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A Sharp-Interface Active Penalty Method for the Incompressible Navier–Stokes Equations

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Abstract

The volume penalty method provides a simple, efficient approach for solving the incompressible Navier–Stokes equations in domains with boundaries or in the presence of moving objects. Despite the simplicity, the method is typically limited to first order spatial accuracy. We demonstrate that one may achieve high order accuracy by introducing an active penalty term. One key difference from other works is that we use a sharp, unregularized mask function. We discuss how to construct the active penalty term, and provide numerical examples, in dimensions one and two. We demonstrate second and third order convergence for the heat equation, and second order convergence for the Navier–Stokes equations. In addition, we show that modifying the penalty term does not significantly alter the time step restriction from that of the conventional penalty method.

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Notes

  1. Here \(||\cdot ||_2\) is any appropriate numerical \(L^2(\varOmega _p)\) norm.

  2. The coordinates \(\varvec{\xi }(\mathbf {x})\) and \(s(\mathbf {x})\) are both at least \(C^1\) functions.

  3. Although not shown, a similar result of \(\Delta t < \min \{ N^{-2}, 1.1 \eta \}\) holds for a Fourier scheme.

  4. One can also restrict the forcing \(\tilde{f} = f (1-\chi _s)\) to the physical domain, and obtain similar results.

References

  1. Angot, P.: A unified fictitious domain model for general embedded boundary conditions. C. R. Math. Acad. Sci. Paris 341, 683–688 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angot, P.: A fictitious domain model for the stokes/brinkman problem with jump embedded boundary conditions. C. R. Math. Acad. Sci. Paris 348, 697–702 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Angot, P., Bruneau, C.-H., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81, 497–520 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Angot, P., Caltagirone, J.-P.: New graphical and computational architecture concept for numerical simulation on supercomputers. In: Proceedings of 2nd World Congress on Computational Mechanics, vol. 1, pp. 973–976 (1990)

  5. Arquis, E., Caltagirone, J.-P.: Sur les conditions hydrodynamiques au voisinage d’une interface milieu fluide—milieux poreux: application à la convection naturelle. Comptes Rendus de l’Academie des Science Paris II(299), 1–4 (1984)

    Google Scholar 

  6. Bruneau, C.-H.: Boundary conditions on artificial frontiers for incompressible and compressible Navier–Stokes equations. M2AN Math. Model. Numer. Anal. 34, 303–314 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruneau, C.-H., Fabrie, P.: New efficient boundary conditions for incompressible navier-stokes equations: a well-posedness result. M2AN Math. Model. Numer. Anal. 30, 815–840 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Carbou, G., Fabrie, P.: Boundary layer for a penalization method for viscous incompressible flow. Adv. Diff. Equat. 8(12), 1409–1532 (2003)

    Google Scholar 

  9. Chantalat, F., Bruneau, C.-H., Galusinski, C., Iollo, A.: Level-set, penalization and cartesian meshes: a paradigm for inverse problems and optimal design. J. Comput. Phys. 228, 6291–6315 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Coquerelle, M., Cottet, G.-H.: A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227, 9121–9137 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fedkiw, R., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Folland, G: Introduction to Partial Differential Equations, 2d ed. Princeton University Press, Princeton (1995)

  13. Gibou, F., Chen, L., Nguyen, D., Banerjee, S.: A level set based sharp interface method for the multiphase incompressible Navier–Stokes equations with phase change. J. Comput. Phys. 222, 536–555 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gibou, F., Fedkiw, R.: A Fourth order accurate discretization for the Laplace and Heat equations on arbitrary domains with applications to the Stefan problem. J. Comput. Phys. 202, 577–601 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Henshaw, William D.: A fourth-order accurate method for the incompressible Navier–Stokes equations on overlapping grids. J. Comput. Phys. 113(1), 13–25 (July 1994)

    Google Scholar 

  16. Johnston, H., Liu, J.-G.: Accurate, stable and efficient navier-stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 199, 221–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kadoch, B., Kolomenskiy, D., Angot, P., Schneider, K.: A volume penalization method for incompressible flows and scalar advection-diffusion with moving obstacles. J. Comput. Phys. 231, 4365–4383 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khadra, K., Angot, P., Parneix, S., Caltagirone, J.P.: Fictitious domain approach for numerical modelling of Navier–Stokes equations. Int. J. Numer. Methods Fluids 34, 651–684 (2000)

    Article  MATH  Google Scholar 

  19. Kolomenskiy, D., Schneider, K.: A fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228, 5687–5709 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Koumoutsakos, P., Leonard, A.: High-resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296, 1–38 (1995)

    Article  MATH  Google Scholar 

  21. Le, B., Khoo, B., Peraire, J.: An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries. J. Comput. Phys. 220, 109–138 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Morales, J., Leroy, M., Bos, W., Schneider, K.: Simulation of confined magnetohydrodynamic flows using a pseudo-spectral method with volume penalization. J. Comput. Phys. (under review), hal-00719737, version 1. (2012)

  23. Ng, Y.T., Min, C., Gibou, F.: An efficient fluid–solid coupling algorithm for single-phase flows. J. Comput. Phys. 228, 8807–8829 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Peskin, C.: The immersed boundary method. Acta Numerica 11, 479–517 (2000)

    MathSciNet  Google Scholar 

  25. Ramière, I., Angot, P., Belliard, M.: A general fictitious domain method with immersed jumps and multilevel nested structured meshes. J. Comput. Phys. 225, 1347–1387 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sarthou, A., Vincent, S., Caltagirone, J.P.: Consistent velocity-pressure for second-order \(L^2\)-penalty and direct-forcing methods, hal-00592079, Version 1. (2011)

  27. Sarthou, A., Vincent, S., Caltagirone, J.P., Angot, P.: Eulerian–Lagrangian grid coupling and penalty methods for the simulation of multiphase flows interacting with complex objects. Int. J. Numer. Methods Fluids 56, 1093–1099 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shirokoff, D.I.: A Pressure Poisson Method for the Incompressible Navier–Stokes Equations: II. Long Time Behavior of the Klein–Gordon Equations. PhD thesis, Massachusetts Institute of Technology (2011)

  29. Shirokoff, D., Rosales, R.R.: An efficient method for the incompressible Navier–Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary. J. Comput. Phys 230, 8619–8646 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. von Kármán, T.: Aerodynamics. McGraw-Hill, New York (1963)

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Acknowledgments

The authors would like to thank Kirill Shmakov and Geneviève Bourgeois for additional preliminary computations not currently presented. The authors have also greatly benefited from conversations with Dmitry Kolomenskiy, Kai Schneider, Ruben Rosales and Tsogtgerel Gantumur. This work was supported by an NSERC Discovery Grant and the NSERC DAS.

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  1. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A0B9, Canada

    D. Shirokoff & J.-C. Nave

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  1. D. Shirokoff
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  2. J.-C. Nave
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Correspondence to D. Shirokoff.

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Shirokoff, D., Nave, JC. A Sharp-Interface Active Penalty Method for the Incompressible Navier–Stokes Equations. J Sci Comput 62, 53–77 (2015). https://doi.org/10.1007/s10915-014-9849-6

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  • DOI: https://doi.org/10.1007/s10915-014-9849-6

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