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.
Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.
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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.
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Benítez, M., Bermúdez, A., Fontán, P. (2018). A Second-Order Linear Newmark Method for Lagrangian Navier-Stokes Equations. In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_3
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