More Web Proxy on the site http://driver.im/

article

An efficient semi-implicit immersed boundary method for the Navier-Stokes equations

Authors:

Zuoqiang ShiAuthors Info & Claims

Journal of Computational Physics, Volume 227, Issue 20

Pages 8968 - 8991

https://doi.org/10.1016/j.jcp.2008.07.005

Published: 01 October 2008 Publication History

Abstract

The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier-Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semi-implicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.

References

[1]

M. Abramowitz, I.A. Stegun, Modified Bessel Functions I and K, Dover, New York, 9th printing, 1972.

[2]

Brown, D.L., Cortex, R. and Minion, M., Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys. v168. 464-499.

Digital Library

[3]

Beyer, R.P., A computational model of the cochlea using the immersed boundary method. J. Comput. Phys. v98. 145-162.

Digital Library

[4]

Cortez, R., Cowen, N., Dillon, R. and Fauci, L.J., Simulation of swimming organisms: coupling internal mechanics with external fluid dynamics. Comput. Sci. Eng. v6. 38-45.

Digital Library

[5]

Chang, Y.C., Hou, T.Y., Merriman, B. and Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows. J. Comput. Phys. v124. 449-464.

Digital Library

[6]

Ceniceros, H.D. and Roma, A.M., Study of the long-time dynamics of viscous vortex sheet with a fully adaptive nonstiff method. Phys. Fluid. v16. 4285-4318.

[7]

Fauci, L. and Peskin, C.S., A computational model of aquatic animal locomotion. J. Comput. Phys. v77. 85-108.

Digital Library

[8]

Fauci, L. and Peskin, C.S., A fast numerical method for solving the three-dimensional Stokes equations in the presence of suspended particles. J. Comput. Phys. v79. 50-69.

[9]

Fauci, L.J., Interaction of oscillating silaments - a computational study. J. Comput. Phys. v86. 294-313.

Digital Library

[10]

Fogelson, A.L., A mathematical model and numerical methods for studying platelet adhesion and aggregation during blood clotting. J. Comput. Phys. v56. 111-134.

[11]

E. Givelberg, Modeling elastic shells immersed in fluid, PhD Thesis, Courant Institute of Mathematical Sciences, New York University, 1997.

[12]

Hou, T.Y., Lowengrub, J. and Shelley, M., Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. v114. 312-338.

Digital Library

[13]

Hou, T.Y., Lowengrub, J. and Shelley, M., The long-time motion of vortex sheets with surface tension. Phys. Fluid A. v9. 1933-1954.

[14]

Hopkins, M. and Fauci, L.J., A computational model of the collective fluid dynamics of motile microorganisms. J. Fluid Mech. v455. 149-174.

[15]

T.Y. Hou, Z. Shi, Removing the stiffness of elastic force from the immersed boundary method for the 2D stokes equations, J. Comput Phys., in press.

[16]

Jung, E.N. and Peskin, C.S., Two-dimensional simulations of valveless pumping using the immersed boundary method. SIAM J. Sci. Comput. v23. 19-45.

[17]

Mayo, A.A. and Peskin, C.S., An implicit numerical method for fluid dynamics problems with immersed elastic boundaries. Contemp. Math. v141. 261-277.

[18]

McQueen, D.M., Peskin, C.S. and Yellin, E.L., Fluid dynamics of the mitral valve: physiological aspects of a mathematical model. Am. J. Physiol. v242. H1095-H1110.

[19]

McQueen, D.M. and Peskin, C.S., Computer assisted design of pivoting-disc prosthetic mitral valves. J. Thorac. Cardiovasc. Surg. v86. 126-135.

[20]

McQueen, D.M. and Peskin, C.S., Computer assisted design of butterfly bileaflet valves for mitral position. Scand. J. Thorac. Cardiovasc. Surg. v19. 139-148.

[21]

McQueen, D.M. and Peskin, C.S., A three-dimensional computational method for blood flow in the heart: (II) Contractile fibers. J. Comput. Phys. v82. 289-297.

[22]

D.M. McQueen, C.S. Peskin, Heart simulation by an immersed boundary method with formal second order accuracy and reduced viscosity, in: Mechanics for a New Millennium, Proceedings of the International Conference on Theoretical and Applied mechanics (ICTAM), Kluwer Academic Publishers, 2000.

[23]

Miller, L.A. and Peskin, C.S., When vortices stick: and aerodynamic transition in tiny insects. J. Exp. Biol. v207. 3073-3088.

[24]

Miller, L.A. and Peskin, C.S., A computational fluid dynamics of 'clap and fling' in small insects. J. Exp. Biol. v208. 195-212.

[25]

Mori, Y. and Peskin, C.S., Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Engrg. v197. 2049-2067.

[26]

Newren, E.P., Fogelson, A.L., Guy, R.D. and Kirby, R.M., Unconditionally stable discretizations of the immersed boundary equations. J. Comput. Phys. v222. 702-719.

Digital Library

[27]

Peskin, C.S., Numerical study of blood flow in the heart. J. Comput. Phys. v25. 220-252.

[28]

Peskin, C.S. and McQueen, D.M., A three-dimensional computational method for blood flow in the heart: (I) Immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. v81. 372-405.

Digital Library

[29]

Peskin, C.S., The immersed boundary method. Acta Numer. v11. 479-517.

[30]

Peyret, R. and Taylor, T.D., Computational Methods for Fluid Flow. 1983. Springer-Verlag, New York.

[31]

Rosar, M.E. and Peskin, C.S., Fluid flow in collapsible elastic tubes: a three-dimensional numerical model. New York J. Math. v7. 281-302.

[32]

Stockie, J.M. and Wetton, B.R., Stability analysis for the immersed fiber problem. SIAM J. Appl. Math. v55. 1577-1591.

[33]

Stockie, J.M. and Green, S.I., Simulating the motion of flexible pulp fibers using the immersed boundary method. J. Comput. Phys. v147. 147-165.

Digital Library

[34]

Stockie, J.M. and Wetton, B.R., Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes. J. Comput. Phys. v154. 41-64.

Digital Library

[35]

Tu, C. and Peskin, C.S., Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods. SIAM J. Sci. Stat. Comput. v13. 1361-1376.

Digital Library

[36]

Tauber, W., Unverdi, S.O. and Tryggvason, G., The nonlinear behavior of a sheared immiscible fluid interface. Phys. Fluid. v14. 2871-2885.

[37]

Wang, N.T. and Fogelson, A.L., Computational methods or continuum models of platelet aggregation. J. Comput. Phys. v151. 649-675.

Digital Library

[38]

Zhu, L. and Peskin, C.S., Simulation of a flexible flapping filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. v179. 452-468.

Digital Library

Cited By

Wang QPan MTseng YHe D(2023)An Energy Stable Immersed Boundary Method for Deformable Membrane Problem with Non-uniform Density and ViscosityJournal of Scientific Computing10.1007/s10915-022-02092-394:2Online publication date: 3-Jan-2023
https://dl.acm.org/doi/10.1007/s10915-022-02092-3
Zhang TYuan J(2022)Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes EquationsJournal of Scientific Computing10.1007/s10915-021-01681-y90:1Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1007/s10915-021-01681-y
Miao SHendrickson KLiu Y(2017)Computation of three-dimensional multiphase flow dynamics by Fully-Coupled Immersed Flow (FCIF) solverJournal of Computational Physics10.1016/j.jcp.2017.08.042350:C(97-116)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1016/j.jcp.2017.08.042
Show More Cited By

Index Terms

An efficient semi-implicit immersed boundary method for the Navier-Stokes equations

Recommendations

An immersed boundary method for interfacial flows with insoluble surfactant

In this paper, an immersed boundary method is proposed for the simulation of two-dimensional fluid interfaces with insoluble surfactant. The governing equations are written in a usual immersed boundary formulation where a mixture of Eulerian flow and ...
A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations

In this paper, we develop a hybrid immersed boundary (IB) and immersed interface method (IIM) to simulate the dynamics of a drop under an electric field in Navier-Stokes flows. Within the leaky dielectric framework with piecewise constant electric ...
A new immersed boundary method for compressible Navier–Stokes equations

This paper presents an immersed boundary method for compressible Navier–Stokes equations in irregular domains, based on a local radial basis function approximation. This approach allows one to define a reconstruction of the radial basis functions on ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 227, Issue 20

October, 2008

239 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © Elsevier Inc. © 2008.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 October 2008

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

13
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 09 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Wang QPan MTseng YHe D(2023)An Energy Stable Immersed Boundary Method for Deformable Membrane Problem with Non-uniform Density and ViscosityJournal of Scientific Computing10.1007/s10915-022-02092-394:2Online publication date: 3-Jan-2023
https://dl.acm.org/doi/10.1007/s10915-022-02092-3
Zhang TYuan J(2022)Unconditional Stability and Optimal Error Estimates of Euler Implicit/Explicit-SAV Scheme for the Navier–Stokes EquationsJournal of Scientific Computing10.1007/s10915-021-01681-y90:1Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1007/s10915-021-01681-y
Miao SHendrickson KLiu Y(2017)Computation of three-dimensional multiphase flow dynamics by Fully-Coupled Immersed Flow (FCIF) solverJournal of Computational Physics10.1016/j.jcp.2017.08.042350:C(97-116)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1016/j.jcp.2017.08.042
Wiens JStockie J(2015)An efficient parallel immersed boundary algorithm using a pseudo-compressible fluid solverJournal of Computational Physics10.1016/j.jcp.2014.10.058281:C(917-941)Online publication date: 15-Jan-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.10.058
Strychalski WCopos CLewis OGuy R(2015)A poroelastic immersed boundary method with applications to cell biologyJournal of Computational Physics10.1016/j.jcp.2014.10.004282:C(77-97)Online publication date: 1-Feb-2015
https://dl.acm.org/doi/10.1016/j.jcp.2014.10.004
Guy RPhilip BGriffith B(2015)Geometric multigrid for an implicit-time immersed boundary methodAdvances in Computational Mathematics10.1007/s10444-014-9380-141:3(635-662)Online publication date: 1-Jun-2015
https://dl.acm.org/doi/10.1007/s10444-014-9380-1
Thomas Beale J(2012)Partially implicit motion of a sharp interface in Navier-Stokes flowJournal of Computational Physics10.1016/j.jcp.2012.05.018231:18(6159-6172)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1016/j.jcp.2012.05.018
Xu S(2011)A boundary condition capturing immersed interface method for 3D rigid objects in a flowJournal of Computational Physics10.1016/j.jcp.2011.05.019230:19(7176-7190)Online publication date: 1-Aug-2011
https://dl.acm.org/doi/10.1016/j.jcp.2011.05.019
Ceniceros HFisher J(2011)A fast, robust, and non-stiff Immersed Boundary MethodJournal of Computational Physics10.1016/j.jcp.2011.03.037230:12(5133-5153)Online publication date: 1-Jun-2011
https://dl.acm.org/doi/10.1016/j.jcp.2011.03.037
Ceniceros HNós RRoma A(2010)Three-dimensional, fully adaptive simulations of phase-field fluid modelsJournal of Computational Physics10.1016/j.jcp.2010.04.045229:17(6135-6155)Online publication date: 1-Aug-2010
https://dl.acm.org/doi/10.1016/j.jcp.2010.04.045
Show More Cited By

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents