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Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction
a. | Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13,120 00 Prague, Czech Republic |
b. | Department of Heat Engineering and Environment Protection, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow, Poland |
In this article, we deal with the numerical immersed boundary-lattice Boltzmann method for simulation of the fluid-structure interaction problems in 2D. We consider the interaction of incompressible, Newtonian fluid in an isothermal system with an elastic fiber, which represents an immersed body boundary. First, a short introduction to the lattice Boltzmann and immersed boundary method is presented and the combination of these two methods is briefly discussed. Then, the choice of the smooth approximation of the Dirac delta function and the discretization of the immersed body is discussed. One of the significant drawbacks of immersed boundary method is the penetrative flow through the immersed impermeable boundary. The effect of the immersed body boundary discretization is investigated using two benchmark problems, where an elastic fiber is deformed. The results indicate that the restrictions placed on the discretization in literature are not necessary.
References:
[1] |
F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action: By comparing the theoretical and measured forms of drops of fluid, Cambridge University Press, 1883.
![]() |
[2] |
S. R. Blair and Y. W. Kwon, Modeling of fluid–structure interaction using lattice Boltzmann and finite element methods, J. Pressure Vessel Technol., 137 (2015), 9pp.
doi: 10.1115/1.4027866. |
[3] |
S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Phys. Rev. Lett., 97 (2006), 4pp.
doi: 10.1103/PhysRevLett.97.010201. |
[4] |
B. S. H. Connell and D. K. P. Yue,
Flapping dynamics of a flag in a uniform stream, J. Fluid Mech., 581 (2007), 33-67.
doi: 10.1017/S0022112007005307. |
[5] |
D. d'Humières, Generalized lattice-Boltzmann equations, Prog. Astronautics Aeronautics, 159 (1994).
doi: 10.2514/5.9781600866319.0450.0458. |
[6] |
R. Fučík, P. Eichler, R. Straka, P. Pauš, J. Klinkovský and T. Oberhuber,
On optimal node spacing for immersed boundary lattice Boltzmann method in 2D and 3D, Comput. Math. Appl., 77 (2019), 1144-1162.
doi: 10.1016/j.camwa.2018.10.045. |
[7] |
M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Phys. Rev. E, 73 (2006).
doi: 10.1103/PhysRevE.73.066705. |
[8] |
M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk,
The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl., 70 (2015), 507-547.
doi: 10.1016/j.camwa.2015.05.001. |
[9] |
Z. Guo and C. Shu, Lattice Boltzmann Method and its Applications in Engineering, Advances in Computational Fluid Dynamics, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8806. |
[10] |
B. Kallemov, A. P. S. Bhalla, B. E. Griffith and A. Donev,
An immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11 (2016), 79-141.
doi: 10.2140/camcos.2016.11.79. |
[11] |
I. Karlin, F. Bösch and S. Chikatamarla, Gibbs' principle for the lattice-kinetic theory of fluid dynamics, Phys. Rev. E, 90 (2014).
doi: 10.1103/PhysRevE.90.031302. |
[12] |
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. |
[13] |
T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The Lattice Boltzmann Method: Principles and Practice, Graduate Texts in Physics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-44649-3. |
[14] |
T. Krüger, F. Varnik and D. Raabe,
Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method, Comput. Math. Appl., 61 (2011), 3485-3505.
doi: 10.1016/j.camwa.2010.03.057. |
[15] |
H. Lewy, K. Friedrichs and R. Courant,
Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.
doi: 10.1007/BF01448839. |
[16] |
C. S. Peskin,
The immersed boundary method, Acta Numer., 11 (2002), 479-517.
doi: 10.1017/S0962492902000077. |
[17] |
K. N. Premnath and S. Banerjee, Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments, Phys. Rev. E, 80 (2009).
doi: 10.1103/PhysRevE.80.036702. |
[18] |
K. V. Sharman, R. Straka and F. W. Tavares,
New Cascaded Thermal Lattice Boltzmann Method for simulations of advection-diffusion and convective heat transfer, Internat. J. Thermal Sciences, 118 (2017), 259-277.
doi: 10.1016/j.ijthermalsci.2017.04.020. |
[19] |
M. C. Sukop and D. J. Thorne Jr., Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers, Springer, Berlin, Heidelberg, 2006.
doi: 10.1007/978-3-540-27982-2. |
[20] |
K. Taira and T. Colonius,
The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.
doi: 10.1016/j.jcp.2007.03.005. |
[21] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[22] |
F.-B. Tian, H. Luo, L. Zhu, J. C. Liao and X.-Y. Lu,
An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230 (2011), 7266-7283.
doi: 10.1016/j.jcp.2011.05.028. |
[23] |
J. Tölke,
Implementation of a Lattice Boltzmann kernel using the Compute Unified Device Architecture developed by nVIDIA, Comput. Visualization Science, 13 (2010), 29-39.
doi: 10.1007/s00791-008-0120-2. |
[24] |
V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002. |
[25] |
X. Yang, X. Zhang, Z. Li and G.-W. He,
A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput. Phys., 228 (2009), 7821-7836.
doi: 10.1016/j.jcp.2009.07.023. |
[26] |
Z.-Q. Zhang, G. R. Liu and B. C. Khoo,
Immersed smoothed finite element method for two dimensional fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 90 (2012), 1292-1320.
doi: 10.1002/nme.4299. |
show all references
References:
[1] |
F. Bashforth and J. C. Adams, An attempt to test the theories of capillary action: By comparing the theoretical and measured forms of drops of fluid, Cambridge University Press, 1883.
![]() |
[2] |
S. R. Blair and Y. W. Kwon, Modeling of fluid–structure interaction using lattice Boltzmann and finite element methods, J. Pressure Vessel Technol., 137 (2015), 9pp.
doi: 10.1115/1.4027866. |
[3] |
S. Chikatamarla, S. Ansumali and I. Karlin, Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Phys. Rev. Lett., 97 (2006), 4pp.
doi: 10.1103/PhysRevLett.97.010201. |
[4] |
B. S. H. Connell and D. K. P. Yue,
Flapping dynamics of a flag in a uniform stream, J. Fluid Mech., 581 (2007), 33-67.
doi: 10.1017/S0022112007005307. |
[5] |
D. d'Humières, Generalized lattice-Boltzmann equations, Prog. Astronautics Aeronautics, 159 (1994).
doi: 10.2514/5.9781600866319.0450.0458. |
[6] |
R. Fučík, P. Eichler, R. Straka, P. Pauš, J. Klinkovský and T. Oberhuber,
On optimal node spacing for immersed boundary lattice Boltzmann method in 2D and 3D, Comput. Math. Appl., 77 (2019), 1144-1162.
doi: 10.1016/j.camwa.2018.10.045. |
[7] |
M. Geier, A. Greiner and J. G. Korvink, Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Phys. Rev. E, 73 (2006).
doi: 10.1103/PhysRevE.73.066705. |
[8] |
M. Geier, M. Schönherr, A. Pasquili and M. Krafczyk,
The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl., 70 (2015), 507-547.
doi: 10.1016/j.camwa.2015.05.001. |
[9] |
Z. Guo and C. Shu, Lattice Boltzmann Method and its Applications in Engineering, Advances in Computational Fluid Dynamics, 3, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013.
doi: 10.1142/8806. |
[10] |
B. Kallemov, A. P. S. Bhalla, B. E. Griffith and A. Donev,
An immersed boundary method for rigid bodies, Commun. Appl. Math. Comput. Sci., 11 (2016), 79-141.
doi: 10.2140/camcos.2016.11.79. |
[11] |
I. Karlin, F. Bösch and S. Chikatamarla, Gibbs' principle for the lattice-kinetic theory of fluid dynamics, Phys. Rev. E, 90 (2014).
doi: 10.1103/PhysRevE.90.031302. |
[12] |
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. |
[13] |
T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Orest and E. Viggen, The Lattice Boltzmann Method: Principles and Practice, Graduate Texts in Physics, Springer, Cham, 2017.
doi: 10.1007/978-3-319-44649-3. |
[14] |
T. Krüger, F. Varnik and D. Raabe,
Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method, Comput. Math. Appl., 61 (2011), 3485-3505.
doi: 10.1016/j.camwa.2010.03.057. |
[15] |
H. Lewy, K. Friedrichs and R. Courant,
Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.
doi: 10.1007/BF01448839. |
[16] |
C. S. Peskin,
The immersed boundary method, Acta Numer., 11 (2002), 479-517.
doi: 10.1017/S0962492902000077. |
[17] |
K. N. Premnath and S. Banerjee, Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments, Phys. Rev. E, 80 (2009).
doi: 10.1103/PhysRevE.80.036702. |
[18] |
K. V. Sharman, R. Straka and F. W. Tavares,
New Cascaded Thermal Lattice Boltzmann Method for simulations of advection-diffusion and convective heat transfer, Internat. J. Thermal Sciences, 118 (2017), 259-277.
doi: 10.1016/j.ijthermalsci.2017.04.020. |
[19] |
M. C. Sukop and D. J. Thorne Jr., Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers, Springer, Berlin, Heidelberg, 2006.
doi: 10.1007/978-3-540-27982-2. |
[20] |
K. Taira and T. Colonius,
The immersed boundary method: A projection approach, J. Comput. Phys., 225 (2007), 2118-2137.
doi: 10.1016/j.jcp.2007.03.005. |
[21] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
[22] |
F.-B. Tian, H. Luo, L. Zhu, J. C. Liao and X.-Y. Lu,
An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments, J. Comput. Phys., 230 (2011), 7266-7283.
doi: 10.1016/j.jcp.2011.05.028. |
[23] |
J. Tölke,
Implementation of a Lattice Boltzmann kernel using the Compute Unified Device Architecture developed by nVIDIA, Comput. Visualization Science, 13 (2010), 29-39.
doi: 10.1007/s00791-008-0120-2. |
[24] |
V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002. |
[25] |
X. Yang, X. Zhang, Z. Li and G.-W. He,
A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations, J. Comput. Phys., 228 (2009), 7821-7836.
doi: 10.1016/j.jcp.2009.07.023. |
[26] |
Z.-Q. Zhang, G. R. Liu and B. C. Khoo,
Immersed smoothed finite element method for two dimensional fluid-structure interaction problems, Internat. J. Numer. Methods Engrg., 90 (2012), 1292-1320.
doi: 10.1002/nme.4299. |










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