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# A Second-Order Linear Newmark Method for Lagrangian Navier-Stokes Equations

• Marta Benítez
• Alfredo Bermúdez
• Pedro Fontán
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)

## Abstract

In this paper we propose a second-order pure Lagrange-Galerkin method for the numerical solution of free surface problems in fluid mechanics. We consider a viscous, incompressible Newtonian fluid in a time dependent domain which may present large deformations but no topological changes at interfaces. Pure-Lagrangian methods are useful for solving these problems because the convective term disappears, the computational domain is independent of time and modelling and tracking of the free surface is straightforward as far as there is no solid walls preventing the free motion of surface particles. Unfortunately, for moderate to high-Reynolds number flows and as a consequence of high distortion of the moved mesh, it can be necessary to re-mesh and re-initialize the motion each certain time. In this paper, a Newmark algorithm is considered for both, the time semi-discretization of equations in Lagrangian coordinates and the computation of initial conditions. The proposed scheme is pure-Lagrangian and can be written in terms of either material velocity and pressure or material acceleration and pressure or material displacement and pressure. The three formulations are stated. In order to assess the performance of the overall numerical method, we solve different problems in two space dimensions. In particular, numerical results of a dam break problem and a flow past a cylinder are presented.

## Keywords

free-surface problems Lagrangian Navier-Stokes equations second-order schemes linear Newmark algorithm Lagrange-Galerkin methods

## Notes

### Acknowledgements

Authors wish to thank the referee for his/her valuable comments.

This work was partially funded by FEDER and the Spanish Ministry of Science and Innovation under research projects ENE2013-47867-C2-1-R and MTM2013-43745-R, and by FEDER and Xunta de Galicia under research project GRC2013/014.

## References

1. 1.
Ewing, R.E., Wang, H.: A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Appl. Math. 128, 423–445 (2001)
2. 2.
Douglas, J. Jr., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)
3. 3.
Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982)
4. 4.
Süli, E.: Stability and convergence of the Lagrange-Galerkin method with non-exact integration. In: The Mathematics of Finite Elements and Applications, VI, pp. 435–442. Academic, London (1988)Google Scholar
5. 5.
Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math. 92, 161–177 (2002)
6. 6.
Bermúdez, A., Nogueiras, M.R., Vázquez, C.: Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part I: Time discretization. SIAM. J. Numer. Anal. 44, 1829–1853 (2006)
7. 7.
Bermúdez, A., Nogueiras, M.R., Vázquez, C.: Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part II: fully discretized scheme and quadrature formulas. SIAM. J. Numer. Anal. 44, 1854–1876 (2006)
8. 8.
Benítez, M., Bermúdez, A.: A second order characteristics finite element scheme for natural convection problems. J. Comput. Appl. Math. 235, 3270–3284 (2011)
9. 9.
Idelsohn, S., Oñate, E., Del Pin, F.: The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int. J. Numer. Meth. Eng. 61, 964–989 (2004)
10. 10.
Idelsohn, S., Oñate, E., Del Pin, F., Calvo, N.: Fluid-structure interaction using the particle finite element method. Comput. Methods Appl. Mech. Eng. 195, 2100–2123 (2006)
11. 11.
Idelsohn, S., Marti, J., Limache, A., Oñate, E.: Unified Lagrangian formulation for elastic solids and incompressible fluids: application to fluid-structure interaction problems via the PFEM. Comput. Methods Appl. Mech. Eng. 197, 1762–1776 (2008)
12. 12.
Del Pin, F., Idelsohn, S., Oñate, E., Aubry, R.: The Ale/Lagrangian particle finite element method: a new approach to computation of free-surface flows and fluid-object interactions. Comput. Fluids 36, 27–38 (2007)
13. 13.
Oñate, E., Idelsohn, S., Celigueta, M.A., Rossi, R.: Advances in the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows. Comput. Methods Appl. Mech. Eng. 197, 1777–1800 (2008)
14. 14.
Radovitzky, R., Ortiz, M.: Lagrangian finite element analysis of newtonian fluid flows. Int. J. Numer. Meth. Eng. 43, 607–619 (1998)
15. 15.
Benítez, M., Bermúdez, A.: Numerical Analysis of a second-order pure Lagrange-Galerkin method for convection-diffusion problems. Part I: time discretization. SIAM. J. Numer. Anal. 50, 858–882 (2012)
16. 16.
Benítez, M., Bermúdez, A.: Numerical Analysis of a second-order pure Lagrange-Galerkin method for convection-diffusion problems. Part II: fully discretized scheme and numerical results. SIAM. J. Numer. Anal. 50, 2824–2844 (2012)
17. 17.
Benítez, M., Bermúdez, A.: Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of convection-diffusion problems. Int. J. Numer. Anal. Mod. 11, 271–287 (2014)
18. 18.
Benítez, M., Bermúdez, A.: Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier-Stokes equations. Appl. Numer. Math. 95, 62–81 (2015)
19. 19.
Benítez, M., Bermúdez, A.: Second order pure Lagrange-Galerkin methods for fluid-structure interaction problems. SIAM J. Sci. Comput. 37, B744–B777 (2015)
20. 20.
Gurtin, M.E.: An Introduction to Continuum Mechanics, vol. 158. Academic, San Diego (1981)
21. 21.
Badía, S., Codina, R.: Unified stabilized finite element formulations for the stokes and the darcy problems. SIAM J. Num. Anal. 47, 1971–2000 (2009)
22. 22.
Raviart, P.A., Thomas, J.M.: A mixed-finite element method for second order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics. Springer, New York (1977)Google Scholar
23. 23.
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
24. 24.
Ramaswamy, B., Kawahara, M.: Lagrangian finite element analysis applied to viscous free surface fluid flow. Int. J. Numer. Meth. Fluids 7, 953–984 (1987)
25. 25.
Walhorn, E., Kölke, A., Hübner, B., Dinkler, D.: Fluid–structure coupling within a monolithic model involving free surface flows. Comput. Struct. 83(25), 2100–2111 (2005)
26. 26.
Hansbo, P.: The characteristic streamline diffusion method for the time-dependent incompressible Navier-Stokes equations. Comput. Method. Appl. M. 99(2–3), 171–186 (1992)
27. 27.
Wall, W.A., Genkinger, S., Ramm, E.: A strong coupling partitioned approach for fluid–structure interaction with free surfaces. Comput. Fluids 36(1), 169–183 (2007)
28. 28.
Hirt, C.W., Nichols, B.D.: Volume of fluid (vof) method for the dynamics of free boundaries. J. Comput. Phys. 39(1), 201–225 (1981)
29. 29.
Martin, J.C., Moyce, W.J.: Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philos. Trans. R. Soc. A 244(882), 312–324 (1952)Google Scholar

## Copyright information

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## Authors and Affiliations

• Marta Benítez
• 1
• Alfredo Bermúdez
• 2
Email author
• Pedro Fontán
• 3
1. 1.Departamento de MatemáticasUniversidade da CoruñaFerrolSpain
2. 2.Departamento de Matemática AplicadaUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
3. 3.Instituto tecnológico de matemática industrialSantiago de CompostelaSpain