    ## 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

* Corresponding author: eichlpa1@fjfi.cvut.cz

Received  January 2019 Revised  December 2019 Published  May 2020

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.

Citation: Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020349
##### References:
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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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  H. Lewy, K. Friedrichs and R. Courant, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.  Google Scholar  C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002. Google Scholar  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.  Google Scholar  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.  Google Scholar

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##### References:
  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.   Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  D. d'Humières, Generalized lattice-Boltzmann equations, Prog. Astronautics Aeronautics, 159 (1994). doi: 10.2514/5.9781600866319.0450.0458. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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  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.  Google Scholar  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.  Google Scholar  H. Lewy, K. Friedrichs and R. Courant, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32-74.  doi: 10.1007/BF01448839.  Google Scholar  C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  V. S. Vladimirov, Methods of the Theory of Generalized Functions, Analytical Methods and Special Functions, 6, Taylor & Francis, London, 2002. Google Scholar  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.  Google Scholar  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.  Google Scholar The setup of the computational domain in the case of the Kármán vortex flow problem. Two obstacles are immersed in the fluid. The first one is a rigid cylinder with diameter $d$. The last one is an elastic fiber with one fixed end (filled circle) and one free end (empty circle) The study of the elastic fiber deformation in the Kármán vortex street region. In Figs. 2a and 2b, the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with those in  (black circles). In Fig. 2c, the superposition of the elastic fibers is shown The study of the elastic fiber deformation in the Kármán vortex street region with neglected inertial forces of the elastic fiber. In Figs. 3a and 3b the free end trajectory is shown for two methods: (A) Forward Euler method (red), (B) The third order Runge-Kutta method (blue), and the results are compared with results in  (black triangles). In Fig. 3c, the superposition of the elastic fibers is shown Evolution of $E_{rkl}$ w.r.t. ${\sigma}/{h}$ for different values of the stiffness coefficient $k_S$ and for different mesh spacings $h$: $h_1 = 1.6 \cdot 10^{-4}\;\mathrm{m}$, $h_2 = 7.84\cdot 10^{-5}\;\mathrm{m}$, and $h_3 = 5.24\cdot 10^{-5}\;\mathrm{m}$. The results computed at the final time $T = 10\;\mathrm{s}$ when the steady state is reached Schematic visualization of the fluid penetration through the flexible fiber for three values of ${\sigma}/{h}$. In the figures, the velocity vectors (white arrows) and values of the fluid velocity magnitude (color scale) are shown. The results are at the final time $T = 10\;\mathrm{s}$ when the steady state is reached. All results are computed using $k_S = 100\;\mathrm{kg\, s^{-2}}$ and $h = 5.24\cdot 10^{-5}\; \mathrm{m}$ Schematic visualization of the fiber deformation at the final time $T = 10\, \mathrm{s}$ and for $h = 5.24 \cdot 10^{-5}$ m. Different colors of the fibers represent different values of ${\sigma}/{h}$ The setup of the computational domain in the case of elastic bump deformation Investigation of fluid flow around the bump. In Figs. (A) and (B), the relation between the surface area $\mu\left(\Omega_u\right)$ and ${\sigma}/{h}$ is shown. Figs. (C) and (D) illustrate the evolution of $E_{rkl}$ with respect to ${\sigma}/{h}$. For $U_\infty = 0.5\, \mathrm{m\, s^{-1}}$, the values are time-averaged over the time interval of $(1, 9.5)$ s. For $U_\infty = 0.5\, \mathrm{m\, s^{-1}}$, the values are computed at the final time $T = 10\; \mathrm{s}$ Schematic visualization of the fluid flow around an impermeable flexible fiber in time $t = 9.5\, \mathrm{s}$. In the figures, the velocity vectors (white arrows) and the values of the fluid velocity magnitude (color scale) are shown
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