June  2012, 17(4): 1175-1184. doi: 10.3934/dcdsb.2012.17.1175

An augmented immersed interface method for moving structures with mass

1. 

Department of Mathematics, Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, North Carolina State University, Raleigh, NC 27695, United States, United States, United States

Received  February 2011 Revised  August 2011 Published  February 2012

We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.
Citation: Jian Hao, Zhilin Li, Sharon R. Lubkin. An augmented immersed interface method for moving structures with mass. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1175-1184. doi: 10.3934/dcdsb.2012.17.1175
References:
[1]

R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow,, J. Comput. Phys., 169 (2001), 363.  doi: 10.1006/jcph.2000.6542.  Google Scholar

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J. Hao, T.-W. Pan, R. Glowinski and D. D. Joseph, A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach,, J. Non-Newtonian Fluid Mech., 156 (2009), 95.  doi: 10.1016/j.jnnfm.2008.07.006.  Google Scholar

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J. Hao, T.-W. Pan, S. Čanić, R. Glowinski and D. Rosenstrauch, A fluid-cell interaction and adhesion algorithm for tissue-coating of cardiovascular implants,, Multiscale Model. Simul., 7 (2009), 1669.  doi: 10.1137/080733188.  Google Scholar

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J. Hao and L. Zhu, A lattice Boltzmann based implicit immersed boundary method for fluid-structure interaction,, Comp. Math. Appl., 59 (2010), 185.  doi: 10.1016/j.camwa.2009.06.055.  Google Scholar

[5]

J. Hao and L. Zhu, A lattice Boltzmann based implicit immersed boundary method in three dimensions,, submitted., ().   Google Scholar

[6]

T. Y. Hou and Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations,, J. Comput. Phys., 227 (2008), 8968.  doi: 10.1016/j.jcp.2008.07.005.  Google Scholar

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K. Ito, M.-C. Lai and Z. Li, A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains,, J. Comput. Phys., 228 (2009), 2616.  doi: 10.1016/j.jcp.2008.12.028.  Google Scholar

[8]

S. W. Jung, K. Mareck, M. Shelley and J. Zhang, Dynamics of a deformable body in a fast flowing soap film,, Phys. Rev. Lett., 97 (2006).  doi: 10.1103/PhysRevLett.97.134502.  Google Scholar

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Y. Kim and C. S. Peskin, Penalty immersed boundary method for an elastic boundary with mass,, Phys. Fluids, 19 (2007).  doi: 10.1063/1.2734674.  Google Scholar

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Y. Kim, L. Zhu, X. Wang and C. S. Peskin, On various techniques for computer simulation of boundaries with mass, in "Computational Fluid and Solid Mechanics" (eds. K. J. Bathe),, Elsevier, (2003), 1746.   Google Scholar

[11]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension,, SIAM J. Sci. Comput., 18 (1997), 709.  doi: 10.1137/S1064827595282532.  Google Scholar

[12]

Z. Li, "The Immersed Interface Method-A Numerical Approach for Partial Differential Equations with Interfaces,", Ph.D thesis, (1994).   Google Scholar

[13]

Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains,", Frontiers in Applied Mathematics, 33 (2006).   Google Scholar

[14]

Z. Li and M.-C. Lai, The immersed interface method for the Navier-Stokes equations with singular forces,, J. Comput. Phys., 171 (2001), 822.  doi: 10.1006/jcph.2001.6813.  Google Scholar

[15]

Z. Li and M.-C. Lai, New finite difference methods based on IIM for inextensible interfaces in incompressible flows,, East Asian J. Applied Math., 1 (2011), 155.   Google Scholar

[16]

Y. Mori and C. S. Peskin, Implicit second-order immersed boundary methods with boundary mass,, Comput. Methods Appl. Mech. Engrg., 197 (2008), 2049.  doi: 10.1016/j.cma.2007.05.028.  Google Scholar

[17]

T.-W. Pan and R. Glowinski, Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow,, J. Comput. Phys., 181 (2002), 260.  doi: 10.1006/jcph.2002.7123.  Google Scholar

[18]

C. S. Peskin, "Flow Patterns Around Heart Valves: A Digital Computer Method for Solving the Equations of Motion,", Ph.D thesis, (1972).   Google Scholar

[19]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput. Phys., 25 (1977), 220.  doi: 10.1016/0021-9991(77)90100-0.  Google Scholar

[20]

C. S. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479.  doi: 10.1017/S0962492902000077.  Google Scholar

[21]

K. Shoele and Q. Zhu, Flow-induced vibrations of a deformable ring,, J. Fluid Mech., 650 (2010), 343.  doi: 10.1017/S0022112009993697.  Google Scholar

[22]

X. S. Wang, An iterative matrix-free method in implicit immersed boundary/continuum methods,, Computers and Structures, 85 (2007), 739.  doi: 10.1016/j.compstruc.2007.01.017.  Google Scholar

[23]

J. Zhang, S. Childress, A. Libchaber and M. Shelley, Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind,, Nature, 408 (2000).  doi: 10.1038/35048530.  Google Scholar

[24]

L. Zhu, Scaling laws for drag of a compliant body in an incompressible viscous flow,, J. Fluid Mech., 607 (2008), 387.   Google Scholar

[25]

L. Zhu and C. S. Peskin, Interaction of two flexible filaments in a flowing soap film,, Phys. Fluids, 15 (2003), 1954.  doi: 10.1063/1.1582476.  Google Scholar

[26]

L. Zhu and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method,, J. Comput. Phys., 179 (2002), 452.  doi: 10.1006/jcph.2002.7066.  Google Scholar

show all references

References:
[1]

R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph and J. Périaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow,, J. Comput. Phys., 169 (2001), 363.  doi: 10.1006/jcph.2000.6542.  Google Scholar

[2]

J. Hao, T.-W. Pan, R. Glowinski and D. D. Joseph, A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach,, J. Non-Newtonian Fluid Mech., 156 (2009), 95.  doi: 10.1016/j.jnnfm.2008.07.006.  Google Scholar

[3]

J. Hao, T.-W. Pan, S. Čanić, R. Glowinski and D. Rosenstrauch, A fluid-cell interaction and adhesion algorithm for tissue-coating of cardiovascular implants,, Multiscale Model. Simul., 7 (2009), 1669.  doi: 10.1137/080733188.  Google Scholar

[4]

J. Hao and L. Zhu, A lattice Boltzmann based implicit immersed boundary method for fluid-structure interaction,, Comp. Math. Appl., 59 (2010), 185.  doi: 10.1016/j.camwa.2009.06.055.  Google Scholar

[5]

J. Hao and L. Zhu, A lattice Boltzmann based implicit immersed boundary method in three dimensions,, submitted., ().   Google Scholar

[6]

T. Y. Hou and Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations,, J. Comput. Phys., 227 (2008), 8968.  doi: 10.1016/j.jcp.2008.07.005.  Google Scholar

[7]

K. Ito, M.-C. Lai and Z. Li, A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains,, J. Comput. Phys., 228 (2009), 2616.  doi: 10.1016/j.jcp.2008.12.028.  Google Scholar

[8]

S. W. Jung, K. Mareck, M. Shelley and J. Zhang, Dynamics of a deformable body in a fast flowing soap film,, Phys. Rev. Lett., 97 (2006).  doi: 10.1103/PhysRevLett.97.134502.  Google Scholar

[9]

Y. Kim and C. S. Peskin, Penalty immersed boundary method for an elastic boundary with mass,, Phys. Fluids, 19 (2007).  doi: 10.1063/1.2734674.  Google Scholar

[10]

Y. Kim, L. Zhu, X. Wang and C. S. Peskin, On various techniques for computer simulation of boundaries with mass, in "Computational Fluid and Solid Mechanics" (eds. K. J. Bathe),, Elsevier, (2003), 1746.   Google Scholar

[11]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension,, SIAM J. Sci. Comput., 18 (1997), 709.  doi: 10.1137/S1064827595282532.  Google Scholar

[12]

Z. Li, "The Immersed Interface Method-A Numerical Approach for Partial Differential Equations with Interfaces,", Ph.D thesis, (1994).   Google Scholar

[13]

Z. Li and K. Ito, "The Immersed Interface Method. Numerical Solutions of PDEs Involving Interfaces and Irregular Domains,", Frontiers in Applied Mathematics, 33 (2006).   Google Scholar

[14]

Z. Li and M.-C. Lai, The immersed interface method for the Navier-Stokes equations with singular forces,, J. Comput. Phys., 171 (2001), 822.  doi: 10.1006/jcph.2001.6813.  Google Scholar

[15]

Z. Li and M.-C. Lai, New finite difference methods based on IIM for inextensible interfaces in incompressible flows,, East Asian J. Applied Math., 1 (2011), 155.   Google Scholar

[16]

Y. Mori and C. S. Peskin, Implicit second-order immersed boundary methods with boundary mass,, Comput. Methods Appl. Mech. Engrg., 197 (2008), 2049.  doi: 10.1016/j.cma.2007.05.028.  Google Scholar

[17]

T.-W. Pan and R. Glowinski, Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow,, J. Comput. Phys., 181 (2002), 260.  doi: 10.1006/jcph.2002.7123.  Google Scholar

[18]

C. S. Peskin, "Flow Patterns Around Heart Valves: A Digital Computer Method for Solving the Equations of Motion,", Ph.D thesis, (1972).   Google Scholar

[19]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput. Phys., 25 (1977), 220.  doi: 10.1016/0021-9991(77)90100-0.  Google Scholar

[20]

C. S. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479.  doi: 10.1017/S0962492902000077.  Google Scholar

[21]

K. Shoele and Q. Zhu, Flow-induced vibrations of a deformable ring,, J. Fluid Mech., 650 (2010), 343.  doi: 10.1017/S0022112009993697.  Google Scholar

[22]

X. S. Wang, An iterative matrix-free method in implicit immersed boundary/continuum methods,, Computers and Structures, 85 (2007), 739.  doi: 10.1016/j.compstruc.2007.01.017.  Google Scholar

[23]

J. Zhang, S. Childress, A. Libchaber and M. Shelley, Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind,, Nature, 408 (2000).  doi: 10.1038/35048530.  Google Scholar

[24]

L. Zhu, Scaling laws for drag of a compliant body in an incompressible viscous flow,, J. Fluid Mech., 607 (2008), 387.   Google Scholar

[25]

L. Zhu and C. S. Peskin, Interaction of two flexible filaments in a flowing soap film,, Phys. Fluids, 15 (2003), 1954.  doi: 10.1063/1.1582476.  Google Scholar

[26]

L. Zhu and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method,, J. Comput. Phys., 179 (2002), 452.  doi: 10.1006/jcph.2002.7066.  Google Scholar

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