We consider the incompressible Navier-Stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, and S. Turek, Internat. J. Numer. Methods Fluids, 22 ( 1996), pp. 325-352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, and A. Quarteroni, SIAM J. Numer. Anal., 40 (2002), pp. 376-401, and A. Veneziani and C. Vergara, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 803-816). However, both of these approaches present some drawbacks. For the flow rate problem, the former resorts to nonstandard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yields exact solutions only in very special cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions. In this paper, we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional. This approach allows us to treat the two kinds of problems (flow rate and mean pressure) successfully within the same framework, which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries. Moreover, it is more versatile, being prone to be extended to other kinds of defective conditions. We analyze the problems obtained with this approach and propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.

A new approach to the numerical solution of defective boundary problems in incompressible fluid dynamics

FORMAGGIA, LUCA;VENEZIANI, ALESSANDRO;VERGARA, CHRISTIAN
2008-01-01

Abstract

We consider the incompressible Navier-Stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed. There are basically two strategies for solving these defective boundary problems: the variational approach (see J. Heywood, R. Rannacher, and S. Turek, Internat. J. Numer. Methods Fluids, 22 ( 1996), pp. 325-352) and the augmented formulation (see L. Formaggia, J. F. Gerbeau, F. Nobile, and A. Quarteroni, SIAM J. Numer. Anal., 40 (2002), pp. 376-401, and A. Veneziani and C. Vergara, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 803-816). However, both of these approaches present some drawbacks. For the flow rate problem, the former resorts to nonstandard functional spaces, which are quite difficult to discretize. On the other hand, for the mean pressure problem, it yields exact solutions only in very special cases. The latter is applicable only to the flow rate problem, since for the mean pressure problem it provides unfeasible boundary conditions. In this paper, we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional. This approach allows us to treat the two kinds of problems (flow rate and mean pressure) successfully within the same framework, which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries. Moreover, it is more versatile, being prone to be extended to other kinds of defective conditions. We analyze the problems obtained with this approach and propose some algorithms for their numerical solution. Several numerical results are presented supporting the effectiveness of our approach.
2008
NavierStokes equations; flow rate boundary condition; mean pressure boundary condition; computational hemodynamics
File in questo prodotto:
File Dimensione Formato  
Formaggia_Veneziani_Vergama_SNA08.pdf

Accesso riservato

: Post-Print (DRAFT o Author’s Accepted Manuscript-AAM)
Dimensione 625.6 kB
Formato Adobe PDF
625.6 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/529720
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 37
  • ???jsp.display-item.citation.isi??? 33
social impact