait dans la formation du prompt splash ; ici
nous proposons de même de considérer que l’épaisseur h(r) du film à une distance r du
point d’impact est lié à une longueur visqueuse et on prend donc pour ce modèle :
C $ [12]
h(r) ∼
%
45
νr
U0
ce qui donne pour l’angle α la courbe représentée figure (2.12), le paramètre C $ étant
déterminé par un fit entre la théorie et les résultats expérimentaux. L’accord est raisonnable
étant donné les approximations faites dans ce modèle très simplifié et une étude détaillée
du profil du film d’étalement d’une goutte reste à faire dans ce contexte.
2.3.2
Impacts sur surfaces super-hydrophobes : étalement, rétraction et
rebond
L’impact d’une goutte sur surface sèche conduit ainsi à des résultats très différents
suivant l’état de la surface (rugosité) mais aussi suivant les propriétés de cette surface.
La compréhension de l’influence des propriétés de la surface sur la dynamique de l’impact est particulièrement importante dans les chambres à combustion (augmentera-t-on
l’atomisation du liquide, en recouvrement des surfaces (impression notamment) ou dans la
détermination de pare-brise plus efficace à évacuer la pluie (en Formule 1 essentiellement
actuellement) par exemple. Cela se manifeste notamment au travers d’un angle de contact
formé par l’interface liquide/gaz par rapport à l’interface liquide/solide à la frontière gazliquide-solide. La théorie de cette ligne de contact reste encore sujette à débat[28] et la
dissipation proche de cette ligne reste encore mal comprise. Cependant, on peut remarquer
que la dynamique de l’angle de contact peut en général se ramener à une dépendance entre
cet angle et la vitesse locale de la ligne de contact. On définit donc un angle de contact statique (ou d’équilibre) alors que l’angle d’avancée est souvent plus grand que l’angle de recul
et que l’on peut observer des comportements hystérétiques. Une modélisation numérique
de cette dynamique est difficile et reste dans une grande mesure à faire malgré des avancées récentes[99]. On distingue les propriétés des surfaces par hydrophile lorsque l’angle
d’équilibre est inférieur à 90◦ et hydrophobe s’il est supérieur. Depuis quelques années des
surfaces super-hydrophobes (dont l’angle de contact d’équilibre est quasiment de 180◦ ont
pu être réalisées[73] : ces surfaces combinent à la fois des propriétés fortement hydrophobes
à une texture à très faible échelle que l’on nomme de manière imagée ”fakir”. L’interface se
trouve en quelque sorte ”posée” sur la surface micro-strucutrée en pointes ! Comme nous
l’avons discuté dans l’introduction, la comparaison entre des simulations numériques avec
une méthode imposant un angle de contact de 180◦ et des expériences d’impact sur surface
super-hydrophobes sont très bonnes. La goutte se déforme fortement lors de l’impact et
la rétraction due au forces capillaires conduit au rebond total de la goutte. Cependant,
on observe dans les simulations numériques que lorsque la vitesse d’impact est suffisamment élevée, le film central de la goutte peut ”s’assécher” et on pourrait alors observer
le démouillage de la goutte à partir du point d’impact (voir figure (2.13). Notons que la
méthode numérique ne permet pas de traiter continûment cette transition entre film mince
et démouillage. Ce démouillage n’a pas été observé expérimentalement et est certainement
dû en fait à la condition de glissement libre prise sur la surface solide. Cette condition au
bord est justifiée dans une certaine mesure par la structure en fakir de la surface texturé
et le séchage est déterminé numériquement par un rapprochement des deux interfaces en
dessous d’une fraction de taille de maille. Cependant, on peut imaginer que la condition au
bord plus réaliste serait un glissement partiel de Navier, le liquide pouvant en effet glisser
facilement entre les pointes rugueuses de la surface où là une condition de non glissement
46
locale devrait s’appliquer, ce qui conduirait à une longueur de glissement effective. Notons
cependant que l’obtention analytique d’une condition aux limites ”macroscopique” pour
une surface texturée hydrophobe reste à faire.
Fig. 2.13 – Profils de la goutte à des temps successifs lors de l’impact sur une surface superhydrophobe obtenu par simulation numérique avec la méthode marqueur. On observe le
séchage du centre de la goutte dans les dernières images.
Cette transition de démouillage observée numériquement suit une loi remarquable :
We =
1590
+ 3.62
Re1.49
(2.6)
qui distingue deux régions dans l’espace des paramètres entre une zone où la goutte
rebondit sans qu’une transition de démouillage ne puisse avoir lieu et une région où un
démouillage apparaı̂t (figure 2.14) :
Cette loi obtenue numériquement correspond bien à la notion intuitive que le démouillage devrait apparaı̂tre pour des vitesses élevées, mais, à ce jour aucune théorie
convaincante ne permet d’expliquer cette loi de transition(2.6).
47
Fig. 2.14 – Diagramme de transition en nombres de Reynolds et de Weber entre zone
où une partie sèche peut croı̂tre au centre du film lors de l’impact et une région sans
démouillage possible et où le rebond de la goutte sera observé. Les cercles correspondent
au point de transition obtenu par simulation numérique et la courbe représente le fit (2.6).
2.3.3
Etalement et rétraction lors d’un impact sur surface hydrophobe
L’impact sur une surface hydrophobe sans être super-hydrophobe, i.e. dont l’angle
de contact d’équilibre est supérieur à 90◦ (dans le cas présenté ici, l’angle de contact se
situe aux alentours de 110◦ ), donne lieu également à une dynamique d’étalement puis
de rétraction. Cette rétraction peut conduire à nouveau à un rebond de la goutte, qu’on
appelle en général rebond partiel car seul une fraction de la goutte initiale se détache
de la surface. Nous nous sommes intéressés aux lois d’échelles que l’on peut observer
expérimentalement lors de la rétraction. On identifie notamment deux régimes distincts :
le premier provient d’une balance capillaro-inertiel où la rétraction de la goutte correspond
simplement au rappel du bourrelet liquide par la tension capillaire. L’autre régime est
dominé par un équilibre entre viscosité et tension capillaire qui sélectionne une vitesse de
rétraction proportionnelle à la vitesse capillaire via la dynamique de retour vers l’équilibre
de l’angle de contact. Si l’on tient compte maintenant de la dynamique d’étalement que l’on
peut caractériser par le rayon maximal d’étalement de la goutte, on obtient un diagramme
de phase pour l’impact de goutte sur surface sèche hydrophobe qui se décompose en quatre
régions. En effet, on trouve également que le rayon maximal d’étalement présente deux
régimes distincts : un régime où l’inertie et la capillarité dominent et pour lequel le rapport
48
entre le rayon maximal et le rayon initial dépend du nombre de Weber. Dans le cas où la
viscosité devient primordiale, ce rapport dépend alors du nombre de Reynolds. Nous avons
notamment retrouvé pour ces deux régimes les lois d’échelles des lois compatibles avec
celles proposées précédemment (voir notamment [17]) soit pour le cas capillaro-inertiel :
Rmax
∝ We1/4
R0
bien que ces données soient aussi compatibles avec une loi du type :
Rmax
− 1 ∝ We1/2
R0
qui correspond à un bilan global entre inertie et forces capillaires.
D’autre part, dans le cas visqueux, on trouve :
Rmax
∝ Re1/5
R0
qui peut se déduire d’une balance globale entre forces visqueuses et inertie.
Finalement, les quatre régions dans le plan We-Oh (nombres de Weber et d’Ohnesorge)
peuvent s’expliciter en fonction de chacun des deux comportements (balance inertiecapillaire ou inertie-viscosité) pour chaque régime (étalement ou rétraction). La figure
(2.15) montre dans un diagramme log-log dans le plan We-Oh ces quatre domaines ainsi
que les différentes expériences réalisées.
1000
We
IV-CI
IV-CV
100
IC-CV
IC-CI
10
0.001
0.01
0.1
1
Oh
Fig. 2.15 – Diagramme de phase dans le plan nombres de Weber et Ohnesorge séparant quatre régions distinctes correspondant à des régimes d’impacts différents : IV et IC
pour caractériser la dynamique d’étalement (balance inertielle-visqueuse (IV) ou inertiellecapillaire (IC)), CI et CV pour la rétraction équilibre capillaro-inertiel (CI) ou capillarovisqueux (CV).
On peut remarquer sur cette figure que le domaine d’étalement capillaro-inertielle (IC)
avec une rétraction capillaro-visqueuse (CV) n’est pas atteint par les conditions expéri-
49
mentales de l’étude. Il aurait fallu notamment pouvoir faire varier de manière importante
la valeur de la tension de surface. Dans ce genre de situation, les simulations numériques
peuvent se révéler très utiles car elles permettent justement de s’affranchir des contraintes
expérimentales. Dans le cas présent, il est intéressant de remarquer que les simulations
nous ont été finalement quasiment d’aucune aide. En effet, les codes à notre disposition
ne prenant pas en compte la dynamique de la ligne de contact, il n’est pas possible de
retrouver le régime capillaro-visqueux qui s’appuye justement sur l’évolution lente de cet
angle.
50
Chapitre 3
Quelques singularités d’interface
3.1
Singularités et auto-similarités
La présence de singularités dans un problème est en général la source de nombreux enseignements : cela permet tout d’abord d’obtenir des comportements extrêmes et violents
des variables en jeu, mais aussi de tester en profondeur les propriétés des équations qui
modélise le système. Les comportements singuliers sont en fait au coeur de notre compréhension de la physique et sont présents dans quasiment tous les domaines (trous noirs en
astrophysique, fracture-fissure en mécanique du solide, condensation de Bose-Einstein en
physique atomique, singularités de l’équation d’Euler et chocs en mécanique des fluides par
exemple). Les singularités apparaissent en général via des lois auto-similaires et au delà
des singularités elles mêmes, ce sont ces comportements auto-similaires que nous cherchons à caractériser dans les écoulements en présence d’interface. On peut citer comme
exemples classiques le détachement d’une goutte d’une colonne liquide[42] ou la rétraction capillaire[71]. Dans le chapitre précédent, nous avons montré comment la pression
lors des temps courts de l’impact avait un comportement auto-similaire et le modèle que
nous avons développé pour expliquer la transition éclaboussures-étalement est également
auto-similaire en l’épaisseur du jet formé. Dans ces deux cas, nous décrivons un comportement auto-similaire partant d’une singularité (formellement t = 0 est une singularité
pour le champ de pression et l’épaisseur du jet). On peut aussi obtenir un comportement
auto-similaire pour les temps longs dans une limite asymptotique qui traduit à nouveau
formellement une singularité en temps infini. Ce régime est observé dans le cadre de l’instabilité de Rayleigh-Taylor (RT) à deux dimensions en écoulement potentiel en l’absence
de tension de surface et pour un rapport de densité infini. On considère donc une interface
séparant un fluide parfait au dessus du vide sous un cham de gravité. Cette situation
est
$
instable et on observe la formation de bulles remontant à vitesse constante (vk = g/(3k)
où k est le nombre d’onde du mode linéaire de l’instabilité) et de pointes (spikes en anglais)
qui ”tombent” quasiment en chute libre (voir figure (3.1).
La vitesse et la forme des bulles montantes sont assez bien décrites par des études
modales où on décompose le profil de la bulle en modes de Fourier multiples de k (remarquons que lorsque le rapport de densité n’est plus infini, cette approche se révèle moins
précise). En revanche, la forme de la pointe et le champ de vitesse voisin sont mal décrits
par un développement en modes de Fourier, ce qui est assez naturel étant donné le caractère fortement non-linéaire de la pointe. Ce cas simplifié de l’instabilité RT possède une
solution asymptotique auto-similaire du potentiel vitesse et de la surface libre proche de la
52
Fig. 3.1 – Profils d’interface (gauche) et champ de vitesse (droite) pour un fluide soumis
à la gravité initialement occupant le demi-plan supérieur. L’interface est perturbé initialement avec un mode sinusoı̈dal de nombre d’onde 1. Le calcul se fait par une$
méthode
d’intégrales de frontière. La figure de droite représente la vitesse (normalisée par g/(3k))
en fonction du temps pour un ensemble de points répartis
régulièrement sur l’interface.
$
On observe que la bulle remonte à vitesse constant g/(3k) tandis que la pointe est
uniformément accélérée.
pointe. Le point crucial tient dans l’hypothèse que l’ordre dominant de la vitesse verticale
se déduit de l’équation de Bernoulli stationnaire, ce qui revient également à considérer
un écoulement parallèle (où la vitesse ne dépend que de y). On prend donc la vitesse
√
verticale v ∼ 2gy, où g est la gravité, ce qui pour une particule dont la trajectoire est
en chute libre (y ∼ 12 gt2 ) donne une vitesse cohérente avec une dynamique de particule
libre (v ∼ gt). Le développement du champ de vitesse proche de la pointe autour de cette
solution permet de déduire le comportement asymptotique de la surface libre proche de la
pointe qui s’écrit sous la forme :
!
"
t
y = α(x, t) = t
− θ(x · t)
2
On obtient ainsi que la courbure de la pointe suit une loi en :
$
κ = k 2 g gkt3 θ$$ (0).
L’ordre suivant du développement permet de caractériser la correction asymptotique
3.1. Singularités et auto-similarités
53
à l’accélération de la gravité subie par la particule :
d 2 ys
2
= g + 2 √ 5 $$
2
dt
gk gkt θ (0)
(3.1)
L’accélération de la particule converge donc vers la chute libre sous l’accélération de
la gravité avec une correction qui décroı̂t en 1/t5 . Cette augmentation de l’accélération
de la pointe, comparée à la chute libre, appelée ”overshoot” est due au champ de pression
du fluide et est connu dans le cas de l’instabilité de Rayleigh-Taylor. Il est important de
remarquer que cette correction n’a pas de paramètre ajustable, une fois la courbure de la
pointe déterminée par la loi auto-similaire précédente. Ce calcul a été proposé tout d’abord
par F. Williams et P. Clavin[19] avant d’être repris dans le cadre d’une comparaison avec un
calcul numérique par intégrale de frontière avec un raffinement de maillage adaptatif proche
de la pointe[34]. On obtient alors un très bon accord de cette théorie avec la simulation
numérique. Notamment, l’overshoot de l’accélération est retrouvé sans paramètre ajustable
et la forme autosimilaire de la surface libre proche de la pointe est bien vérifiée (voir figure
(3.2) et la publication jointe à ce document L. Duchemin, C. Josserand and P. Clavin,
”Asymptotic behavior of the Rayleigh-Taylor instability”, Phys. Rev. Lett. 94, 224501
(2005)[34]).
0
acceleration overshoot
10
10
!1
10
!2
10
!3
10
!4
10
!5
1
10
t!t0
Fig. 3.2 – (gauche) Overshoot de l’accélération en fonction du temps obtenus par simulation numérique avec la méthode d’intégrales de frontière. La droite en pointillée correspond
à la loi (3.1) où le temps de décalage t0 et la courbure auto-similaire θ$$ (0) ont été déterminés par la trajectoire de la pointe et sa courbure en fonction du temps. La figure de
droite représente les profils de l’interface à la pointe pour différents temps (partie droite)
et les même profils en fonction des variables autosimilaire (t · x, t · α(x, t)), ce qui illustre
bien la structure auto-similaire de la pointe.
On peut remarquer que ces résultats se transposent assez simplement au cas de l’instabilité de Richtmyer-Meshkov pour laquelle on retrouve une courbure en t3 et une
accélération asymptotique (pour cette instabilité, la vitesse du spike converge vers une
constante) en 1/t5 . Une théorie générale de ces déformations a été proposée récemment
par L. Duchemin[32]. D’autre part, l’instabilité de Rayleigh-Taylor est cruciale dans notre
compréhension de la fusion par confinement inertiel où l’accélération est couplée à l’ablation de l’interface[18]. On observe alors que la pointe autosimilaire disparaı̂t sous l’action
de l’ablation alors qu’une singularité de courbure à temps fini apparaı̂t[1].
54
On peut aussi observer des singularités à temps fini dans le cadre d’une dynamique
d’interface plus générale, notamment pour laquelle la tension de surface et la viscosité sont
prises en compte. Bien que la viscosité soit dissipative pour la dynamique, elle n’empêche
nullement l’existence de solution auto-similaire et de singularités à temps fini, comme le
montre le pincement d’une goutte[42]. Dans les exemples avec viscosité que nous allons
discuter ici cependant celle-ci n’interviendra pas directement dans la dynamique singulière
mais jouera plutôt le rôle de sélection paramétrique de la singularité. Le premier exemple
a pour motivation l’éclatement des bulles à la surface d’un volume liquide, que l’on peut
observer dans notre cuisine tous les jours. Cet éclatement joue un rôle important dans
les échanges air-océan via les bulles formées par les différents mouvements de l’interface
(vagues, déferlement, action du vent). Ces bulles concentrent en général à leur interface
de nombreuse molécules qui peuvent être rejetées dans l’atmosphère lors de l’éclatement.
L’éclatement intervient lorsque le drainage de la surface supérieure de la bulle est suffisamment important pour faire éclater ce film. Cet éclatement très rapide projette d’une
part de multiples gouttelettes dans l’atmosphère alors que la partie inférieure de la goutte
(qu’on peut considérer comme une partie de sphère) s’effondre sous l’action de la tension
de surface. En effet, la surpression à l’intérieur de la bulle qui la maintenait à l’équilibre
à la surface a disparue lors de l’éclatement. Cet effondrement se traduit par la formation
d’ondes capillaires qui se propagent sur l’interface et en général par la création d’un jet
liquide vertical plus ou moins intense (voir figure (3.3).
La formation de ce jet n’est pas a priori du à une singularité à temps fini bien que sa
dynamique puisse certes montrer des comportements autosimilaires, que nous n’avons pas
étudié ici mais qui pourraient se révéler finalement assez proche de ceux décrit ci-dessus
dans le cas de l’instabilité de Rayleigh-Taylor (plus précisément Richtmyer-Meshkov). En
revanche, la variation de la vitesse maximale de ce jet formé en fonction du seul paramètre
de contrôle pertinent pour ce problème, le nombre d’Ohnesorge :
Oh =
ρν 2
,
σR0
où R0 est le rayon initial de la bulle, montre la présence d’une singularité pour au
moins une valeur de ce nombre (voir figure (3.4).
Proche de la singularité on peut observer que le jet formé est bien plus fin que pour les
valeurs plus éloignées du Ohnesorge (voir figure (3.5)), pour lesquelles le jet est de taille
constante une fois les distances redimensionnées par la taille de la bulle,
De même, entre les deux lignes verticales sur la figure (3.4), la simulation numérique
montre l’emprisonnement d’une bulle (voir figure (3.6)), ce qui empêche la simulation
numérique de se poursuivre car la méthode marqueur ne permet pas de décrire le pincement
d’une interface.
Outre cette singularité de pincement que l’on trouve donc entre certaines valeurs du
nombre d’Ohnesorge, on est en présence de singularités paramétriques aux frontières entre
ces zones qui séparent les collapses avec emprisonnement d’une bulle par pincement de ceux
formant un jet sans pincement. En ces points singuliers la dynamique finale du collapse
peut être vue comme la fermeture auto-similaire d’une interface verticale. Cette dynamique
suit en fait une loi auto-similaire provenant d’un équilibre capillaro-inertiel[71] que l’on
peut retrouver en étudiant l’équation de Bernoulli sur la surface libre, en l’absence de
viscosité et de gravité :
55
Fig. 3.3 – Profils d’interface expérimentaux (haut pris dans la ref. [78]) et obtenus par
simulation numérique avec la méthode marqueur (bas) pour une bulle de 2.57 µl qui éclate
à la surface libre de l’eau.
∂Φ 1
σ
+ (∇Φ)2 = κ.
∂t
2
ρ
où Φ est le potentiel vitesse et κ la courbure de l’interface de la solution recherchée.
Une solution auto-similaire peut être trouvée sous la forme :
r
)
tβ
où les variables d’espace suivent une loi d’échelle en tβ . La comparaison des différents
termes de l’équation de Bernoulli permet d’obtenir β = 2/3 et α = 1/3 qui correspond à la
solution auto-similaire capillaro-inertielle, décrite précédemment par Keller et Miksis[71].
La figure (3.7) montre justement les profils de l’interface à des temps précédents la singularité pour une valeur du nombre de Ohnsesorge proche de la singularité paramétrique,
sans emprisonnement de bulle. Le redimensionnement des profils en |t|2/3 (où t = 0 à la
singularité) montre bien la dynamique auto-similaire au moment du collapse (voir aussi
la publication L. Duchemin, S. Popinet, C. Josserand and S. Zaleski, ”Jet formation in
bubbles bursting at a free surface”, Phys. Fluids 14, 3000-3008 (2002)).
Φ = tα F (
56
Fig. 3.4 – √
Vitesse maximale du jet en fonction du nombre d’Ohnesorge. La droite indique
la loi en 1/ Oh obtenue par analyse dimensionnelle en supposant que la viscosité peut être
négligée. On observe une déviation à cette loi pour les petites valeurs de Oh. En particulier
autour de Oh = 1000 la formation du jet s’accompagne de la formation d’une bulle, qui
indique la présence d’une singularité. Les droites verticales indiques des transitions entre
zones où une bulle peut être emprisonnée de zones sans formation de bulles.
Cette singularité a été observée dans plusieurs situations expérimentales, notamment
pour des ondes de Faraday de grande amplitude[120] et de manière surprenante pour des
impacts de gouttes à faible vitesse sur surface hydrophobe (voir la publication D. Bartolo,
C. Josserand and D. Bonn, ”Singular jets and bubbles in drop impact”, Phys. Rev. Lett.
96, 124501 (2006)[8]). Dans ce dernier cas, on obtient des jets très violents lors du rebond
de la goutte via le collapse des ondes capillaires créées par l’impact (voir figure (3.8)).
Lorsque l’on mesure la vitesse du jet en fonction de la vitesse d’impact, on trouve une
dépendance qualitativement similaire à celle observée pour le collapse de bulle (voir la
figure (3.9)).
On remarque notamment deux singularités paramétriques séparant trois zones bien
distinctes. Ces singularités sont de nature très différentes : à la frontière entre zone I et
II, le jet singulier est formé par le collapse inertiel d’une cavité cylindrique, comme cela
avait été remarqué par Burton, Waldrep et Taborek précédemment[15] et pour lequel les
lois d’échelles se distinguent des lois capillaro-inertielles précédentes. En effet, la dynamique d’une caité cylindrique de rayon R(t) (le fluide étant à l’extérieur) peut s’écrire (en
négligeant notamment les viscosité de chaque fluide et la gravité) :
57
$"!
%&'()*+%,(
%&-.'/))
!"!
!!"#
!"#
Fig. 3.5 – Profil du jet pour deux rayons de bulle initiale R0 distincts (1 µm et 2 mm),
redimensionné par R0 .
' R 1
P (r) − P0 &
σ
= R̈R + Ṙ2 ln + Ṙ2 −
.
ρ
r
2
ρR
P (r) est la pression dans le liquide à la distance r du centre, P0 étant la pression du gaz
à l’interface. On trouve alors une loi d’échelle de collapse purement inertiel R(t) ∼ |t|1/2
en annulant exactement le terme en facteur du logarithme et en supposant qu’on peut
négliger les autres termes. Cette loi est en bon accord avec les mesures expérimentales du
rayon de la cavité en fonction du temps[8]. De nombreuses corrections (notamment en log)
ont été apportées dernièrement aux lois du collapse inertiel cylindrique (voir notamment
[45]), en lien avec le caractère non universel de la dynaique (qui dépend ainsi faiblement
des conditions initiales par exemple)
La seconde singularité paramétrique (frontière entre zone II et III) est de la même
nature que la singularité de collapse d’une bulle et correspond donc à un collapse autosimilaire capillaro-inertiel. On retrouve notamment la dépendance du profil en |t−tc |2/3 où
tc est l’instant du collapse et l’emprisonnement de bulles d’une part de la frontière (dans
la zone II).
Finalement, le pincement d’une interface emprisonnant une bulle peut aussi être observé dans une situation dynamique très différente, la fusion de deux gouttes que l’on
prendra de rayon identique R pour simplifier. On se place dans la limite inviscide et on
s’intéressera au cas asymptotique où le rapport entre le rayon initial de contact et le rayon
58
Fig. 3.6 – Profil de l’interface pour des valeurs de Ohnesorge ne permettant pas d’obtenir
un jet numériquement à cause de l’emprisonnement d’une bulle lors du collapse.
des gouttes rb /R tend vers zéro. On utilisera alors la méthode par intégrale de frontières en
géométrie axisymmétrique (et en l’absence gravité) pour laquelle le raffinement de maillage
adaptatif proche de la zone de contact dans ce cas là se révèle crucial. La question qui
se pose alors est de savoir si, dans ces conditions, la rétraction capillaire due au contact
entre les deux gouttes se fait en pinçant l’interface ou non, et donc en emprisonnant des
bulles (dans le cas cylindrique de notre calcul, il s’agit de tores qui seraient instables et
formeraient ensuite des bulles). Remarquons encore que dans le cadre de ce calcul la densité du fluide externe (et donc éventuellement emprisonné dans le tore) est nulle. La figure
(3.10) montre ainsi que la rétraction capillaire en géométrie cylindrique pour un rapport
initial rb (0)/R = 10−5 (et donc une pression capillaire de l’ordre de 1010 fois la tension de
surface) bien un pincement en temps fini via la croissance d’une onde capillaire.
Ce résultat permet d’établir une théorie de rétraction par le pincement autosimilaire de
tores (qui se déstabiliseront ensuite en bulles) qui crée donc un chapelet de bulles emprisonnées autosimilaires (voir la figure (3.11) qui illustre cette théorie). Cette théorie repose
sur l’analyse d’échelle qui montre simplement que la rétraction dans la limite rb (0)/R → 0
est équivalente à la situation 2D et donc que seule la longueur initiale rb (0) est pertinente
pour chaque pincement. Si on suppose qu’après chaque pincement la dynamique de pincement reprend mais pour une valeur actualisée de rb (0), sans être influencée par la bulle
enfermée, on obtient le même pincement avec la nouvelle valeur de rb comme unique chan-
59
!"%
!"#
!"$
!
!!"$
!!"#
!!"$
!
!"$
!"#
Fig. 3.7 – Profil de l’interface à différents temps précédents la singuarité (gauche), redimensionné par le temps à l’interface suivant t2/3 (droite).
gement d’échelle. Cette théorie auto-similaire conduit à une dynamique de fusion discrète
dont l’approximation continue donne :
√
rb (t) ∝ t
pour plus de détails, voir la publication L. Duchemin, J. Eggers and C. Josserand, ”Inviscid
coalescence of drops”, J. Fluid Mech. 487, 167-178 (2003)[33]. Cette solution auto-similaire
est illustrée sur la figure (3.11) qui montre les profils successifs à chaque pincement pour des
valeurs de rb /R d’ordre 10−2 environ. Dans cette simulation numérique, le calcul est arrêté
à chaque pincement puis repris après avoir enlevé la bulle emprisonnée du domaine de
calcul. D’autre part, l’évolution temporelle de rb (t), qui se fait par une série de pincement
√
successifs, épouse bien une loi en t jusqu’à des valeurs de rb /R de l’ordre de 0.1 au delà
de laquelle plus aucun pincement n’a lieu (voir figure (3.12)). L’évolution du profil de la
goutte devient suffisamment rapide pour empêcher le pincement de l’onde capillaire (on
ne peut plus considérer que le pincement se fait à épaisseur d’interstice constant).
60
Fig. 3.8 – Profils de gouttes lors d’impacts sur une surface hydrophobe pris à l’aide d’une
caméra rapide pour un rayon initial de 1 mm des vitesses d’impacts de 0.45 (a) et (b), 0.56
(c) et 0.68 m · s−1 (d). On observe ainsi la formation d’une cavité cylindrique ainsi que son
collapse qui donne naissance à un jet plus ou moins violent. Remarquons l’emprisonnement
d’une bulle dans le cas (c).
50
I
II
(a)
III
1
/R
apex
impact
20
R
/V
30
apex
impact
40
0
V
0
V
(m/s)
impact
1,2
10
0
0,2
0,4
0,6
V
impact
0,8
1
1,2
(m/s)
Fig. 3.9 – Vitesse du jet (adimensionnée par la vitesse d’impact) en fonction de la vitesse
d’impact. On observe trois régions différentes séparées par les traits pointillés verticaux.
Des bulles sont emprisonnées dans la région II uniquement. En inset, on représente le
rayon du jet en fonction de la vitesse d’impact.
61
!(%
&-.!&&
grid spacing
"-.!&&
!($
)*+,
/-.!&&
!(#
!("
0+,.123144256+*1'+5,
!
!
"
#
$
%
&!
'
Fig. 3.10 – Profils d’interface (gauche) et épaisseur minimal de l’interstice en fonction du
temps (droite) lors de la rétraction capillaire pour une valeur initiale du rapport rb /R =
10−5 . On obtient un pincement de l’interstice en temps fini par l’onde capillaire créée par la
rétraction comme le montre la figure de droite lorsque le maillage de calcul est augmenté.
!!"!!#
!
! !!#
Fig. 3.11 – Profils d’interface pour rb (0)/R = 0.08 pour chaque pincement successif.
Le calcul est repris après chaque pincement en considérant que la présence de la bulle
emprisonnée n’a pas d’influence sur la rétraction.
62
Fig. 3.12 – Eolution temporelle de rb (t) calculé à chaque pincement et comparée à la
loi auto-similaire déduite du modèle. En inset, nous caractérisons la dynamique autosimiulaire en comparant la vitesse moyenne entre deux pincement avec l’inverse du rayon
rb , comme prédit par la théorie.
Chapitre 4
Conclusion-Perspectives
Les perspectives générales de mon travail s’inscrivent d’une part dans la continuité de
mes recherches en milieux diphasiques, plus particulièrement pour des écoulements où la
déformation de l’interface est cruciale, et d’autre part vers d’autres domaines d’activités
récents pour lesquels je vais décrire succinctement les enjeux qui m’intéressent.
4.1
4.1.1
Ecoulements granulaires
Un liquide granulaire ?
Les matériaux granulaires malgré leur apparente simplicité résistent encore dans une
grande mesure à notre capacité de compréhension et de modélisation. Pourtant, on en
rencontre partout et ils jouent un rôle dans la plupart des activités humaines (construction, agriculture, pharmacologie, énergie, transports, loisirs...). Il s’agit d’ailleurs de la
seconde forme de matériaux échangés au monde (derrière les liquides)[53, 36]. Une grande
difficulté qui apparaı̂t lorsque l’on veut modéliser ces systèmes provient de leur caractère
athermique : en effet, de par leur taille et leur poids, la température thermodynamique ne
joue aucun rôle ici et on ne peut donc appliquer les modèles classiques de physique statistique (théorie cinétique des gaz par exemple). De plus, il n’y a pas de distinction claire
pour les milieux granulaires entre une échelle microscopique (les grains) et une échelle de
description macroscopique ce qui empêche de pouvoir appliquer les hypothèses classiques
de la mécanique des milieux continus (pour une description ”fluide” d’un écoulement granulaire, à la fois l’absence de mouvement brownien et l’absence d’un découplage micro-macro
empêche de pouvoir développer simplement un modèle de mécanique des fluides granulaires). On pourrait se féliciter de l’absence de ces effets thermiques puisqu’on peut donc
considérer que la dynamique d’un milieu granulaire se ramène donc à des problèmes de
mécanique du solide avec des contacts multiples. En fait, au contraire, la complexité des
écoulements et des configurations granulaires est telle qu’il est illusoire de vouloir la décrire par un ensemble d’équations (cela est cependant fait numériquement dans des calculs
de type mécanique des contacts[98] ou dynamique moléculaire[24]). D’autre part, la complexité intrinsèque des lois de frictions et de contact rend même périlleux l’approche de ce
problème de mécanique à N corps[5] ! Cela peut se manifester par un comportement granulaire assez similaire aux systèmes vitreux : ainsi, on a pu montrer sur des expériences de
compaction granulaire la dépendance du milieu à son ”histoire”, i.e. dans ce cas précis aux
types de secousses qui lui avaient été appliquées précédemment (voir la publication non
64
jointe C. Josserand, A. Tkachenko, D. Mueth and H. Jaeger, ”Memory effects in granular
materials”, Phys. Rev. Lett. 85, 3632 (2000)).
Et pourtant, lorsque l’on fait vibrer fortement un ensemble de grains[81, 43], lorsqu’on
le fait s’écouler le long d’un plan incliné[97] ou qu’on le soumet à un cisaillement[83],
l’observateur a l’impression d’avoir face à lui un gaz ou un liquide ! Cette impression s’appuie notamment sur le mouvement désordonné des grains qui semble pouvoir décrire en
moyenne un gaz de molécule ou un liquide. Les fluctuations des mouvements des grains,
dues à un forçage plus ou moins externe (le cisaillement, les vibrations, etc...), ”mimeraient” alors l’équivalent d’un mouvement brownien[27] et on serait alors amené à décrire
une température granulaire. Il s’agit alors de construire une approche statistique de la
thermodynamique granulaire où l’on cherche à définir l’équivalent d’une entropie granulaire (on utilise en fait la compactivité introduite par S. Edwards)[41, 6]. Dans le cas dilué,
l’équivalent d’une théorie cinétique a aussi été déduite, pour des collisions inélastique entre
les grains et permet également de définir une température granulaire[60]. En écoulement
dense (en moyenne, les grains ont plus d’un contact solide avec d’autres grains) cisaillé
(obtenu par exemple sur plan incliné), les fluctuations de vitesse proviennent des impacts
multiples et de la géométrie complexe des contacts entre grains. Cela se traduit par une
rhéologie complexe qui reste encore dans une grande mesure à expliciter. Dernièrement, un
certain consensus a émergé des nombreuses réalisations expérimentales faisant apparaı̂tre
pour les écoulements bidimensionnels une viscosité effective fonction du taux de cisaillement adimensionnalisé µ(I) défini par analogie avec l’écoulement d’un liquide sur un plan
incliné[97]. Différentes situations expérimentales font alors apparaı̂tre une fonction µ(I)
bien définie et similaire d’une expérience à l’autre[83]. Cependant une faible dépendance
de cette fonction avec la pression de confinement existe et sa description repose encore essentiellement sur des arguments empiriques : la généralisation de cette viscosité à des écoulements 3D quelconques est encore mal comprise (voir cette généralisation dans [62]) ; de
plus, le lien entre la rhéologie ”moyenne” observée lors de ces écoulements et la mécanique
régissant les interactions entre grains reste à élucider. Dans la publication jointe ”Granular pressure and the thickness of a layer jamming on a rough incline” (C. Josserand, P.-Y.
Lagrée and D. Lhuillier, Europhys. Lett. 73, 363-369 (2006)[66]), nous décrivons la modélisation du tenseur des contraintes pour un milieu granulaire bidimensionnel dense cisaillé
en écoulement stationnaire, dans l’esprit d’une publication antérieure[65]. Nous décrivons
l’écoulement par la donnée du champ de vitesse longitudinal V (z) et la fraction volumique
Φ(z). Le tenseur des contraintes se décompose alors en une partie indépendante du taux de
cisaillement pour laquelle la pression est déduite d’un argument entropique et une partie
due au cisaillement pour laquelle nous utilisons la forme introduite par Bagnold[3], qui
peut s’obtenir également par analyse dimensionnelle. A partir d’hypothèses simples sur
la dépendance du coefficient de friction de Coulomb nous proposons une interprétation
rhéologique de l’épaisseur résiduelle de grains lors d’un écoulement sur plan incliné.
4.1.2
Méandres laminaires
La méandres de rivières, les dunes ou les structures érosives en milieu naturel sont parmi
les exemples les plus connus et fascinants de l’interaction complexe qui existe entre les
milieux granulaires (plus ou moins cohésifs) et un écoulement fluide (vent, rivière, marée,
etc...). Alors que ces écoulements sont principalement turbulents (le nombre de Reynolds
dans une rivière est de l’ordre du million), il est intéressant de remarquer que toutes ces
4.1. Ecoulements granulaires
65
structures peuvent être obtenues en laboratoire pour des écoulement a priori laminaire,
comme de récentes études expérimentales l’ont montré[80, 82]. Dans le cadre de la thèse
d’Olivier Devauchelle, nous nous sommes intéressés à ce régime d’écoulement encore peu
étudié analytiquement (à cause de l’absence d’expérience). L’écoulement liquide peut être
considéré comme quasi-classique et s’adaptant donc très rapidement aux changements
lents du lit granulaire. La dynamique de l’écoulement est prise en compte simplement
dans la limite où les équations de Saint-Venant sont pertinentes (ce qui est le cas ici).
L’érosion est modélisée via la présence d’un flux de sédiments. Une loi empirique déduite
de résultats expérimentaux[16] permet de relier le cisaillement du fluide sur le lit granulaire
au flux de sédiments. La dynamique temporelle n’intervient donc alors que via l’équation
de conservation de la masse de sédiments. La stabilité linéaire de ce système d’équations
couplant la dynamique de l’écoulement avec l’érosion permet de décrire plusieurs modes
d’instabilités : en effet, il manque encore à cette description une approche précise des
conditions aux bords. Par exemple, pour un film liquide sur un domaine infini (ou avec des
conditions périodiques), les modes les plus instables se développent en chevrons similaires
à ceux observés lors de l’écoulement d’un mince film liquide sur un lit granulaire[26].
On peut notamment observer ces chevrons sur la plage lorsque le retrait des vagues est
suffisamment lent pour obtenir ce régime, comme sur la figure (4.1).
Fig. 4.1 – Structures en chevrons observées sur une plage après le retrait d’une vague (ici
plage de Goleta en Californie).
La prise en compte de berges mobiles se révèle autrement plus complexe car il faut
notamment décrire l’érosion le long des berges où des avalanches ont en général lieu.
De plus, à la berge même, l’équation de Saint-Venant à une écriture singulière qui rend
difficile à la fois l’approche numérique et un calcul analytique de stabilité linéaire par
exemple. Toutefois, à partir d’arguments sur les propriétés quasi-statiques de l’écoulement
près de la berge, on peut déduire des conditions aux bords décalées maintenant à la ligne
séparant la zone d’avalanche de la zone d’érosion sans avalanches. La stabilité linéaire
d’un écoulement droit est donc réalisable et permet de décrire les modes les plus instables.
Notamment, le cas où on considère les berges fixes (écoulement dans un canal), qui simplifie
66
grandement le problème des conditions aux bords, se révèle très instructif : l’instabilité
linéaire se manifeste donc au travers de la déformation du fond de la rivière. On peut ainsi
montrer que suivant les valeurs du rapport d’aspect entre la largeur et la profondeur de la
rivière, les modes les plus instables comportaient un ou plusieurs noeuds, préfigurant les
instabilités classiques des rivières en méandre (un noeud) ou en tresses (plusieurs noeuds).
Pour plus de détails, voir la publication jointe O. Devauchelle, C. Josserand, P.-Y. Lagrée
and S. Zaleski, ”Morphodynamic modeling of erodible laminar channels”, Phys. Rev. E 76,
056318 (2007)[30].
4.1.3
Perspectives
La modélisation des écoulements granulaires, comme nous l’avons vu, reste encore parcellaire : ainsi, malgré des généralisations des approches 2D récentes[62, 75], les équations
générales décrivant la dynamique d’un milieu granulaire dense restent un sujet de débat
scientifique important. Une des difficulté qui apparaı̂t, dans l’esprit du modèle 2D que
nous avons développé[65, 66] vient du fait que la modélisation du tenseur des contraintes
tient compte d’une direction particulière du tenseur de cisaillement (facilement identifiable
à 2 dimensions). La généralisation 3D de cette approche, prenant en compte les directions
propres du tenseur de cisaillement reste à ma connaissance à être entreprise (projet en
cours avec Olivier Devauchelle). Dans le cas 2D, la déduction des équations de SaintVenant granulaire, obtenues non par analogie avec les liquides, mais par intégration du
tenseur des contraintes granulaire devrait aussi permettre une description intéressante des
ondes de Kapitza observées en écoulement granulaire[49]. Le développement d’un schéma
numérique d’écoulements granulaires en partant de ces équations serait un premier pas
important vers un code générique d’écoulements granulaires denses (3D instationnaire
notamment).
Nous avons montré pendant la thèse d’Olivier Devauchelle que la modélisation des
dynamiques d’érosion mettant en jeu un écoulement laminaire et le mouvement des grains
était accessible par un système d’équations simplifiées et permettait alors de retrouver les
instabilités pertinentes présentes en général dans ces écoulements. Les expériences réalisées
à l’Institut de Physique du Globe de Paris en écoulement laminaire ont mis en avant ces
analogies fortes avec les dynamiques érosives générales (notamment turbulentes)[82, 80].
Plusieurs problèmes demeurent, notamment lorsque les berges de l’écoulement sont mobiles, question-clé pour pouvoir obtenir un modèle ”géophysique” complet de la dynamique
géomorphologique. La déduction des conditions aux limites au niveau des berges mobiles,
obtenues lors de la thèse d’O. Devauchelle fait apparaı̂tre de manière cruciale le manque
dans notre modélisation de la prise en compte des effets transverses dus au tenseur des
contraintes (notamment la contribution transverse des forces visqueuses). La déduction
de ces termes dans le cadre des équations de Saint-Venant (i.e. intégrées sur la hauteur
d’eau) reste encore à faire car elle échappe à l’heure actuelle à une écriture simple. La
description des structures érosives formées après la croissance des modes de l’instabilité
linéaire et notamment leur profil final demandent la résolution du problème nonlinéaire
associé. La structure de chevrons, périodiques que l’on observe dans les simulations numériques et les expériences est une première étape importante pour la prise en compte des
effets nonlinéaires car elle doit pouvoir se résoudre sous la forme d’une somme de modes
de Fourier dont les amplitudes interagissent entre elles.
Pour ces deux pistes de travail sur les écoulements granulaires (écoulements cisaillés
4.2. Turbulence d’ondes.
67
denses et écoulements érosifs), il me semble qu’un but important à atteindre soit l’obtention d’un système d’équations pertinentes, c’est à dire qui soirnt suffisamment complexes
pour rendre compte des propriétés cruciales de ces écoulements (ondes granulaires, effets
transverses, etc....) mais qui dans le même temps restent accessible à des approches analytiques et à une modélisation numérique simple (mais 3D malgré tout).
4.2
Turbulence d’ondes.
Alors que la turbulence 3D reste encore dans une grande mesure mal comprise[50], on
peut décrire des systèmes turbulents plus simples pour lesquels une théorie peut être dérivée : il s’agit de problèmes d’ondes dont l’interaction nonlinéaire (qui permet de ”mélanger”
ces ondes) peut être considérée comme petite par rapport aux termes linéaires. On nomme
alors cette dynamique turbulence faible d’ondes et on en trouve des exemples fameux dans
le cas des ondes de surface sur un fluide (ondes de gravités dans l’océan[107, 57] ou ondes
capillaires en laboratoire[113, 54, 13]). La figure (4.2) montre justement la forme de la
surface liquide pour des systèmes d’ondes turbulents dans ces deux cas.
Fig. 4.2 – Ondes de gravité (gauche) observées à la surface d’un océan et ondes capillaires
(droite) obtenues par vibration.
L’interaction entre ondes de différentes fréquences et donc de nombres d’onde différents
provient du terme nonlinéaire. En décomposant la solution en modes de Fourier, on peut
alors écrire la dynamique de l’amplitude de chaque mode sous la forme d’un développement
perturbatif en fonction d’un petit paramètre provenant du faible terme nonlinéaire (d’où
le nom de turbulence faible) et dans lequel le mélange entre les modes s’opère. On obtient
ensuite une dynamique hiérarchique liant les différents cumulants de ces modes. On détermine une fermeture asymptotique en considérant la moyenne aux temps longs (comparés
aux fréquences des modes en jeu) de ces équations hiérarchiques (voir par exemple [89]
pour une discussion détaillée des hypothèses liées à cette fermeture). On obtient finalement
une équation cinétique pour le carré de l’amplitude de chaque mode (appelé dans certains
contextes densité de particules) où le terme de collision est déduit du terme nonlinéaire.
Cette équation cinétique admet en général plusieurs types de solutions stationnaires. Certaines correspondent simplement à l’équipartition entre les modes de quantités conservées
par la dynamique alors que d’autres décrivent un transfert (appelé cascade) de quantités
entre les modes : cascade directe des grandes vers les petites échelles ou cascade inverse
lorsque le flux va des petites vers les grandes échelles. On peut alors observer ces cascades
68
lorsque l’on injecte la quantité concernée à l’échelle adéquate, cette quantité se trouvant
alors dissipé (par des mécanismes ignorés en général dans un premier temps dans le modèle
et introduit ensuite de manière schématique) de l’autre côté de la zone inertielle où se situe la cascade. Ces solutions décrivent donc une turbulence particulière d’interaction faible
d’ondes et ont été étudiées initialement pour plusieurs systèmes d’ondes dispersives, à la
surface d’un fluide ou en turbulence plasma par exemple[116, 10, 118, 117, 9, 119]. Par analogie avec la turbulence, on appelle ces solutions spectres de Zakharov-Kolmogorov. Une
caractéristique remarquable de ces spectres est précisément qu’ils sont solutions exactes
de l’équation cinétique (obtenue elle-même en se limitant au premier ordre non-nulle du
développement hiérarchique) et non obtenues par analyse dimensionnelle uniquement.
Cette dynamique générale permet notamment d’expliquer et de décrire la formation
de structures cohérentes et de condensats dans des modèles d’optique non-linéaire, de
superfluide ou de condensat de Bose-Einstein s’appuyant sur l’équation de Schrödinger
Nonlinéaire (voir les publications non-jointes [63, 68, 64, 21, 69]).
On peut aussi décrire l’interaction des modes de vibrations sur une plaque élastique
dans ce contexte de turbulence d’ondes, comme cela a été fait dans la publication jointe
G. Düring, C. Josserand, and S. Rica, ”Weak Turbulence for a Vibrating Plate : Can
One Hear a Kolmogorov Spectrum ?”, Phys. Rev. Lett. 97, 025503 (2006)[37]. On se place
dans le cadre de l’équation de Föppl-Von Karman qui décrit la dynamique élastique d’une
plaque d’épaisseur h et pour laquelle la dissipation est négligée (on l’introduira de manière
schématique ultérieurement). La figure (4.3) montre l’évolution de la déformation d’une
plaque obtenue par simulation numérique de l’équation de Föppl-Von Karman.
"dens.gnup.0" matrix
8
6
4
2
0
-2
-4
-6
-8
"dens.gnup.60" matrix
8
6
4
2
0
-2
-4
-6
-8
250
250
200
0
150
50
100
100
150
200
50
250 0
200
0
150
50
100
100
150
200
50
250 0
Fig. 4.3 – Evolution de la déformation d’une plaque élastique obtenue par simulation
numérique : (gauche) condition initiale (distribution gaussienne des modes de Fourier) ;
(droite) déformation aux temps longs pour lequel un régime de cascade s’est instauré.
L’analyse de la dynamique dans le cadre de la théorie de turbulence d’ondes permet de
prédire un spectre de Kolmogorov-Zakharov de cascade d’énergie vers les petites échelles
qui peut-être observé numériquement en forçant le système par l’injection d’énergie à
grande échelle et la dissipation aux petites échelles (voir figure (4.4) et la publication
jointe pour plus de détails sur cette cascade directe dans les plaques élastiques).
Les problèmes de turbulence d’ondes et/ou de turbulence faible restent d’actualité pour
plusieurs raisons : la dérivation d’un système d’équations hiérarchiques et donc l’obtention
d’une théorie cinétique pour laquelle des solutions d’échelles peuvent être trouvées en font
une classe de problème d’un intérêt fondamental indéniable pour notre compréhension de la
4.3. Dynamiques d’interfaces
!"!""!
k
69
2
1
Out
In
1e-05
-4
0.0001
1e-10
!"!""!
k
2
k4
1e-05
1e-15
1/3
1e-06
1e-20
0.01
1e-25
0.1
1
0.1
10
1
log(k * /k)
10
k
Fig. 4.4 – Spectre moyen de déformation d’une plaque élastique soumise à des vibrations
à grande échelle (zone marquée in) et à une dissipation aux petites échelles (out) obtenu
par simulation numérique. Le spectre en 1/k 4 avec un préfacteur logarothmique prédit par
la théorie de turbulence faible est bien retrouvé par le calcul numérique.
turbulence. La déduction mathématique de ces équations soulève d’ailleurs encore quelques
questions. Depuis quelques années, ces approches de turbulence faible sont appliquées
plus systématiquement à d’autres domaines que les classiques ondes de gravité, ondes
de surface en mécanique des fluides. On peut citer notamment l’optique non-linéaire, la
physique des lasers, les condensats de Bose-Einstein ou les plaques élastiques. Ces deux
derniers exemples ont plus particulièrement été abordés dans ce document (voir la liste
de publications de l’auteur). Pour toutes ces situations, les expériences mettant en jeu
ces dynamiques restent finalement assez rares (voir dernièrement [47]). Les interactions
d’ondes sur des plaques élastiques semblent offrir alors un cadre expérimental pour la
caractérisation des spectres
4.3
Dynamiques d’interfaces
Les dynamiques d’interface se révèlent complexes comme nous l’avons vu sur des problèmes d’impacts de gouttes, de coalescence ou de collapse de bulles. Les déformations
de ces interfaces mettent notamment en jeu des structures intenses tels que des jets, des
nappes liquides aux comportements souvent auto-similaires, signatures de singularités présentes dans les équations. Malgré les nombreuses études existantes sur ces particularités,
l’intérêt pour la formation et la description de ces structures auto-similaires est croissant
et constitue une source de recherches futures importante. Au delà des aspects techniques
indéniables (provenant principalement de l’accroissement de la précision des instruments
de mesures, notamment les caméras rapides et de l’augmentation continue des capacités
de calcul) qui nous permettent d’avoir une description plus précise de ces structures, le
70
plus important est ce qu’elles nous apprennent sur la physique de ces écoulements dans
des situations extrêmes et sur les propriétés des équations en jeu. De même la stabilité de
ces écoulements ou plutôt le développement d’instabilités sur ces interfaces restent encore
un problème globalement mal compris : la dynamique non-linéaire bien sûr mais aussi
l’étirement dynamique de l’interface compliquent les approches linéaires classiques. Les
perspectives sur la dynamique des interfaces seront développées en gardant à l’esprit ces
deux motivations principales et en prenant pour exemple le plus souvent des situations
d’impacts.
4.3.1
Scénario global d’impact
Nous sommes en effet encore loin d’avoir une description globale de la dynamique globale de l’impact. Par description globale, il faut comprendre savoir caractériser simplement
la dynamique. Prenons l’exemple de l’impact d’une goutte sur une surface solide sèche.
Certes, nous savons que l’impact va se caractériser par une phase d’étalement puis la rétraction du film liquide. Prédire si l’impact engendrera des éclaboussures, soit par prompt
splash soit par le développement d’instabilités de la corolle liquide reste encore sujet à
débat, malgré quelques lois empiriques qui restent incomplètes. En déduire la répartition
de la taille des gouttelettes éjectées ainsi que leur vitesse tient encore de nos jours de la
gageure. Pourtant, nous sommes capables de décrire plus ou moins bien de nombreuses
briques de la dynamique de cette goutte : c’est le cas par exemples pour les temps courts
dominés par le champ de pression, le rôle de la viscosité dans la formation des nappes fines,
l’instabilité du bourrelet de la corolle, le détachement de la goutte par instabilité RayleighPlateau. Cependant, il n’existe pas à l’heure actuelle de théorie convaincante qui permette
de relier ces différentes briques entre elles, ce qui rend tout prédiction quantitative hasardeuse. Une telle théorie paraı̂t encore inaccessible mais aller vers une meilleure description
de l’enchaı̂nement entre ces différents régimes semble crucial pour notre compréhension
globale de l’impact. Notamment, cela plaide fortement pour un développement important
de la capacité de calcul numérique sur ces problèmes, développement qui doit aller au-delà
de la simple augmentation de la puissance des machines et se traduire par des méthodes
numériques plus performantes (résolution 3D, maillage adaptatif, calcul parallèle). Par
exemple, la figure (4.5) montre le résultat d’une simulation numérique 3D d’impact d’une
goutte sur un film liquide réalisée sur une machine parallèle par la méthode VOF avec un
maillage fixe en 2004.
Finalement, notre compréhension des différentes étapes de l’impact d’une goutte reste
encore lacunaire : les travaux sur le comportement de la pression aux temps courts (à comparer aux prédictions obtenues par une théorie de pression impulsive) devraient permettre
d’une part de mieux comprendre la balance des forces lors de la formation des films minces.
Mais on doit aussi chercher à déterminer aux temps plus longs les équations d’évolution
du film liquide formé par l’étalement de la goutte, notamment dans le but de décrire la dynamique de rétraction de la goutte dans les cas d’impacts sur surfaces hydrophobes. Pour
déterminer la taille des gouttes éjectées lors de l’impact il semble important de pouvoir
estimer l’importance des différentes instabilités pertinentes durant l’évolution de l’impact.
En effet, de nombreuses instabilités interviennent, Richtmyer-Meshkov, Rayleigh-Taylor,
Kelvin-Helmoltz, Rayleigh-Plateau... à des étapes différentes dans la dynamique. Même si
la taille des gouttes finales semblent principalement dépendantes des épaisseurs du bourrelet et de la corolle formés, il convient de pouvoir estimer quantitativement les autres
71
Fig. 4.5 – Simulation numérique d’impact d’une goutte sur un film liquide.
effets. Cela demande notamment de pouvoir suivre ces instabilités lorsque la goutte s’étale
et donc que la corolle s’élargit par exemple. Suivre la croissance des modes au fur et à
mesure de l’évolution de l’impact et de la variation des types d’instabilités mises en jeu
requiert de pouvoir tout d’abord connaı̂tre avec plus de précision les caractéristiques de
l’impact en l’absence de perturbations (l’état de base). Bien que la position de la corolle au
cours du temps soit bien comprise pour l’impact sur film liquide, ce n’est pas le cas pour
l’impact sur surface sèche, même parfaitement hydrophobe. De même, l’épaisseur de la
corolle et son profil, quantités très importantes pour déterminer sa dynamique d’étirement
et probablement aussi l’instabilité du bourrelet, restent mal connus au cours d’un impact.
4.3.2
Sélection d’épaisseurs caractéristiques
La formation de structures intenses et fines est une caractéristique majeure des écoulements interfaciaux sur laquelle bute la plupart des théories. Il peut s’agir de jets, de
nappes liquides ou de corolles par exemple, comme on l’a vu illustré dans la plupart des
problèmes discutés dans ce manuscrit. Une meilleure compréhension des échelles sélectionnées lors de la dynamique est donc cruciale pour décrire les mécanismes de régularisations
des singularités lorsqu’elles sont présentes ou le développement d’instabilités comme nous
l’avons montré dans le paragraphe précédent. Cette sélection peut provenir de la formation
d’une couche limite instationnaire ou de la formation d’une structure capillaire (bourrelet)
par exemple. Ces deux exemples me semblent particulièrement pertinents car ils illustrent
une difficulté récurrente de ces problèmes qui mettent en compétition l’inertie, les forces
visqueuses et les effets capillaires. Lorsque des singularités dynamiques sont présentes, les
longueurs sélectionnés in fine par les termes d’ordre supérieurs peuvent être particuliè-
72
rement petites, comme dans le cas de la singularité visqueuse par entraı̂nement[61, 38].
En général cependant, on se contente d’introduire ces longueurs caractéristiques par une
analyse dimensionnelle prenant en compte les mécanismes qui semblent pertinents pour
la sélection de cette longueur. Par exemple, c’est exactement ce qui a été proposé pour
déterminer l’épaisseur de la nappe liquide éjecté lors de l’impact d’une goutte sur un film
liquide[70]. Une fois le rôle de la viscosité identifié, la loi pour l’épaisseur est déterminé
en prenant celle d’une couche limite visqueuse pour laquelle un temps doit être ”choisi” :
on observe d’ailleurs que les différents choix possibles de ce temps (le temps t depuis le
début de l’impact ou le temps propre de l’impact D/U0 ) ne changent pas la conclusion
finale ! Cette approche, classique dans la littérature, est également celle utilisée par Yarin
et Weiss dans leur étude expérimentale de l’impact[115]. Améliorer notre connaissance
de la sélection de ces longueurs, en prenant le cas précis des impacts comme illustration,
reste un problème difficile. En effet, les simulations numériques sont d’une aide restreinte
dans ces cas-là car les structures à caractériser sont des échelles trop proches des tailles
des mailles. Là encore, le développement de méthodes à raffinement de maillage adaptatif
puissantes représente une importante source de perfectionnement de l’analyse numérique
de ces écoulements. Expérimentalement, il est aussi très délicat et cela a donc été très rare
jusqu’à présent de pouvoir mesurer avec précision l’épaisseur de ces structures. Dans le cas
de l’impact sur film liquide, cela a pu être réalisé par Thoroddsen[106] grâce à l’utilisation
d’une camera ultra-rapide, en bon accord qualitatif avec le modèle proposé (voir figure
(4.6)).
Il est donc particulièrement important de rechercher des situations pour lesquelles on
pourrait déterminer analytiquement ou mesurer avec précision les longueurs caractéristiques de ces structures fines. On peut s’interroger notamment sur la croissance et la formation de couches limites visqueuses lors des problèmes d’impact et/ou lorsque l’interface
est fortement déformée. Cette question est difficile à traiter analytiquement, notamment à
cause de la complexité géométrique de l’impact (la zone d’impact croı̂t avec le temps dans
le cas général d’une goutte sur un film ou une surface). Une manière de s’affranchir de cette
contrainte géométrique consisterait à réaliser un impact d’une surface plane. Ceci est en
fait réalisable en s’inspirant d’expériences récentes où l’impact était transmis à l’interface
par l’impact du récipient contenant l’interface[2]. La figure (4.7) montre justement une
expérience d’impact d’un ensemble goutte-tige sur le sol qui peut donc s’analyser comme
l’impact d’un hémisphère liquide sur un disque solide. On observe l’éjection d’un film liquide mince sous la forme d’une cloche liquide. Cet impact offre l’avantage de pouvoir
être traitée analytiquement dans le cadre de la théorie de pression impulsive[22] pour laquelle on peut chercher à déterminer le développement de la couche limite visqueuse en
considérant les termes correctifs dus à la viscosité. Cette configuration représente ainsi un
système idéal pour caractériser les paramètres pertinents de l’épaisseur de la nappe liquide
éjectée, notamment en faisant varier la viscosité du liquide. D’autre part, une approche
numérique de l’écoulement après l’impact peut être envisagé en prenant comme champ de
vitesse initial celui obtenu par la théorie impulsive, afin de déterminer le rôle de la tension
de surface dans l’évolution de la nappe liquide.
Finalement, on peut justement conclure sur les perspectives nombreuses illustrées par
ce type d’études qui combinent des approches expérimentales, théoriques et numériques
complémentaires.
73
Fig. 4.6 – Visualisation de la corolle formée lors de l’impact d’une goutte sur un film
liquide mince (empruntée à [106]).
74
Fig. 4.7 – Une demi-sphère liquide posée sur une tige est lachée d’une hauteur donnée sur
le sol. Cette situation est équivalente à l’impact de la demi-sphère sur la tige. Dans cette
expérience préliminaire, réalisée par Arnaud Antkowiak, on observe l’éjection d’une nappe
formant une cloche liquide.
Troisième partie
Publications jointes
77
– C. Josserand and S. Zaleski, ”Droplet splashing on a thin liquid film”, Physics of
Fluids 15, p 1650 (2003).
– C. Josserand, L. Lemoyne, R. Troeger and S. Zaleski, ”Droplet impact on a dry
surface : triggering the splash with a small obstacle”, Journal of Fluid Mechanics
524, p 47-56 (2005).
– Y. Renardy, S. Popinet, L. Duchemin, M. Renardy, S. Zaleski, C. Josserand, M.
Drumright-Clarke, D. Richard, C. Clanet and D. Quéré, ”Pyramidal and toroidal
water drops after impact on a solid surface”, Journal of Fluid Mechanics 484, pp
69-83 (2003).
– D. Bartolo, C. Josserand and D. Bonn, ”Retraction dynamics of aqueous drops upon
impact on nonwetting surfaces.” J. Fluid Mech. 545, 329-338 (2005).
– L. Duchemin, C. Josserand and P. Clavin, ”Asymptotic behavior of the RayleighTaylor instability”, Phys. Rev. Lett. 94, 224501 (2005).
– L. Duchemin, S. Popinet, C. Josserand and S. Zaleski, ”Jet formation in bubbles
bursting at a free surface”, Phys. Fluids 14, 3000-3008 (2002).
– D. Bartolo, C. Josserand and D. Bonn, ”Singular jets and bubbles in drop impact”,
Phys. Rev. Lett. 96, 124501 (2006).
– L. Duchemin, J. Eggers and C. Josserand, ”Inviscid coalescence of drops”, J. Fluid
Mech. 487, 167-178 (2003).
– C. Josserand, P.-Y. Lagrée and D. Lhuillier, ”Granular pressure and the thickness of
a layer jamming on a rough incline”, Europhys. Lett. 73, 363-369 (2006).
– O. Devauchelle, C. Josserand, P.-Y. Lagrée and S. Zaleski, ”Morphodynamic modeling of erodible laminar channels”, Phys. Rev. E 76, 056318 (2007).
– G. Düring, C. Josserand, and S. Rica, ”Weak Turbulence for a Vibrating Plate : Can
One Hear a Kolmogorov Spectrum ?”, Phys. Rev. Lett. 97, 025503 (2006).
78
PHYSICS OF FLUIDS
VOLUME 15, NUMBER 6
JUNE 2003
Droplet splashing on a thin liquid film
Christophe Josseranda) and Stéphane Zaleskib)
Laboratoire de Modélisation en Mécanique, CNRS and Université Pierre et Marie Curie (Paris VI),
8 rue du Capitaine Scott, 75015 Paris, France
!Received 17 May 2002; accepted 14 March 2003; published 5 May 2003"
We propose a theory predicting the transition between splashing and deposition for impacting drops.
This theory agrees with current experimental observations and is supported by numerical
simulations. It assumes that the width of the ejected liquid sheet during impact is precisely
controlled by a viscous length l # . Numerous predictions follow this theory and they compare well
with recent experiments reported by Thoroddsen $J. Fluid Mech. 451, 373 !2002"%. © 2003
American Institute of Physics. $DOI: 10.1063/1.1572815%
I. INTRODUCTION
A raindrop splashing on the ground, the impact of a fuel
droplet on the walls of a combustion chamber, pesticide
sprays, ink-jet printing, all involve the same complex dynamics. These impacts arise in many different contexts, and
have important industrial applications as well as relevance to
the natural sciences, such as soil erosion. Photographs of
splashing droplets, starting with Edgerton’s classic,1 have become media icons. Splashing can occur at widely different
scales, from the astronomical when a comet impacts a planet
to the microscopic in laboratory experiments. Since the pioneering work of Worthington,2 many experimental, theoretical, and numerical works have been performed. Nevertheless
the problem is far from being fully understood !for a review
see Refs. 3– 6".
The present study concentrates on the early times of
droplet impact on a thin preexisting liquid layer. The two
principal outcomes are splashing and deposition. Splashing
occurs for large Weber and Reynolds numbers and involves
many different dynamics. In many cases, a thin liquid sheet
jets almost immediately after the impact. It arises in a small
‘‘impact neck’’ region located at radius r K , at the intersection between the almost spherical drop and the upper boundary of the liquid layer. Starting at early times the sheet grows
into a corolla and propagates radially from the drop. The end
rim that grows at the edge of this corolla is unstable and
develops fingers of liquid.7 The fingers eventually break up
into small droplets by the Rayleigh–Plateau instability. This
whole process, shown on well-known photographs, has been
described in numerous experimental papers !for instance, see
Refs. 8 –11" and is generally called ‘‘corona’’ or ‘‘crown’’
splash. Such crown splash can still be produced until fairly
low Weber and Reynolds number although it may not have
enough time to break up and create secondary droplets. Also,
usually at very high impact velocity or for impact on rough
surfaces, the crown-type splash is not observed but a
‘‘prompt splash’’ is seen, in which secondary droplets are
emitted immediately after the impact without any observable
smooth sheet or corolla at the base. Below the velocities at
which all these kinds of splashes are present, the drop only
spreads gently on the surface without emitting jets or secondary droplets.
An empirical relation has been established experimentally for the cross-over between spreading and deposition
behaviors through the dimensionless ‘‘Sommerfeld parameter’’ K!We1/2Re1/4, where We is the Weber number and Re
the Reynolds number !as defined below".10,12,13 When K is
smaller than a threshold value K c then only deposition is
observed, while for K"K c a splash develops. A reasonable
estimate is K c &50 although the exact value of K c depends
on the roughness of the solid surface and on the thickness of
the liquid layer.
Direct numerical simulations for drops impacting on liquid layers are a relatively new topic. It has attracted the
attention of many researchers,14 –18 but only recently. Several
of the simulations are based on a potential flow model with
surface tension in axisymmetric geometry. They exhibit corolla formation and the r K &t 1/2 spreading law.14,16,17 In particular, simulations have shown that for strong impacts !large
Weber numbers" a jet was formed in the neck region defined
as the region of the interface where the drop and the film
meet.16,17 However, potential flow calculations cannot exhibit the effect of viscosity while the Sommerfeld law includes it to predict the splashing-deposition transition. The
theory that follows indeed shows why. Our theory investigates the splash dynamics with special emphasis on jet formation at short times.
In simulations, we included viscosity and surface tension
and solved the Navier–Stokes equations with sharp interfaces between a liquid and a gas phase. Momentum balance
was solved on a very fine square grid. Interface tracking and
conditions on the interface use volume of fluid !VOF" tracking as in Refs. 19 and 20.
The theory to follow starts by assuming potential flow,
which is then matched to a small viscous region at the impact
neck.
II. NUMERICAL METHOD
We consider a liquid drop of diameter D, density ' L ,
and dynamic viscosity ( L . It impacts at speed U onto a thin
a"
Electronic mail: [email protected]
b"
1070-6631/2003/15(6)/1650/8/$20.00
1650
© 2003 American Institute of Physics
Downloaded 31 Mar 2004 to 134.157.34.8. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 15, No. 6, June 2003
1651
FIG. 2. Geometry at early times. The unperturbed droplet and the unperturbed surface of the liquid layer intersect at distance r J from the origin. The
asymptotic analysis is performed by assuming the flow perturbed in region
II only.
FIG. 1. Representation of the simulation setup for the impact of a drop on a
liquid layer.
liquid layer of the same liquid of thickness h. The surrounding gas has density ' G and viscosity ( G , the tension of the
interface is ) . These initial conditions are shown in Fig. 1.
We solve the axisymmetric incompressible Navier–
Stokes equation with surface tension. Written in the one-fluid
formulation, the equations read
'
!
"
*u
#u"“u !$“ p#“"! 2 ( D" # ) + , s n,
*t
!1"
where u!(u, v ) is the fluid velocity, ' the fluid density, D
! 12 (“u#(“u) T ) is the rate of deformation tensor, and p is
the pressure. n denotes the normal to the interface and , s is
the bidimensional delta function restricted to the interface.
The viscosity and density are constant in each phase. Gravity
is neglected and the continuity equation is
“"u!0.
!2"
Later on we will consider only axisymmetric dynamics.
The discretization is performed on a marker and cell grid and
pressure is solved by the explicit projection method making
use of multigrid convergence. The interface is followed by
the volume of fluid/piecewise linear interface calculation
method of Ref. 21 and the capillary force is computed
through a variant of the continuous surface stress and continuous surface force methods19,20,22 adapted to axisymmetric
geometry. A full description of the method can be found in
Ref. 19 except for the adaptation to axisymmetric geometry
which is described in Ref. 23.
In this paper we investigate various grids from 128
%128 to 512%512. We have found that 256%256 grid points
are needed to obtain a fair description of the impact, although we sometimes perform control runs on the finest grid
(512%512).
The VOF calculation is started with a droplet traveling at
a prescribed velocity toward the fluid layer. At some time the
two fluid regions reconnect. This involves various very small
spatial scales. A thin gas layer forms between the two liquid
regions and must be expelled. Under certain conditions voids
or gas bubbles are actually entrapped during the impact be-
cause the thin layer cannot retract fast enough.14 Molecular
forces, acting on very small length,would also affect the motion of the interfaces in reality. These effects are below the
grid scale and are not simulated. For the moment these very
early times are beyond the scope of our study. However since
the numerical method is based on the precise resolution of
the momentum balance and the mass conservation equation,
we argue that the situation just after reconnection offers a
realistic starting point for the simulation run.
Here the Weber and Reynolds numbers for the liquid are
We!
' LU 2D
,
)
Re!
' L UD
.
(L
Additional dimensionless numbers are the ratio h/D and the
two numbers ' G / ' L and ( G / ( L describing the gas. The effects of the gas are not considered in any detail here, although they are present and add realism to the simulations.
The characteristic time of impact is - !D/U. In the present
study, we neglect compressibility and gravity. This implies
that the Mach number Ma!U/c s and Froude number Fr
!gD/U 2 are small. For instance, for the impact of a waterdrop of D!2 mm from a height of one meter, we obtain
Ma!3%10$3 and Fr!10$3 . At very short times and scales,
it is clear that gravity has an even smaller effect, but compressibility could be relevant. However even there, Lesser
and Field24 have shown that compressibility plays no significant role at the velocities considered in the following.
III. THEORY
We attempt to analyze the early instants of impact in the
large Reynolds and Weber number regime, when the descending sphere, if it were unperturbed by the impact, would
intersect the upper plane of the liquid layer at a point J at a
small distance r J (t)#(DUt) 1/2 from the origin !Fig. 2". The
z axis is the vertical while the x – y plane is the upper surface
of the initial layer.
Our theory is based on the principle that at these early
times, most of the liquid layer and the impacting drop are
unperturbed. Thus there is an outer region !called region I"
for $ r$ &r J in which the velocity is nearly constant in the
falling fluid !region Ia" so u#$Uey , and nearby zero in the
liquid layer !region Ib". Region I matches to an inner region
1652
C. Josserand and S. Zaleski
FIG. 3. Two types of solution, without jet !a" and with jet !b". If mass does
not flow in from, or out to, infinity the areas marked # and $ must be equal.
II where the flow is perturbed. We first investigate the potential flow problem of impact. The horizontal length scale for
that flow is r J (t), while the vertical distance between the
free surfaces is r 2J /D!Ut. It is likely that multiple solutions
of the free surface potential flow problem exist, as is the case
in other problems of this type.25 In particular we may distinguish between solutions with and without jet !Fig. 3". The
solution without jet is reminiscent of the solution for the
water entry of a wedge, which was found by conformal mapping techniques in Ref. 25. It consists there to find a selfsimilar solution defined by a geometrical length equivalent to
r J . Similar problems have also been considered for free surface flows.26,27
The base of the jet is on the horizontal length scale at
distance r K !Cr J from the origin, where C is a numerical
constant given by the full solution of the potential flow problem. This constant will be taken equal to 1 in what follows,
and so will other dimensionless constants, to avoid carrying
on too many unknown numerical constants. The jet thickness
e J must match the vertical length scale at the neck so that we
find e J !Ut.
The velocity v J of the jet may be obtained by a mass
conservation argument. We assume that no mass radiates to
or from infinity, an assumption that will be discussed in the
following. Then the mass that comes from the impacting
sphere per unit time is
q! ' U . r K2 ,
!3"
which has to be equal to the mass flux through the jet,
q#2 .' r K e J v J ,
!4"
v J # ! t/ - " $1/2U,
!5"
thus
where - !D/U.
When the jet starts forming it ends with a rim which
tends to recede through surface tension effects. The receding
or Taylor–Culick velocity28,29 is given by
v C#
! "
2)
'eJ
1/2
!6"
for a sheet of thickness e J , thus
v C # ! t/ - " $1/2U We$1/2.
!7"
If v C ' v J the length of the jet increases, but if on the other
hand, v C " v J the jet cannot form. Thus a necesssary condi-
tion for the formation of the jet is that the Weber number is
large. In that case, potential flow theory predicts that the jet
may form at any time after impact. It remains to be explained
why a potential flow solution with jet is selected over a solution without jet. In particular one needs to explain how the
jet can appear in the free surface potential flow. We just
notice that in the solution without jet the acceleration of the
free surface at point K in Fig. 3 points from the gas to the
liquid, which indicates that the interface is unstable with respect to the Rayleigh–Taylor instability in the frame of reference moving with the fluid, and for two-dimensional perturbations. These perturbations would naturally lead to the
formation of a jet.
When viscous effects are included, we observe that the
inviscid theory is not valid for short time. Indeed the Reynolds number of the jet can be estimated as U(DUt) 1/2/ #
asymptotically small for short time. We may though introduce a viscous length scale of the order of l # #( # t) 1/2. This
length scale is larger than the vertical length scale Ut of the
potential solution for t't # where t # ! - Re$1 . This defines a
region III of size l # at the base of the jet where the vorticity
is concentrated. Indeed vorticity is created in the highly
curved regions of the interface at the base of the jet, then
diffuses to distances of order l # . The solution in region III
merges to the jet-like flow outside it. The analysis of the
dynamics in this region presents a peculiar property since
both viscous length l # and geometrical intersection r J follow
the same regime in square-root of the time. Thus the viscous
effects appear as a self-similar correction embedded in the
geometrical dynamics. The jet has naturally a thickness of
the order of the size of region III, e J !l # . When this happens
the velocities of the jet and the receding tip are modified: the
above-presented argument now yields
v J #Re1/2 U,
!8"
while
v C #U ! t/ - " $1/4 We$1/2 Re1/4.
!9"
The condition v C ' v J for jet formation now implies
Re1/2 We
!"
t
-
1/2
"1.
!10"
This condition gives a time t J after which the jet can form,
t J # - K 4c Re$1 We$2 .
!11"
This time is always shorter than t # for We&1, thus jet formation at t J is consistent with the assumption of a viscous
regime. Whenever t J / - is not small, the above may be seen
as an approximation in the limit of large Re and We for the
time of jet formation. Jet formation occurs early if t J ' - that
is when
We1/2 Re1/4/K c .
!12"
We thus recover Sommerfeld’s law. The factor K c has been
introduced to account for all the numerical constants omitted
until now. Additionally, the exact determination of this factor
would need a precise description of the flow inside the jet.
Further predictions may be made if we assume a scaling
form for the solution. For distances larger than ! # the solu-
tion ought to be potential. For t"t # this involves all length
scales !"Ut, for t't # the potential flow is limited to !
"l # . These conditions define region II, inside which we
assume the scaling form for the velocity potential
0 !UD ! t/ - " n f ! X,Y " ,
!13"
where X,Y are the rescaled variables X!1$r/r J , Y
!z/r J , and n is an exponent to be determined. The potential
must satisfy boundary conditions on a free surface. The free
surface must be determined as part of the solution but asymptotes to the spherical surface of the drop at large distance
from region II.
In region II the pressure also scales as
p! ' U 2 ! t/ - " m g ! X,Y " ,
!14"
where g is another scaling function and m has to be determined. This may be done by a momentum conservation argument. The downward momentum of the falling sphere is
reduced by impact. This effect is felt in region II, so vertical
momentum ‘‘lost’’ during the impact is of the order
M #C ' r K3 U,
!15"
where C is a numerical constant of order 1. It must be
stressed that the entire hatched region in Fig. 2 is affected by
impact. The pressure balances this lost momentum by
dM
#2 .
dt
thus
! "
DU
C
t
%
rJ
!16"
pr dr,
0
1/2
U#4 . U
2
!"%
t
-
m
1
! 1$X " g ! X,0" dX.
!17"
0
Assuming the integral converges, it is a constant term and we
find m!$1/2. Our reasoning is similar to the one expressed
in Batchelor30 albeit in a different geometry. The maximum
reached by the pressure depends on the behavior of g as X
→0, X!0 being the location of the base of the jet. Expression !14" is valid until the smallest length scale in region II is
reached. For t"t # this length is Ut so $ X $ &Ut/r K
!(t/ - ) 1/2. Thus
p max& ' U 2 ! t/ - " $1/2g !! t/ - " 1/2,0" .
!18"
On the other hand, when t't # the minimum length is l # ,
$ X $ &Re$1/2, and
p max& ' U 2 ! t/ - " $1/2g ! Re$1/2,0" .
!19"
On this basis it is possible to show that the pressure field is
harmonic. The Bernoulli theorem for potential flow implies
1
p
* t 0 # ! “ 0 " 2 # !0,
2
'
!20"
where the “ operator is in the original !unscaled" variables.
We shall see that at short times the first and third terms
dominate the second. Balancing the third with the first term,
we find that n!1/2. The velocity may then be found by
differentiating Eq. !13" to yield
“ 0 !U“X f !UV! X,Y " ,
!21"
1653
where “X is the gradient with respect to the scaled variables
and V is a dimensionless function. We expect “X V to vanish
for large X,Y and thus V has a limit independent of time at
large x,y. This matches the constant velocity U of the falling
fluid in the outer region Ia. Thus our hypothesis is confirmed,
namely that the first and third terms of the Bernouilli equation, being of order t $1/2, dominate the second. Thus
p
* t 0 #$ .
'
!22"
Since 1 0 !0 we also must have
1p!0,
!23"
so the pressure field is harmonic as in the pressure-impulse
theory. We retrieve here a property of the problem of pressure impact studied in Refs. 30 and 31.
IV. NUMERICAL RESULTS AND DISCUSSION
For more insight into the splashing dynamics we now
turn to the numerical simulation results. A series of simulations has been performed, for a 2-mm-diam droplet of a
water-like liquid impacting a layer of the same liquid, 0.3
mm deep with a velocity of 10 m s$1 . The gas is taken twice
denser than air at atmospheric pressure ( ' L / ' G !500) and
the surface tension has been taken as a fraction of the airwater surface tension ( ) !0.025 kg s$2 ). The viscosity of
the gas is taken slightly higher than for air ( ( g !5%10$4
kg m$1 s$1 ). We only varied the viscosity of the liquid,
from 0.02 to 0.5 kg m$1 s$1 , with intermediate values 0.05,
0.1, and 0.2. The Weber number is thus 8000 for all cases
while the Reynolds number evolves from 1000 to 40. The
choice of these values was determined mainly by numerical
stability and convenience to allow a large range of liquid
Reynolds numbers. For instance higher density ratios would
lead to stronger numerical instability at the interface by spurious currents. Time t!0 is set at the instant of impact. In
addition, the liquid in the droplet and in the layer have been
marked with two different colors, so that their specific evolution can be observed for various Reynolds numbers in
Figs. 4– 6.
These three simulations show important qualitative differences. The two less viscous impacts !Figs. 4 and 5" are
splashing. The less viscous the fluid, the thinner the corolla
and the larger the angle between the liquid sheet and the
liquid layer. For the highest Reynolds number the impact is
rapidly followed by secondary droplet break-up. Because of
axisymmetry the liquid patches seen in Fig. 4 are actually
toroidal. Further three-dimensional calculations would be
needed to account properly for the evolution of the crown
once non-axisymmetric perturbations grow.15,32 On the other
hand, for Re!40, the droplet spreads gently on the surface.
The impact creates a radially expanding surface wave. Jet
formation is visualized in Fig. 7!a", where the interface profiles are shown near the neck of the impact as the jet is
created, for Re!1000.
The spreading radius r K is detected automatically in the
simulations. For this purpose it is defined as the radius of the
point where the velocity of the fluid was maximal at a given
1654
FIG. 4. Density fields at t!0.01, 0.05, 0.1, 0.2, 0.7, and 1.5 unit time for
We!8000 and Re!1000.
instant. We have checked that this definition agrees with the
visually determined spreading radius. In particular the basis
of the corolla is well captured at large times and for high
Reynolds number with this method. At short time this point
almost coincides with the intersection between the drop and
the liquid layer. Indeed we first observe that r K and r J follow
the same scaling behavior:
r K #r J ! t " ! !DUt.
FIG. 5. Density fields at t!0.01, 0.05, 0.1, 0.2, 0.7, and 1.5 unit time for
We!8000 and Re!100.
panding liquid coming from the drop with the liquid of the
layer at rest. It has been shown10 that the dynamics is there
controlled by a Burgers-type equation which leads to the
same scaling !24". There is here a remarkable coincidence:
the physical principles that lead to the square-root behavior
at short times are completely different from those leading to
the scaling at long times. However in our range of simulations law !24" is valid with the same prefactor from short
!24"
Actually, a careful analysis of the data in Fig. 8 gives
r K !1.1r J (t). It accounts thus for the mass conservation correction to the geometric law r J . The above-presented theory
predicts a constant ratio r K /l # with time. This is observed in
Fig. 7!b" where the width of the jet is captured by two
straight lines as r K increases.
Figure 8 shows r K for We!8000 and Re!100, 200, 400,
and 1000. The geometric relation !24" is well verified for
each curve and no significant dependence on the viscosity is
found. We can remark that the geometric scaling law !24" is
valid even at large times while the argument was only valid
at short times. It is, however, a well-established result for the
impact radius.10 At larger times the dynamics of expansion is
indeed imposed by an effective collision between the ex-
FIG. 6. Density fields at t!0.5 and t!1.5 unit time for We!8000 and
Re!40.
FIG. 7. !a" Interface shapes near the neck of the impact, for We!8000 and
Re!1000 !same case as Fig. 4". The four profiles correspond to the times
t/ - !0.07, 0.11, 0.15, and 0.19. We observe the formation of a jet coming
out of the neck. The last profile shows the breakup of the jet. Numerically,
it happens when the structure size is of the order of the grid size. !b" The
evolution of the jet width is contained by two straight dashed lines as the
splash develops. The extremity of the jet is also followed by the straight
solid line !courtesy of Denis Gueyffier—Ref. 23".
1655
FIG. 9. Pressure field near the neck of the impact for t/ - !0.1, We!8000,
and !a" Re!100 and !b" Re!1000. The isoline curves are distributed
equally on each curve.
times to moderate times !t/- #3). No explanation has yet
been found for this fact.
A pressure peak is observed in the simulation at the impact neck !Fig. 9" as predicted in our theory. Figure 10 shows
the maximum pressure in the liquid as a function of time for
We!8000 and Re!100, 200, 400 and 1000. For these Reynolds numbers a jet is always created. The scale ' U 2 (t/ - ) 1/2
is also shown on the figure and gives the correct behavior of
the pressure peak at short times (t/ - '1). Indeed, for large
Reynolds and small ratio t/ - the scaling !19" suggests that
the pressure peak should follow this scale. However the dependence on Re1/2 that we have determined, Eq. !19", does
not appear here. The pressure drops rapidly as expected in
our theory for t& - . The observation of the pressure field
shows also that regions of equal pressure fan around the neck
!see Fig. 9". Moreover, Fig. 11 exhibits the pressure profile
near the neck at short times after the impact. The normalized
pressure is exactly shown as a function of the normalized
distance 2 (2! $ r$r K er$ /r K ) along the vertical from the
neck. The pressure drops spatialy from the impact center.
The profiles however do not superpose, indicating a poor
agreement with our similarity theory. However, the numerical accuracy of these curves is very weak and a special study
is needed for the short time pressure profiles.
Eventually, we analyze the influence of viscosity on the
vorticity field. The vorticity fields for Re!1000 and Re!100
are shown in Fig. 12 for t/ - !0.1. It corresponds to the third
snapshot of !4" and !5". The vorticity is concentrated near the
neck and forms two counter-rotating vortex rings which expel the liquid toward the jet. Figure 13!a" follows the absolute mean amplitude of the vortex dipole as a function of
time for the different Reynolds numbers explored already.
The amplitude of the vortex dipole is computed through the
difference between the highest !positive" and the lowest
!negative" values of the vorticity in the liquid bulk. The four
FIG. 8. Log–log plot of the spread factor r/D as a function of Ut/D for the
same Weber number !8000" and different Reynolds numbers !Re!100, 200,
400, and 1000". The straight line corresponds precisely to the power law
r J ! !DUt.
FIG. 10. Log–log plot of the dimensionless pressure peak P max /('U2) as a
function of t/ - for the same Weber number !8000" and different Reynolds
numbers !Re!20, 100, 200, 400, and 1000". The pressure scale (t/ - ) $1/2 is
represented by the straight line above the curves.
1656
FIG. 13. !a" Intensity of the vortex rings as a function of the time t/ - for
We!8000 and Re!100, 200, 400, and 1000 !curves from bottom to top";
!b" same curves but where the vorticity is rescaled by !Re so that the curves
are now on the same scale.
time at which the jet forms grows linearly with viscosity $Eq.
!11"%. In addition the scaling law v J & !ReU $Eq. !8"% corresponds exactly to the measured dependence of the initial
sheet velocity with viscosity reported in Ref. 8.
FIG. 11. Pressure profiles from the neck center for t/ - !0.01 !crosses",
t/ - !0.05 !squares", and t/ - !0.09 !diamonds" for the less viscous case
considered, Re!1000 and We!8000. The pressure has been normalized by
the pressure peak. The profiles are taken along the vertical direction from
the neck and are shown as a function of 2 the relative distance to the
pressure peak.
curves show a similar behavior but at different scale, and we
observe that the larger the viscosity, the smaller the amplitude. The vorticity is indeed diffused through viscous diffusion over a distance l # (t) from the interface. Thus the vorticity scale should evolve like 1/!Re when viscosity varies.
Indeed, in Fig. 13!b" we present the same quantity rescaled
by the factor !Re. We observe that all the curves converge to
the same one, particularly at short times. This result validates
the assumption of our theory that the length scale involved at
the jet basis is selected by the viscous length l # .
There are little quantitative experimental data to compare our theory with. The scaling of the experimental law
!12" is recovered with the critical number K c estimated to be
225 in our numerics. Experimentally, K c has been found to
vary between few tens and few hundreds, decreasing with the
surface roughness. In our numerics, the surface has no
roughness so that the observed K c is consistent with experiments. Besides the Sommerfeld law there are few other experimental results with the exception of the recent work of
Thoroddsen.8 As in our theory, experiments show that the
FIG. 12. Vorticity field near the neck of the impact for t/ - !0.1, We!8000,
and !a" Re!100 and !b" Re!1000. The color scale shows high vorticity
region in dark color.
V. CONCLUSION
Our theory, supported by numerical simulations, predicts
the scaling of the transition between splashing and deposition. It is in agreement with current experimental observations and in particular recovers the splashing-deposition criterion !12". Agreement is found with recent experimental
observations such as the initial sheet velocity.8 To summarize
our theory, it constructs a potential flow everywhere except
in a small neck region. Viscosity is shown to play a major
role in that region, selecting the width of the jet that develops
into the crown. Surface tension then comes in to allow or
prevent the formation of the jet.
Several points remain to be investigated in future experiments and simulations. A numerical investigation of the time
of jet formation has not been possible, because the jet and
the neck region that scales with it become too thin as the
formation time recedes to zero.
However, one should note a contradiction with results
from calculations done with BIM methods.16 There, a welldefined jet width is selected, while the asymptotic limit of
our self-similar theory would suggest that a jet should be
present at arbitrarily small times, with a vanishing thickness.
It is possible that the BIM numerical method produces a
short length scale cutoff which limits the minimum size of
the jet. This amounts to the problem of jet formation in a
potential flow as pointed to earlier. A numerical confirmation
of this hypothesis would clearly be interesting. In any case a
physical cut-off of microscopic size will always be present in
a real system. In our investigations the highest Reynolds
number was actually restricted by the numerical cut-off of
the finest jet width that could be resolved !here the thinnest
jet is 2 (m width". Further numerical developments such as
adaptive meshed refinement for instance would greatly improve this issue.
Another point of interest would be to investigate the
shape of the emerging jet as it evolves into the corolla. This
shape can be found by integration of the equations for a thin
sheet with surface tension. Without surface tension the problem is even simpler, as each sheet particle follows a ballistic
path. The shapes depend on the angle of ejection of the jet
!which our theory does not give". A recent paper has investigated the corolla dynamics following these rules.33 Moreover the location of the tip of the jet and the width of the
associated rim depend on the time of appearance of the jet.
Knowing the shape of the jet and the location of the end rim
would allow one to compute its stability with respect to perturbations, thus addressing the long-standing problem of
crown formation. These unresolved issues regarding the
shape of the interface and the angle of the jet could probably
be resolved through a detailed analysis of potential flow in
the neck region, which is the focus of ongoing investigations.
1
H. E. Edgerton and J. R. Killian, Flash !Branford, Boston, 1954".
A. M. Worthington, ‘‘On the form assumed by drops of liquids falling
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A. Prosperetti and H. Og̃uz, ‘‘The impact of drops on liquid surfaces and
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10
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11
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12
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13
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2
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14
H. N. Og̃uz and A. Prosperetti, ‘‘Bubble entrainment by the impact of
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15
D. Gueyffier and S. Zaleski, ‘‘Formation de digitations lors de l’impact
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Astron. 326, 839 !1998".
16
D. A. Weiss and A. L. Yarin, ‘‘Single drop impact onto liquid films: Neck
distortion, jetting, tiny bubble entrainment, and crown formation,’’ J. Fluid
Mech. 385, 229 !1999".
17
M. R. Davidson, ‘‘Boundary integral prediction of the spreading of an
inviscid drop impacting on a solid surface,’’ Crit. Rev. Solid State Mater.
Sci. 55, 1159 !2000".
18
M. Bussman, S. Chandra, and J. Mostaghimi, ‘‘Modeling the splash of a
droplet impacting a solid surace,’’ Phys. Fluids 12, 3121 !2000".
19
D. Gueyffier, A. Nadim, J. Li, R. Scardovelli, and S. Zaleski, ‘‘Volume of
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20
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23
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!2002".
90
* 0.035mm
o 0.07mm
+ 0.2mm
80
90
measurements
theory
numeric
80
sheet angle (degrees)
Angle (°)
70
60
50
70
60
50
40
30
20
40
10
30
0
0.5
1
1.5
Time (t*U/D)
2
2.5
3
0
0.2
0.4
0.6
0.8
1
dimensionless obstacle distance dobs/D
1.2
1.4
1.2
0.035mm
0.07mm
0.2mm
Length (l/D)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Time (t*U/D)
2
2.5
3
(a)
(b)
(c)
3
3
3
2
1
0
z
4
z
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z
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r
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0
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1
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r
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1
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r
3
x
Vi,j+1/2
Δxj
Ui-1/2,j
r
Δr i
(a)
P i,j
C i,j
Vi,j-1/2
(b)
Ui+1/2,j
tB
B
ρu
Ω
tA
A
x
no slip & non-wetting on walls
Lx
V
g
0
0
Lr
r
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
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−0.2 0 0.2
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0.5
0.5
0.5
0.5
0
0
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0
0
0
−0.2 0 0.2
−0.2 0 0.2
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−0.2 0 0.2
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−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
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Kinetic Energy
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10
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2
0
0.002
0.004
0.006
t units
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0.01
0.012
0.3
0.2
0.1
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0
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0.2
0.3
0.25
0.2
0.1
0
0.001
0.1
0.2
0.299
0.1
0.2
0.3
0.1
0.2
0.3
0.25
0.2
0.1
0
0.001
0.2
0.1
0
0.001
0.5
0.5
0.5
0.5
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0
0
0
0
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−0.2 0 0.2
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−0.2 0 0.2
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0
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0
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−0.2 0 0.2
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−0.2 0 0.2
0.5
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0.5
0.5
0.5
0.5
0
0
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0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
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0.5
0.5
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0
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0
−0.2 0 0.2
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0.5
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0.5
0
−0.2 0 0.2
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Kinetic Energy
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0
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0.002
0.004
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0.01
t units
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0.014
0.016
0.018
0.5
0.5
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0
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−0.2 0 0.2
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0
−0.2 0 0.2
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0
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0.3
0.3
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0
0.3 0
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0.3 0
0.2
0.2
0.1
0.1
0
0.3 0
0.1
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0
0.3 0
0.2
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512x768
256x512
Kinetic Energy
14
12
10
384x576
8
6
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t units
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9
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x 10
0.3
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0.2
0.3
0
0.3 0
0.2
0.2
0.1
0.1
0
0.3 0
0.1
0.2
0.3
0
0.3 0
0.2
0.2
0.1
0.1
0
0.3 0
0.1
0.2
0.3
0
0.3 0
0.2
0.2
0.1
0.1
0
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0.1
0.2
0.3
0
0
0.1
0.2
0.3
0
18
16
14
Kinetic Energy
12
10
8
6
4
2
0
0
0.005
0.01
0.015
0.02
t units
0.025
0.03
0.035
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.1
0.2
0.3
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0
0.1
0.2
0.3
0
0
0.1
0.2
0.3
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
−0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
−0.2 0 0.2
0.5
0
0
0
0
0
0
−0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
−0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2
0.5
0.5
0.5
0.5
0.5
0.5
0
0
0
0
−0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2 −0.2 0 0.2
0.5
0.5
0.5
0.5
0
−0.2 0 0.2
0
−0.2 0 0.2
0
−0.2 0 0.2
0
−0.2 0 0.2
0
−0.2 0 0.2
0
−0.2 0 0.2
9
1590/Re^1.49+3.62
Numerical simulation
8
Weber
7
6
Dry
5
4
Wet
3
0
200
400
600
Reynolds
800
1000
t=4.59694
t=5.00101
t=7.19185
t=6.00097
t=8.09174
t=6.79424
t=6.81488
t=7.21545
t=8.10929
t=4.60597
t=5.00396
t=6.01979
t=6.39092
t=2.80142
t=6.40985
t=2.80972
t=6.19045
t=0
t=6.20524
t=0
Under consideration for publication in J. Fluid Mech.
1
Retraction dynamics of aquous drops upon
impact on nonwetting surfaces.
By DENIS BARTOLO1 CHRISTOPHE JOSSERAND2 and
DANIEL BONN1,3
1
Laboratoire de Physique Statistique de l’ENS, 24 Rue Lhomond, 75231 Paris cedex 05, France
2
Laboratoire de Modélisation en Mécanique, CNRS-UMR 7606, Case 162, 4 place Jussieu,
75252 Paris Cédex 05-France
3
van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE
Amsterdam, The Netherlands
(Received 10 November 2006)
We study the impact and subsequent retraction dynamics of liquid droplets upon highspeed impact on hydrophobic surfaces. Performing extensive experiments, we show that
the drop retraction rate is a material constant and does not depend on √
the impact
velocity. We show that when increasing the Ohnesorge number, Oh = η/ ρRI γ, the
retraction, i.e. dewetting, dynamics crosses over from a capillaro-inertial regime to a
capillaro-viscous regime. We rationalize the experimental observations by a simple but
robust semi-quantitative model for the solid-liquid contact line dynamics inspired by the
standard theories for thin film dewetting.
1. Introduction: Drop Impact on Solid Surfaces
Drops impacting onto solid surfaces are important for a large number of applications:
for instance, almost all spray coating and deposition processes rely ultimately on the
interaction of a droplet with a surface. A large variety of phenomena can be present
during drop impacts, from splashes to spreading, and from large wave surface deformation
to rebound (see (Rein 1993) and references therein).
Research on drop impacts has a long history, starting with the pioneering studies
of Worthington and later on with the famous photographs of Edgerton(Worthington
1876; Edgerton & Killian 1954). Most of the previous work on drop impact focused on
determining the maximum diameter a drop is capable of covering upon impact (Fukai
et al. 1993; Roisman et al. 2002; Clanet et al. 2004). However, the practical problem
of deposition can be very different if one wants to efficiently deposit some material on
the surface. This is especially grave when the surface is not wetted by the liquid, as is
illustrated by the high-speed video pictures in Fig.1 for the impact of a water droplet.
It can be observed that the drop expands rapidly, due to the large speed with which it
arrives at the surface. However, due to the hydrophobicity of the surface, subsequently
the drop retracts violently, leading to the ejection of part of the droplet from the surface:
we observe droplet rebound. It is this ”rebound” that is the limiting factor for deposition
in many applications, for instance for the deposition of pesticide solutions on hydrophobic
plant leaves (Bergeron et al. 2000). We study here the impact and subsequent retraction
of aqueous drops onto a hydrophobic surface, and seek to understand the dynamics of
expansion and retraction of the droplets.
2
Denis Bartolo, Christophe Josserand and Daniel Bonn
(a)
(b)
Figure 1. Temporal evolution of the contact radius of droplets upon impact and retraction.
The radii are normalized by those of the spherical droplets before impact. The pictures show
the shape of the droplets at the different stages of retraction. Droplet radius is 1 mm, impact
speed is 2 m · s−1 : a) pure water, b) viscous water-glycerol mixture, viscosity 50 mPa · s.
In general, these problems are difficult because for most practical and laboratory situations, three forces play an important role: the capillarity and viscous forces, and the
inertia of the droplets. We try and disentangle the effects of the three forces here by
performing systematic experiments, varying both the importance of viscous and inertial
forces.
We provide experimental evidence for the existence of two distinct retraction regimes.
In both regimes, capillary forces are the motor behind the droplet retraction, and are,
for the first regime countered by inertial forces. In the second regime the main force
slowing down the retraction is viscous. We also show that, perhaps surprisingly, the drop
Drop Retraction upon impact
3
retraction rate (the retraction speed divided by the maximum radius) does not depend on
the impact velocity for strong enough impacts. The dimensionless number
that governs
√
the retraction rate is found to be the Ohnesorge number, Oh = η/ ρRI γ, with η the
viscosity, ρ the liquid density, RI the impacting drop radius, and γ the surface tension.
The Ohnesorge number therefore compares the dissipative (viscous) forces to the nondissipative (capillary and inertial) forces. The crossover between the two regimes is found
to happen at a critical Ohnesorge number on the order of 0.05 .
In order to develop a better understanding for the different regimes that are encountered, particularly the retraction dynamics in these regimes, we propose two simple hydrodynamic models inspired by the standard description of thin film dewetting dynamics.
These simple models provide a simple but quite robust picture that allows us to rationalize the retraction rate in both regimes.
In order to be able to say something about the speed of retraction, one also needs to
understand the maximum radius to which the droplet expands. Combining our results
with those obtained by (Clanet et al. 2004) for the maximum radius, we propose a phase
diagram delimiting four regions for the spreading and retraction dynamics of impacting
drops.
2. Drop retraction dynamics: Generic Features
As the impact dynamics of liquid droplets on a solid surface happens usually in a few
tens of milliseconds, we use a high-speed video system (1000 frames/second, Photonetics)
to analyze the drop-impact events. When necessary, we use an ultrahigh-speed system
allowing to go up to 120,000 frames/second (Phantom V7). We study aqueous drops
impacting on a solid surface; the surface we used is Parafilm, which provides us with a
hydrophobic surface (receding contact angle for water θR ≈ 80◦ ). In addition, the surface
has a low contact angle hysteresis with water, and allows us to obtain highly reproducible
results. The liquids we used are different water-glycerol mixtures. Varying the glycerol
concentration, we vary the liquid viscosity, keeping the liquid density and its surface
tension almost constant. For the highest concentration of glycerol, the surface tension
has decreased from 72 (pure water) to 59 mN m−1 , whereas the density has increased
to 1150kg/m3. The viscosity is varied between 1 and 205 mPas. Viscosity, density and
surface tension were measured before each impact experiment. Drops were produced
using precision needles, and the initial radius of the drops R I have been systematically
measured on the images (1.1 < R I < 1.4 mm). From the high-speed images such as the
ones shown in Fig.1, we follow the contact radius R in time. This section summarizes
the results of more than 80 different drop impact experiments, each of which have been
repeated at least two times.
Two series of experiments were performed: first, letting the droplets fall from a fixed
height, but increasing the viscosity, we increase the Ohnesorge number while keeping the
inertial forces constant. The second series of experiments is performed at fixed viscosity
and upon increasing the height from which the droplets falls; the droplet turns out to be
in free fall (as is verified in the experiment to within a few√percent) and so the relation
between fall height h and impact velocity is simply V I = gh, with g the gravitational
acceleration. Increasing the impact velocity increases the Weber number, keeping the
Ohnesorge number fixed, where the Weber number, We, compares the inertial forces to
the capillary forces, We ≡ ρRI V 2I /γ.
In all that follows, we restrain ourselves to high-speed impact conditions. More precisely, the Weber and Reynolds numbers are chosen so that We > 10 and Re > 10, where
Re ≡ ρRI V I /η is the Reynolds number. This implies that inertial forces are at least one
4
(a)
(b)
Figure 2. Temporal evolution of the contact radius for a water-glycerol drop Oh = 9.1 10−2 ,
R I = 1.2 mm (a) contact radius vs. time, (b) contact radius normalized by the maximum spreading radius vs. time Impact velocities : ×: VI = 2.4ms−1 , +: VI = 2.2ms−1 , ◦: VI = 1.9ms−1 ,!:
VI = 1.7ms−1 , ∆: VI = 1.4ms−1 , #: VI = 1ms−1
order of magnitude larger than both the capillary and the viscous forces. Such conditions
imply large deformations of the drop when the liquid impinges on the solid substrate.
On the other hand, we also restrain our experiments to impact speeds that are far from
the ’splashing’ regime in which the drop disintegrates after impact to form a collection
of much smaller droplets (Mundo et al. 1995).
The pictures in Fig.1 show that two distinctly different regimes exist for the shape of
the droplets after impact. For low fluid viscosity, we typically obtain the images shown
in Fig.1(a). At the onset of retraction, almost all of the fluid is contained in a donutshaped rim, with only a thin film of liquid in the center. On the other hand, for high
viscosities the deformation of the drop is less important, and the pancake-shaped droplet
of Fig.1(b) results. These visual observations allow to distinguish the capillary-inertial
and the capillary-viscous regimes that are described in detail below directly.
2.1. Drop Retraction Rate: influence of fall height and viscosity
Fig. 2 summarizes the most important findings of this study. The temporal evolution of
the drop contact radius R(t) for different impact velocities, shown in (a), is normalized in
(b) by its maximal value at the end of the spreading R max . Two important observations
are made. (i) A well defined retraction velocity Vret can be extracted from each experiment; this is a non-trivial observation that will be rationalized below. (ii) Independently
of the impact speed, all the R(t)/R max curves collapse onto a single curve for different
impact velocities. This shows that the retraction rate, rather than the retraction speed is
the natural quantity to consider, and that this rate is independent of the impact velocity.
These results hold for all the viscosities tested in our experiments.
In Fig. 3 we have plotted the retraction rate %̇ ≡ V ret /Rmax versus the impact Weber
5
%̇
Figure 3. Retraction Rate plotted versus Impact Weber number for various water glycerol
droplets. ×: Oh = 2.510−3 , +: Oh = 3.910−3 , ◦: Oh = 1.510−2 , ": Oh = 1.610−2 , !:
Oh = 2.310−2 , #: Oh = 7.110−2
number, where V ret is defined by V ret ≡ max [−Ṙ(t)]. Clearly, the drop retraction rate
does not depend on the impact velocity. One might think that the explanation for this
observation is rather obvious: the initial kinetic energy of the droplet is transformed
into surface energy (which fixes R max /RI ∝ We 1/2 ), and is then transformed back into
kinetic energy (which in turn fixes Vret ∝ VI ). This naive explanation is unfortunately
wrong fro the following reasons. First, it has been observed recently that, at the onset
of retraction, low viscosity liquids undergo vortical motion in the drop (Clanet et al.
2004). This residual flow in the drop reveals that a part of the initial kinetic energy is
still available then, and thus that a simple energy balance argument cannot work. This
was indeed already suggested by previous observations of a clear disagreement between
experiments and the Rmax /RI ∝ We 1/2 law(Fukai et al. 1993; Roisman et al. 2002;
Okumura et al. 2003). The second reason why the simple energy-balance argument does
not work follows directly from Fig. 3, where it is shown that the retraction rate depends
on the viscosity and consequently that the previous inviscid picture is not correct.
We therefore performed experiments that elucidate the role of the viscosity, or, equivalently, of the Ohnesorge number. For what follows, it is convenient to define two intrinsic
time scales for the droplet: a viscous one and an inertial one. The viscous time is the
relaxation time of a large-scale deformation of a viscous drop: τv ≡ (ηR I )/γ, whereas the
inertial time scale: τi = ( 34 πρR 3I /γ)1/2 corresponds to the capillary oscillation period of
a perturbed inviscid droplet. Since τi is independent of V I and η, this quantity is almost
constant for all tested drops.
Fig. 4 shows the retraction rate, made dimensionless using the inertial time, as a function of the Ohnesorge number. It can be observed in the figure that two different regimes
exist for the retraction rate. The first region where the retraction rate %̇ is independent of
the viscosity points to an inertial regime and %̇ ∝ τi−1 . The retraction rate is consequently
found not to depend on the impact speed, a result similar to that obtained recently by
(Richard et al. 2002) who show that the contact time is independent of the impact speed.
For higher viscosities, typically Oh > 0.05, the retraction rate decreases strongly. In this
regime, capillary and viscous forces govern the dynamics: we find %̇ ∝ τv−1 .
6
10
1
e
%̇τi
0.1
0.01
0.001
0.01
0.1
1
Oh
Figure 4. Circles: Normalized retraction rate !τ
˙ i plotted versus the Ohnesorge number, experimental values. Error bars represent the maximum deviation from the mean value. Full line:
(left) !τ
˙ i evaluated using Eq. 3.2, (right) !τ
˙ i evaluated using Eq. 3.6. Dashed line :(left) Fit
obtained taking the mean value of the five first experimental points,(right) Best fit according to
the predicted 1/Oh power law.
3. Two simple models for the drop retraction dynamics
We have consequently established the existence of two different regimes for the retraction rate: a viscous one and an inertial one. We now develop some simple arguments
allowing for a semi-quantitative description of the dynamics, using ideas already existing
for the dynamics of dewetting, a problem closely related to the current one.
3.1. Inertial regime
We employ a Taylor-Culick approach commonly used for the inertial dewetting of thin
films (Taylor 1959; Culick 1960; Buguin et al. 1999) to describe the drop retraction rate.
For high-velocity drop impacts, liquid spreads out into a thin film of thickness h and
radius R max . The liquid subsequently dewets rapidly the surface, and in doing so forms
a rim that collects the liquid that is initially stored in the film. The shape of the drop
surface shape is therefore never in a steady state and consists of a liquid film formed
during the spreading stage and a receding rim. The contact angle at the outer side of the
rim is taken to be very close to the receding contact angle (θR ) since viscous effects can
be neglected small (Buguin et al. 1999). The dynamics is consequently determined by a
competition between capillary tension coming from the thin film and the inertia of the
rim. If we write down momentum conservation for the liquid rim:
!
"
d
dR(t)
m
= FC
(3.1)
dt
dt
with m the mass of the liquid rim and FC the capillary force acting on it, FC ∼
2πγR(t) [1 − cos(θR )]. The stationary solution of Eq.3.1 can be obtained writing ṁ(t) =
2πρRV ret h, and gives: V ret
4 3 −2
3 R I R max , it follows that:
7
#
= γ[1 − cos(θR )]/(ρh). Using volume conservation, h ∼
#
V ret
∼ τi−1 π [1 − cos θR )]
R max
(3.2)
Which is the final result. Comparing with the experimental data, it turns out that this
equation not only gives the correct scaling behavior for the retraction in this regime
rate but also provides a rather accurate estimate of the numerical prefactor (see Fig 4).
Indeed, the ratio between the experimental and the predicted numerical prefactors is
found to be 0.6 Repeating the experiment for water on a polycarbonate surface, which
changes the contact angle value to 60◦ , we retrieve exactly the same ratio of 0.6.
3.2. Viscous regime
In the opposite limit of very viscous liquids, the drops adopt pancake shapes upon impact. During the first stages of retraction, the pancake shape rapidly relaxes towards a
roughly spherical cap, and the drop shape remains like this during the retraction since
the capillary number is small. During the retraction, it is only the contact angle that
varies slowly: it is mainly this slow contact angle dynamics that dictates the drop evolution during the retraction. Contrary to the previous analysis, the slow receding velocity
allows to assume a quasi-static dynamics for the surface shape during the retraction.
In this regime, it is then natural to assume that the work done by the capillary force
FC is dissipated through viscous flow near the contact line. Since we focus our study
on high-speed impacts, R max is always much larger that RI which justifies a small θ(t)
approximation at the onset of retraction. The viscous effects near the contact line then
lead to the well-known linear force-velocity relation (DeGennes 1985):
! "
6πη
Λ
˙
R(t)R(t)
(3.3)
FV = −
ln
θ
λ
where Λ and λ are respectively a macroscopic and a microscopic cutoff lengths. Λ is
typically of the same order as the drop size ∼ 1mm. λ is a microscopic length, and is
usually taken to be on the order of λ ∼ 1nm (DeGennes 1985). On the other hand, the
capillary force drives the retraction. Near the contact line it can be written:
FC = 2πR(t)γ [cos θ(t) − cos θR ]
(3.4)
Volume conservation gives: 43 πR 3I ∼ π4 θ(t)R3 (t), where we have taken the small angle
limit. Eqs. 3.3 and 3.4 together with the volume constraint leads to the following relation
for the variation of the contact radius:
$
%
1 − 21 θ2 (t) − cos(θR ) θ(t)4/3 −1
Ṙ(t)
τv
(3.5)
=−
R(t)
(144)1/3 ln(Λ/λ)
the above equation is obtained in the small angle limit and is only valid for short time
after the onset of retraction. We estimate the retraction rate %̇ as the maximum value of
Ṙ(t)/R(t) so that:
! "1/3
V ret
(1 − cos θR )5/3 −1
3
≈
τv
(3.6)
Rmax
25
5 ln(Λ/λ)
Comparing again to the experiments, good agreement is found: the retraction rate is
solely set by the viscous relaxation time τv and consequently %̇τi ∝ Oh −1 . Beyond this
correct scaling prediction, Eq. 3.6 provides a quite accurate estimate for the numerical
prefactor as is shown in Fig 4. Indeed, the ratio between the experimental and the
8
Figure 5. (a) Normalized maximum spreading radius plotted vs. the impact number. (b) R max
(normalized by the radius before impact) plotted vs. Weber number for small values of the impact number. Full line: power-law fit. (c) R max (normalized by the radius before impact) plotted
vs. Reynolds number for large values of the impact number. Full line: predicted power-law dependence with power 0.2. ∗: η = 10−1 P a.s, +: 9.510−2 , ◦: η = 4.810−2 P a.s, ": η = 2.810−2 P a.s,
!: η = 10−2 P a.s
predicted numerical prefactors is found to be 1.5. Again, repeating the experiment on a
polycarbonate surface, this ratio changes only slightly from 1.5 to 1.8.
4. Conclusions and perspectives
Our experiments reveal that the retraction rate is independent of the impact speed.
To account for the retraction speed, the maximum radius to which the droplet expands,
has to be known also. A number of studies have been devoted to the understanding of
the maximum spreading radius (see for instance (Fukai et al. 1993; Roisman et al. 2002;
Clanet et al. 2004)). However, no clear and unified picture emerges from previous experimental investigations. A recent experimental study of Rmax , combined with recent
theoretical ideas in the same spirit of the ones presented here was done by Clanet et al.
(2004). They obtain a zeroth order (asymptotic) description of the spreading stage, compare it with experiments and suggest that two asymptotic regimes exist for R max . The
first is given by a subtle competition between the inertia of the droplet and the capillary
forces; if only these two are important, it follows that R max /RI ∝ We 1/4 . In the second
regime, R max is given by a balance between inertia and viscous dissipation in the expanding droplet, leading to R max /R I ∝ Re 1/5 . Consequently, a single dimensionless number
is defined that discriminates between the two regimes: P = WeRe −4/5 referred to as the
Impact number. The crossover between the two regimes happens at a P of order unity.
Our experimental data are in qualitative agreement with their prediction, as is shown in
Fig. 5.a. At low P, the scaling Rmax /R I ∼ We 1/4 is clearly observed. However, for impacts
corresponding to P > 1, we observe only a very slow variation of the maximum spreading
radius as a function of P . Therefore, the relation between R max and the Reynolds number
is not very clear from our data (Fig 5. c). Although the main trend is not in strong
contradiction with the prediction R max /R I ∝ Re 1/5 , a power-law fit of our data gives
exponents that are always smaller than the predicted value of 0.2. Perhaps even more
important- in view of the small range of the maximal expansion R max that we cover- is
that the different water-glycerol mixtures do not appear to collapse on a single master
9
curve, as would be predicted by the above argument. However, since the maximum value
of P that we reach is on the order of 10, it may be that we have not reached the
purely viscous regime. In that case, the capillary, inertial and viscous forces are still
of comparable amplitude and have to be taken into account together. Note also that
the more sophisticated models reviewed in Ukiwe & Kwok (2004) do not provide better
agreement with our experimental measurements.
Despite this small problem, we are now able to develop a simple unified picture for drop
impact dynamics accounting for both the spreading and the retraction dynamics. The two
natural dimensionless numbers that have been identified are the impact number P, that
quantifies the spreading out of the droplet, and the Ohnesorge number Oh that quantifies
the retraction. We can thus construct a phase diagram in the experimentally explored
(Oh, We) plane, which is shown on Fig. 6. The experimentally accessible plane is divided
in four parts, where the main mechanisms at work during the impact process are different.
These four parts are separated by the curves Oh = 0.05 and We = Oh −4/3 . They are
labeled as follows: ICCI the drop dynamics is given by a competition between inertia and
capillarity both for the spreading and the retraction. IVCV: inertia and viscous forces
dominate the spreading, capillary and viscous forces dominate the retraction. These two
regimes have been studied in detail here. The two more intriguing regions are the IVCI
(viscous spreading, inertial retraction) and ICCV (capillary spreading, viscous retraction)
that are unfortunately difficult to explore in detail. For the IVCI- regime, the large inertia
at impact, combined with a small surface tension, will make the droplets undergo large
non-axisymetric deformations and they will eventually splash and disintegrate. On the
other end of the phase diagram, the ICCV region corresponds to very low impact speeds
and important capillary forces, implying very small deformations of the droplets. If the
deformations are small, pinning of the contact line of the droplets will become important,
and all our simple scaling arguments for both the maximum radius and the retraction
rate are invalidated.
A numerical investigation of droplet impact would be very helpful for two reasons.
First, numerics would allow to vary R I while keeping all the other physical parameters
constant. This would allow to check the robustness of our results, since experimentally
it is not easy to vary R I over a wide range. Second, as emphasized above, the viscous
regime for the maximum radius is difficult to characterize precisely due to the smallness
of the variation of R max for viscous drops. If precise numerical simulations could be done,
these different remaining problems could be resolved.
In sum, we have studied the retraction dynamics of liquid droplets upon high-speed
impact on non-wetting solid surfaces. Perhaps the strongest conclusion from our investigation is that the rate of retraction of the droplet is a drop constant which does not
depend on the impact velocity. Two regimes for the retraction rate have been identified: a viscous regime and an inertial regime. We have in addition shown here that
simple hydrodynamic arguments can be formulated that give very reasonable agreement
with experiments in the two different regimes. Acknowledgments: Benjamin HelnannMoussa is acknowledged for help with the experiments. Denis Bartolo is indebted to the
CNRS for providing a post-doctoral fellowship. LPS de l’ENS is UMR 8550 of the CNRS,
associated with the universities Paris 6 and Paris 7.
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1000
We
IV-CI
IV-CV
100
IC-CV
IC-CI
10
0.001
0.01
0.1
1
Oh
Figure 6. Phase diagram in the (We, Oh) plane for the impact and retraction dynamics of
droplets. The four regions are discussed in the text, and the symbols represent the parameters
of the data reported in this paper. Different symbols have been assigned for each region.
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model for bouncing drop. Europhys. Lett. 62, 237–243.
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811–811.
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model for spreading and receding. Proc. R. Soc. Lond. 458, 1411–1430.
Taylor, G. I. 1959 The dynamics of thin sheets of fluid iii. disintegration of fluid sheets. Proc.
Roy. Soc. London A 253, 253–313.
Ukiwe, C. & Kwok, D. Y. 2004 On the maximum spreading diameter of impacting droplets
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PRL 94, 224501 (2005)
week ending
10 JUNE 2005
PHYSICAL REVIEW LETTERS
Asymptotic Behavior of the Rayleigh-Taylor Instability
Laurent Duchemin,1 Christophe Josserand,2 and Paul Clavin3
1
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
2
Laboratoire de Modélisation en Mécanique, UPMC-CNRS UMR 7607, 4 place Jussieu, 75252 Paris CEDEX 05, France
3
IRPHE, Universités d’Aix-Marseille I and II-CNRS, 49 rue Joliot-Curie, BP 146, 13384 Marseille CEDEX, France
(Received 19 December 2004; published 8 June 2005)
We investigate long time numerical simulations of the inviscid Rayleigh-Taylor instability at Atwood
number one using a boundary integral method. We are able to attain the asymptotic behavior for the spikes
predicted by Clavin and Williams for which we give a simplified demonstration. In particular, we observe
that the spike’s curvature evolves as t3 , while the overshoot in acceleration shows good agreement with the
suggested 1=t5 law. Moreover, we obtain consistent results for the prefactor coefficients of the asymptotic
laws. Eventually we exhibit the self-similar behavior of the interface profile near the spike.
DOI: 10.1103/PhysRevLett.94.224501
PACS numbers: 47.20.Ma, 47.11.+j, 47.20.Ky
Introduction.—The Rayleigh-Taylor (RT) instability appears when, under gravity, a heavy liquid is placed over a
lighter one [1]. This instability is crucial for our understanding of different phenomena in fluid mechanics: mixing, thermal convection ([2], and references therein), and
also finger number selection in splashes [3]. It is also
important in inertial confinement fusion (ICF) where the
mass ablation provides a stabilizing effect to the interface
instability [4]. Without ablation, after the exponential
growth of the perturbations due to the linear RT instability,
nonlinear profiles develop through the formation of bubbles of lighter fluid rising into the heavier one and falling
spikes of the heavier liquid penetrating the lighter one. In
the general situations of viscous fluids, which are immiscible and/or have Atwood number not equal to unity [AT !
"!h # !l $="!h % !l $, with !h and !l being the density of
the heavier and lighter fluids, respectively], famous mushroomlike structures grow for larger times [2,5,6]. The limit
of an inviscid fluid above a vacuum (AT ! 1) without
surface tension plays a specific role since no stabilizing
effects are present in the linear dynamics. It is important to
understand ICF in the limit of high density ratio, and it is
also the most challenging case for the numerics. Most
theoretical and numerical work have focused on this idealized limit in order to track insights into the instability itself
[7–12]. It has been shown using a conformal mapping that
a finite time singularity might appear in the conformal
plane [13], and it is also suspected that for some sufficiently irregular initial conditions finite time singularities
should also be observed in the physical plane. However,
starting with sufficiently smooth initial conditions, the
asymptotic dynamics [8,11,12] presents a constant velocity
rising bubble separated by free falling tiny spikes as displayed in Fig. 1. Although the rising bubble motion has
been described using local properties of the flow [14], the
asymptotic dynamics of the spikes is far from being well
understood. The single mode approach gives a fair description of the constant velocity of the rising bubble [vb !
p!!!!!!!!!!!!!!
g="3k$, where g is the acceleration of the gravity and k
0031-9007=05=94(22)=224501(4)$23.00
the wave number of the perturbation] but gives only partial
results for the spike [8]. The fluid there obeys free fall
dynamics to a good approximation, and the pressure field
of the flow leads to an overshoot in the acceleration. The
accelerated motion of the liquid stretches the spike geometry and one expects self-similar behavior of the tip of the
spikes.
Recently, an asymptotic theory using a parallel flow
description of the velocity field near the spikes has been
constructed [15]. The interface dynamics is nonlinear for
large time and can be described using the theory of characteristics which gives rise to finite time singularity solutions. In the case of regular dynamics, a self-similar
description of the peak is obtained for large time: the
maximal curvature of the interface at the peak tip is found
to behave as the cubic power of time t3 . Moreover, the
−π
−2
−1
0
π
0
1
t = 6.5
2π
2
t = 7.5
3
4
3π
5
6
4π
5π
6π
−π
t = 9.5
0
π0
5
7
8
10
FIG. 1. Snapshots of the interface subject to the RayleighTaylor instability for time ranging from t ! 0 to 10, starting
with a small amplitude sine mode (left). On the right is shown
the velocity of several points along the interface, nondimensionp!!!!!!!!!!!
alized with the stationary bubble rising velocity g=3k, as a
function of time.
224501-1
 2005 The American Physical Society
PRL 94, 224501 (2005)
spike position, following the free fall 12 gt2 at leading order,
is shown to converge to the constant acceleration g with an
overshoot in acceleration decreasing as t#5 . In this Letter,
we present a numerical study of the Rayleigh-Taylor instability which focuses on the large time dynamics of the
spikes in order to investigate the self-similar dynamics
predicted in [15], where no numerical studies were performed. We consider the dynamics for an inviscid liquid
(heavy) with an exterior fluid of zero density (At ! 1) and
no surface tension. The numerics use a boundary integral
method. Because of strong numerical instabilities, a careful
treatment of the interface using conformal mapping is
needed as explained below. The results are then shown
and compared with the theory.
Asymptotic analysis and numerical method.—We consider the two-dimensional motion of an inviscid fluid
above a vacuum, subject to a negative acceleration #g.
A periodic sine perturbation of the interface of wave number k is implemented as initial conditions. Neglecting
surface tension, the equations of motion have no control
parameter after rescaling the time, the position,
p!!!!!!!!!! and the
p!!!!!!
velocity potential ’ by factors gk, k, and k3 =g, respectively. The interface is described by y ! ""x; t$, where y is
the direction along the gravity and x orthogonal to it (see
Fig. 2). The velocity field U ! "u; v$ satisfies the dimensionless Euler equation
dU
! #rP % ey ;
dt
where P"x; y; t$ is the pressure, ey the nondimensional
acceleration due to gravity, and the fluid density ! ! 1.
The kinetic equation for the interface reads
@""x; t$
@""x; t$
! v;
%u
@x
@t
with the velocity field "u; v$ evaluated at the interface
&x; ""x; t$'. Starting at time t ! 0 with a small sine amplitude interface, we observe for large time that the fluid
particles located in the vicinity of the tiny spikes come
from an almost free fall from the initial interface region.
Therefore, following [15], we assume quasiparallel steady
flow for the velocitypfield
!!!!! which gives then in the tip region
juj ( jvj and v ) 2y with y ) 12 t2 for large time. Writing a perturbation expansion of the velocity field in the tip
Ω
x
week ending
10 JUNE 2005
f (z ) = e− iz
M
ζ = f (z )
y
FIG. 2. Conformal map used to transform the physical periodic
plane ! into a closed domain M.
p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
region jxj ( y, we in fact consider v ! 2&y % f"x; y; t$',
with f"x; y; t$ ( y. Taking a Taylor expansion in x of the
p!!!!! 0 "y;t$
!!!! %
perturbation f, we obtain by symmetry v ! 2y % fp
2y
x2 fp
2 "y;t$
!!!! % O"x4 $. We limit
2
2y
our expansion to the second order
in x for the velocity field
! later on. Incompressibility gives
q!!!! @"f "y;t$=p!!!
2y$
0
1
u ! #" 2y %
$x % O"x3 $. At the leading order
@y
[where we neglect even the perturbation f"x; y; t$], we
obtain the following expression for the interface location:
x
@""x; t$ p!!!!!!!!!!!!!!!!
@""x; t$
! 2""x; t$;
# p!!!!!!!!!!!!!!!!
@t
2""x; t$ @x
which can be solved using the methods of characteristics
(see [15]). Writing ""x; t$ ! t&2t # #"x; t$' and noting that
#"x; t$ ( t=2 in the spike region, we obtain, after linearization,
@#"x; t$ x @#"x; t$
! 0;
#
t @x
@t
which has the self-similar solution of the form #"x; t$ !
$"xt$. A first conclusion can be drawn about the curvature
of the interface at the tip, % ! #@2 "=@x2 jx!0 , which is
thus found to increase as the cubic power of time [16]:
% ! t3 $00 "0$:
(1)
1 df "y ; t$
2
d2 ys
p!!!!!!! 0 s ! 1 % 5 00 ;
!
1
%
dt
2ys
t $ "0$
dt2
(2)
The next order terms of the expansion allow the determination of the function f0 "y; t$ near the tip. Using the
constant value of the pressure at the interface, we use the
projection of the Euler equation at the interface on its lo@""x;t$ dv
@""x;t$
cal tangent: du
dt % @x
dt ! @x , since on the interface dP!x; ""x; t$; t"=dx ! 0. We develop this equation at
first nonzero order (which will end up the first order in x)
with the expansion $"xt$ ! $"0$ % x2 t2 $00 "0$=2 % O"x4 $.
Remembering that jfj (py,
also the large
!!!!!we can neglect
p!!!!!
2 !f "y; t$= 2y"=@t@y and
scale
terms
@
2y
@2 !f0 "y; t$=
0
p!!!!!
2
2y"=@y with respect to the others. We finally obtain
p!!!!!!! "ys ;t$
"ys ;t$
for the tip position y ! ys : @f0@t
% 2ys @f0@y
!
q!!!!
df0 "ys ;t$
! y2s %1 .
dt
p!!!!!!! f0 "ys ;t$
s
p!!!!! , we obtain, for the tip
Recalling that dy
dt ! 2ys %
2ys
acceleration at leading order,
which corresponds to an overshoot in the spike acceleration decreasing as the fifth power of time. An overshoot in
the acceleration was observed in numerical simulation already in [17], but with no explicit scaling laws.
The numerical method is elaborated using the incompressible and potential properties of the flow. The velocity
field can thus be evaluated everywhere when the velocity
potential is known on the interface thanks to Cauchy’s
theorem, in the spirit of other pioneering works [17–20].
The nondimensional Bernoulli equation on the free surface
224501-2
week ending
10 JUNE 2005
PRL 94, 224501 (2005)
20
reads
100
1
@’
! # "r’$2 % y;
2
@t
(3)
10
"’ ! 0:
spike position y
where the velocity potential ’ is a harmonic function in the
fluid domain !:
(4)
1
10
0.1
The kinematic condition on the free surface expresses
the fact that fluid particles move with the same normal
velocity as the free surface itself:
0
(5)
Knowing ’ on the free surface at a given time step, we
search for the solution of Eq. (4) that satisfies this boundary
condition (5). We use the complex potential &"z$ ! ’ %
i and the conformal map f"z$ ! exp"#iz$ (cf. Fig. 2),
where z ! x % iy and is the stream function. The conformal map transforms the periodic domain ! into the
closed domain M. Since is harmonic inside !, &"z$ is
analytic inside ! and therefore #"'$ ! &!f"z$" is analytic
inside M. Using Cauchy’s theorem, we obtain a Fredholm
equation of the second kind for the stream function ,
which is solved using discretization of the free surface
(@! and thus @M). This linear system of equations is
solved using a LU decomposition. Once we know on
each point on @M, the complex velocity of each marker in
the physical plane is given by
d&
! u # iv;
dz
(6)
where u and v are the horizontal and vertical velocities,
respectively. This complex velocity is computed with a
finite difference scheme using the values of the complex
potential on the collocation points on @!.
The position of the surface markers (kinematic condition) and the value of the velocity potential on each of these
markers (Bernoulli equation) are then updated in time
using a fourth order Runge-Kutta method. Finally, an
adaptive mesh refinement technique is used in order to
concentrate markers on the spike.
Results and discussions.—We have performed numerical simulations of the Rayleigh-Taylor instability using the
numerical method described above. We start with a sinemode deformation of the interface of amplitude a ( 1.
The unavoidable numerical noise cannot be damped by the
numerics and the calculations always end up subject to
numerical instabilities. Nevertheless, we emphasize that
the numerical scheme used here is remarkably robust and
can be accurately evolved to reach the large time where the
scalings predicted by the theory [15] are valid. Comparing
our simulations with recent numerical works [5,6,9], we
have been able to run the dynamics at least twice as far,
which corresponds roughly to an increase of a factor of 8 in
the tip’s curvature.
0
5
time
10
FIG. 3. Position of the spike ys "t$ as a function of time. The
inset shows in a log-log plot of the spike position (solid curve) as
function of time t # t0 , with t0 ! 3:74 obtained by a second
order polynomial fit of ys . The dashed line shows the expected
behavior 12 t2 .
The position of the spike is shown on Fig. 3 as function
of time. We observe that the asymptotic dynamics are very
well approximated by the relation ys ! 12 g"t # t0 $2 , as
shown in the inset of the figure with t0 ! 3:74 for the
amplitude of the initial perturbation a ! 0:01. This remarkable behavior, in good agreement with the free fall
hypothesis, suggests that t0 is the time delay accounting for
the initial exponential development of the instability.
Indeed, the linear growth rate of the perturbation is precisely one, and, varying a, we observe that the time delay
correspond to aet0 ! 0:42, which corresponds in the numerical simulation to a constant amplitude of the spike of
0.22 to be compared to 0.21 obtained with the linear
instability only. The time and amplitude correspond thus
roughly to the transition between the linear and the nonlinear regime. We will therefore present further data on the
curvature dependence and the acceleration of the tip as
functions of this delayed time t # t0 instead of t.
10000
spike curvature
dx
* n ! r’ * n:
dt
t−t0
100
1
0.01
0.01
0.1
1
10
t−t0
FIG. 4. Spike curvature %s calculated at the tip y ! ys as
function of the delayed time t # t0 in a log-log plot. The dashed
line displays the cubic law (1) with $00 "0$ ! 1:5.
224501-3
PRL 94, 224501 (2005)
10
0
−1
acceleration overshoot
10
−2
10
−3
10
−4
10
−5
10
1
10
t−t0
FIG. 5. Overshoot in acceleration, defined as the difference
between the tip acceleration and the gravity. The plot is in loglog scale and with the delayed time t # t0 . The dashed line
shows the theoretical prediction (2) using the value of $00 "0$
obtained from Fig. 4.
The curvature %s at the tip is then shown in Fig. 4. The
large time asymptotic behavior is similarly found to follow
the cubic law [see Eq. (1)] with $00 "0$ ! 1:5. We have not
been able to deduce analytically this value of the selfsimilar curvature $00 "0$ ! 1:5 using the characteristic dynamics, which is valid for large times only.
In addition, the acceleration of the tip is computed by
finite differences on the tip velocity and the overshoot in
the acceleration is presented on Fig. 5. We observe that the
results look noisier than the two previous ones. Two factors
can explain such noise: first, we are looking to a finite
difference which decreases to zero so that the numerical
errors are relatively more important. However, we note that
the overshoot in acceleration shows good agreement with
the 1=t5 law, noting that no adjustable parameter is used in
FIG. 6. Self-similar structure of the tip: the interface profiles
around the spike have been superimposed on the right side of the
figure for different time t ranging from 4 to 12. The left side of
the figure shows the same curves rescaled by factor 1="t # t0 $
and "t # t0 $ for the x and y coordinates, respectively, following
the scaling behavior predicted by the theory.
week ending
10 JUNE 2005
this comparison. Finally, the self-similar structure of the
interface near the tip has been exhibited on Fig. 6. We
observe after the proper rescaling on the left part of the
figure that the interface profiles collapse onto a single
curve near the spike.
We have thus exhibited large times numerical simulations of the Rayleigh-Taylor instability which present
asymptotic scaling behavior in agreement with theoretical
predictions using Taylor expansions of the free fall velocity
field at the spike [15]. Although our numerics always stops
due to numerical instability, we have been able to reach
large time enough to exhibit the cubic power in time
dependence for the spike curvature and the inverse of the
quintinc power of time decreasing of the overshoot in
acceleration. Moreover, the numerical methods used here
and the analytical description of the flow in the vicinity of
the spike offer a powerful tool to investigate RayleighTaylor and Richtmyer-Meshkov instabilities for any density ratio.
It is our pleasure to thank J. Ashmore for useful comments. We acknowledge also the support of CEA through
Contract No. CEA/DIF N 4600051147/P6H29.
[1] Lord Rayleigh, Scientific Papers II (Cambridge University
Press, Cambridge, UK, 1900), p. 200.
[2] B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff,
A. Libchaber, S. Thomae, X. Wu, S. Zaleski, and
G. Zanetti, J. Fluid Mech. 204, 1 (1989).
[3] D. Gueyffier and S. Zaleski, C.R. Acad. Sci. Paris, Ser. IIb
326, 839 (1998).
[4] J. Sanz, J. Ramirez, R. Ramis, R. Betti, and R. P. J. Town,
Phys. Rev. Lett. 89, 195002 (2002).
[5] S.-I. Sohn, Phys. Rev. E 67, 026301 (2003).
[6] S.-I. Sohn, Phys. Rev. E 69, 036703 (2004).
[7] D. Layzer, Astrophys. J. 122, 1 (1955).
[8] Q. Zhang, Phys. Rev. Lett. 81, 3391 (1998).
[9] G. Hazak, Phys. Rev. Lett. 76, 4167 (1996).
[10] S. I. Abarzhi, Phys. Rev. Lett. 81, 337 (1998).
[11] K. O. Mikaelian, Phys. Rev. Lett. 80, 508 (1998).
[12] N. Inogamov, Astrophys. Space Phys. Rev. 10, 1 (1999).
[13] S. Tanveer, Proc. R. Soc. A 441, 501 (1993).
[14] V. N. Goncharov, Phys. Rev. Lett. 88, 134502 (2002).
[15] P. Clavin and F. Williams, J. Fluid Mech. (to be
published).
[16] It is interesting to notice that this cubic property is also
found to be valid for the Richtmyer-Meshkov instability
where the interface is only impulsively accelerated. There,
the asymptotic vertical velocity field behaves as y=t near
the spike, which is located around y ) v0 t. Numerical
simulations are also found in correct agreement with this
scaling.
[17] G. R. Baker, D. I. Meiron, and S. A. Orszag, Phys. Fluids
23, 1485 (1980).
[18] T. Vinje and P. Brevig, Adv. Water Resour. 4, 77 (1981).
[19] M. S. Longuet-Higgins and E. D. Cokelet, Proc. R. Soc. A
350, 1 (1976).
[20] R. Menikoff and C. Zemach, J. Comput. Phys. 51, 28
(1983).
224501-4
PHYSICS OF FLUIDS
VOLUME 14, NUMBER 9
SEPTEMBER 2002
Jet formation in bubbles bursting at a free surface
Laurent Duchemin, Stéphane Popinet, Christophe Josserand,
and Stéphane Zaleskia)
Laboratoire de Modélisation en Mécanique, CNRS and Université Pierre et Marie Curie (Paris VI),
8 rue du Capitaine Scott, 75015 Paris, France
!Received 6 June 2001; accepted 24 May 2002; published 2 August 2002"
We study numerically bubbles bursting at a free surface and the subsequent jet formation. The
Navier–Stokes equations with a free surface and surface tension are solved using a marker-chain
approach. Differentiation and boundary conditions near the free surface are satisfied using
least-squares methods. Initial conditions involve a bubble connected to the outside atmosphere by a
preexisting opening in a thin liquid layer. The evolution of the bubble is studied as a function of
bubble radius. A jet forms with or without the formation of a tiny air bubble at the base of the jet.
The radius of the droplet formed at the tip of the jet is found to be about one tenth of the initial
bubble radius. A series of critical radii exist, for which a transition from a dynamics with or without
bubbles exist. For some parameter values, the jet formation is close to a singular flow, with a conical
cavity shape and a large curvature or cusp at the bottom. This is compared to similar singularities
investigated in other contexts such as Faraday waves. © 2002 American Institute of Physics.
#DOI: 10.1063/1.1494072$
I. INTRODUCTION
meter. Depending on their mass and initial velocity, the droplets will either fall back into water or evaporate.
The topic of this paper is the investigation of the bubble
evolution after the initial film rupturing, including the jet
formation. A numerical method solving the Navier–Stokes
equations and describing the free surface with high precision
is used. Previous numerical studies of these phenomena have
been made postulating mostly inviscid fluids; however, a
modified boundary element method taking into account small
viscous effects was also used.2– 4 A Navier–Stokes simulation was shown in Ref. 5, with a VOF-type method in a
regime where the bubble is very deformed.
In most previous studies the effect of film atomization on
jet birth was assumed to be negligible. Few comparisons
were made with experimental data. Some experimental studies were also conducted to measure quantities such as jet
velocity,6 – 8 size of the first ejected droplet, height at which
the droplet detaches from the jet, or height reached by the
droplet. These experiments are fairly difficult to conduct, because of surface contamination which modifies significantly
the free-surface boundary condition and the surface tension
coefficient.
As our numerical results will demonstrate, the jet formation is in many cases singular and self-similar. Singular jets
forming at a free surface have already been observed and
studied in different contexts. Indeed, in the bubble-bursting
problem as well as in several other free-surface flows, one
observes the formation of a conical cavity, with a very high
curvature or cusp at its base. In some cases a small bubble is
trapped at the bottom of the cavity. A thin narrow jet subsequently forms in a self-similar manner. This phenomenon
was observed experimentally in Faraday waves by
Longuet-Higgins9 and Lathrop,10 in the development of the
jet inside a bubble containing a sink flow in the numerical
Bubbles bursting at the water surface are a familiar everyday occurrence. They also take part in important processes of transport and exchange across liquid/gas interfaces,
caused by the ejection of jets and various kinds of small
droplets. These are involved in the transfer of heat, mass and
various contaminants between the oceans and the
atmosphere.1 Indeed, breaking waves cause the formation of
a large number of bubbles beneath the water level. The efficiency of the resulting mass transfer, including the transfer of
CO2 depends on the initial properties of the ejected droplets
!size, initial velocity".
The phenomena producing aerosols during the bursting
of a bubble are of two kinds: the first is the rupture of the
film separating the bubble from the atmosphere. This film
atomization can produce several hundred droplets of around
a micrometer in diameter which probably represent a large
fraction of the transfers.1 Since the scales involved during
this rupture are of the order of 100 nm, a physical description
is outside the scope of continuum fluid mechanics. Indeed,
long-range molecular forces such as van der Waals forces or
electrostatic repulsion must be taken into account.2
The small cavity remaining after the film rupture collapses under the effect of both surface-tension and buoyancy.
This collapse gives birth to a narrow vertical jet which eventually breaks into one or several droplets !see Fig. 1". This
phenomenon constitutes the second aerosol production process and is the principal topic of this paper.
These aerosols are of a different kind: they are ejected
vertically—which is not the case for film aerosols—and their
diameter is about one tenth of the size of the initial cavity,
i.e., about 100 %m for a typical bubble radius of one millia"
1070-6631/2002/14(9)/3000/9/$19.00
3000
© 2002 American Institute of Physics
Downloaded 14 Aug 2002 to 134.157.34.8. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 14, No. 9, September 2002
Jet formation in bubbles
3001
!(+t2/,)1/3 where + is the surface tension and , the density.
Then the similarity variable is - !x/r c and the flow velocity
diverges like t "1/3 near the singularity for a fixed value of the
similarity variable -. This idea is at the basis of 2/3 exponents found for instance by Miksis and Keller.16 This type of
scaling, was applied by Zeff et al.10 to observations of jet
formation in Faraday waves. The leading order term for the
velocity potential is of the form
&!C ! t"t 0 ! 1/3r 1/2P 1/2! cos ' " ,
FIG. 1. Jet produced by the collapse of a spherical cavity. The end droplet
will eventually detach due to the Savart–Plateau–Rayleigh instability.
study of Ref. 11. For bursting bubbles the conical cavity may
be seen in the experiments and simulations but the singular
character of the jet formation has not been investigated to
our knowledge. The phenomenon may also be seen in the
cavity formed by falling raindrops.12–15
The evolution of the conical cavity has been studied by
Longuet-Higgins13 as a special case of a family of hyperbolic
surfaces: conical surfaces were shown to be a special case of
the hyperbolic surfaces of Ref. 9. These conical surfaces are
preserved by the velocity potential
2
&!A ! t " r P 2 ! cos ' " ,
!1"
where r is the spherical radial distance, P 2 the Legendre
polynom of second degree and ' the polar angle measured
from the north pole, which yields the velocity field
( ) !A ! t "! "x,"y,2z " ,
!2"
where A(t) is an arbitrary function. Indeed any conical free
surface in this flow remains conical. For positive A the cavity
opens in time as in the experiment.
Of course the actual flow is not exactly conical. The
bottom of the cone is rounded, and oscillates in shape as
capillary waves converge towards the bottom of the conical
surface. At some instant in time the bottom may develop a
cusp, followed by jet formation. This process is obviously
singular at least for some values of the parameters, but there
is no agreement among the above cited publications on the
exact nature of the singularity.
Indeed one may inquire into the specific scaling form of
the singularity. The Euler equations without surface tension
and gravity will in principle admit self-similar solutions of
the form
& ! x,t " ! ! t"t 0 ! * ! x! t"t 0 !
m
"n
",
!3"
where * is an arbitrary scaling function and t 0 is the singularity time. The solution may be valid before and/or after the
singularity time. The exponents should satisfy m!2n"1.
Indeed with this condition all the terms in the Bernoulli
equation balance. However, when surface tension is added,
the only way to form a self-similar solution that balances
inertia and surface tension is by selecting n!2/3. This is
because the only length scale that can be built is r c
!4"
where P 1/2 is the Legendre function of order 1/2.
However, a series of alternate theories for singular freesurface flows and in particular the conical cavity and jet formation was proposed by Longuet-Higgins. He has shown
that the type of flow described by Eq. !2" had a divergent
velocity with A(t). ! t"t 0 ! "1/3 thus a t "1/3 divergence for a
fixed value of the real !unscaled" distance r.13 In this solution
the scaling is not fixed by a balance with surface tension.
Instead, surface tension is added as a perturbation to the
conical solution, in the form of a sink flow.13 The LonguetHiggins solution yields an angle for the conical cavity of
2'!109°5, in good agreement with the numerical observations of Ref. 12. Another self-similar solution for jet formation was found numerically by Ref. 11 obtaining yet another
scaling, for the case of jet formation inside a bubble. The
potential is then approximated by
&!A ! t " r 1/4P 1/4! cos ' " ,
!5"
where P 1/4 is the Legendre function of order 1/4.
This paper is organized as follows. We first describe the
general context of this study, the nondimensional numbers
controlling the problem and the scaling laws deduced from
dimensional analysis. We then briefly introduce the numerical method we use and its main advantages. A first comparison with experimental profiles is presented. Finally, a detailed parametric study is conducted using a simple initial
shape for the cavity and neglecting gravity. We measure the
volume of the first ejected droplet, the velocity of the jet and
the maximum pressure encountered on the axis of symmetry
and discuss the results. In some circumstances, a tiny bubble
is formed at the base of the jet. The self-similar flow occurring when the conical cavity and the cusp form is investigated.
II. INITIAL CONDITIONS AND EXPECTED SCALING
LAWS
Given the small size of the bubbles we are interested in
!diameter is around one millimeter", some assumptions can
be made regarding the parameters governing jet birth. The
first idea is to suppose that the cavity is motionless at the
initial time. Experiments have shown that, even in the absence of surfactants, the bubble can stay at the free surface in
a quasistatic equilibrium for a few seconds.17 The bubble is
then separated from the atmosphere by a thin liquid film, the
cavity being subject to surface tension and buoyancy forces.
A model for this static configuration is a more or less deformed bubble adjacent over part of its surface to a film of
3002
Duchemin et al.
Ohnesorge number Z! ,/ 2 / + R governs the phenomenon
and dimensional analysis gives velocity in the form
V
!
" #
,R
,/ 2
!F
,
+
+R
!6"
where F is an unknown function.
Whenever R / #R#R c , we also expect that viscosity
plays no role. The only way to eliminate viscosity is to suppose that the function F has a finite, nonzero limit C when /
goes to zero. The nondimensional velocity of the jet then
behaves like
" #
V
R
$C
V/
R/
"1/2
,
!7"
where V / ! + / ,/ .
Similar arguments lead to a scaling law for the nondimensional pressure of the form
" #
R
P
.C !
P/
R/
FIG. 2. The initial configuration in the ‘‘large rim’’ case. The grid is a 5122
Cartesian grid.
"1
,
!8"
where P / ! + 2 / ,/ 2 .
III. NUMERICAL METHOD
negligible thickness. This configuration may be computed, or
obtained from the experimental data as in the case reported
in Ref. 8.
When the film reaches a critical thickness !about 100
nm" after draining slowly, it breaks more or less rapidly !depending on the presence of surface contaminants". It is then
possible to run simulations by taking the current, static configuration and removing the thin film. While we do this in
one case, the drawback is that a sharp corner exists at the rim
of the neck or juncture between the film and the bulk liquid.
The small length scales involved may create numerical convergence problems. Moreover, as we show below, small
length scales are generated independently of initial conditions by the steepening of capillary waves and jet formation.
Keeping the small length scales in the initial conditions
makes it more difficult to observe the intrinsically generated
small scales. We thus decided to drastically smooth the rim
of the neck. In most calculations, the initial shape was defined as follows. A spherical cavity is separated from the
atmosphere by a circular hole, the border of the hole being a
circular rim !see Fig. 2".
The collapse behavior depends only on four physical parameters: the kinematic viscosity /, +, , and the acceleration
due to gravity g. Out of the four physical parameters only
two length scales can be defined, the capillary length R c
!( + / , g) 1/2 and the viscous-capillary length R / ! ,/ 2 / + . In
pure water R c !2.7 mm and R / !0.014 % m, respectively. If
the radius of the bubble R#R c , capillary effects are predominant compared with the gravity effects; if R$R / , viscous effects are expected to be negligible compared to the
capillary ones. For R / #R#R c the phenomenon is dominated by surface tension and inertia.
We also decided to neglect the effect of gravity which is
a correct approximation for R#R c . Therefore, only the
The choice of the numerical method is conditioned by
the terms we need to solve accurately. In our problem, the
first term of interest is surface tension: being the main driving force in the parameter range we consider, it is important
to model it correctly. Given the large density ratio between
water and air we can moreover assume that the influence of
the gas phase is negligible.
According to these two assumptions, we used a numerical method which solves the full axisymmetric Navier–
Stokes equations in a fluid bounded by a free surface while
allowing an accurate description of the interfacial terms such
as surface tension. This method has been documented
elsewhere18,19 and has been shown to produce accurate qualitative and quantitative results when compared with both theoretical and experimental data.
In short, a regular Cartesian fixed grid is used. Massless
particles !markers" advected by the flow define the position
of the interface. Linked by cubic splines, they describe accurately the geometry of the free surface. For cells which are
not cut by the free surface, a classical finite-volume scheme
is applied. For the cells in the vicinity of the interface, finite
differences cannot be computed since velocities are not defined in the ‘‘gas’’ phase. Therefore, an extrapolation of the
velocity field near the free surface on the other side is necessary. This extrapolation must take into account the boundary conditions on the free surface !in particularly the nullity
of the tangential stress". This is done by using a least-meansquare procedure constrained by the condition of vanishing
tangential stress. Comparisons with theoretical results show
that this approach gives an accurate description of the viscous dissipative terms associated with the boundary conditions.
The pressure on the boundary is obtained as follows. The
local curvature is estimated from the spline reconstruction.
The local normal viscous stress is estimated from the above
3003
FIG. 3. Time sequence of the jet formation in a 2.57 %l bubble bursting at
a free surface. Top, experimental !Ref.
8" and bottom, computational results.
Profiles are 1/6000 s apart.
least-squares procedure. Then the pressure is obtained from
the normal-stress boundary condition. The pressure on the
boundary serves as a boundary condition for the Poisson
equation for the pressure. This equation is in turn solved
using a multigrid algorithm.
Most computations have been made on a 5122 grid, except the comparison with the experimental profiles from
MacIntyre, which has been made on a 10242 grid and some
selected computations which were refined to 10242 grids.
IV. COMPARISON OF THE NUMERICAL RESULTS
WITH EXPERIMENTAL PROFILES
We have first initialized the calculation with a realistic
shape, and taken into account all the physical parameters,
i.e., capillarity, viscosity and gravity. The goal was to compare the results with a series of shapes published by
MacIntyre.8 The initial shape of the free surface has been
obtained from a numerical calculation based on the works of
Toba.20
Figure 3 illustrates the experimental and the computational results. The numerical parameters are / !10"6 m2 /s,
+!0.072 kg/s2, ,!1000 kg/m3 and the volume of the bubble
is the same as the one given by MacIntyre: 2.57 %l. The
computational time is about one day on the 10242 grid. The
overall agreement is very satisfactory. In particular capillary
waves are well described, in contrast to the earlier published
results using boundary integral methods.2,3 We believe that
this lack of capillary waves is due to the strong smoothing
needed to avoid numerical instabilities in boundary integral
techniques !and probably also to an insufficient spatial resolution, which is also limited by numerical stability". In our
method, real, molecular viscosity is present and the fine grid
we use allows in principle to solve the small spatial scales of
the capillary waves.
The time interval between images is the same as the one
given by MacIntyre, i.e., 1/6000 s. A difference in time between profiles can be seen, even if the shape is very similar.
A possible explanation is the presence of surface contaminants in the MacIntyre experiment. These contaminants
could change the surface tension, even modify its value locally, therefore changing the behavior of the free surface
through generation of Marangoni currents. They could also
make the interface partially or entirely rigid, changing the
free-surface boundary condition.
Solving the full Navier–Stokes equations, we have ac-
3004
FIG. 4. Vorticity isolines during the collapse of the bubble for the same
conditions as in Fig. 3 at time t!1.12 ms. The maximum vorticity isoline
value is 4.80 s"1 and the level difference between the isolines is about 1.07
s"1.
cess to vorticity which can, as we will see later, have an
important effect even on very small structures in a low viscosity fluid such as water. Figure 4 shows the vorticity isolines during the collapse of the cavity. The vorticity is confined to a thin boundary layer before the jet birth. Later on
however !see Figs. 5–7", a vorticity cone is entrained below
the jet and the shear stress there is comparable to that in the
narrow jet. This detachment of vorticity illustrates the formation of a downward jet, already observed by Boulton–Stone
and Blake with their modified boundary integral method.2,3
V. RESULTS OF THE PARAMETRIC STUDY
A set of computations have been made for radii between
10"6 m and 10"2 m (102 %R/R / %106 ) with the initial shape
described above.
The evolution of the profiles is very similar to that
shown on Fig. 3. A conical cavity forms with a train of
capillary waves converging to the axis. The number of capillary waves depends strongly on the Ohnesorge number: the
higher this number, the higher the number of capillary waves
converging to the base of the cavity. Figure 8 shows a largeradius case with a large number of waves !see also Fig. 11".
In some cases, especially near R/R / !103 the jet became
s"1.
Duchemin et al.
s"1.
very thin !Fig. 9" and the local radius of curvature smaller
than the grid size. The calculation then becomes inaccurate
and has to be stopped.
For some parameter values we observe a tendency to
trap a bubble on the axis of symmetry just before the formation of the jet. We have searched systematically for bubble
entrapment. There are two competing changes of shape: the
jet formation is heralded by a change of curvature at the base
of the cavity, while the bubble pinching is preceded by the
formation of an overhang in the interface, i.e., the height
h(r) of the interface becomes multivalued. Thus our criterion for incipient bubble formation is as follows: !a" The
height h(r) becomes steep, then multivalued, and !b" the
curvature at the base remains positive. This is only an indication that a bubble will be trapped before the jet forms as
shown in Fig. 10, but we need such a crude criterion because
the bubbles are very small for the kinds of grids we have. We
found a first bubble entrapment region for 576%R/R /
%2016, the second one between 57600%R/R / %288 000.
Other such regions at higher values of R/R / are likely, but
difficult to observe numerically. One indication is the existence of large trains of capillary waves at large R/R / as
shown on Fig. 11.
The topology of the interface changes when a bubble is
trapped. This pinching is a singular event akin to the pinching of a gas cylinder by the Savart–Plateau–Rayleigh instability. We shall call it a pinching singularity to distinguish it
s"1.
FIG. 8. Capillary waves for R/R / !144 000. The times of the successive
profiles are, respectively, t!0.137, 0.859, 1.858, 2.752, 3.449, 3.983, 4.387,
4.689, 4.832 ms.
from other free surface singularities. To pursue the calculation numerically beyond a pinching singularity, one should in
principle perform surgery on the marker chain and continue
the simulation. This is however difficult because the problem
slightly changes in nature: the pressure inside the small
trapped bubble cannot be set to atmospheric pressure but
should in principle depend on the bubble volume through
some equation of state. This changes markedly the nature of
the calculation. Moreover the trapped bubbles are extremely
FIG. 9. The initial phase of jet formation as seen in two simulations. For the
large bubble !dashed line, R/R / !144 000" the jet is relatively wide and well
resolved numerically. For smaller bubbles !solid line, R/R / !720" the jet
may become extremely thin.
3005
FIG. 10. Beginning of the entrapment of a bubble by the collapsing cavity,
for R/R / !105 !1.4 mm bubble".
small and very difficult to resolve without mesh adaptation.
Thus in most cases we continued the simulation without
marker surgery. When the trapped bubble is very small, the
marker chain reorganizes itself spontaneously and the calculation proceeds. In some cases, as in the rightmost bubble
entrapment region, it seems that the effect on the dynamics is
small. In other cases, as in the leftmost entrapment region,
FIG. 11. The velocity of the interface on the axis for R/R / !2.88&105 . The
oscillations correspond to the arrival of a train of capillary waves. For this
large value of R/R / capillary waves are numerous and of short wavelength.
The very large excursion in velocity may be due to the existence of a further
bubble entrapment region, however the very small scales involved make
numerical resolution difficult.
3006
Duchemin et al.
FIG. 12. Non-dimensional jet velocity as a function of the nondimensional
bubble radius. The two regions between vertical straight lines correspond to
the radii for which a bubble is trapped at the base of the jet.
FIG. 13. Maximum pressure on the axis of symmetry when the jet reaches
the mean water level. As in the previous figure the bubble entrapment regions are marked.
B. Maximum pressure on the axis of symmetry
the calculation has to be stopped or provides unreliable results which were removed from the quantitative analyses below.
We have redone all the calculations for a different initial
condition. The overall configuration is the same as on Fig. 2
but the rim thickness is halved. All the above qualitative
results are identical. In particular, we do not observe any
steepening of the capillary waves or thinner jets as we reduce
the rim size. This is a clear indication that the small length
scales we observe form spontaneously, independently of initial conditions.
We have computed the maximum pressure on the axis of
symmetry when the jet reaches the mean water level. Figure
13 shows this pressure and a fit in (R/R / ) "1 . Once more, the
numerical result is in good agreement with the scaling law
for radii between 2&104 and 106 times the viscous-capillary
length. We also remark a small jitter about the straight line
on the right-hand side of the curve, perhaps as a result of the
singular behavior in the bubble entrapment region. Note
again that in the left-hand side of the curve we could not
reliably calculate pressure.
A. Jet velocity
C. Radius of the first ejected drop
A first quantity of interest is the velocity of the jet, or the
ejection speed of the first drop. Figure 12 shows the nondimensional velocity of the jet, measured when the top of the
jet reaches the mean water level. Circle symbols correspond
to the larger rim thickness as on Fig. 2 while triangle symbols correspond to thinner rims. Apart from a vertical shift,
the measured velocities are very similar. This shift may in
part be explained by the fact that we measure the jet velocity
at the mean water level for both cases, which is at a different
distance from the base of the two cavities.
For a large range of radii !between 2&104 and 106 times
the viscous-capillary length R / ", the numerical results are in
good agreement with the inviscid scaling. For small radii the
velocity starts decreasing as R decreases. For the smallest
radii we have investigated the cavity relaxes to a flat surface
shape without jet formation.
The regions where bubbles form at the base of the jet are
indicated as vertical lines in Fig. 12. In the leftmost region,
around R/R / !103 , for the reasons discussed above, there is
a gap in data points. It is thus possible that much higher jet
velocities may be reached in that region.
Experimental data obtained by Spiel et al.7 tend to show
that the radius of the first ejected drop is about one-tenth the
radius of the initial bubble.
We have obtained this radius from the numerical simulations as follows. The computation stops when the jet thickness reaches the size of one computational cell. The jet rupture will occur soon thereafter. The volume enclosed by the
free-surface between this point of minimum thickness and
the tip of the jet is then a good approximation of the volume
of the ejected droplet. The equivalent radius R d is defined as
the radius of a spherical droplet with the same volume.
Figure 14 shows R d /R. For large R/R / we obtain a linear trend R d 00.13R which is consistent with the experimentally observed value of R/10. This linear behavior is consistent with the viscosity-independent regime of Eqs. !7" and
!8" in which the only length scale is R. On the other hand,
there is a large fraction of the data where this regime does
not hold and the ejected drop radius is much smaller than
R/10. Notice again the gap in values around R/R / !103 .
There the jet was too thin to be well-resolved numerically,
and the actual droplet size may be much smaller. Varying the
FIG. 14. Ratio of the radius of the first ejected drop and the radius of the
initial bubble as a function of R/R / .
initial condition has little effect, except at small radii where
the thinner rim leads to a larger droplet.
VI. SINGULAR JET FORMATION BY CURVATURE
REVERSAL
The formation of a thin, high-velocity jet in and around
the first bubble entrapment region leads to suspect the existence of a singularity. The scaling !3" yields
h ! u,t " ! ! t 0 "t " 2/3 f ! u ! t 0 "t " "2/3" ,
3007
FIG. 15. Comparison between successive profiles made non-dimensional
using the scaling - !x/(t 0 "t) 2/3 described in the text. Left, unrescaled profiles. Right, rescaled profiles. There the opening angle of the cavity is
around 73°. Since the numerical method uses adaptive time stepping, the
profiles are not separated by equal time intervals. The first time distance
from the singularity is t 0 "t!32.2 ms, the last one is 7.67 ms.
our case the picture seems different. The self-similar solution
!10" is observed in the entire first bubble entrapment band.
On the other hand, this solution is not seen in or around the
second band of bubble entrapment, where we should in principle also have a singularity. However the shape of the interface is very different in that case !Fig. 16" and a superposi-
!9"
& ! u, v ,t " ! ! t 0 "t " 1/3* ! u ! t 0 "t " "2/3, v! t 0 "t " "2/3" , !10"
where h is the surface elevation and u!r s sin ' the distance
to the axis of symmetry !r s being the spherical radius". We
rescaled the radial and vertical coordinates of the surface
points by (t 0 "t) 2/3 for R/R / !720. We determined t 0 by
fitting two of the rescaled profiles onto one another. The
results are shown on Fig. 15.
All the profiles have been translated vertically in order
for the point on the axis of symmetry to be at the same
vertical coordinate. The rescaled profiles superimpose well at
small values of the similarity variable -. The shape of the
profiles closely resembles the experimental and numerical
profiles in other types of flow.10,12,15 However at a large distance from the singularity the cone angle is about 73°. This
should be compared with the angle of the cavity seen in the
McIntyre data shown on Fig. 3. There, on profile 6 we measure an angle of 68°, a small difference with our calculations.
In contrast, the other physical processes discussed in the introduction yield relatively larger angles.
The finite viscosity should also introduce a discrepancy
with the theoretical similarity solution. It seems however that
its effects are small in that case.
In Ref. 9 it was shown that for Faraday waves there was
a connection between bubble entrapment and singularities. In
FIG. 16. Shape of the interface on the edge of the second bubble formation
region at R/R / !57 600. In that case a rescaling of the type shown on Fig.
15 could not be found. As in the previous figure profiles are not separated by
uniform time intervals. The time of the first profile is t!1.284 ms, the last
one is t!1.298 ms.
3008
Duchemin et al.
tion using the above rescaled variables could not be found. A
possible explanation is that the conical cavity associated with
the singularity is present here only on the large scale as seen
on Fig. 10. The convergence of small-scale capillary waves
is not able by itself to generate a self-similar conical flow.
Thus the conical flow would be at least as important as
bubble formation in producing the surface-tension driven
self-similar scaling.
A tantalizing possibility is the existence of further bands
of bubble entrapment and singularities to the right of the
second band. While bubble entrapment is observed in some
cases, details of the dynamics are not well-enough resolved.
It is likely that bubble entrapment and cusp singularities are
related to the amplitude of the converging capillary waves.
As viscosity is reduced, ever more capillary waves are observed to converge on the axis. For very large values of
R/R / , waves having both short wavelength and small amplitude are formed.
VII. CONCLUSIONS
We have presented a numerical study of the bursting
process of bubbles at a free surface. The scheme used was
based on an accurate description of the free surface with the
help of a markers chain. This method has shown good capabilities to resolve small capillary waves. The large scale features of the dynamics, the pressure and final droplet radius
may be predicted with accuracy, except near the first bubble
entrapment region near R/R / !103 . The predictions are
quantitatively in agreement with experiment: the angle of
opening of the cavity is similar to the angle observed in the
experiments of MacIntyre and the size of the droplet at the
tip of the jet is close to the experimentally reported size.
The measurements of jet velocity near R/R / !103 show
a surprisingly large velocity. The interface shape scales with
a characteristic length r. ! t"t 0 ! 2/3 predicted by the balance
of surface tension and inertia. The shape of the interface
resembles shapes found in other jet-forming flows and cusp
singularities, but has quantitative differences such as the
opening angle of the conical cavity.
The connection of this scaling with bubble entrapment is
less clear. We found the scaling in a wide region. The occurrence of self similar flow and an approximate singularity is
not connected to the exact boundary of a bubble entrapment
band. We also found bubble entrapment transitions which
were not associated to the ! t"t 0 ! 2/3 scaling. Finally the angle
of the conical cavity agrees with the experimental data for
bursting bubbles, but not with the angles seen or predicted in
other flows. This indicates that other types of singularities,
corresponding to different topologies or initial conditions,
may be observed. Further work should explore in detail the
nature of these singularities using for instance mesh refinement.
Also of interest would be a study of the influence of the
initial shape of the bubble. We have shown that a factor of
two change in the rim thickness had no qualitative effect, and
very little quantitative effect on the collapse process. However other changes in the initial condition may cause a
change in the position of the various singularities. In other
words, for a given radius, it would be possible to reach a
singularity by changing the shape of the bubble.
1
M. Coantic, ‘‘Mass transfer across the ocean-air interface: small scale
hydrodynamic and aerodynamic mechanisms,’’ Physico Chemical Hydrodyn. 1, 249 !1980".
2
J. M. Boulton-Stone and J. R. Blake, ‘‘Gas bubbles bursting at a free
surface,’’ J. Fluid Mech. 254, 437 !1993".
3
J. M. Boulton-Stone, ‘‘The effect of surfactants on bursting gas bubbles,’’
J. Fluid Mech. 302, 231 !1995".
4
S.-C. Georgescu, J.-L. Achard, and E. Canot, ‘‘Jet drops ejection in bursting gas bubble processes,’’ Eur. J. Mech. B/Fluids 21, 265 !2002".
5
M. Rudman, ‘‘A volume-tracking method for incompressible multi-fluid
flows with large density variations,’’ Int. J. Numer. Methods Fluids 28, 357
!1998".
6
D. C. Blanchard, ‘‘The size and height to which drops are ejected from
bursting bubbles in seawater,’’ J. Geophys. Res. 94, 10999 !1989".
7
D. E. Spiel, ‘‘On the births of jet drops from bubbles bursting on water
surfaces,’’ J. Geophys. Res. 100, 4995 !1995".
8
F. MacIntyre, ‘‘Flow patterns in breaking bubbles,’’ J. Geophys. Res. 77,
5211 !1972".
9
M. S. Longuet-Higgins, ‘‘Bubbles, breaking waves and hyperbolic jets at a
free surface,’’ J. Fluid Mech. 127, 103 !1983".
10
B. W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, ‘‘Singularity dynamics in curvature collapse and jet eruption on a fluid surface,’’ Lett. Nature
403, 401 !2000".
11
M. S. Longuet-Higgins and H. Oguz, ‘‘Critical microjets in collapsing
cavities,’’ J. Fluid Mech. 290, 183 !1995".
12
H. N. Oguz and A. Prosperetti, ‘‘Bubble entrainment by the impact of
drops on liquid surfaces,’’ J. Fluid Mech. 203, 143 !1990".
13
M. S. Longuet-Higgins, ‘‘An analytic model of sound production by raindrops,’’ J. Fluid Mech. 214, 395 !1990".
14
M. Rein, ‘‘The transitional regime between coalescing and splashing
drops,’’ J. Fluid Mech. 306, 145 !1996".
15
D. Morton, L. Jong-Leng, and M. Rudman, ‘‘An investigation of the flow
regimes resulting from splashing drops,’’ Phys. Fluids 12, 747 !2000".
16
J. B. Keller and M. J. Miksis, ‘‘Surface tension driven flows,’’ SIAM !Soc.
Ind. Appl. Math." J. Appl. Math. 43, 268 !1983".
17
D. C. Blanchard and L. D. Syzdek, ‘‘Film drop production as a function of
bubble size,’’ J. Geophys. Res. 93, 3649 !1988".
18
S. Popinet and S. Zaleski, ‘‘A front tracking algorithm for the accurate
representation of surface tension,’’ Int. J. Numer. Methods Fluids 30, 775
!1999".
19
S. Popinet and S. Zaleski, ‘‘Bubble collapse near a solid boundary: A
numerical study of the influence of viscosity,’’ J. Fluid Mech. !in press".
20
Y. Toba, ‘‘Drop production by bursting of air bubbles on the sea surface
!ii" theoretical study on the shape of floating bubbles,’’ J. Oceanogr. Soc.
Jpn. 15, !1959".
PRL 96, 124501 (2006)
week ending
31 MARCH 2006
Singular Jets and Bubbles in Drop Impact
Denis Bartolo,1,* Christophe Josserand,2,† and Daniel Bonn1,3,‡
1
Laboratoire de Physique Statistique de l’ENS, 24 Rue Lhomond, 75231 Paris Cédex 05, France
Laboratoire de Modélisation en Mécanique, CNRS-UMR 7606, Case 162, 4 place Jussieu, 75252 Paris Cédex 05, France
3
van der Waals-Zeeman Institute, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
(Received 21 November 2005; published 27 March 2006)
2
We show that when water droplets gently impact on a hydrophobic surface, the droplet shoots out a
violent jet, the velocity of which can be up to 40 times the drop impact speed. As a function of the impact
velocity, two different hydrodynamic singularities are found that correspond to the collapse of the air
cavity formed by the deformation of the drop at impact. It is the collapse that subsequently leads to the jet
formation. We show that the divergence of the jet velocity can be understood using simple scaling
arguments. In addition, we find that very large air bubbles can remain trapped in the drops. The surprising
occurrence of the bubbles for low-speed impact is connected with the nature of the singularities, and can
have important consequences for drop deposition, e.g., in ink-jet printing.
PACS numbers: 47.55.D!, 47.20.Cq, 47.20.Dr
Droplets impacting on solid surfaces have fascinated
scientists and even artists ever since the first pictures
became available [1]. From a practical point of view, the
control of the impact dynamics is of prime importance:
droplet deposition is a key factor in many industrial processes. Pertinent examples are ink-jet printing [2], sputtering, spray painting [3], deposition of pesticides on plant
leaves [4], etc. For most, if not all, of the applications, one
wants to efficiently deposit a droplet, without spilling over
any of the droplet material and without trapping air
bubbles.
From a fundamental point of view, the last ten years have
witnessed an enormous progress in the understanding of
free-surface flows. In particular, the description of the surface singularities that are ubiquitous in these problems has
greatly advanced [5]. Experimental and numerical studies
have taken advantage of the recent progress in high-speed
imaging techniques and computational power, respectively, to determine the self-similar behavior close to the
singularities. Examples of much recent interest are drop
formation [6–8], surface driven flows [9], and bubble collapse [10]. However, complete understanding of the drop
impact problem remains nontrivial due to a complicated
multiscale interplay between capillarity, viscous forces,
inertia, and the properties of the solid surface [3,11].
In this Letter, we find that a surprising outcome of this
interplay is that an ultrafine, high-speed jet can emerge
from the drop, sending out part of the drop material elsewhere. In addition, for a certain range of impact velocities,
the jet formation is accompanied by entrapment of air
bubbles. Both the jet formation and the bubble entrapment
are understood as a consequence of the formation of an air
cavity that subsequently closes up in a singular (selfsimilar) fashion. We find that the cavity can close in two
different ways, and, consequently, identify two distinct collapse singularities. A detailed analysis and theoretical interpretation of the singularities allow an understanding of
0031-9007=06=96(12)=124501(4)$23.00
the main features of these high-speed and ultrathin emitted
jets.
We study the impact of water droplets on a hydrophobic
surface. The water drops (of density ! " 1000 kg m!3 ,
surface tension " " 72 mN=m) are made using a precision
needle which allows for a controlled release of spherical
drop of radius RI " 1 mm. The impact velocity Vimpact of
the droplets is varied simply by increasing the fall height.
We use a superhydrophobic solid surface with a contact
angle for water of # # 160$ [12]. The results described
below are robust against viscosity changes (at least up to
10 times the water viscosity using water-glycerol mixtures), small variations of the drop radius or hydrophobicity: we obtain in fact very similar results for less hydrophobic, but still nonwetting surfaces (Parafilm and PDMS
elastomer). However, due to contact line pinning on the
solid, there is more scatter in the data; we will therefore
restrict our discussion here to the superhydrophobic surfaces. Since the impact and subsequent jet formation are
extremely rapid, we follow the drop impact using an ultrahigh speed video system (Phantom V7), and use frame
rates of 60 000 and 100 000 fps. Figure 1 depicts three
series of snapshots of drops impacting at three different
velocities (0.45, 0.56, and 0:68 m % s!1 ). For all the cases
studied here, the general dynamics of the impact consists in
the spreading of the drop deformed by capillary waves
followed by a retraction phase leading to the formation
of a jet and finally to the total rebound of the droplet.
Since we use a low-viscosity liquid, viscous effects
can be neglected in our experiments; the drop dynamics
is therefore dictated by the Weber number We, which
compares the inertial to the capillary forces, We "
2
=". In the experiments We ranges between 0.6
!RI Vimpact
and 16: this corresponds to small deformations on the scale
of the droplets, as is indeed observed in the experiment.
The jets, on the contrary, correspond to very large deformations, but have a very small characteristic size, Fig. 1.
124501-1
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As the drop hits the surface, capillary waves are excited,
propagate along the surface, and deform the drop into a
pyramidal shape. The oscillations of the capillary waves
then lead to the formation a cylindrical air cavity located at
the center of the drop along the vertical axis [Fig. 1(a)]. For
the lowest impact speed, Fig. 1(b), the cavity retains its
cylindrical shape all the way until its collapse. Conversely,
for intermediate impact speeds, the bottom-up symmetry is
broken almost immediately [see Fig. 1(c)]: the top of the
cavity retracts faster than the bottom. As the liquid surface
reconnects at the top, a bubble is entrapped. In Fig. 1(d),
for higher impact speeds, the bottom of the cavity detaches
itself from the solid surface prior to the cavity collapse,
which leaves no bubble behind. By additionally filming
some of the droplets from above, we determine whether
dewetting due to the cavity formation occurs. We observe
in fact rupture of the liquid film between the air cavity and
the solid surface for 0:37 < Vimpact < 0:55 m=s. The drop
then adopts a toroidal shape. In all cases, the collapse of the
air cavity is immediately followed by the violent ejection
of a thin jet. For a low-viscosity fluid, the only intrinsic
length scale in the problem is the wavelength of the capillary waves excited by the impact (i.e., the characteristic
step size of the pyramidal drop). This length is on the order
2
of $ # "=&!Vimpact
' [13], an order of magnitude larger than
the radius of the smallest jets.
We therefore need to understand the anomalously small
size and large velocity of the jets. The velocity is determined by measuring the position of the top of the jet on
four successive images when the jet emerges from the
drop. We find that this velocity, Vjet can be 40 times larger
than the impact speed: Vjet " 18 m % s!1 for Vimpact "
0:45 m % s!1 . The variation of the jet speed, Vjet , and of
the jet radius, Rjet , are nonmonotonic and highly nontrivial,
40
I
III
II
1.2
R jet /R I
FIG. 1. Rapid camera snapshots. Impact of water drops (radius
RI " 1 mm) on a superhydrophobic surface. (a) Formation of
the cylindrical air cavity: impact velocity Vimpact " 0:45 m % s!1 .
Pictures taken 2.3, 2.7, and 3.4 ms after the impact time.
(b) Vimpact " 0:45 m % s!1 . Pictures taken 3.8, 4.0, and 4.3 ms
after the impact. (c) Impact velocity Vimpact " 0:56 m=s.
Pictures taken 3.8, 4.2, and 4.4 ms after the impact. Note the
presence of a trapped air bubble. (c) Impact velocity Vimpact "
0:68 m % s!1 . Pictures taken 3.7, 4.2, and 4.9 ms after the impact.
as shown in Fig. 2. We identify three main regions, separated by dotted lines and denoted I, II, and III.
Region I.—At small impact velocities, when increasing
the impact velocity, the jet radius decreases rapidly
whereas its velocity appears to diverge as the impact
velocity approaches VI-II . In parallel, the jet radius, measured just after its formation, becomes very small.
Region II.—This region corresponds to very small values of the jet radius and high jet velocities, and a rather
complicated behavior of the jet velocity is observed as a
function of Vimpact . The latter is due to a subtle interplay
between the rupture of the liquid film and entrapment of air
bubbles. When approaching VII-III , we observe bubble entrapment as shown in Fig. 1(b) for Vimpact ( 0:55 m=s, as
indicated by the black arrow in Fig. 2. The entrapment is
clearly due to the closing of the drop top surface before the
air can escape from the cavity, neither from the top before
the cavity closes, nor through the bottom since the liquid
film does not open [14]. Our experiment thus suggests that
two conditions need to be fulfilled for entrapment: the
cavity closes at the top and the air is not evacuated from
the bottom.
Region III.— For ‘‘high’’ impact velocities a decrease in
radius and a diverging jet velocity are found upon decreasing the impact velocity. As Vimpact > VII-III air bubbles are
not trapped in the liquid anymore.
The high-speed movies (and Fig. 1) clearly show that the
jet is formed due to the collapse of the air cavity. To
understand what fixes the jet’s length and velocity scale,
we thus need to consider the collapse dynamics in detail.
This allows us to show that the jet structure with its
anomalous length and velocity scales is fully determined
by the collapse dynamics of the cavity.
The jets ejected for Vimpact close to VI-II result from the
pinch-off singularity of an air cylinder immersed in a low
viscosity liquid. Investigating the collapse dynamics in
detail [between images 1 and 3 on Fig. 1(b)], we observe
that the air thread exhibits self-similar behavior characteristic of a finite-time singularity, with a character-
30
Vjet /Vimpact
PRL 96, 124501 (2006)
20
0.8
0.4
0.0
0.2
10
0.7
VI (m/s)
1.2
0
0.2
0.4
0.6
Vimpact
0.8
1
1.2
(m/s)
FIG. 2. The jet velocity normalized by the impact velocity is
shown as a function of Vimpact . The inset shows the jet radius.
Three regions are identified separated by dotted lines at VI-II "
0:45 and VII-III " 0:65 ms!1 . Open symbols correspond to the
same experiments as in Figs. 1(a)–1(c). Inset: Jet radius normalized by the initial drop radius plotted versus the impact speed.
124501-2
PRL 96, 124501 (2006)
week ending
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istic exponent for the vanishing of the cavity radius R "
A&tc ! t'0:5)0:01 , with A " 0:02 ) 0:003 m % s!1=2 , see
Fig. 3. This dynamics is found almost but not completely
up to the singularity point. Rather, very close to the singularity the air cylinder suddenly breaks down at some critical radius Rc . This loss of the cylindrical symmetry of
the cavity depends in fact on the initial velocity Vimpact .
This is an incredibly rapid process: even when filming at
100 000 fps, the air cavity disappears in a single image.
Similar cylindrical pinch-off was described recently in
bubble pinch-off in different liquids [15] and for cavity
closure after impacts of steel balls on dilute granular
materials [16]. For the latter, the pinch-off is followed by
jet formation as is also the case here. For the former, a
similar and equally surprising deviation from the selfsimilar behavior close to the singularity was reported.
The pinch-off dynamics for all three cases is given by the
Rayleigh-Plesset equation [6,17] for cylindrical freesurface flows. With the assumption that the air viscosity
can be neglected it reads for an inviscid fluid:
! "
R
1
"
P&r' ! P0
2
!
_
" &RR * R ' ln
* R_ 2 !
; (1)
r
2
!R
!
The relation between the jet radius and the jet velocity can
be inferred using a second conservation law. The continuity
equation for the kinetic energy fluxes at the onset of the jet
3
shooting reads: 12 !R_ 3 &t'-2%R&t'h. " 2 12 !Vjet
-%R2jet ., using Eq. (2), we obtain
! "
A2 Rjet !1
:
(3)
Vjet +
2RI RI
Equations (2) and (3) correctly predict the scaling relations
between Vjet , Rjet , and Rc : Vjet / &Rc =RI '!2 and Vjet /
&Rjet =RI '!1 , in good agreement with the experimental
data. These results are strongly substantiated by the complete absence of free fitting parameters, since A is known
2
10
V
jet
(m/s)
where P is the pressure, R&t' is the radius of the cavity, and
r is a large-scale cutoff, on the order of the drop radius. For
low-viscosity fluids, the pinch-off dynamics is dominated
by inertia and the first term on the right-hand side. If indeed
all other terms can be neglected, this implies that the
prefactor of the logarithm is zero, leading immediately to
R&t' / A&tc ! t'1=2 for the radius of the cylinder [18], in
excellent agreement with the experimental data. In addition to the correct scaling, the prefactor A can be estimated
using simple dimensional analysis: A + &"RI =!'1=4 . This
leads to A + 1:6 , 10!2 m % s!1=2 , in good agreement
with the experimental value.
We can now use our understanding of the cavity dynamics to account for the jet velocity. In Fig. 4 we show Vjet as a
function of the radius of the air cavity at rupture Rc and
similarly as a function of the jet radius Rjet . Clear power
law relations over more than one decade of Vjet are observed: Vjet / &Rc =RI '!1:9)0:2 , where Rc is the cavity radius just prior to the collapse, and Vjet / &Rjet =RI '!0:9)0:2 .
To understand these relations between the jet velocity
and radius on the one hand, and the critical cavity radius on
the other hand, we propose a simple semiquantitative
model based on mass and kinetic energy fluxes, neglecting
both viscous and capillary effects. At the onset of cavity
rupture, the pinch-off dynamics described above leads to a
_ "
velocity of collapse of the air cylinder equal to R&t'
1=2
2
A=-2&tc ! t' . " A =&2Rc '. Recalling that the bulk velocity field is steady for inertia dominated cylindrical
collapse, we can use a balance equation through the boundary defined by the radius Rv and the height of the deformed
drop, taken here to be on the order of the initial drop radius
_ "
RI . Then, mass conservation gives: -2%R&t'h.R&t'
2
2-%Rc .Vjet , and h # RI which leads to
! "
A2 Rc !2
Vjet "
:
(2)
2RI RI
1
10
0
10
-1
10
-2
10
-1
10
(R ,R )/R
c
FIG. 3. Radius of the cylindrical cavity R plotted versus
jt ! tc j. Triangles: drop impact speed Vimpact " 0:43 m % s!1 .
Circles: Vimpact " 0:46 m % s!1 . Squares: Vimpact " 0:52 m % s!1 .
Lines: power law fits.
jet
0
10
I
FIG. 4. Open circles: Jet velocity plotted versus the jet radius
normalized by the initial drop radius, Rjet =RI . Dashed line:
theoretical prediction obtained from Eq. (2). Filled circles: Jet
velocity plotted versus the cylindrical cavity radius before collapse normalized by the initial drop radius Rc =RI . Full line:
theoretical prediction obtained from Eq. (3).
124501-3
PRL 96, 124501 (2006)
0.8
0.14
week ending
31 MARCH 2006
FIG. 5. Self-similar structure of the jet. Impact velocity:
Vimpact " 0:687 m % s!1 . Left: twelve superimposed jet profiles
for different times (time interval 15 &s). First profile taken
0.17 ms after the collapse of the air bubble. Axes units: mm.
Right: the same curves have been rescaled by a factor of &t !
tc '2=3 for the x and y coordinates, following the scaling behavior
expected for self-similar dynamics in a capillary-inertial regime
(arbitrary units for the rescaled profile plots).
features of this are the anomalously small lengths and high
velocities: very narrow jets are formed that shoot out with
speeds up to 40 times the drop impact velocity. As is the
case for all singular jets in liquids, granular media, for
Faraday waves, etc., the jet formation is preceded by the
collapse of an air cavity. It is this singularity that is
responsible for the selection of the anomalous length and
velocity scales of the jets. We have related here, for the first
time, the collapse dynamics of the cavity to the divergence
of the speed of the jet. An open question remains what
intrinsic length scale provides the cutoff mechanism for the
singularity. In our experiment, close to VI-II the maximum
velocity of the jet is bound by the minimum radius the
cavity can reach. This turns out to be rather different from a
microscopic length scale: the cylindrical cavity destabilizes at + 50 &m, the reason for which remains unclear.
from the void collapse dynamics. Figure 3 compares the
theoretical predictions with the experimental results; given
the simplicity of our model, the agreement with the experimental data is quite satisfactory not only for the scaling
exponents but also for the numerical prefactors.
The last observation that needs to be explained is then
the divergence of the jet velocity upon approaching VII-III .
Experimentally, the transition between regions II and III
can also be identified by the topological change from a
simply connected topology to a multiply connected topology, revealed by the bubble trapping. The divergence of the
velocity and the topological change bear a striking similarity to the capillary-inertial singularities observed for
driven Faraday waves [19] and for bubbles bursting at
the free surface of a liquid [10]. For these two cases, the
singular behavior can be accounted for by balancing capillary and inertial forces. It was shown, see, e.g., [19], that
all length scales behave self-similarly as jt ! tc j2=3 , with tc
the collapse time. Our camera is not fast enough to follow
the ultrafast collapse dynamics of the air cavity prior to the
jet formation [Fig. 1(d)]. However, we did manage to
monitor the shape of the jet profile for Vimpact slightly
above VII-III , and, consequently, probe the dynamics after
the singularity. Theoretically there is no reason why the
length scales should not follow the same jt ! tc j2=3 behavior after the singularity. However, this post-singular regime
has hardly ever been investigated in detail [20]. Figure 5
shows the rescaling of the different jet profiles taken at a
regular time interval. The different profiles converge reasonably well onto a single master curve, in agreement with
the idea that the dynamics after the singularity is still
governed by capillary-inertial self-similar dynamics. The
rather surprising conclusion is, therefore, that although the
jet velocity diverges upon approaching both VI-II and
VII-III , the underlying collapse singularities leading to the
jet divergence is fundamentally different in the two cases.
In summary, we have shown here that gentle drop impacts can lead to singular behavior. The most surprising
*[email protected]
†
[email protected]
‡
[email protected]
[1] A. Worthington, Proc. R. Soc. London 25, 261 (1876).
[2] D. B. Dam and C. Le Clerc, Phys. Fluids 16, 3403 (2004).
[3] R. Rioboo, M. Marengo, and C. Tropea, Atomization and
Sprays 11, 155 (2001).
[4] V. Bergeron, D. Bonn, J.-Y. Martin, and L. Vovelle, Nature
(London) 405, 772 (2000).
[5] J. Eggers, Rev. Mod. Phys. 69, 865 (1997).
[6] A. Prosperetti and H. N. Oguz, Annu. Rev. Fluid Mech. 25,
577 (1993).
[7] I. Cohen, M. P. Brenner, J. Eggers, and S. R. Nagel, Phys.
Rev. Lett. 83, 1147 (1999).
[8] R. F. Day, E. J. Hinch, and J. R. Lister, Phys. Rev. Lett. 80,
704 (1998).
[9] J. Keller and M. Miksis, SIAM J. Appl. Math. 43, 268
(1983).
[10] L. Duchemin, S. Popinet, C. Josserand, and S. Zaleski,
Phys. Fluids 14, 300 (2002).
[11] M. Rein, Fluid Dyn. Res. 12, 61 (1993).
[12] F. Bouamrirene (private communication).
[13] Y. Renardy et al., J. Fluid Mech. 484, 69 (2003).
[14] This is very different from the bubble formation considered in Mehdi-Nejad et al., where the bubble is entrapped
in a very early stage of the impact due to the drop deformation.V. Mehdi-Nejad, J. Mostaghimi, and S. Chandra, Phys. Fluids 15, 173 (2003).
[15] J. C. Burton, R. Waldrep, and P. Taborek, Phys. Rev. Lett.
94, 184502 (2005).
[16] D. Lohse, R. Bergmann, R. Mikkelsen, C. Zeilstra, R. van
der Meer, M. Versluis, J. van der Weele, M. van der Hoef,
and H. Kuipers, Phys. Rev. Lett. 93, 198003 (2004).
[17] M. S. Plesset and A. Prosperetti, Annu. Rev. Fluid Mech.
9, 145 (1977).
[18] This scaling is valid up to logarithmic corrections; see
J. M. Gordillo et al., Phys. Rev. Lett. 95, 194501 (2005).
[19] B. Zeff, B. Kleber, J. Fineberg, and D. Lathrop, Nature
(London) 403, 401 (2000).
[20] J. E. Hogrefe, Physica D (Amsterdam) 123, 183 (1998).
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EUROPHYSICS LETTERS
1 February 2006
Europhys. Lett., 73 (3), pp. 363–369 (2006)
DOI: 10.1209/epl/i2005-10398-1
Granular pressure and the thickness of a layer jamming
on a rough incline
C. Josserand, P.-Y. Lagrée and D. Lhuillier
Laboratoire de Modélisation en Mécanique, UMR CNRS 7607
Boı̂te 162, Université Pierre et Marie Curie - 75252 Paris cedex 05, France
received 6 June 2005; accepted in final form 30 November 2005
published online 21 December 2005
PACS. 45.70.Ht – Avalanches.
PACS. 45.70.-n – Granular systems.
PACS. 83.80.Fg – Granular solids.
Abstract. – Granular media with a compaction between the random loose and random close
packings have very specific features. Among them is the “disorder” pressure which strongly
depends on the solid fraction and is pertinent for moving media only. The concept of a granular
pressure depending on the solid fraction is not unanimously accepted because dense granular
media are often in a frozen state which prevents them from exploring all their possible microstates, a necessary condition for defining a pressure and a compressibility unambiguously.
While periodic tapping or cyclic fluidization have already been used for that exploration, we
here suggest that a succession of flowing states with velocities slowly decreasing down to zero
can also be used for that purpose. And we propose to deduce the pressure in dense and flowing
granular media from experiments measuring the thickness of the granular layer that remains
on a rough incline just after the flow has jammed.
Introduction. – The existence of a pressure in granular media is the simplest way to
represent their stiffness. When the concentration of grains is above the random close packing,
the granular medium acts as a poro-elastic solid, its pressure is a function of the solid fraction
and it involves the elastic constants of the material the grains are made of (see, e.g., [1]).
When the granular medium has a smaller compaction, in the range between the random loose
and random close packings, the grains can be considered as rigid and the expression of the
granular pressure is far less evident. The difficulty comes from the glassy behaviour which
makes it quite usual to find these granular media in a frozen state concerning their compaction.
Hence the feeling that the granular pressure is largely dependent on the way the medium was
prepared. However, experiments have been conducted which aim at allowing the granular
medium to reach a steady and quasi-equilibrium state concerning its solid fraction. These
experiments relied on regular tappings [2,3] or cyclic fluidization [4] favouring the exploration
of a maximum of microstates. A systematic exploration of the microstates can also be achieved
starting from a flowing granular medium, and slowly reducing its velocity down to rest. As a
consequence, we suggest that some of the experiments on rough plates (inclined with angle θ)
c EDP Sciences
!
364
EUROPHYSICS LETTERS
which led to define the thickness hstop (θ) which remains after the flow has jammed [5], can
also be used to infer the relation between granular pressure and solid fraction.
The main features of the elastic stress tensor will be reviewed, then those of the “disorder” pressure. How these two contributions combine to build the total granular pressure is
debated thereafter. We then deduce the link with the hstop experiments and discuss how the
experimental data must be handled to deduce an expression for the granular pressure.
The stress generated by interparticle forces and velocity fluctuations. – Consider a large
" α (t) and velocity V
" α = dR
" α /dt, submitted to
number of particles with mass mα , position R
αβ
"
the forces F
exerted by all other particles β. The stress tensor of this granular assembly is
known to be [6]
!" " R
#
"β − R
"α
" α) −
⊗ F" αβ δ("x − R
2
α β!=α
#
!"
"α−V
" ) ⊗ (V
"α−V
" ) δ("x − R
" α) ,
mα (V
−
σ=
(1)
α
" α ) is a Dirac function located
where the brackets # $ represent a statistical average while δ("x − R
"
at the center of particle α and V is the mean velocity of the grains. For a non-cohesive granular
medium the interparticle forces are null except when particle β is in contact with particle α.
The above general expression was used in numerical simulations to obtain the stress tensor
of a moving granular medium. In a steady shear flow with a shear ∂Vx /∂z, the components
σxz and σzz of the above stress tensor were found to be rate dependent, with a (∂Vx /∂z)2
behaviour (see the second contribution in eq. (4) for example). However, above some minimal
concentration of the grains, a rate-independent contribution was also found. According to
Aharonov and Sparks [7], Silbert et al. [8], O’Hern et al. [9] and Head and Doi [10], this
rate-independent contribution, called pelastic henceforth, vanishes below a value φc which is
slightly smaller than the random close packing φM and which depends on the microscopic
friction between the grains [9] as well as on the orientation of the flow relative to gravity [8].
Above this threshold packing needed to observe the elastic stiffness of the granular assembly,
the elastic pressure was found to depend on the volume fraction φ and to obey the scaling law
pelastic ∼ E(φ − φc )α ,
(2)
where E is the bulk modulus of the material the particles are made of, while α = 1 for
Hookean contact forces and α = 3/2 for Hertzian ones. Due to the very large value of E, the
compressibility of the granular medium is almost infinite for volume fractions less than the
threshold φc and vanishingly small above it. It is thus not surprising that numerical simulations based on expression (1) of the stress tensor predict an almost uniform volume fraction
φ ≈ φc all over the granular material. However, many experiments exist on surface flows of
dense granular materials in which the solid fraction profile was observed to be non-uniform
(see, for example, [11, 12] and [13]) and extended in the range between the random loose
packing φm at the surface and some higher volume fraction (presumably φc ) far from it. Is it
possible that a second contribution exists to the rate-independent pressure, giving the flowing
granular material a finite compressibility for solid fractions in the range between φm and φc ?
The configuration pressure and the disorder pressure. – Consider a large volume containing many rigid spheres with a high enough volume fraction φ̃ for a contact network to
invade the whole volume. Let Ω(φ̃)dφ̃ be the number of different spatial configurations of
C. Josserand et al.: Granular pressure and the thickness etc.
365
these spheres in the range between φ̃ and φ̃ + dφ̃. To belong to Ω(φ̃), a configuration must
display a large enough number of contacts, but no forces at the contact points, hence giving
no contribution to the mechanical stress (1). The density of micro-states Ω(φ̃) is thus a purely
geometric concept. If instead of rigid spheres we were considering soft ones, we would say we
are counting the number of configurations with zero energy (“incipient” contacts), yet able
to resist an infinitesimal external pressure load. This density of states presumably vanishes
below a minimum compaction φmin ' 0.40 (the gel threshold) and above the maximum compaction φmax ' 0.74 (the most compact crystalline configuration). At some intermediate
volume fraction the density of microstates displays a large maximum value. This intermediate compaction with the maximum number of microstates is likely to be the loosest random
packing φm . According to Onoda and Liniger [14] its value for spherical grains is φm = 0.555.
Introduce now a number P which represents a non-dimensional measure of the configuration
pressure and define the partition function
Z(P ) =
$
φmax
eP φ̃ Ω(φ̃) dφ̃.
φmin
The mean volume fraction is then related to the configuration pressure in the form φ =
d
dP Log Z. This relation can be used to obtain both P (φ) and the variance of the density
fluctuations #(φ̃ − φ)2 $ = dφ/dP . Another useful quantity is the configuration entropy S =
Log Z − P φ . This configuration entropy is a function of the mean volume fraction and
the non-dimensional configuration pressure is nothing but P = − dS
dφ . Many works starting
from Kanatani [15] and revived by Edwards and Oakeshott [16] strived to find an explicit
form for the configuration entropy. The general trends of their results are the following:
for a vanishing pressure, the mean volume fraction is the one with the maximum number
of microstates, i.e. φm , while for an infinite pressure, the volume fraction is φmax and the
compressibility dφ/dP vanishes. The simplest expression meeting these conditions is the one
deduced from the entropy of the lattice-gas model (see, e.g., [17])
P ∼ Log
φmax − φm
.
φmax − φ
Note that the above pressure stems from the total number of different configurations, including
both random and crystalline ones. It is also possible to select random configurations only. In
this case one introduces the random close packing φM above which all configurations display
some cristalline order. For spheres, it is generally admitted that φM = 0.635. A plausible
expression for the disorder pressure is pdisorder ∼ Log(φM − φm )/(φM − φ) an expression
restricted to the range φm < φ < φM . The gradient of the disorder pressure acts as a diffusion
force which pushes the grains towards regions of smaller volume fractions, those with a larger
number of microstates belonging to Ω(φ̃). For the grains to have a chance to explore all
microstates, the best solution is a steady flow. This is why the concept of a disorder granular
pressure is pertinent for dynamic situations only. The disorder pressure confers the granular
medium a compressibility which decreases when the volume fraction increases. With the
above lattice-gas expression for the pressure, the compressibility is proportional to φM − φ, in
agreement with the experimental results of Nowak et al. [2] but not with those of Schroeter et
al. [4], which display a minimum of the compressibility for a compaction between φm and
φM . Besides its φ-dependence, a second issue to be considered is the scaling of the disorder
pressure. Since neither the grain elastic properties nor the Brownian motion is involved in
pdisorder , one must discard any elastic modulus or the thermal energy as candidates. We are
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EUROPHYSICS LETTERS
thus led to write the disorder pressure as
pdisorder = P0 Log
φM − φm
,
φM − φ
(3)
where P0 is some yet undetermined characteristic pressure. This constitutive relation for
the disorder pressure will hold in all circumstances. The characteristic pressure is expected
to compare with the self-weight pressure that exists under several granular layers and which
leads to volume fractions below φM . In contrast, the pressure load considered in soil mechanics
compares with the elastic bulk modulus and leads to volume fractions higher than the random
close packing. In that domain, the notion of disorder pressure is irrelevant.
The rate-independent granular pressure. –
σzz has been modelled as [18–20]
In a steady shear ∂Vx /∂z the normal stress
σzz = p(φ) + ρp D2 µN (φ)(∂Vx /∂z)2 ,
(4)
granular pressure
where µN (φ) is a function of the solid fraction which depicts the relative importance of dilatancy effects, ρp is the mass density of the grains and D is the grain size. We are here
interested in the rate-independent contribution to the normal stress. We would like the granular pressure p(φ) to represent as far as possible all what was said above about the disorder
pressure and the elastic pressure. It is clear that p(φ) would coincide with pdisorder if φc were
equal to φM . Unfortunately, this equality appears to hold only for horizontal flows [8] and
frictionless grains [9]. In all other cases, φc is slightly smaller than φM and because of the
tremendous increase of pelastic above φc , the granular pressure will strongly increase (and for
us will diverge) at φc instead of φM (see fig. 1). For this reason we will test two different
40
hstop /D
35
30
25
20
15
10
5
φm
Fig. 1
φ
φc φM
30
31
32
33
θ
34
35
36
37
Fig. 2
Fig. 1 – Schematic representation of the elastic pressure (dashed line) and disorder pressure (plain
line) as a function of the compaction in the range between the random loose packing φm and the
random close packing φM . The disorder pressure vanishes below φm and diverges at φM while the
elastic pressure vanishes below φc and strongly increases above.
Fig. 2 – Dependence of hstop /D on the inclination θ. Comparison between experimental results
(points) for sand over carpets (from [22]) and fitting curves deduced from the granular pressure (6)
(plain) and (5) (dashed). Experimental results with hstop /D ≤ 3.6 were discarded and the value
θmax ≈ 36.1 was adopted.
367
expressions for the granular pressure:
φc − φ m
φc − φ
(5)
(φ − φm )n
,
(φc − φ)m
(6)
p(φ) = P0 Log
and
p(φ) = P0
with positive but yet unknown exponents m and n for the second expression. Note that these
expressions for the rate-independent granular pressure hold in the very small solid fraction
range φm < φ < φc , and for flowing media only. Expression (5), with φM instead of φc , was
already adopted in previous works [18–20]. We now acknowledge this replacement of φc by
φM was not justified and we want to test the pertinence of the granular pressure (5) or (6) in
an unsteady flow that leads to jamming.
The maximum thickness of a granular layer that jams on a rough incline. – Consider
a layer of granular material flowing down a rough inclined plate. Upon gently reducing the
inclination with a constant thickness h, the layer ultimately stops at some angle θ. Since the
flow velocity was slowly reduced to zero, the granular layer had time to explore a lot (if not
all) of the microstates involved in pdisorder . It is thus likely that the peculiar jammed state
which the layer arrives at is described by the rate-independent pressure p defined above. The
mechanical equilibrium of the freshly jammed layer is thus expressed by
0=−
∂p
+ φρp g cos θ and
∂z
tan θ = min [µ(z)] ,
(7)
where the z-axis is orthogonal to the free surface of the layer and points downwards while µ is
the macroscopic friction coefficient (which is possibly non-uniform over the layer thickness),
and g is the acceleration of gravity. Substituting expression (5) or (6), one deduces that the
solid fraction profile φ(z) increases from φm at the free surface up to values close to φc at a
depth of order L with
P0
.
(8)
L=
φc ρp g cos θ
Concerning the particular expression (5), the whole profile is exponential-like and given by
φ(z) =
1+
φc
z .
− 1)e− L
( φφmc
(9)
A similar exponential profile was already observed in experiments [11–13] and was also obtained in simulations of a frustrated lattice-gas model [21]. On the contrary, recent numerical
simulations [8] which do not see any rate-independent pressure below φc predicted a flat profile
at this volume fraction. We come back to this discrepancy in the concluding section.
Knowing the compaction profile, let us now focus on the layer thickness h. Da Cruz [22]
has deduced from numerical simulations of dense flows a very important (and a bit counterintuitive) result: the macroscopic friction coefficient decreases almost linearly with the solid
fraction and can be written as
µ = µmax − (µmax − µmin )
φ − φm
.
φc − φm
(10)
Since the compaction increases with the distance from the free surface, the minimum value of
µ happens very close to the rough plate, that is to say for z ' h. Just after jamming, we thus
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EUROPHYSICS LETTERS
φ(h
)−φ
m
have: tan θ = µmax − (µmax − µmin ) φstop
. For the special expression (5) and related
c −φm
profile (9), one obtains a rather simple expression,
%
&
hstop
φc µmax − tan θ
= Log 1 +
.
(11)
L
φm tan θ − µmin
When the granular pressure is given by (6) there is no simple analytical result for hstop but
two asymptotic results:
'
(m
'
(n
µmax − µmin
hstop
φc µmax − tan θ
hstop
≈
≈
and
,
(12)
L
tan θ − µmin
L
φm µmax − µmin
which hold when tan θ is slightly larger than µmin and slightly smaller than µmax , respectively.
Experimental results. – Systematic measurements of the layer thickness were initiated
by Pouliquen [5] who obtained the thickness hstart (θ) for an initially static layer and hstop (θ)
for an initially flowing layer. Since the disorder pressure is relevant to moving media only, we
are interested in hstop exclusively. The experimental results were fitted with [5, 22]
%
&
µmax − µmin
µmax − tan θ
hstop
hstop
= Log
=
,
(13)
or
BD
tan θ − µmin
BD
tan θ − µmin
where D is the grain size while B is a number to be deduced from experiments. To deduce the
bulk granular pressure from these experiments on rough inclines, one must discard experiments
performed with relatively smooth plates, and more generally those for which the curves hstop (θ)
are strongly modified upon changing the plate roughness. And concerning those with a high
enough roughness, we must exclude some boundary layer of thickness δ and consider h−δ as the
relevant thickness for bulk behaviour. Accordingly, we were led to discard all the experiments
performed with glass beads because the friction generated by the beads glued on the incline
is only slightly larger than the friction in the bulk. But we considered as significative the
experiments with sand flowing on carpets of various roughnesses. And for these experiments
with sand we discarded a boundary layer with thickness estimated to δ ' 4D. Because of the
scarcity of data, we could not fully discriminate between expression (5) and expression (6)
with m = 1 and n = 1 (see fig. 2). However, it was possible to give an order of magnitude to
the reference pressure P0 . If it can be taken for granted that hstop scales with the grain size
D, then the reference pressure must scale with ρp gD and in fact, the fit with experimental
data of sand over carpets was obtained with
P0 ≈ 5 ρp gD ,
together with µmax ≈ 0.7 and µmin ≈ 0.5.
(14)
Conclusions. – A rate-independent granular pressure depending on solid fraction was
proposed long ago for poro-elastic media [1]. It corresponds to pelastic and vanishes below
the threshold solid fraction φc . The statistical properties of granular materials in the range
between the random loose packing φm and the random close packing φM [15, 16] can be represented by a second rate-independent pressure pdisorder . Since φm < φc ≤ φM , we proposed
that granular media with solid fraction in the range φm < φ ≤ φc can be endowed with a
granular pressure p(φ) similar to pdisorder but diverging at φc instead of φM . It must be
stressed that for pdisorder to be observed, the granular material must be able to explore all its
microstates. This is certainly the case of steady flows and in fact, with expression (5) for the
369
granular pressure, we have found already very good agreement with the experimental velocity
and solid fraction profiles observed in steady free surface or confined flows [19, 20]. In the
present letter we assumed that the exploration of microstates can also be achieved in unsteady
flows provided their velocity is smoothly decreasing down to zero and we proposed to infer the
granular pressure from measurements of hstop (θ). With expression (5) we predicted result (11)
which fits quite well with the experimental results. Moreover the scaling of hstop with the
grain size D has confirmed our previous guess [19, 20]: the granular pressure (and pdisorder )
scales with ρp gD. We have thus many reasons to believe in the existence of a granular pressure
of entropic origin in the range φm < φ ≤ φc . Thereby we do not understand why up to date
simulations are unable to observe pdisorder and the related finite compressibility for solid fractions between φm and φc . We have, however, a suggestion. As observed in some experiments
(e.g., [23]), the mean velocity and the granular temperature reach their steady state much more
rapidly than the solid fraction. It is thus possible that the number of time-steps performed
in the simulations is not large enough to let the solid fraction reach its true steady profile.
∗∗∗
The authors appreciate interactions and debates with R. Ball, F. Chevoir, D. Dean,
J. Gollub, T. Harsley, D. Head, V. Kumaran, M. Nicodemi, J. N. Roux, G. Tarjus
and P. Viot. DL thanks the Kavli Institute for Theoretical Physics for hospitality and
support through the Granular Physics Program GRANULAR05.
REFERENCES
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57 (1998) 1971.
[3] Philippe P. and Bideau D., Europhys. Lett., 60 (2002) 677.
[4] Schroeter M., Goldman D. I. and Swinney H. L., Phys. Rev. E, 71 (2005) 030301.
[5] Pouliquen O., Phys. Fluids, 11 (1999) 542-548.
[6] Irving J. H. and Kirkwood J. G., J. Chem. Phys., 18 (1950) 817.
[7] Aharonov E. and Sparks D., Phys. Rev. E, 60 (1999) 6890.
[8] Silbert L. E., Ertas D., Grest G. S., Halsey T. C., Levine D. and Plimpton S. J., Phys.
Rev. E, 64 (2001) 051302.
[9] O’Hern C. S., Silbert L. E., Liu A. J. and Nagel S. R., Phys. Rev. E, 68 (2003) 011306.
[10] Head D. and Doi M., preprint (2005).
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[12] Rajchenbach J., Adv. Phys., 49 (2000) 229.
[13] Ancey C., Phys. Rev. E, 65 (2001) 011304.
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[15] Kanatani K., Lett. Appl. Eng. Sci., 18 (1980) 989.
[16] Edwards S. F. and Oakeshott R. B., Physica A, 157 (1989) 1080.
[17] Witten T. A., Structured Fluids (Oxford University Press) 2004, sect. 2.
[18] Savage S. B., J. Fluid. Mech., 377 (1998) 1.
[19] Josserand C., Lagrée P.-Y. and Lhuillier D., Eur. Phys. J. E, 14 (2004) 127.
[20] Josserand C., Lagrée P.-Y. and Lhuillier D., Trends in Applications of Mathematics in
Mechanics, edited by Wang Y. and Hutter K. (Shaker Verlag, Aachen) 2005, pp. 157-164.
[21] Nicodemi M., Coniglio A. and Herrmann H. J., J. Phys. A, 30 (1997) L379.
[22] Da Cruz F., Ecoulements de grains secs: frottement et blocage, PhD Thesis, ENPC 2004.
Available from http://pastel.paristech.org/archive/00000946/.
[23] Charru F., Mouilleron H. and Eiff O., J. Fluid Mech., 519 (2004) 55.
Morphodynamic modeling of erodible laminar channels
Olivier Devauchelle, Christophe Josserand,
Pierre-Yves Lagrée and Stéphane Zaleski
Institut Jean le Rond d’Alembert, Université Pierre et Marie Curie - Paris 6,
CNRS-UMR 7190, Case 162, 4 place Jussieu, 75252 Paris Cédex 05-France
A two-dimensional model for the erosion generated by viscous free-surface flows, based on the
shallow-water equations and the lubrication approximation is presented. It has a family of selfsimilar solutions for straight erodible channels, with an aspect ratio that increases in time. It is
also shown, through a simplified stability analysis, that a laminar river can generate various bar
instabilities very similar to those observed in natural rivers. This theoretical similarity reflects the
meandering and braiding tendencies of laminar rivers indicated by F. Métivier and P. Meunier,
(Journal of Hydrology, 27 1, pp. 22-38 (2003)). Finally, we propose a simple scenario for the
transition between patterns observed in experimental erodible channels.
PACS numbers:
I.
INTRODUCTION
Natural rivers seldom form straight beds. Instead, they
usually develop braids or meanders as a consequence of
current-induced sediment transport. The understanding
of such river sedimentation mechanisms can also help to
characterize the spatial heterogeneity of alluvial rocks,
which is a key parameter when simulating aquifer flows
or oil traps in petroleum reservoirs [1]. The theoretical
work of [2], [3] and [4] introduced among geomorphologists the fruitful idea that such patterns may originate in
the linear instability of the flow, bed and bank system.
Two-dimensional turbulent shallow water equations associated with a simple sediment transport law are able
to predict the formation of alternate bars in channels of
constant width. Such bars have been commonly accepted
as a key phenomenon for braids and meander formation
[5]. Numerous refinements of this theory may be found in
the literature: [6] performed the bar stability analysis in
three dimensions, while [5] focused on the differentiation
between braids and meanders. Later [7] and [8] relaxed
the rigid-banks hypothesis. [9] and more recently [10]
modeled the non-linear evolution of free bars. All these
works (and to our knowledge, every study in this field)
considered turbulent flows, which is entirely legitimate as
far as natural rivers are concerned (the average Reynolds
number of the Seine river in Paris is about 106 ). However,
one should not conclude from this ubiquity of turbulence
that braiding and meandering are inherently turbulent
phenomena. [11] very recently accumulated experimental
evidence showing that laminar flumes may also generate
many patterns created by real rivers. In particular, the
constant flow of a thin liquid film down an homogeneous
granular bed initially crossed by a straight channel exhibits rather complex pattern dynamics as the flume is
deformed by erosion (see [12]). First, the channel widens
while remaining straight. Then a meander-like instability develops, which deforms both the bed and the banks.
Eventually, more bars develop in the middle of the channel and the river starts to braid [32]. This behavior is
qualitatively comparable to the one of larger channels, at
higher Reynolds number (see the two meters-wide experiment of [13]). To our knowledge, no quantitative experimental results have been published about river erosion
instabilities in the laminar regime. As a consequence, the
results presented here can only be compared to the qualitative evolution described in [12]. Reference to turbulent
experiments can only illustrate the sound similarity with
the laminar case.
Our objective is to comfort the idea that micro-rivers
can be an intermediate step toward the understanding of
natural rivers morphodynamics. We do not claim that
quantitative results from micro-rivers could be extrapolated to field results (see section II A). We are rather
convinced that such small-scale experiments share with
larger ones many features still under investigation (nonlinearity of the flow-sediment interaction, equilibrium
shape of the bed, behavior and influence of the bank).
Such laminar flow approach can also help to disentangle
the role of the turbulence in the river morphodynamics.
Moreover, theoretical as well as numerical river models
could be easily tested against micro-rivers data, before
adding the complexity of turbulence and switching to
larger experiments and natural rivers.
In a first section, a two-dimensional evolution model
for laminar flumes is presented. It is based on the assumption that the velocity profile is close to Nußelt’s one.
A rather general erosion law is then discussed and compared to the experiments of [14]. The following section
is devoted to the study of a straight river widening process, and an analytical solution is proposed in a simple
case. In the third section, the linear stability analysis of
a straight laminar flume with solid banks is presented.
II.
A TWO-DIMENSIONAL MODEL
Let us consider an experiment during which an initial channel incised into a uniform and non-cohesive sand
layer is eroded by a viscous flow. If the slope of the sand
bed remains small enough, one may use two-dimensional
2
d
v
(1) allows us to define the horizontal shear stress vector
τ at the bed surface:
"
u
∂!
u ""
(2)
≈ 3ρw ν
τ = ρw ν
"
∂z z=h
d
u
z
x
h
y
FIG. 1: Sketch of a riverbed: h is the elevation of the sand
surface and d is the water depth. The axes x and z are tilted
with respect to horizontal. The Saint-Venant approximation
is used for the velocity field u = (u, v).
equations to model both the water flow and the sediment
transport. A rather general assumption (commonly used
in river mechanics) consists in the time-scale separation
between the flow and erosion process: the bed evolves
slowly enough for the flow to be quasi-static (see [5], [4],
[6] and [8]). Of course, this hypothesis fails during such
violent events as roll-waves .
In the present article the following notations are used
(see also figure 1):
• x and y are the coordinates in the plane of the
experiment, the first aiming toward the main slope.
z is the coordinate normal to the plate;
• h is the elevation of the sand surface and d is the
water depth (η = h + d is thus the water level);
• u = (u, v) is the vertically averaged water velocity,
the horizontal water flux components being ud and
vd;
• S is the plate tilt;
• g is the magnitude of gravity, and ν is the kinematic
viscosity of water.
Secondary currents are thus neglected, although many
authors believe they strongly influence erosion in developed meanders (see for example [17]). The effect of secondary currents is sometimes taken into account in the
shallow water framework by mean of an ad hoc term in
sediment transport equations (see [8]). Since the present
study is restricted to straight channels, we will hereafter assume that the curvature of the flow remains small
enough for the secondary currents to remain negligible
(this argument is developed by [18]). This approximation is actually correct for any curvature, provided the
Reynolds number is low enough.
The vertical integration of the Navier-Stokes equations, associated with (1), leads to the viscous shallow
water equations (sometimes named the Saint-Venant’s
equations):
3ν
6
(u · ∇)u = g (−∇(d + h) + Sex ) − 2 u,
5
d
(3)
∇ · (ud) = 0,
(4)
where ex is the unit vector parallel to the x-axis. These
equations are very similar to those used for turbulent
rivers. The only differences lie in the coefficient 6/5 which
becomes 1 in the turbulent case, and in the friction term
−3νu/d2 which becomes −Cf $u$u/d (Cf being a friction coefficient, related to the Chézy coefficient). One
cannot thus expect micro-rivers to be scaled models for
natural ones, since the laminar flow equations cannot be
reduced to the classical turbulent ones. On the other
hand, it is interesting to point out similarities and differences between these two different (although not too far)
cases, turbulent and laminar.
B.
A.
The present micro-river model requires that the water flow is laminar, so that it can be approximated by a
vertical velocity profile of Nußelt type. For this assumption to hold, the Reynolds number Re = u0 d0 /ν must
remain low enough (d0 and u0 are the typical height and
! is thus
velocity scales respectively). The water velocity u
approached by a parabolic velocity profile which adapts
instantaneously to the topography:
! (x, y, z, t) ≈
u
Sediment transport
Water flow
3
ξ(2 − ξ)(u(x, y, t), v(x, y, t), 0),
2
(1)
where ξ = (z − h)/d. This method corresponds to the
lubrication approximation. Different approaches may be
found in [15] or [16], though in one dimension. Equation
The river bed evolves under the influence of both erosion and avalanches. In the present context erosion consists of flow-induced bed-load transport of sand grains.
On the other hand, avalanches are collective phenomena
triggered by an excess slope of the sand surface. In the
continuous model developed here, we can only handle
the average effects of erosion and avalanches. This approximation allows for the definition of a total volumic
sediment flux q(x, y, t) integrated along the vertical direction. Assuming a strong time scale separation between
erosion and avalanches, one may consider the associated
fluxes (respectively qe and qa ) as independent. The continuity equation for sand then reads:
∂h
= −∇ · q,
∂t
(5)
3
where q = qe + qa . Finally, closure relations have to
be deduced, either on dimensional, physical or empirical
grounds in order to link (5) to the flow equations.
Erosion contribution. Most of the relations between
the sediment flux and the flow are proposed in the literature as functions of the Shields number θ, which expresses
the ratio between hydrodynamic forces exerted on a grain
to its apparent weight :
θ=
$τ$
,
ds (ρg − ρw )g
(6)
where ds , ρw , ρg and τ are respectively the typical particle diameter, the density of water, the density of a grain
and the bottom shear stress. As suggested by [19], we
propose the following expression as a classical relationship (see the review of [20]) for small slope:
#
$
τ
qe = φ(θ)
− G · ∇h ,
(7)
$τ$
where φ is a growing function that may include a threshold value, and G is a diagonal operator.
To determine a plausible form for φ(θ) we shall use recent experimental results obtained by Charru, Mouilleron
and Eiff for grain transport in the viscous flow regime[14].
Their results on grain transports are partially reproduced
on figure 2 and these authors suggest then the following
transport law :
Np d2s
= 0.85 θ(θ − 0.12)H(θ − 0.12),
Vs
(8)
where Np is the particle flux, and Vs is the Stokes settling velocity of a particle (Vs = gd2s (ρs − ρw )/(18νρw )).
H is the Heavyside function. Np is linked to q through
q = Np V, where V is the volume filled by a particle in
the sediment layer. According to this expression, no sediment is transported at Shields number values below a
threshold. However, [14] indicates that some particles
remain in motion at Shields numbers lower than 0.12 during a transition regime, and will eventually settle after an
“armoring time [. . . ] very large compared to the hydrodynamic time scales”. Maybe due to this armoring time ta ,
their measurements of the sediment flux do not exactly
vanish below the threshold (see figure 2). According to
[14], a typical dimensionless value for ta is 105 . In the
present notations, the ratio between the erosion typical
time scale T defined in section III A reads
#
$β
ρw
d0
ta
5 γV
S
= 10
,
(9)
T
ds d20 ρs − ρw ds
which is typically much larger than one. The armoring
phenomenon is a possible explanation for thresholds in
transport laws. Since it occurs at time scales much larger
than the erosion ones, it is tempting to use a pure power
law function instead of formula (8), as already proposed
earlier to model sediment transport under turbulent flow
(such as [10] for instance). Such a law may be adjusted
to fit the data of [14] (see again figure 2) and it gives:
Np d2s
= 5.13 θ3.75 .
Vs
(10)
Relations (8) and (10) cannot be in fact clearly separated
by the experiments of [14]. Thus, for simplicity reasons,
we will use the second one in what follows. This choice
will be discussed again in sections III and IV.
The above discussion suggests that the sediment transport measurements proposed by [14] should be used with
great care when dealing with erosion patterns formation
by laminar flows: equilibrium state may not be reached
if erosion is intense enough. This question also arises in
the study of real rivers, but transient sediment transport
is far out the scope of the present study. The general
form of the erosion law is then taken as:
φ(θ) = φ0 θβ ,
(11)
reminding that φ0 ≈ 5.13 VVs /d2s and β ≈ 3.75 to fit the
data of [14]. These values are fixed only as an illustrative
case in the sequel.
The second term in (7) reproduces the slope-induced
deviation of the sediment flux. [10] sets G = γI where γ
is a constant of order one. This isotropic approximation
is questionable, but should not influence qualitatively the
results. This term is mathematically essential to cut-off
short wavelength instabilities (see Section IV B). According to the definition of the bottom shear stress (2), the
sediment transport equation (7) becomes
qe = Ee
#
$u$
d
$β #
$
u
− γ∇h ,
$u$
(12)
with Ee = φ0 (3ρw ν/(2gr(ρg − ρw )))β .
Avalanches. The full dynamics of avalanches is far out
the scope of this study. Instead, we may propose a simple
model which reproduces the following features:
• the sand mass is conserved through the avalanche
process;
• there are no avalanches under a critical slope α;
• above the critical angle, qa is directed toward the
main slope and increases with the slope value.
Considering these criteria, we propose the following expression :
qa = −Ea F ($∇h$)
∇h
,
$∇h$
(13)
where F (·) = (·−α)H(·−α) and Ea is a constant. Indeed,
a similar law has been successfully employed for aeolian
dunes by [21].
Finally, it is important to notice here that these fluxes
qe and qa do not account for the saltating grain dynamics. In a simplified approach, the grains motions would
4
0.03
0.025
0.02
Np ds 2 0.015
""""""""""""""""""""""
Vs
0.01
0.005
0
0.1
0.05
0.15
Θ
0.2
0.25
FIG. 2: Different transport laws compared with the experimental results obtained by [14]. The grains are transported by a viscous flow in a circular Hele-Shaw cell. Np is
the particle flux, and Vs is the settling velocity of a particle. Dashed : threshold law proposed by [14], Np ds /Vs =
0.85 θ(θ − 0.12)H(θ − 0.12). Solid line : power-law fit,
Np ds /Vs = 5.13 θ3.75 . This law will be used as an illustrative
case in the present study.
end up into a settling distance at which the fluxes develop
(see [22] and references herein for a discussion of these
terms). It manifests in the dynamics through a phase
shift between the shear stress and the fluxes. By sake
of simplicity, we do not take into account such a term
although it could be implemented easily. Such approximation corresponds somehow to a limit where the density
ratio between grains and water is high. In the following,
it is in fact remarkable that the instability exists without
such phase shift.
C.
Boundary conditions
Flow equations (3) and (4) together with sediment
transport equations (5), (12) and (13) form a closed system. To solve this system in the fixed domain Ω, conditions must be specified on its boundary ∂Ω. Their general
form writes
In the above equations, c is the normal velocity of the
frontier ∂Ω+ (t), the subscripts + and − denotes quantities evaluated respectively inside and outside Ω+ (t). d/dt
is the convective derivative at a point of ∂Ω+ (t) moving
with velocity cn. The first boundary condition reflects
the time scale separation between flow and erosion (so
that the condition for the normal velocity of the boundary is zero instead of c). The following ones correspond
to the sediment mass conservation equations integrated
over a small domain crossed by ∂Ω+ (t). A special case
of (15) has been derived by [23].
The classical conditions for non-erodible and impermeable banks are obtained from (15) by setting c = 0. In
that case, and for turbulent flow, bar instabilities may
develop (see [10] for stability analysis and weakly nonlinear theory of bars). The present paper shows that
bar instabilities of the same nature may also develop in
laminar conditions. To switch from bars to meanders
and braids, the condition c = 0 must be relaxed. In a
seminal paper, [8] used an empirical estimation of c as
a function of the additional stress induced by secondary
flows. They also implicitly assumed that the bank material input due to erosion had no influence on the bed
evolution (they set q+ · n = 0 despite a finite value for c).
They showed that meandering results from the interaction between alternate bars instabilities and the so-called
bend instability, which results from the curvature of the
bank. For micro-rivers, their hypothesis would not hold,
since bed and banks are of the same granular material.
The elaboration of a bank evolution law able to model
the effect of avalanches is the subject of on-going work.
The present stability analysis (section IV) is restricted
to channels with rigid banks (c = 0), as were the first
equivalent studies in the case of real rivers (see [4]). On
the other hand, in the case of a prismatic river (section
III), equations are solved on the whole Ω domain, and
thus no boundary conditions are required. For the full
determination of the solution moreover one has to prescribe global boundary conditions on the upstream and
downstream fluxes of water.
III.
λu d + µu u · n = πu ,
λh h + µh q · n = πh ,
where λu , µu , πu , λh , µh and πh are functions to be
specified. n is the 2D unit vector normal to ∂Ω, aiming
outward. In the general case, Ω may include sub-domains
where q = 0. In such domains, the evolution equation
becomes ∂h/∂t = 0.
If one wants to restrict the analysis to the active subdomain Ω+ (t) where q %= 0, the conditions to be imposed
on its mobile boundary ∂Ω+ (t) are
u · n = 0,
PRISMATIC CHANNELS
(14)
q+ · n = c(h+ − h− ),
dh+
dh−
+
= −∇ · q+ + c (n · ∇h+ + n · ∇h− ) . (15)
dt
dt
For a straight, x-invariant river, the equations derived
in section II turn into a one dimensional non-linear diffusion equation which admits self-similar solutions. The
reader interested in the problem of real turbulent river
cross-section, a complex two-dimensional problem in the
general case, may refer to [23–27] among others.
A.
A non-linear diffusion equation
For a prismatic river, any quantity only depends on
time and the transverse coordinate y. The flow equations
5
(3) and (4) thus become
u(y, t) =
gS
d(y, t)2 ,
3ν
x-axis falls before any equilibrium can be reached, due
to bar instabilities ([7] and [29] removed the meandering tendency by using a half-river). To our knowledge,
no river-widening experiment where carried out in the
laminar regime at a fixed water level.
v = 0,
d(y, t) + h(y, t) = η(t).
(16)
B.
The water discharge
Qw =
%
∞
(ud)dy
(17)
−∞
is usually fixed in experiments, and thus governs the evolution of η(t). For the sake of simplicity, we will consider
a different case in what follows. If η instead is fixed, this
arbitrary constant may be set to zero (and thus h = −d).
This case represents a river supplied by an infinite reservoir. The sediment transport equations (5), (12) and (13)
lead to
#
$
∂
∂h∗
∂h∗
=
(−h∗ )β
∂t∗
∂y∗
∂y∗
# #
$
#
$$
1 ∂
∂h∗
∂h∗
−
F $
$ sign
. (18)
-a ∂y∗
∂y∗
∂y∗
In the above equation, the starred quantities are dimensionless. The initial depth d0 is the characteristic length.
The typical widening time scale is defined by
T =
d20
γφ0
#
ds (ρg − ρw )
ρw Sd0
$β
.
(19)
The non-dimensional number -a = d20 /(T Ea ) compares
typical avalanches flux to erosion ones. It will be considered very small in what follows.
The non-linear diffusion equation (18) may be solved
numerically. A classical first-order finite-volume scheme
leads to the solution presented in figure 3 at different
times. The initial width is w∗,0 = 2.5, and -a is fixed
to 0.1. The value of -a has a weak influence on the
result, provided it is small (the same computation performed with -a = 0.01 gives similar results). However,
the Courant, Friedrichs, and Levy condition imposes a
numerical time step smaller and smaller as the value of -a
is reduced. The erosion law is fixed by setting β = 3.75.
The influence of any other parameter of the problem,
such as the Froude number or the channel slope is embedded in the definition of the time and space scales.
Trough erosion, the river widens and gets shallower,
while its cross-section area remains constant. This is
in qualitative accordance with experiments during which
the water outflow was fixed, instead of the water level
in the present theory (see [12], [28], [7] and [29]). Since
the erosion law (10) presents no threshold, this widening
process will never stop in the framework of this model,
which may seem unreasonable. Some authors ([7], [29])
managed to reach an equilibrium width, but in most experiments ([12], [13]) the channel invariance along the
Self-similar solutions
If avalanches are neglected, or if the transverse slope
of the channel ∂h/∂y can remain always smaller than
the critical slope α (so that no avalanche occurs), the
last term of (18) drops. This particular case has simple
self-similar solutions of the form
h∗ (y∗ , t∗ ) =
1
1/(β+2)
t∗
f (χ).
(20)
1/(β+2)
. Then (18) leads to
#
$
χf
∂
β ∂f
(−f )
= 0.
+
∂χ
∂χ β + 2
where χ = y∗ /t∗
(21)
If fs is a symmetrical solution to (21), dfs /dχ = 0 at
χ = 0, and thus integration of (21) gives
fs (χ) =
 #
$1/β
)

β
−
χ2
if χ ∈ [0, A 2(β+2)
]
β
A − 2(β + 2)

0
elsewhere,
(22)
where A is a constant. Let A∗ be the (non-dimensional)
area of the cross-section. Then
% ∞
A∗ =
h∗ (y∗ , t∗ )dy∗
0
1/β+1/2
= −A
%
0
q
2(β+2)
β
#
β
ξ2 − 1
2(β + 2)
$1/β
dξ.
(23)
Thus when avalanches can be neglected, (18) admits a
set of self-similar solutions parameterized by their crosssection area. The solution corresponding to A∗ = −2.5
is represented on figure 3. Despite its rectangular initial
cross-section, the numerical solution converges towards
its self-similar counterpart. This behavior seems quite
general: it very weakly depends on the initial conditions
or the value of β.
Only for β = 1 (that is for an unrealistic linear erosion
law) does the self-similar solution behave regularly at the
banks. In that case, the river cross-section is a parabola.
It width increases as t1/3 while it shallows as t−1/3 . If the
initial shape is flat enough to avoid avalanches, this condition holds at any time. Unfortunately this case cannot
model erosion patterns formation, since it is unconditionally stable (see section IV).
On the other hand, if β > 1 the picture is quite different. The continuous widening process still holds: the
6
consider a rectangular base state with solid-wall boundaries, of width w0 and depth d0 (its aspect ratio is thus
R = w0 /d0 ). The boundary conditions at the bank are
u · n = 0 and q · n = 0. The basic water velocity is uniform and parallel to the x-axis (u0 = gSd20 /(3ν)), and so
is the basic sediment flux (q0 = Ee (u0 /d0 )β ). Let us seek
traveling-wave perturbations of this base state:
# $
y
ϕ(x, y, t) = ϕ0 + -ϕ∗
ei(kx/w0 −ωt/(γT R)) , (24)
w0
0
$0.2
t# %1000
$0.4
t# %100
d#
$0.6
t# %10
$0.8
t# %1
t# %0
$1
2
4
6
8
y#
FIG. 3: Widening of a straight laminar channel through erosion, modeled with (18). Parameter values are "a = 0.1,
α = 0.8, and β = 3.75. Solid lines: numerical solutions of (18)
at different times (with avalanches). Dashed line: self-similar
solution (without avalanches, see section III B) at t∗ = 10,
t∗ = 100 and t∗ = 1000. The presence of avalanches seems to
influence only a small zone near the bank. The main part of
the river section tends to the self-similar solution in any case.
width increases as t1/(β+2) , while the depth decreases as
t−1/(β+2) . However, in that case the bed slope dh∗ /dy∗
diverges at the banks. Thus avalanches must occur at the
banks, and the self-similar solution fails. This tendency is
observed in laboratory experiments (see [30] among others), and was already pointed out by [23]. The effect of
bank avalanches is difficult to quantify analytically. According to numerical simulations in the case β = 3.75
however (see figure 3), they do not seem to influence
strongly the bed evolution far enough from the banks.
Consequently one may still consider the results of the
self-similar theory as a good approximation of the full
system solutions.
IV.
LINEAR STABILITY
Experimental channels such as the one of [12] or [13]
often remain stable for a while, then develop meanders
which in turn are followed by more complex braided-like
patterns. This scenario of transitions (sometimes called
ageing) may be interpreted as the successive dominance
of different unstable modes. If the widening process presented in the previous section is slow enough, a straight
river may be chosen as a quasi static base state for a
stability analysis. This is what we will assume in the
following, so we will disregard any interaction between
widening and instabilities.
A.
Derivation of the dispersion relation
In order to present the simplest stability model which
keeps the essential features of channel stability, we will
where ϕ = (u, v, h, d, qx , qy ). The base state corresponds to ϕ0 = (u0 , 0, −d0 , d0 , q0 , 0) and the perturbation is ϕ∗ = (u0 u∗ , u0 v∗ , d0 h∗ , d0 d∗ , q0 qx,∗ , q0 qy,∗ ). T =
d0 w0 /(γRq0 ) is the characteristic erosion time defined in
section III A, and - is a small dimensionless amplitude of
the perturbation. k is a real dimensionless wave-number
whereas ω is complex in the general case. (3), (4), (12)
and (5) lead to the following system:
$
#
6 2
F ik + RS u∗ + ikh∗ + (ik − 2RS)d∗ = 0, (25)
5
#
$
6 2
dd∗
dh∗
F ik + RS v∗ +
+
= 0,
5
dy
dy
dv∗
= 0,
dy
ik(d∗ + u∗ ) +
−iωh∗ + ikqx,∗ +
qx∗ = βu∗ −
dqy,∗
= 0,
dy
ikγ
h∗ − βd∗ ,
R
qy∗ = v∗ −
γ dh∗
.
R dy
(26)
(27)
(28)
(29)
(30)
In the above system, F = u0 /(gd0 )1/2 is the Froude number of the unperturbed channel. Parameters F , S and R
may be varied independently in experiments. Indeed, if
Qw is the water outflow of the river, then
w0 = R
*#
Qw = 3R
3F ν
S
#
$2 +1/3
1
,
g
(31)
$1/3
(32)
9F 8 ν 5
gS 5
.
The full (F, R, S)-space may be explored by tuning the
slope of the apparatus, the initial width of the channel
and the water outflow. Of course, the validity of the
present theory requires the parameters to satisfy some
7
conditions. First, the flow has to be laminar. The low
Reynolds number condition can be easily checked:
3F 2
u0 d0
=
.
(33)
ν
S
Capillarity can also cause the failure of the theory[33].
Near the banks of the channel, capillarity generates a
meniscus of characteristic size lc (lc = (Γ/(ρg))1/2 , where
Γ is the surface tension). The quantity of water flowing
through the meniscus zone should remain negligible as
compared to the total outflow. As a crude approximation, the outflow in the meniscus zone Qw,m is evaluated
by Poiseuille’s formula : Qw,m ∼ gSlc4 /ν. The condition
Qw ( Qw,m thus reads
# $4
lc
R(
.
(34)
d0
Re =
The ratio of the water depth versus the capillary length
is given by
, ρ -1/2 # F ν $2/3
d0
= 32/3 g 1/6
.
(35)
lc
Γ
S
Consequently, (34) may be satisfied for any values of
R, F and S provided the viscosity of the fluid is high
enough[34]. Typical parameters values during the experiment of [12] (carried on with pure water) are Qw =
13 · 10−6 m3 s−1 , S = 0.088 and w0 = 0.1 m. The nondimensional number of the experiment thus are R ≈ 130,
F ≈ 2, Re ≈ 130 and d0 /lc ≈ 0.3. Condition (34) was
not satisfied in this experiment. However, we expect that
the error resulting from this failure should only affect the
evaluation of non-dimensional parameters from the experimental data, but the qualitative behavior predicted
by the theory should hold.
If the linearized transport equations (29) and (30) are
used to remove qx,∗ and qy,∗ from the mass conservation
equation (28), equations (25) to (28) become a system
of four equations with unknowns u∗ , v∗ , d∗ and h∗ . The
velocities u∗ and v∗ may then be expressed as functions
of d∗ and h∗ through the momentum equations (25) and
(26). The conservation of water and sediment mass thus
become a system with unknowns d∗ and h∗ , which in turn
can be reduced to
d4 h∗
d2 h∗
+
A
+ Bh∗ = 0.
dy 4
dy 2
(36)
In the above equation,
.
.
A = 36F 4 k 3 γ + 30F 2 k − 2k 2 γ
/
.
− ikR(1 + β + 4Sγ) + iRω + 25RS 2ik 2 γ
// . .
/ /
+ kR(−3 + β − 3Sγ) + Rω / 5 6F 2 k − 5iRS γ ,
# # #
$
6F 2 γ
1
3
k k γ−
+ ik 2 R(2β + 3Sγ)+
B=
γ
5
$$
/
1 .
2
2
i − 5 + 6F kRω + 3R Sω
. (37)
5
The boundary conditions state that both the water velocity and the sediment flux vanishe at the bank. Thus
equation (30) implies that
# $
#
$
dh∗
1
dh∗ 1
=
−
= 0.
(38)
dy 2
dy
2
Equation (26) then leads to dd∗ /dy = 0 at the banks.
Equation (25) implies in turn that du∗ /dy = 0 at the
banks. Due to equations (29) and (27) respectively,
dqx,∗ /dy = 0 and d2 v∗ /dy 2 = 0 at the banks. The
sediment mass conservation (28) finally imposes that
d2 qy,∗ /dy 2 = 0 at the banks. One may then deduce from
the second derivative of equation (30) that
# $
#
$
d3 h∗ 1
d3 h∗
1
=
−
= 0.
(39)
dy 3 2
dy 3
2
Let s1 , s2 , −s1 and −s2 be the four solutions of the
characteristic equation attached to (36), namely
s4 + As2 + B = 0.
(40)
Then, if C1,+ , C1,− , C2,+ and C2,− are four independant
constants,
h∗ = C1,+ es1 y + C1,− e−s1 y
+ C2,+ es2 y + C2,− e−s2 y
(41)
is a solution of (36). Such a solution must satisfy the
boundary conditions (38) and (39), that is
/
 .
s1 C.1,+ es1 /2 − C1,− e−s1 /2 +/




s.2 C2,+ es2 /2 − C2,− e−s2//2 = 0




s1 C.1,+ e−s1 /2 − C1,− es1 /2 +/



s.2 C2,+ e−s2 /2 − C2,− es2//2 = 0
.
(42)
s31 C.1,+ es1 /2 − C1,− e−s1 /2 +/



 s3 C2,+ es2 /2 − C2,− e−s2 /2 = 0

2


 s3 .C e−s1 /2 − C es1 /2 / +

1,−

1
.1,+
/

s32 C2,+ e−s2 /2 − C2,− es2 /2 = 0
The determinant of the system 42 reads
.
/2
4s21 s22 s21 − s22 sinh(s1 ) sinh(s2 ),
(43)
and vanishes only if s = inπ, where n is an integer (provided s21 and s22 are distinct). The integer n corresponds
to the number of roots of h∗ in the width of the river.
One may then derive the dispersion relation from (36):
.
.
.
/
ω = 5kR 5iRS − n2 π 2 (−3 + β) + 2k 2 β
.
//
.
/
− 6F 2 k 2k 2 β + n2 π 2 (1 + β) − i k 2 + n2 π 2
. 2
/..
/
6F k − 5iRS
− 5 + 6F 2 k 2 − 5n2 π 2
/ /1. . 2
/
− 15ikRS γ
R 6F k − 5iRS
..
/
//
− 5 + 6F 2 k 2 − 5n2 π 2 − 15ikRS . (44)
If A2 = 4B, then the roots s1 and s2 are equal. Similarly,
if A = 0, s1 is the opposite of s2 . In both cases however,
the boundary conditions impose again s = inπ, and the
dispersion equation (44) is still valid.
8
0.5
'1
0.0
Σ $0.5
$1.0
n%1
$1.5
n%2
$2.0
0.0
0.2
0.4
0.6
0.5
0.8
'2
0.0
Σ $0.5
$1.0
n%1
n%2
$1.5
$2.0
0.0
0.2
0.4
0.6
0.5
0.8
'3
0.0
Σ $0.5
$1.0
n%1
n%2
$1.5
$2.0
0.0
0.2
0.4
0.6
0.8
k
FIG. 4: Linear growth rate σ of bed instability in a laminar
river, versus the corresponding non-dimensional wave number k. The fixed parameters values are β = 3.75, γ = 1,
S = 0.0875. The Froude number and aspect ratio are varied
according to a straight river widening (see section IV B and
points Σi on figure 5). Above : R1 = 20.3 and F1 = 3.94;
middle : R2 = 35.0 and F2 = 3.21; below : R3 = 55.0 and
F3 = 2.71. For each set Σi = (Ri , Fi ), the solid curve corresponds to the mode n = 1, whereas the dashed one corresponds to the mode n = 2. The successive dominance of
modes provides an interpretation for the transition from alternate bars to braids observed experimentally by [12].
B.
Results interpretation
The linear stability of a channel depends on the sign of
the maximum growth rate over n and k, respectively the
transverse and longitudinal wave-numbers. We will thus
focus on the imaginary part σ of ω in what follows. Let
σm be the maximum growth rate, and km and nm the
corresponding wave-numbers (i.e. σm = σ(km , nm ) =
maxk∈R,n∈N (σ)). The transverse wave-number n characterizes the instability pattern: n = 0 for y-invariant
dunes (this mode can also initiate step-pool instability),
n = 1 for meanders and n > 1 for braided patterns. The
present theoretical framework fails to predict the steppool instability often observed in narrow channels [31],
as σ is always negative for n = 0. This is not surprising for the phase-shift between the bed deformation and
the water shear stress is neglected here (this phase shift
controls sand ripple formation, see for instance [22]). For
higher modes, on the other hand, a positive growth rate
is possible (see figure 4), despite the lubrication approximation. This indicates that the instability mechanism
governing bars formation is different than the phase shift
induced by the advection term in the case of dunes and
ripples.
The fluid and sediment choices determine parameters
γ and β. Both parameters are crucial to the present
model. The diffusion term which is proportional to γ
stabilizes the high n modes. Without it, the higher n,
the higher σm . As in [10], we take γ = 1 in the following.
If β = 1, that is if the sediment flux is proportional to the
shear stress, then no instability ever appears (again σ is
always negative in that case). Instability may occur only
if β > 1. β = 3.75 is chosen hereinafter as an illustrative
case (see section II B).
Figure 4 illustrates the transition to bed instability as
the aspect ratio is increased, for constant tilt and Froude
number. A deep and narrow channel is stable, as for no
values of n and k can σ be negative. A shift to a larger aspect ratio value allows for the n = 1 mode to be unstable.
For a still wider channel, both n = 1 and n = 2 modes
are unstable, but the latter grows faster. These transitions can be summarized in a three-dimensional phase
diagram, with coordinates R, F and S. A constant S
slice of this diagram is presented in figure 5. The borders between domains are characterized by the following
relations (σm,n is the maximum growth rate corresponding to mode n):
• σm,1 = 0 between the stable domain and the mode
1 domain;
• σm,1 = σm,2 between the mode 1 domain and the
mode 2 domain;
• σm,2 = 0 between the stable domain and the mode
2 domain.
Each point of the curves represented on figure 5 was obtained by numerical maximization of the dispersion equation.
The most surprising feature appearing on the diagram
of figure 5 is that bars can be unstable even for vanishing
Froude number (and thus for vanishing Reynolds number). In that case, inertia is completely neglected. In
other words, bars may develop in a purely viscous flow,
which is impossible for dunes and ripples. Since a purely
viscous flow can present no transverse recirculation, the
above statement proves that neither turbulence nor recirculation are inherently linked to bar formation.
The same diagram also provides a crude interpretation for the aging of laminar laboratory rivers. Let us
consider for example the case of section III, for which
the mean water level is fixed, while its bed and banks are
freely eroded. If we assume a quasi-static evolution of
the bed width so that the stability analysis for fixed wall
can be roughly used, we can draw a schematic scenario
for the river deformation. Thus, the tilt S remains constant throughout the experiment whereas, in accordance
9
with (22), the Froude number and aspect ratio evolve as
follows:
R ∝ t2/(β+2) ,
F ∝ t3/(2(β+2)) .
(45)
This parameterized curve correspond to F
=
F0 (R/R0 )3/4 in the stability diagram (the subscript 0
denotes initial conditions). In most cases this curve
comes successively through the three stability domains
of figure 5, allowing for the successive development of
different bars modes . If the water output is conserved
instead of the water level (this condition is more common in experiments), the straight channel evolution is
characterized by
−3/8
F = F0 (R/R0 )
.
(46)
Again, for realistic initial conditions (R0 = 20.3 and
F0 = 3.21 in the experiment of [12]), the river undergoes
different instability regimes as it ages. The three points
Σi drawn on figure 5 would then represent three different
states of the same experiment, extrapolated from the initial condition using (46). The corresponding growth rate
are plotted in Figure 4. When the highest growth rate of
the first mode crosses zero, alternate bars appear, eventually replaced by higher order modes, leading to braided
patterns.
If a threshold is introduced in the erosion law, the river
eventually reaches an equilibrium state. The position of
this equilibrium in the stability diagram is an indication
about the instability patterns the river will preferentially
develop. For instance, we may expect that a river will develop meanders if its equilibrium state lies in the domain
where the n = 1 mode is the most unstable one.
V.
mode (n = 1) at small (or even null) Froude number.
This illustrates the sound difference between bars and
dunes or ripples, which need inertia to grow.
The use of a fluid more viscous than water in experiments would allow to reach very low Froude numbers,
4
(
'1
Mode 1
(
'2
3
(
'2
R $3!8
F 2
Larger Modes
R 3!4
1
Stable
Mode 1
0
0
50
100
150
R
FIG. 5: Stability diagram for a laminar channel. The domains
(separated by solid lines) are named after the most unstable
mode between n = 1 and n = 2. The parameters values
are β = 3.75, γ = 1 and S = 0.0875. The dashed lines
represent the evolution of a straight river when the water level
is imposed (F = F0 (R/R0 )3/4 ) or when the outflow is imposed
(F = F0 (R/R0 )−3/8 ). The three points Σi correspond to the
three cases presented in figure 4.
CONCLUSION
The present paper demonstrates that the equations
governing the evolution of laminar micro-rivers are very
similar to their counterpart in the turbulent case. Experimental evidence of this similarity are collected in
[11]. This results suggests that micro-rivers could facilitate the examination of some remaining difficulties of
river morphodynamics, such as non-linearities or bank
evolution. In a first attempt to develop viscous channel widening and stability theory, we presented a two
dimensional shallow-water model. A very simplified analytical approach based on this model was sufficient to
describe qualitatively the aging process observed in some
experiments. A diagram presenting the dominant unstable modes with respect to the channel tilt, Froude
number and aspect ratio was obtained (figure 5), which
shows a large domain of existence for the meandering
[1] G. de Marsily, F. Delay, G. J., P. Renard, V. Teles, and
S. Violette, Hydrogeol. J. 13, 161 (2005).
while reducing the perturbing effect of capillarity. The
consecutive reduction of the Reynolds number would prevent recirculation, thus allowing the experimental separation between the effects of recirculation and bars instability.
The relaxation of the rigid banks hypothesis requires
the development of bank erosion models, able to take
avalanches into account. Such an improvement, associated with numerical simulation, would allow to test
the laminar Shallow-water theory against experiments in
conditions closer to natural rivers.
It is our pleasure to thank B. Andreotti, P. Claudin,
A. Fourrière, E. Lajeunesse, D. Lhuillier, L. Malverti,
F. Métivier and G. Parker for inspiring and stimulating
discussions.
[2] A. Reynolds, J. Fluid Mech. 22, 113 (1965).
10
[3] E. Hansen, Basic Research Report 13, Copenhagen Technical University of Danemark (1967).
[4] R. A. Callander, J. Fluid Mech. 36, 465 (1969).
[5] G. Parker, J. Fluid Mech. 76, 457 (1976).
[6] F. Engelund and O. Skovgaard, J. Fluid Mech. 57, 289
(1973).
[7] S. Ikeda, G. Parker, and K. Saway, J. Fluid Mech. 112,
363 (1981).
[8] P. Blondeaux and G. Seminara, J. Fluid Mech. 157, 449
(1985).
[9] M. Colombini, G. Seminara, and M. Tubino, J. Fluid
Mech. 181, 213 (1987).
[10] R. Schielen, A. Doelman, and H. de Swart, J. Fluid Mech.
252, 325 (1993).
[11] L. Malverti, G. Parker, L. Armstrong, P. Lancien,
E. Lajeunesse, F. Métivier, S. Coleman, C. E. Smith,
T. Davies, and A. Cantelli, Sedimentology (2007), (to
appear).
[12] F. Métivier and P. Meunier, Journal of Hydrology 271,
22 (2003).
[13] B. Federici and C. Paola, Water Resour. Res. 39, 1162
(2003).
[14] F. Charru, H. Mouilleron, and O. Eiff, J. Fluid Mech.
519, 55 (2004).
[15] C. Ruyer-Quil and P. Manneville, Eur. Phys. J. B 15,
pp. 357–369 (2000).
[16] F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and
J.-P. Vilotte, C. R. Acad. Sci. Paris 336, pp. 531–536
(2003).
[17] K. Blankaert and H. de Vriend, J. Fluid Mech. 498, 353
(2004).
[18] S. Ikeda and G. Parker, eds., River meandering (AGU
Water Ressources Monograph, Washington DC, 1989),
vol. 12, chap. Observation on several recent theories of
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
resonance and overdeepening in meandering channels,
pp. 379–415.
N. J. Balmforth and A. Provenzale, eds., Geomorphological Fluid Mechanics (Springer, 2001), chap. 16, pp.
394–402.
L. v. Rijn, ed., Handbook of sediment transport by current and waves (Delft hydraulics, Delft, The Netherlands,
1989), vol. 12, pp. 379–415.
P. Hersen, K. Andersen, H. Elbelrhiti, B. Andreotti,
P. Claudin, and S. Douady, Physical Review 69 (2004).
K. K. J. Kouakou and P.-Y. Lagrée, European journal of
Mechanics B-Fluids 25, 348 (2006).
A. Kovacs and G. Parker, J. Fluid Mech. 267, 153 (1994).
G. Parker, J. Fluid Mech. 89, 109 (1978).
G. Parker, J. Fluid Mech. 89, 127 (1978).
A. Paquier and S. Khodashenas, Journal of Hydraulic
Research 40 (2002).
C. Stark, Geophysical Research Letters 33, L04402
(2006).
L. Armstrong, Ph.D. thesis, Universitı̈¿ 12 Paris 7 Denis
Diderot (2003).
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Research 24, 713 (1988).
T. Nakagawa, Sedimentology 30, 117 (1983).
R. Giménez, Ph.D. thesis, Katholieke Universiteit Leuven (2003).
Personal communication from E. Lajeunesse and F.
Métivier
This was observed during preliminary experiments carried out with L. Malverti and E. Lajeunesse.
The idea of using highly viscous fluid in micro-rivers was
first suggested by B. Andreotti.
PRL 97, 025503 (2006)
week ending
14 JULY 2006
Weak Turbulence for a Vibrating Plate: Can One Hear a Kolmogorov Spectrum?
Gustavo Düring,1 Christophe Josserand,2 and Sergio Rica1,3
1
2
Departamento de Fı́sica, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
Laboratoire de Modélisation en Mécanique, CNRS UMR 7607, Case 162, 4 place Jussieu, 75005 Paris, France
3
LPS-ENS, CNRS UMR 8550, 24 Rue Lhomond, 75231 Paris Cedex 05, France
(Received 24 December 2005; published 14 July 2006)
We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so
that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully
applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at
long times with a slow transfer of energy from one mode to another. We derive a kinetic equation for the
spectral transfer in terms of the second order moment. We show that such a theory describes the approach
to an equilibrium wave spectrum and represents also an energy cascade, often called the KolmogorovZakharov spectrum. We perform numerical simulations that confirm this scenario.
PACS numbers: 62.30.+d, 05.45.!a, 46.40.!f
Introduction.—For more than 40 years it has been established that long-time statistical properties of a random
fluctuating wave system possess a natural asymptotic closure because of the dispersive nature of the waves and of
the weakly nonlinear interaction between them [1,2]. This
‘‘weak turbulence theory’’ has been shown to be a powerful
method for studying the evolution of nonlinear dispersive
wave systems [3,4]. It follows that the long-time dynamics
is driven by a kinetic equation for the distribution of
spectral densities. This method has been applied to surface
gravity waves [1,5], capillary waves [6], plasma waves [7],
and nonlinear optics [8] for instance. The actual kinetic
equation has nonequilibrium properties similar to the usual
Boltzmann equation for dilute gases, conserving energy
and momentum, and it exhibits an H theorem driving the
system to equilibrium, characterized by the Rayleigh-Jeans
distribution. Most important, besides the elementary equilibrium (or thermodynamic) solution, Zakharov has shown
[7] that power-law nonequilibrium solutions also arise,
namely, the Kolmogorov-Zakharov (KZ) solutions or KZ
spectra, which describe the exchange of conserved quantities (e.g., energy) between large and small length scales.
Experimental evidence of KZ spectra have been found in
the ocean surface [9] and in capillary surface waves [10 –
12]. Numerical simulations have also shown the existence
of KZ spectra in weak turbulent capillary waves [13] and,
more recently, in gravity waves [14].
In this Letter an oscillating thin elastic plate is considered [15]. Adding inertia to the well known (static) theory
of thin plates, one finds the existence of ballistic dispersive
waves [16]. They interact via the nonlinear terms that are
weak if the plate deformations are small. Understanding
the interaction between these waves is thus crucial to
describe acoustical properties of the plates. In fact, nonlinear solitary waves have been observed on the surface of
a cylindrical shell that show balance between nonlinear
effects and dispersion [17]. However, we develop here the
first weak turbulence theory for the surface deflection on
0031-9007=06=97(2)=025503(4)
plate dynamics. We find that the bending waves travel
randomly through the system and interact resonantly between each other via the weak nonlinearities. The mathematics behind the resonant condition is formally identical
to the conservation of energy and momentum in a classical
gas. In this sense an elastic plate is formally equivalent to a
2D gas of classical particles interacting with a nontrivial
scattering cross section. An isolated system evolves from a
random initial condition to a situation of statistical equilibrium as a gas of particles does. In addition to statistical
equilibrium for isolated systems, the weak turbulence theory predicts here an energy cascade from a source of
energy (a driving forcing) to a dissipation scale typical of
irreversible processes.
More precisely, we have in mind an elastic thin plate
under an external low frequency (few times the slowest
plate mode) random forcing. Typically the gravest mode
for a 10 " 10 cm2 free bounded steel sheet of 0.1 mm thick
is about 50 Hz and is a bit higher for a clamped sheet.
Internal resonance among modes buildup an energy cascade from the injection scale to small scales where it is
ultimately dissipated mostly because of the boundaries, the
air entraintement, viscoelastic flows, or heat transfer. A
genuine cascade should setup if dissipation occurs at small
scales only. One needs to be careful concerning the heat
transfer since the damping coefficient does not depend on
the oscillation frequency there. However, heat loss is a
weak effect and can be in general neglected (see section 30
in Ref. [18]): indeed, for the above example, the heat loss
time scale is about 15 times smaller than the slowest
oscillation at room temperature. As in fluids, viscosity in
solids acts only at small scale. Finally, in a real experiment
the boundaries play an important role because of the finite
value of the experimental setup impedance. Such a damping coefficient grows linearly with wave number and is
probably the most relevant source of dissipation.
Therefore, it seems possible that energy cascades from
the scale of the plate to the dissipation scale.
025503-1
Moreover, while there is often a lack of direct observations of weak turbulence predictions, we exhibit numerically relaxation to equilibrium and energy cascade for the
plate dynamics, confirming the scenario presented above.
The plate dynamics is illustrated in Fig. 1 for an isolated
system where the plate deformations are shown at initial
time and after a long evolution.
Theory.—The starting point is the dynamical version of
the Föppl–von Kármán equations [18] for the plate deformation !#x; y; t$ and the stress function "#x; y; t$:
#
@2 !
Eh2
!2 ! & f!; "g;
%
!
@t2
12#1 ! $2 $
1
1 2
! " % ! f!; !g;
2
E
(1)
(2)
where h is the thickness of the elastic sheet. The material
has a mass density #, a Young’s modulus E, and a Poisson
ratio $. ! % @xx & @yy is the usual Laplacian and the
bracket f'; 'g is defined by ff; gg ( fxx gyy & fyy gxx !
2fxy gxy , which is an exact divergence, so Eq. (1) preserves
theR momentum of the center of mass, namely
@tt !#x; y; t$dxdy % 0. The first term on the right-hand
side of (1) represents the bending while the second one
f!; "g, together with Eq. (2), represents the stretching [19].
Despite the complexity of Eqs. (1) and (2) the system
presents a Hamiltonian structure that is straightforward in
Fourier
Defining the Fourier transforms as !k #t$ %
R space.
1
ik'x
) ), then one gets from
!#x; t$e d2 x (with !k % !!k
2%
E
Eq. (2): "k #t$ % ! 2jkj4 f!; !gk , where f!; !gk is the Fourier
transform of f!; !g. The final equation is a nonlinear oscillator with the usual ballistic dispersion relation of bending
p!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
waves !k % hcjkj2 % hck2 [c % E=12#1 ! $2 $# has
the dimension of a velocity] [16,18]:
Z
@2 !
Eh2 k4
# 2k % !
!
V!k;k2 ;k3 ;k4 !k2 !k3 !k4 &#2$ #k
!
k
@t
12#1 ! $2 $
2
! k2 ! k3 ! k4 $d k2;3;4 ;
2
2
jk1 "k2 j jk3 "k4 j
E
and d2 k2;3;4 (
where V12;34 % 2#2%$
2
jk1 &k2 j4
d2 k2 d2 k3 d2 k4 . The Hamiltonian structure becomes evident if we define as canonical variables the deformation
!k #t$ and the momentum pk #t$ % #@t !k #t$. Finally, the
canonical transformation !k % pX!!2k #Ak & A)!k $ and pk %
8
ζ (x,y)
8
1 !
! p!!2iX #Ak ! A)!k $ with Xk % p!!!!!!
!k # allows us to write the
k
k
wave equation in a diagonalized form: dA
dt & i!k Ak %
iN3 #Ak $, where N3 #'$ is the cubic nonlinear term.
Weak turbulence theory.—This nonlinear oscillator has
two distinct time scales, the rapid oscillation i!k Ak and the
weak nonlinearity: iN3 #Ak $. Then, following the approach
of [4], one changes Ak % ak #t$e!i!k t which removes the
rapid linear oscillating term:
X Z
dask
% !is
J!kk1 k2 k3 eit#s!k !s1 !k1 !s2 !k2 !s3 !k3 $
dt
s1 s2 s3
" as11 as22 as33 &#2$ #k1 & k2 & k3 ! k$d2 k123 ;
" &#2$ #p1 & s1 k1 & s2 k2 & s3 k3 $d2 k123 :
neq
k % T=!k ;
-8
y
10
x
20
-8
y
10
0
20
30
10
x
10
20
30
FIG. 1. Zoom over a portion of the surface plate deflection
!#x; y$. The left-hand image is the initial condition while the
right-hand one represents a long-time evolution of the plate.
(4)
As for the usual Boltzmann equation, Eq. (4) conserves
‘‘formally’’
[20] the total momentum per unit area P %
R
h R knk #t$d2 k and the kinetic energy per unit area E %
2 k, and exhibits an H theorem: let S#t$ %
h
R !k nk #t$d
2
ln#nk $d k be the nonequilibrium entropy, then dS=dt *
0, for increasing time. However, despite the four waves
interaction
type kinetic Eq. (4), the ‘‘wave action’’ N %
R
nk #t$d2 k is not conserved. The kinetic Eq. (4) describes
thus an irreversible evolution of the wave spectrum towards
the Rayleigh-Jeans equilibrium distribution which reads,
when P % 0:
30
-4
30
20
(3)
where we define ask with the two possible choices s % & or
! relative to the propagation direction, such that a&
k ( ak
) . The interaction term reads: J
while a!
(
a
k1 ;k2 ;k3 ;k4 %
k
!k
1
6 Xk1 Xk2 Xk3 Xk4 P 234 Vk1 ;k2 ;k3 ;k4 , where P 234 is the sum over
the six possible permutations between 2, 3, and 4. The next
step consists of writing a hierarchy of linear equations for
the averaged moments: hask11 ask22 i, hask11 ask22 ask33 ask44 i, etc. A
multiscale analysis provides a natural asymptotic closure
for higher moments: the fast oscillations drive the system
close to Gaussian statistics and higher moments are written
in terms of the second order moment: hak1 a)k2 i %
nk1 &#2$ #k1 & k2 $, where nk is called the wave spectrum.
The wave spectrum thus satisfies a Boltzmann-type
kinetic equation describing a slow exchange of energy
from one mode to another through four waves resonance:
"
Z
X
dnp1
1
s
% 12% jJp1 k1 k2 k3 j2
nk1 nk2 nk3 np1
& 1
np1 nk1
dt
s1 s2 s3
#
s
s
& 2 & 3 &#!p1 & s1 !k1 & s2 !k2 & s3 !k3 $
nk2 nk3
0
0
-4
0
ζ(x,y)
4
4
week ending
14 JULY 2006
PRL 97, 025503 (2006)
(5)
here T is called, by analogy with thermodynamics, the
‘‘temperature’’ (with units of energy/length, i.e., a force),
which
is naturally related
to the initialRenergy by E 0 %
R
R
h !k neq d2 k % hT d2 k. The quantity d2 k is the number of degrees of freedom per unit surface. Therefore each
degree of freedom takes the same energy: hT. Naturally,
025503-2
PRL 97, 025503 (2006)
for an infinite system this number diverges (as well as the
energy). This classical Rayleigh-Jeans catastrophe is always suppressed due to some physical cutoff discussed
above. Numerical simulations on regular grid provide
also a natural cutoff kc % %=dx, where dx is the mesh
size, which gives E 0 % %hTk2c for a large plate.
Kolmogorov spectrum.—In addition, isotropic nonequilibrium distribution solutions can arise [7]. They have a
major importance in the nonequilibrium process for the
energy transfer between different scales. These solutions
can be guessed via a dimensional analysis argument but
they are, in fact, exact solutions of the kinetic equation.
Despite some differences with the usual kinetics equation,
the Zakharov method can be applied here. Assuming an
isotropic spectrum nk ( njkj and integrating over the angles the scattering amplitude jJk1 k2 k3 k4 j2 &#2$ #k1 & k2 &
k3 & k4 $, the new scattering amplitude depends only on
the modulus ki % jki j, and it can be written as a function of
R jJk1 k2 k3 k4 j2
the frequencies !ki : S!1 ;!2 ;!3 ;!4 % 16 P 234
d’4 .
jk2 "k3 j
Since the degree of homogeneity of jJj2 in k is zero, S
scales as 1=k2 + 1=!k .
Looking for a power-law solution nk % A!!'
k , the eight
terms of the collisional integral in the right-hand side of (4)
decompose into 3Coll2$2 & Coll3$1 , defined by
Colls %
3%A3 Z S!k ;!1 ;!2 ;!3
#!' & s!'1 ! !'2 ! !'3 $
2#hc$3 "s !'k !'1 !'2 !'3 k
$
" #( " #( " #( %
!
!
!
" 1&s 1 ! 2 ! 3
d!2 d!3 :
!k
!k
!k
Here ( % 3' ! 2. For Coll2$2 one has that s ( 1 and the
integration domain is over "& % f0 , !2 , !k ; !k !
!2 , !3 , !k g and !1 % !2 & !3 ! !k , while for
Coll3$1 one has that s ( !1 and the integration is over
"! % f0 , !2 , !k ; 0 , !3 , !k ! !2 g, with !1 %
!k ! !2 ! !3 . The collisional terms scale as Coll2$2 %
and Coll3$1 % C! #'$!1!3'
. The coeffiC& #'$!1!3'
k
k
cients C- #'$ are pure real functions depending only on
'. Although, the explicit form of the scattering matrix
S!1 ;!2 ;!3 ;!4 is not simple, its value can be bounded in
both domains "- and the collision term converges for
suitable values of ' 2 #0:5; 1:2$ validating the locality
condition. Both coefficients vanish with double degeneracy
1
1
at ' % 1 indicating that the KZ spectrum: nKZ
k + !k + k2
coincides with the Rayleigh-Jeans solution, Eq. (5). It
means, in fact, that the energy flux is zero. The double
degeneracy at ' % 1 reveals the existence of a logarithmic
correction, similarly to the case of the nonlinear
Schrödinger equation (NLS) in two dimensions [8]. As
stated in Ref. [21] the logarithmic correction produces a
divergent result for NLS. In our case, it is possible to show
that all integrals are finite, indicating a finite energy flux
[22]. Thus one has
nKZ
k %C
week ending
14 JULY 2006
hP1=3 #2=3
ln1=3 #k) =k$
;
k2
.12#1 ! $2 $/2=3
(6)
where P is the energy flux. C and k) are real numbers.
For ' % 0 and 3' ! 2 % 0 the collisional part Coll2$2
also vanishes. This solution corresponds to the wave action
equipartition (' % 0) with a second KZ spectrum nk +
related to wave action inverse cascade. However,
1=!2=3
k
this spectrum does not vanish the second part of the collision term Coll3$1 , in agreement with the nonconservation
of the wave action mentioned above. In conclusion there
exist only a single cascade: the energy cascade (6).
Numerical simulation.—Numerical simulations of the
full nonlinear system of Eqs. (1) and (2) are first performed
to validate the formation of the equilibrium spectrum
Eq. (5). In all the presented results c % 1 and h % 1=2 so
that the linear plate size is the only parameter of the
numerics. We have implemented a pseudospectral scheme
using FFT routines [23], with periodic boundary conditions: the linear part of the dynamics is calculated exactly
_
in Fourier space: !k #t & !t$ % !k #t$ cos#!k !t$ & !!k #t$
"
k
sin#!k !t$. The nonlinear terms in (1) and (2) are first
computed in real space and the integration in time is then
performed in Fourier space using an Adams-Bashford
scheme. It interpolates the nonlinear term of (1) as a
polynomial function of time (of order one in the present
calculations). Energy is conserved within a 1=100 relative
error. As initial conditions, we have taken: !k %
2 2
!0 e!k =k0 ei’k with ’k a random phase, and a zero velocity
field !_ k % 0. As time evolves, the random waves oscillate
with a disorganized behavior, as shown in Fig. 1. After a
long time the system builds up an equilibrium distribution
in agreement with the Rayleigh-Jeans nk + T=k2 spectrum.
nk
That is, for the plate deflection hj!k j2 i % Xk2 nk % #!
%
k
T
as shown in Fig. 2.
#h2 c2 k4
Nonequilibrium distributions can also be observed numerically. One requires to input energy and pump wave
action at low wave numbers (k < kin ) and to dissipate
energy at large ones (k > kout ) defining a window of transparency kin < k < kout . This artifact is implemented by
adding a term #Fk ! )k !_ k $ to the plate Eq. (1).
Following [14] the forcing term Fk is a nonzero random
force for k < kin , and )k is a fictitious linear damping for
short length scales. Figure 3 shows a good agreement with
the predicted KZ spectrum Eq. (6) with an exponant for the
logarithmic correction 1=3 (inset of Fig. 3).
Conclusions.—We have successfully applied weak turbulence theory for the new case of elastic thin plates. The
results allow for an analogy between an important property
of fluid dynamics and the mechanics of elastic plates.
Numerical simulations exhibit both the convergence towards statistical equilibrium for a free system and an
energy cascade when forcing and dissipation are introduced, as predicted by the weak turbulence analysis. An
important consequence is the nonexistence of an inverse
025503-3
PRL 97, 025503 (2006)
100
| ζ k|
regimes, as seemingly observed in [27], the expected spectrum should follow j!k j2 + 1=k6 .
The authors are grateful to E. Cerda for attracting our
attention to this problem, A. C. Newell for suggesting the
title, and A. Boudaoud, E. Hamm, and L. Mahadevan for
the communication of their experimental results prior to
publication. This work was supported by FONDECYT and
Anillo de Investigación Act. 15.
2
1
-4
0.01
1×10-4
1×10-6
∫ nk (t) d 2k
0.5
1×10-8
week ending
14 JULY 2006
0.4
0.3
1×10-10
0.2
1×10-12
0.1
1×10-14
0.01
200
0
400
600
800
t
1000
k
0.1
1
10
FIG. 2 (color online). Numerical simulation for a 512 square
plate using 10242 modes with a mesh size dx % 1=2. The initial
condition is with k0 % 1 and !0 % 0:02. We plot the power
spectrum of the mean deflection hj!k j2 i versus wave number k
after 1200 time units. The line plots the Rayleigh-Jeans power
law 7 " 10!6 =k4 which gives T 0 2 " 10!6 in agreement with
the equipartition of the initial energy. The inset plots the evolution of the wave action with time.
cascade nk + 1=k4=3 , as usually found for four wave interaction systems such as gravity waves or NLS. The results
presented here suggest also a new experimental way of
studying weak turbulence through the analysis of the waves
produced by the plate oscillations [24].
For large deformations the elastic plate equations are
still valid, but stretching cannot be longer treated as a weak
perturbation and a ‘‘wave breaking’’ phenomenon is expected: energy focuses into localized structures as ridges
[25] and conical surfaces (named d-cones) [26].
Amazingly, a regime dominated by ridges shows a power
spectrum j!k j2 + 1=k4 similar to the weak turbulence spectrum derived here. On the other hand for d-cone-dominated
1
| ζ k|
2
Out
In
1×10-5
1×10-10
1×10-15
1×10-20
-4
0.0001
| ζk| 2k 4
1×10-5
1/3
1×10-6
0.01
1×10-25
0.1
0.1
1
10
1
log(k* /k)
10
k
FIG. 3 (color online). Average power spectrum hj!k j2 i for the
energy cascade. The injection scale is kin 2 #0:1; 0:25$ while the
dissipation is at kout % 3. The line plots the power law 1=k4 .
Inset plots k4 hj!k j2 i vs log#k) =k$ in logarithmic scale with k) %
kout . The straight line corresponds to z % 1=3.
[1] K. Hasselmann, J. Fluid Mech. 12, 481 (1962); 15, 273
(1963).
[2] D. J. Benney and P. G. Saffman, Proc. R. Soc. A 289, 301
(1966).
[3] V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov
Spectra of Turbulence I (Springer, Berlin, 1992).
[4] A. Newell, S. Nazarenko, and L. Biven, Physica D
(Amsterdam) 152–153, 520 (2001).
[5] V. E. Zakharov and N. N. Filonenko, Dokl. Akad. Nauk
SSSR 170, 1292 (1966).
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(1967).
[7] V. E. Zakharov, Zh. Eksp. Teor. Fiz. 51, 686 (1966).
[8] S. Dyachenko et al., Physica D (Amsterdam) 57, 96
(1992).
[9] P. A. Hwang et al., J. Phys. Oceanogr. 30, 2753 (2000).
[10] W. Wright, R. Budakian, and S. Putterman, Phys. Rev.
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52, 27 (2000).
[12] M. Brazhnikov, et al., Europhys. Lett. 58, 510 (2002).
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3320 (1996).
[14] A. I. Dyachenko, A. O. Korotkevich, and V. E. Zakharov,
Phys. Rev. Lett. 92, 134501 (2004).
[15] Similar calculations could be done for an elastic shell or
for a plate under a homogeneous external tension.
[16] Lord Rayleigh, Theory of Sound (Dover, New York, 1945),
Sec. 217
[17] J. Wu et al., Phys. Rev. Lett. 59, 2744 (1987).
[18] L. D. Landau and E. M. Lifshitz, Theory of Elasticity
(Pergamon, New York, 1959).
[19] Static versions of Eqs. (1) and (2) are valid also for large
deformation, the only limitation of the model is that hj!!j
is small because of the elastic limit.
[20] ‘‘Formally’’ means here that the proof requires convergence of any simple integral to the exchange of integration
order by the Fubinis theorem [4].
[21] V. M. Malkin, Phys. Rev. Lett. 76, 4524 (1996).
[22] The main difference with Ref. [8] arises because in the
present case the scattering matrix vanishes on the lines
!2 % !3 % !4 % 0, ensuring convergence.
[23] FFTW: www.fftw.org.
[24] A. Boudaoud and E. Hamm (private communication).
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Interfaces complexes : gouttes et bulles, milieux granulaires et

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