Abstract
We present a new fast vector penalty-projection method (VPP ε ), issued from noticeable improvements of previous works [7, 3, 4], to efficiently compute the solution of unsteady Navier-Stokes/Brinkman problems governing incompressible multiphase viscous flows. The method is also efficient to solve anisotropic Darcy problems. The key idea of the method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as \(\mathcal{O} (\epsilon \,\delta )\), δ being the time step, with a penalty parameter ε as small as desired until the machine precision, e.g. \(\epsilon = 1{0}^{-14}\), whereas the solution algorithm can be extremely fast and cheap. The method is numerically validated on a benchmark problem for two-phase bubble dynamics where we compare it to the Uzawa augmented Lagrangian (UAL) and scalar incremental projection (SIP) methods. Moreover, a new test case for fluid-structure interaction problems is also investigated. That results in a robust method running faster than usual methods and being able to efficiently compute accurate solutions to sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.
MSC 2010: 35Q30, 35Q35, 65M12, 65M85, 65N12, 65N85, 74F10, 76D05, 76D45, 76M25, 76R10, 76S05, 76T10
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
PH. ANGOT, Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows, Math. Meth. in the Appl. Sci. (M2AS) 22(16), 1395-1412, 1999.
PH. ANGOT, C.-H. BRUNEAU AND P. FABRIE, A penalization method to take into account obstacles in incompressible viscous flows, Nümerische Mathematik 81(4), 497-520, 1999.
PH. ANGOT, J.-P. CALTAGIRONE AND P. FABRIE, Vector penalty-projection methods for the solution of unsteady incompressible flows, in Finite Volumes for Complex Applications V, R. Eymard and J.-M. Hérard (Eds), pp. 169-176, ISTE Ltd and J. Wiley & Sons, 2008.
PH. ANGOT, J.-P. CALTAGIRONE AND P. FABRIE, A new fast method to compute saddle-points in constrained optimization and applications, Appl. Math. Letters, 2011 (submitted).
PH. ANGOT, J.-P. CALTAGIRONE AND P. FABRIE, A fast vector penalty-projection method for incompressible non-homogeneous or multiphase Navier-Stokes problems, Applied Mathematics Letters, 2011 (submitted).
J. BLAIR PEROT, Discrete conservation properties of unstructured mesh schemes, Annu. Rev. Fluid Mech. 43, 299-318, 2011.
J.-P. CALTAGIRONE AND J. BREIL, Sur une méthode de projection vectorielle pour la résolution des équations de Navier-Stokes, C. R. Acad. Sci. Paris, IIb 327, 1179-1184, 1999.
S. HYSING, S. TUREK, D. KUZMIN, N. PAROLINI, E. BURMAN, S. GANESAN AND L. TOBISKA, Quantitative benchmark computations of two-dimensional bubble dynamics, Int. J. Numer. Meth. Fluids, 60, 1259-1288, 2009.
K. KHADRA, PH. ANGOT, S. PARNEIX AND J.-P. CALTAGIRONE, Fictitious domain approach for numerical modelling of Navier-Stokes equations, Int. J. Numer. Meth. in Fluids, 34(8), 651-684, 2000.
A. SARTHOU, S. VINCENT, J.-P. CALTAGIRONE AND PH. ANGOT, Eulerian-Lagrangian grid coupling and penalty methods for the simulation of multiphase flows interacting with complex objects, Int. J. Numer. Meth. in Fluids 56(8), 1093-1099, 2008.
S. VINCENT, A. SARTHOU, J.-P. CALTAGIRONE, F. SONILHAC, P. FéVRIER, C. MIGNOT AND G. PIANET, Augmented Lagrangian and penalty methods for the simulation of two-phase flows interacting with moving solids. Application to hydroplaning flows interacting with real tire tread patterns, J. Comput. Phys. 230, 956-983, 2011.
D.L. YOUNGS, Time-dependent multimaterial flow with large fluid distortion, in Numerical Methods for Fluid Dynamics, K.W. Morton, M.J. Baines (Eds.), Academic Press, 1982.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Angot, P., Caltagirone, JP., Fabrie, P. (2011). A Spectacular Vector Penalty-Projection Method for Darcy and Navier-Stokes Problems. In: Fořt, J., Fürst, J., Halama, J., Herbin, R., Hubert, F. (eds) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20671-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-20671-9_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20670-2
Online ISBN: 978-3-642-20671-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)