Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
Fluid Flows with Complex Free Surfaces 205 Fig. 9. Fingering instabilities. Shape of the liquid region at times t = 0 s (left) and t =0.745 s (right). Fig. 10. Fingering instabilities. Horizontal cuts through the middle of the liquid region at times t =0.119 s, t =0.245 s, t =0.364 s, t =0.49 s (first row) and times t =0.609 s, t =0.735 s, t =0.854 s, t =0.98 s (second row). 6 Conclusions An efficient computational model for the simulation of two-phases flows has been presented. It allows to consider both Newtonian and non-Newtonian flows. It relies on an Eulerian framework and couples finite element techniques with a forward characteristics method. Numerical results illustrate the large range of applications covered by the model. Extensions are being investigated (1) to couple viscoelastic and surface tension effects, (2) to reduce the CPU time required to solve Stokes problems, and (3) to improve the reconstruction of the interface and the computation of surface tension effects. Acknowledgement. The authors wish to thank Vincent Maronnier for his contribution to this project and his implementation support.
206 A. Bonito et al. References [AMS04] [BCP06a] [BCP06b] [BCP07] [BDJ80] [BKZ92] [BPL06] [BPS01] E. Aulisa, S. Manservisi, and R. Scardovelli. A surface marker algorithm coupled to an area-preserving marker redistribution method for threedimensional interface tracking. J. Comput. Phys., 197(2):555–584, 2004. A. Bonito, Ph. Clément, and M. Picasso. Finite element analysis of a simplified stochastic Hookean dumbbells model arising from viscoelastic flows. M2AN Math. Model. Numer. Anal., 40(4):785–814, 2006. A. Bonito, Ph. Clément, and M. Picasso. Mathematical analysis of a simplified Hookean dumbbells model arising from viscoelastic flows. J. Evol. Equ., 6(3):381–398, 2006. A. Bonito, Ph. Clément, and M. Picasso. Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow. Numer. Math., 107(2):213–255, 2007. R. B. Bird, N. L. Dotson, and N. L. Johnson. Polymer solution rheology based on a finitely extensible bead-spring chain model. J. Non- Newtonian Fluid Mech., 7:213–235, 1980. J. U. Brackbill, D. B. Kothe, and C. Zemach. A continuum method for modeling surface tension. J. Comput. Phys., 100:335–354, 1992. A. Bonito, M. Picasso, and M. Laso. Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys., 215(2):691–716, 2006. J. Bonvin, M. Picasso, and R. Stenberg. GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows. Comput. Methods Appl. Mech. Engrg., 190(29–30):3893–3914, 2001. [BRLH02] A. Bach, H. K. Rasmussen, P.-Y. Longin, and O. Hassager. Growth of non-axisymmetric disturbances of the free surface in the filament stretching rheometer: experiments and simulation. J. Non-Newtonian Fluid Mech., 108:163–186, 2002. [Cab05] [Cab06] A. Caboussat. Numerical simulation of two-phase free surface flows. Arch. Comput. Methods Engrg., 12(2):165–210, 2005. A. Caboussat. A numerical method for the simulation of free surface flows with surface tension. Comput. & Fluids, 35(10):1205–1216, 2006. [CPR05] A. Caboussat, M. Picasso, and J. Rappaz. Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas. J. Comput. Phys., 203(2):626–649, 2005. [CR05] A. Caboussat and J. Rappaz. Analysis of a one-dimensional free surface flow problem. Numer. Math., 101(1):67–86, 2005. [DLCB03] D. Derks, A. Lindner, C. Creton, and D. Bonn. Cohesive failure of [FCD + 06] [FF92] thin layers of soft model adhesives under tension. J. Appl. Phys., 93(3):1557–1566, 2003. M. M. Francois, S. J. Cummins, E. D. Dendy, D. B. Kothe, J. M. Sicilian, and M. W. Williams. A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework. J. Comput. Phys., 213(1):141–173, 2006. L. P. Franca and S. L. Frey. Stabilized finite element method: II. The incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg., 99:209–233, 1992.
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206 A. Bonito et al.<br />
References<br />
[AMS04]<br />
[BCP06a]<br />
[BCP06b]<br />
[BCP07]<br />
[BDJ80]<br />
[BKZ92]<br />
[BPL06]<br />
[BPS01]<br />
E. Aulisa, S. Manservisi, <strong>and</strong> R. Scardovelli. A surface marker algorithm<br />
coupled to an area-preserving marker redistribution method for threedimensional<br />
interface tracking. J. Comput. Phys., 197(2):555–584, 2004.<br />
A. Bonito, Ph. Clément, <strong>and</strong> M. Picasso. Finite element analysis of a<br />
simplified stochastic Hookean dumbbells model arising from viscoelastic<br />
flows. M2AN Math. Model. Numer. Anal., 40(4):785–814, 2006.<br />
A. Bonito, Ph. Clément, <strong>and</strong> M. Picasso. Mathematical analysis of a<br />
simplified Hookean dumbbells model arising from viscoelastic flows. J.<br />
Evol. Equ., 6(3):381–398, 2006.<br />
A. Bonito, Ph. Clément, <strong>and</strong> M. Picasso. Mathematical <strong>and</strong> numerical<br />
analysis of a simplified time-dependent viscoelastic flow. Numer. Math.,<br />
107(2):213–255, 2007.<br />
R. B. Bird, N. L. Dotson, <strong>and</strong> N. L. Johnson. Polymer solution rheology<br />
based on a finitely extensible bead-spring chain model. J. Non-<br />
Newtonian Fluid Mech., 7:213–235, 1980.<br />
J. U. Brackbill, D. B. Kothe, <strong>and</strong> C. Zemach. A continuum method for<br />
modeling surface tension. J. Comput. Phys., 100:335–354, 1992.<br />
A. Bonito, M. Picasso, <strong>and</strong> M. Laso. Numerical simulation of 3D viscoelastic<br />
flows with free surfaces. J. Comput. Phys., 215(2):691–716,<br />
2006.<br />
J. Bonvin, M. Picasso, <strong>and</strong> R. Stenberg. GLS <strong>and</strong> EVSS methods for<br />
a three-field Stokes problem arising from viscoelastic flows. Comput.<br />
Methods Appl. Mech. Engrg., 190(29–30):3893–3914, 2001.<br />
[BRLH02] A. Bach, H. K. Rasmussen, P.-Y. Longin, <strong>and</strong> O. Hassager. Growth<br />
of non-axisymmetric disturbances of the free surface in the filament<br />
stretching rheometer: experiments <strong>and</strong> simulation. J. Non-Newtonian<br />
Fluid Mech., 108:163–186, 2002.<br />
[Cab05]<br />
[Cab06]<br />
A. Caboussat. Numerical simulation of two-phase free surface flows.<br />
Arch. Comput. Methods Engrg., 12(2):165–210, 2005.<br />
A. Caboussat. A numerical method for the simulation of free surface<br />
flows with surface tension. Comput. & Fluids, 35(10):1205–1216, 2006.<br />
[CPR05] A. Caboussat, M. Picasso, <strong>and</strong> J. Rappaz. Numerical simulation of<br />
free surface incompressible liquid flows surrounded by compressible gas.<br />
J. Comput. Phys., 203(2):626–649, 2005.<br />
[CR05] A. Caboussat <strong>and</strong> J. Rappaz. Analysis of a one-dimensional free surface<br />
flow problem. Numer. Math., 101(1):67–86, 2005.<br />
[DLCB03] D. Derks, A. Lindner, C. Creton, <strong>and</strong> D. Bonn. Cohesive failure of<br />
[FCD + 06]<br />
[FF92]<br />
thin layers of soft model adhesives under tension. J. Appl. Phys.,<br />
93(3):1557–1566, 2003.<br />
M. M. Francois, S. J. Cummins, E. D. Dendy, D. B. Kothe, J. M. Sicilian,<br />
<strong>and</strong> M. W. Williams. A balanced-force algorithm for continuous <strong>and</strong><br />
sharp interfacial surface tension models within a volume tracking framework.<br />
J. Comput. Phys., 213(1):141–173, 2006.<br />
L. P. Franca <strong>and</strong> S. L. Frey. Stabilized finite element method: II. The<br />
incompressible Navier–Stokes equations. Comput. Methods Appl. Mech.<br />
Engrg., 99:209–233, 1992.
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