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Immersed boundary methods for numerical simulation of confined fluid and plasma turbulence in complex geometries: a review

Part of: ITER International School 2014

Published online by Cambridge University Press:  30 September 2015

Kai Schneider
Kai Schneider*
Affiliation:
M2P2-CNRS, Aix-Marseille Université, 38, Rue Frédéric Joliot-Curie, 13451 Marseille CEDEX 13, France
*
†Email address for correspondence: [email protected]
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Abstract

Immersed boundary methods for computing confined fluid and plasma flows in complex geometries are reviewed. The mathematical principle of the volume penalization technique is described and simple examples for imposing Dirichlet and Neumann boundary conditions in one dimension are given. Applications for fluid and plasma turbulence in two and three space dimensions illustrate the applicability and the efficiency of the method in computing flows in complex geometries, for example in toroidal geometries with asymmetric poloidal cross-sections.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 6 , December 2015 , 435810601
Copyright
© Cambridge University Press 2015 

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References

Angot, P., Auphan, T. & Gues, O. 2014 An optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak. J. Comput. Phys. 261, 1–22.Google Scholar
Angot, P., Bruneau, C.-H. & Fabrie, P. 1999 A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81, 497–520.Google Scholar
Angot, P., Kadoch, B., Kolomenskiy, D. & Schneider, K. 2014 Convergence analysis of a penalty method for scalar transport and mixing in Navier–Stokes flows. Intl J. Numer. Anal. Model.; submitted.Google Scholar
Boiron, O., Chiavassa, G. & Donat, R. 2009 A high-resolution penalization method for large Mach number flows in the presence of obstacles. Comput. Fluids 38 (3), 703–714.Google Scholar
Bos, W. J. T., Neffaa, S. & Schneider, K. 2008 Rapid generation of angular momentum in bounded magnetized plasma. Phys. Rev. Lett. 101, 235003.Google Scholar
Bos, W. J. T., Neffaa, S. & Schneider, K. 2010 Self-organization and symmetry-breaking in two-dimensional plasma turbulence. Phys. Plasmas 17, 092302.Google Scholar
Carbou, G. & Fabrie, P. 2003 Boundary layer for a penalization method for viscous incompressible flow. Adv. Differ. Equat. 8 (12), 1453–1480.Google Scholar
Courant, R. 1943 Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23.Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J. 2013 Numerical simulation of the fluttering instability using a pseudospectral method with volume penalization. Comput. Struct. 122, 101–112.Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J.2014 A numerical study of vortex-induced drag of elastic swimmer models. In Proceedings, 6th International Symposium on Aero-aqua Bio-Mechanics, November 13–16, 2014, Honolulu, Hawaii, USA.Google Scholar
Engels, T., Kolomenskiy, D., Schneider, K. & Sesterhenn, J. 2015 Numerical simulation of fluid–structure interaction with the volume penalization method. J. Comput. Phys. 281, 96–115.Google Scholar
Ferziger, J. & Peric, M. 1996 Numerical Methods in Fluid Dynamics. Springer.Google Scholar
Kadoch, B., Kolomenskiy, D., Angot, P. & Schneider, K. 2012 A volume penalization method with moving obstacles for Navier–Stokes with advection diffusion equations. J. Comput. Phys. 231 (12), 4365–4383.Google Scholar
Keetels, G. H., d’Ortona, U., Kramer, W., Clercx, H. J. H., Schneider, K. & van Heijst, G. J. F. 2007 Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume penalization: convergence of a dipole–wall collision. J. Comput. Phys. 227, 919–945.Google Scholar
Kolomenskiy, D., Engels, T. & Schneider, K. 2013 Numerical modelling of flexible heaving foils. J. Aero-Aqua Bio-Mech. 3 (1), 22–28.CrossRefGoogle Scholar
Kolomenskiy, D., Moffatt, H. K., Farge, M. & Schneider, K. 2011 Two- and three-dimensional numerical simulations of the clap-fling-sweep of hovering insects. J. Fluids Struct. 27, 784–791.Google Scholar
Kolomenskiy, D., Nguyen van yen, R. & Schneider, K. 2015 Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions. Appl. Numer. Maths. 95, 238–249.CrossRefGoogle Scholar
Kolomenskiy, D. & Schneider, K. 2009 A Fourier spectral method for the Navier–Stokes equations with volume penalisation for moving solid obstacles. J. Comput. Phys. 228, 5687–5709.Google Scholar
Kreuzahler, S., Schulz, D., Homann, H., Ponty, Y. & Grauer, R. 2014 Numerical study of impeller-driven von Karman flows via a volume penalization method. New J. Phys. 16, 103001.Google Scholar
Liu, Q. & Vasilyev, O. 2007 A Brinkman penalization method for compressible flows in complex geometries. J. Comput. Phys. 227 (2), 946–966.Google Scholar
Min, M. S. & Gottlieb, D. 2003 On the convergence of the Fourier approximation for eigenvalues and eigenfunctions of discontinuous problems. SIAM J. Numer. Anal. 40, 2254–2269.CrossRefGoogle Scholar
Mittal, R., Dong, H., Bozkurttas, M., Najjar, F. M., Vargas, A. & von Loebbecke, A. 2008 A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227, 4825–4852.CrossRefGoogle ScholarPubMed
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.CrossRefGoogle Scholar
Morales, J. A., Bos, W. J. T., Schneider, K. & Montgomery, D. C. 2012 Intrinsic rotation of toroidally confined magnetohydrodynamics. Phys. Rev. Lett. 109, 175002.Google Scholar
Morales, J., Bos, W. J. T., Schneider, K. & Montgomery, D. C. 2014a The effect of toroidicity on reversed field pinch dynamics. Plasma Phys. Control. Fusion 56, 095024.Google Scholar
Morales, J., Leroy, M., Bos, W. & Schneider, K. 2014b Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization. J. Comput. Phys. 274, 64–94.Google Scholar
Neffaa, S., Bos, W. J. T. & Schneider, K. 2008 The decay of magnetohydrodynamic turbulence in confined domains. Phys. Plasmas 15, 092304.Google Scholar
Nguyen van yen, R., Farge, M. & Schneider, K. 2011 Energy dissipating structures produced by walls in two-dimensional flows at vanishing viscosity. Phys. Rev. Lett. 106, 184502.CrossRefGoogle ScholarPubMed
Nguyen van yen, R., Kolomenskiy, D. & Schneider, K. 2014 Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint. Numer. Math. 128, 301–338.Google Scholar
Nordlund, P. 1994 The structure and stability of the reversed field pinch magnetic equilibrium in Extrap T1. Phys. Scr. 49, 239–244.CrossRefGoogle Scholar
Paccou, A., Chiavassa, G., Liandrat, J. & Schneider, K. 2005 A penalization method applied to the wave equation. C. R. Méc. 333 (1), 79–85.Google Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252.Google Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479–517.Google Scholar
Ramière, I., Angot, P. & Belliard, M. 2007 A fictitious domain approach with spread interface for elliptic problems with general boundary conditions. Comput. Meth. Appl. Mech. Engng 196, 766–781.Google Scholar
Roberts, M., Leroy, M., Morales, J., Bos, W. J. T. & Schneider, K. 2014 Self-organization of helically forced MHD flows in confined cylindrical geometries. Fluid Dyn. Res. 46, 061422.Google Scholar
Sarthou, A., Vincent, S., Caltagirone, J. P. & Angot, P. 2008 Eulerian–Lagrangian grid coupling and penalty methods for the simulation of multiphase flows interacting with complex objects. Intl J. Numer. Meth. Fluids 56 (8), 1093–1099.Google Scholar
Saul’ev, V. K. 1963 Solution of some boundary-value problems on fast computers by the method of fictive domains. Siber. Math. J. 4 (4), 912–925; in Russian.Google Scholar
Schneider, K. 2005 Numerical simulation of the transient flow behavior in chemical reactors using a penalisation method. Comput. Fluids 34, 1223–1238.Google Scholar
Schneider, K. & Farge, M. 2005 Decaying two-dimensional turbulence in a circular container. Phys. Rev. Lett. 95, 244502.Google Scholar
Schneider, K. & Farge, M. 2008 Final states of decaying 2D turbulence in bounded domains: influence of the geometry. Physica D 237, 2228–2233.Google Scholar
Schneider, K., Neffaa, S. & Bos, W. J. T. 2011 A pseudo-spectral method with volume penalisation for magnetohydrodynamic turbulence in confined domains. Comput. Phys. Commun. 182 (1), 2–7.CrossRefGoogle Scholar
Schneider, K., Paget-Goy, M., Verga, A. & Farge, M. 2014 Numerical simulation of flows past flat plates using volume penalization. Comput. Appl. Math. 33 (2), 481–495.Google Scholar
Shirokoff, D. & Nave, J.-C. 2015 A sharp-interface active penalty method for the incompressible Navier–Stokes equations. J. Sci. Comput. 62 (1), 53–77.Google Scholar