Comptes Rendus
no. 11-12
Variational discretizations for the dynamics of fluid-conveying flexible tubes
François Gay-Balmaz1 ; Vakhtang Putkaradze2
1 LMD/IPSL, CNRS, École normale supérieure, PSL Research University, École polytechnique, Université Paris-Saclay, Sorbonne Universités, UPMC Université Paris-6, 24, rue Lhomond, 75005 Paris, France
2 Department of Mathematical and Statistical Sciences, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB T6G 2G1 Canada
Comptes Rendus. Mécanique, Volume 344 (2016) no. 11-12, pp. 769-775.

We derive a variational approach for discretizing fluid–structure interactions, with a particular focus on the dynamics of fluid-conveying elastic tubes. Our method is based on a discretization of the fluid's back-to-labels map and a Lie group discretization of the tube's variables, coupled with an appropriately formulated discrete version of the fluid conservation law. This approach allows the development of geometric numerical schemes for the dynamics of fluid-conveying collapsible tubes, which preserve several intrinsic geometric properties of the continuous system, such as symmetries and symplecticity. In addition, our approach can also be used to derive simplified, but geometrically consistent, low-component models for further analytical and numerical analysis of the system.

Nous proposons une approche variationnelle pour la discrétisation d'interactions fluide–structure, en nous focalisant sur la dynamique de tubes élastiques avec écoulement interne. Notre approche est basée sur une discrétisation des trajectoires inverses du fluide et une discrétisation de type groupe de Lie des variables du tube élastique, couplée à une discrétisation appropriée de la contrainte de préservation du volume de fluide. Notre approche permet le développement de schémas numériques géométriques pour la dynamique des tubes souples avec écoulement interne, qui préservent plusieurs propriétés géométriques intrinsèques du système continu, telles que les symétries et la symplecticité. De plus, notre approche peut être utilisée pour produire des modèles simplifiés et géométriquement consistants, appropriés pour des études analytiques et numériques plus approfondies de ce système.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2016.08.004
Keywords: Variational integrators, Fluid–structure interaction, Fluid-conveying tubes
Mots-clés : Intégrateurs variationnels, Interaction fluide–structure, Tubes avec écoulement interne

François Gay-Balmaz 1; Vakhtang Putkaradze 2

1 LMD/IPSL, CNRS, École normale supérieure, PSL Research University, École polytechnique, Université Paris-Saclay, Sorbonne Universités, UPMC Université Paris-6, 24, rue Lhomond, 75005 Paris, France
2 Department of Mathematical and Statistical Sciences, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB T6G 2G1 Canada 
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François Gay-Balmaz; Vakhtang Putkaradze. Variational discretizations for the dynamics of fluid-conveying flexible tubes. Comptes Rendus. Mécanique, Volume 344 (2016) no. 11-12, pp. 769-775. doi : 10.1016/j.crme.2016.08.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.08.004/

[1] H. Ashley; G. Haviland Bending vibrations of a pipe line containing flowing fluid, J. Appl. Mech., Volume 17 (1950), pp. 229-232

[2] B.T. Benjamin Dynamics of a system of articulated pipes conveying fluid I. Theory, Proc. Roy. Soc. A, Volume 261 (1961), pp. 457-486

[3] B.T. Benjamin Dynamics of a system of articulated pipes conveying fluid II. Experiments, Proc. Roy. Soc. A, Volume 261 (1961), pp. 487-499

[4] R.W. Gregory; M.P. Païdoussis Unstable oscillation of tubular cantilevers conveying fluid I. Theory, Proc. R. Soc. A, Volume 293 (1966), pp. 512-527

[5] M.P. Païdoussis Dynamics of tubular cantilevers conveying fluid, Int. J. Mech. Eng. Sci., Volume 12 (1970), pp. 85-103

[6] M.P. Païdoussis; N.T. Issid Dynamic stability of pipes conveying fluid, J. Sound Vib., Volume 33 (1974), pp. 267-294

[7] M.P. Païdoussis Fluid-Structure Interactions. Slender Structures and Axial Flow, vol. 1, Academic Press, London, 1998

[8] S. Shima; T. Mizuguchi Dynamics of a tube conveying fluid | arXiv

[9] O. Doaré; E. de Langre The flow-induced instability of long hanging pipes, Eur. J. Mech. A, Solids, Volume 21 (2002), pp. 857-867

[10] M.P. Païdoussis; G.X. Li Pipes conveying fluid: a model dynamical problem, J. Fluids Struct., Volume 7 (1993), pp. 137-204

[11] M.P. Païdoussis Fluid-Structure Interactions. Slender Structures and Axial Flow, vol. 2, Academic Press, London, 2004

[12] R.W. Gregory; M.P. Païdoussis Unstable oscillation of tubular cantilevers conveying fluid II. Experiments, Proc. R. Soc. A, Volume 293 (1966), pp. 528-542

[13] S. Kuronuma; M. Sato Stability and bifurcations of tube conveying flow, J. Phys. Soc. Jpn., Volume 72 (2003), pp. 3106-3112

[14] F.C. Flores; A. Cros Transition to chaos of a vertical collapsible tube conveying air flow, J. Phys. Conf. Ser., Volume 166 (2009)

[15] A. Cros; J.A.R. Romero; F.C. Flores, Springer (2012), pp. 15-24

[16] G.X.L.C. Semler; M.P. Païdoussis The non-linear equations of motion of pipes conveying fluid, J. Sound Vib., Volume 169 (1994), pp. 577-599

[17] Y. Modarres-Sadeghi; M.P. Païdoussis Nonlinear dynamics of extensible fluid-conveying pipes supported at both ends, J. Fluids Struct., Volume 25 (2009), pp. 535-543

[18] M. Ghayesh; M.P. Païdoussis; M. Amabili Nonlinear dynamics of cantilevered extensible pipes conveying fluid, J. Sound Vib., Volume 332 (2013), pp. 6405-6418

[19] M.A. Beauregard; A. Goriely; M. Tabor The nonlinear dynamics of elastic tubes conveying a fluid, Int. J. Solids Struct., Volume 47 (2010), pp. 161-168

[20] N. Bou-Rabee; L. Romero; A. Salinger A multiparameter, numerical stability analysis of a standing cantilever conveying fluid, SIAM J. Appl. Dyn. Syst., Volume 1 (2002), pp. 190-214

[21] I. Elishakoff Controversy associated with the so-called “follower forces”: critical overview, Appl. Mech. Rev., Volume 58 (2005), pp. 117-142

[22] M.H. Ghayesh; M. Païdoussis; M. Amabili Nonlinear dynamics of cantilevered extensible pipes conveying fluid, J. Sound Vib. (2013), pp. 6405-6418

[23] F. Gay-Balmaz; V. Putkaradze Exact geometric theory for flexible, fluid-conducting tubes, C. R. Mecanique, Volume 342 (2014), pp. 79-84

[24] F. Gay-Balmaz; V. Putkaradze On flexible tubes conducting fluid: geometric nonlinear theory, stability and dynamics, J. Nonlinear Sci., Volume 25 (2015), pp. 889-936

[25] F. Demoures; F. Gay-Balmaz; M. Kobilarov; T.S. Ratiu Multisymplectic Lie group variational integrator for a geometrically exact beam in R3, Commun. Nonlinear Sci. Numer. Simul., Volume 19 (2014), pp. 3492-3512

[26] F. Demoures; F. Gay-Balmaz; T.S. Ratiu Multisymplectic variational integrators and space/time symplecticity, Anal. Appl. (2015), pp. 1-51

[27] J.E. Marsden; G.W. Patrick; S. Shkoller Multisymplectic geometry, variational integrators and nonlinear PDEs, Commun. Math. Phys., Volume 199 (1998), pp. 351-395

[28] J. Marsden; M. West Discrete mechanics and variational integrators, Acta Numer. (2001), pp. 1-158

[29] J.C. Simó; J.E. Marsden; P.S. Krishnaprasad The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates, Arch. Ration. Mech. Anal., Volume 104 (1988), pp. 125-183

[30] D.D. Holm; V. Putkaradze Nonlocal orientation-dependent dynamics of charged strands and ribbons, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009), pp. 1093-1098

[31] D. Ellis; D.D. Holm; F. Gay-Balmaz; V. Putkaradze; T. Ratiu Symmetry reduced dynamics of charged molecular strands, Arch. Ration. Mech. Anal., Volume 197 (2010), pp. 811-902

[32] J. Marsden; S. Pekarsky; S. Shkoller Symmetry reduction of discrete Lagrangian mechanics on Lie groups, J. Geom. Phys., Volume 36 (1999), pp. 140-151

[33] A.I. Bobenko; Y.B. Suris Discrete Lagrangian reduction, discrete Euler–Poincaré equations, and semidirect products, Lett. Math. Phys., Volume 49 (1999), pp. 79-93

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