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Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. (English) Zbl 1130.76406
Summary: This paper reports the computer simulation of a flapping flexible filament in a flowing soap film using the immersed boundary method. Our mathematical formulation includes filament mass and elasticity, gravity, air resistance, and the two wires that bound the flowing soap film. The incompressible viscous Navier-Stokes equations, which are used to describe the motion of the soap film and filament in our formulation, are discretized on a fixed uniform Eulerian lattice while the filament equations are discretized on a moving Lagrangian array of points which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction between the filament and the soap film is handled by a smoothed approximation to the Dirac delta function. This delta function approximation is used not only to interpolate the fluid velocity and to apply force to the fluid (as is commonly done in immersed boundary computations), but also to handle the mass of the filament, which is represented in our calculation as delta function layer of fluid mass density supported along the immersed filament. Because of this nonuniform density, we need to use a multigrid method for solving the discretized fluid equations. This replaces the FFT-based method that is commonly used in the uniform-density case.
Our main results are as follows:
(i) The sustained flapping of the filament only occurs when filament mass is included in the formulation of the model; within a certain range of mass, the more the mass of the filament, the bigger the amplitude of the flapping.
(ii) When the length of filament is short enough (below some critical length), the filament always approaches its straight (rest) state, in which the filament points downstream; but when the length is larger, the system is bistable, which means that it can settle into either state (rest state or sustained flapping) depending on the initial conditions. This numerical result we observed in computer simulation is the same as that of the laboratory experiment even though the Reynolds number of the computations is lower than that of the laboratory experiment by two orders of magnitude.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
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[1] Zhang, J; Childress, S; Libchaber, A; Shelley, M, Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind, Nature, 408, 835, (2000)
[2] M. Shelley, S. Childress, and, J. Zhang, Inertia dynamics of filaments, manuscript in preparation.
[3] P. Fast, and, W. D. Henshaw, Time-Accurate Computation of Viscous Flow Around Deforming Bodies Using Overset Grids, AIAA paper 2001-2604, 2001.
[4] McQueen, D.M; Peskin, C.S, Shared memory parallel vector implementation of the immersed boundary method for the computation of the blood flow in the beating Mammalian heart, J. supercomput., 11, 213, (1997)
[5] Peskin, C.S, Flow patterns around heart valves: A numerical method, J. comput. phys., 25, 220, (1977)
[6] Peskin, C.S; McQueen, D.M, A general method for the computer simulation of biological systems interacting with fluids, Symp. soc. exp. biol., 49, 265, (1995)
[7] C. S. Peskin, and, D. M. McQueen, Fluid dynamics of the heart and its valves, in, Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology, edited by, H. G. Othmer, F. R. Adler, M. A. Lewis, and J. C. Dallon, Prentice-Hall, Englewood Cliffs, NJ, 1996, p, 309.
[8] McQueen, M.C; Peskin, C.S, A three-dimensional computer model of the human heart for studying cardiac fluid dynamics, Comput. graph., 34, 56, (2000)
[9] Peskin, C.S; McQueen, D.M, Computational biofluid dynamics, Contemp. math., 141, 161, (1993) · Zbl 0786.76108
[10] McQueen, D.M; Peskin, C.S; Zhu, L, The immersed boundary method for incompressible fluid-structure interaction, In computational fluid and solid mechanics: Proceedings first M.I.T. conference on computational fluid and solid mechanics, (June 12-15, 2001)
[11] Peskin, C.S; Printz, B.F, Improved volume conservation in the computation of flows with immersed elastic boundaries, J. comput. phys., 105, 33, (1993) · Zbl 0762.92011
[12] McQueen, D.M; Peskin, C.S; Yellin, E.L, Fluid dynamics of the mitral valve: physiological aspects of a mathematical model, Am. J. physiol., 242, 1095, (1982)
[13] Fauci, L.J, Interaction of oscillating filaments—A computational study, J. comput. phys., 86, 294, (1990) · Zbl 0682.76105
[14] Fauci, L.J; Fogelson, A.L, Truncated Newton methods and the modeling of complex elastic structures, Commun. pure appl. math., 46, 787, (1993) · Zbl 0741.76103
[15] Fauci, L.J; Peskin, C.S, A computational model of aquatic animal locomotion, J. comput. phys., 77, 85, (1988) · Zbl 0641.76140
[16] Beyer, R.P, A computational model of the cochlea using the immersed boundary method, J. comput. phys., 98, 145, (1992) · Zbl 0744.76128
[17] Givelberg, E, modeling elastic shells immersed in fluid, (1997), New York UniversityCourant Institute of Mathematical Sciences
[18] Fogelson, A.L, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. comput. phys., 56, 111, (1984) · Zbl 0558.92009
[19] Fogelson, A.L; Peskin, C.S, A fast numerical method for solving three-dimensional Stokes equations in the presence of suspended particles, J. comput. phys., 79, 50, (1988) · Zbl 0652.76025
[20] Sulsky, D; Brackbill, J.U, A numerical method for suspension flow, J. comput. phys., 96, 339, (1991) · Zbl 0727.76082
[21] Jung, E; Peskin, C.S, 2-D simulation of valveless pumping using the immersed boundary method, SIAM J. sci. comput., 23, 19-45, (2001) · Zbl 1065.76156
[22] Rosar, M.E, A three-dimensional computer model for fluid flow through a collapsible tube, (1994), New York UniversityCourant Institute of Mathematical Sciences
[23] Arthurs, K.M; Moore, L.C; Peskin, C.S; Pitman, E.B; Layton, H.E, Modeling arteriolar flow and mass transport using the immersed boundary method, J. comput. phys., 147, 402, (1998) · Zbl 0936.76062
[24] Bottino, D.C, Modeling viscoelastic networks and cell deformation in the context of the immersed boundary method, J. comput. phys., 147, 86, (1998) · Zbl 0933.74077
[25] Eggleton, C.D; Popel, A.S, Large deformation of red blood cell ghosts in a simple shear flow, Phys. fluids, 10, 1834, (1998)
[26] Stockie, J.M; Green, S.I, Simulating the motion of flexible pulp fibres using the immersed boundary method, J. comput. phys., 147, 147, (1998) · Zbl 0935.76065
[27] Lai, M.C; Peskin, C.S, An immersed boundary method with formal second order accuracy and reduced numerical viscosity, J. comput. phys., 160, 705, (2000) · Zbl 0954.76066
[28] D. M. McQueen, and, C. S. Peskin, Heart simulation by an immersed boundary method with formal second order accuracy and reduced numerical viscosity, in Mechanics for a new millennium: proceedings of the 20th International Congress of Theoretical and Applied Mechanics, Chicago, Illinois, USA, 27 Aug.-2 Sept. 2000, ICTAM 2000, Chicago, edited by, Hassan, Aref and James, W. Phillips, Dordrecht, Norwell, MA, Kluwer Academic Publishers, C2001.
[29] A. L. Fogelson, and, J. Zhu, Implementation of a Variable-Density Immersed Boundary Method, available at, http://www.math.utah.edu/fogelson.
[30] Chorin, A.J, Numerical solution of the navier – stokes equations, Math. comp., 22, 745, (1968) · Zbl 0198.50103
[31] Chorin, A.J, On the convergence of discrete approximations to the navier – stokes equations, Math. comp., 23, 341, (1969) · Zbl 0184.20103
[32] Bell, J.B; Colella, P; Glaz, H.M, A second order projection method for the incompressible navier – stokes equations, J. comput. phys., 85, 257, (1989) · Zbl 0681.76030
[33] Bell, J.B; Colella, P; Howell, L.H, An efficient second-order projection method for viscous incompressible flow, Proceedings of the tenth AIAA computational fluid dynamics conference, June 1991, 360, (1991)
[34] Botella, O, On the solution of the navier – stokes equations using Chebyshev projection schemes with third-order accuracy in time, Comput. fluids, 26, 107, (1997) · Zbl 0898.76077
[35] E., W; Guo Liu, J, Projection method. I: convergence and numerical boundary layers, SIAM J. numer. anal., 32, 1017, (1995) · Zbl 0842.76052
[36] E., W; Guo Liu, J, Projection method. II: godunov – ryabenki analysis, SIAM J. numer. anal., 33, 1597, (1996) · Zbl 0920.76056
[37] Perot, J.B, An analysis of the fractional step method, J. comput. phys., 108, 51, (1993) · Zbl 0778.76064
[38] Brown, D.L; Cortez, R; Minion, M.L, Accurate projection methods for the incompressible navier – stokes equations, J. comput. phys., 168, 464, (2001) · Zbl 1153.76339
[39] Briggs, W.L; Henson, V.E; McCormick, S.F, A multigrid tutorial, (2000), Soc. for Industr. & Appl. Math Philadelphia
[40] Press, W.H; Teukolsky, S.A; Vetterling, W.T; Flannery, B.P, numerical recipes in Fortran: the art of scientific computing, (1992), Cambridge University Press Cambridge · Zbl 0778.65002
[41] Brandt, A, multigrid techniques: 1984 guide with application to fluid dynamics, (1984), Gesellschaft fur Mathematik und Datenverarbeitung St. Augustin/Bonn
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