Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction

Figure 1.  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)

Figure 2.  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 [22] (black circles). In Fig. 2c, the superposition of the elastic fibers is shown

Figure 3.  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 [22] (black triangles). In Fig. 3c, the superposition of the elastic fibers is shown

Figure 4.  The setup of the computational domain for the cavity flow problem

Figure 5.  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

Figure 6.  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} $

Figure 7.  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} $

Figure 8.  The setup of the computational domain in the case of elastic bump deformation

Figure 9.  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} $

Figure 10.  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