# Investigation of Fluid-Structure Interaction Induced Bending for Elastic Flaps in a Cross Flow
## 4 Discussion  
The present investigation focuses on the numerical investigation of flexible flaps in a glycerin environment for different thicknesses and flap orientations. The deformation of the flaps was measured to observe the various flap clamping positions under laminar flow and also the maximum tip-deflection of the flap was observed. For both experimental and simulation conditions, the flaps settle down to a steady-state and the bending patterns attained by current simulation model are very close to that observed in the experiments. The focus of the present FSI simulations is to design a 3D interaction between structure and fluid in order to narrow the literature gap for flexible structures. The results indicate that flap bending increases with the increase in Reynolds number in the range of  
Figure 13: Comparison of flap wake; 10 mm Re = 12 (a) angle 0\({}^{\circ}\); (b) 90\({}^{\circ}\); (c) Inclined 45\({}^{\circ}\); (d) angle 90\({}^{\circ}\) Re = 6; (e) angle 90\({}^{\circ}\) Re = 12; (f) 5 mm Re = 3; (g) Re = 12.  
\(3<\mathrm{Re}<12\). However, for a different clamping position of 0\({}^{\circ}\) to 90\({}^{\circ}\) indicates minimum flap deflection with the highest clamping angle. Correspondingly, for an inclined position of 45\({}^{\circ}\) less deflection is perceived as compared to its undeformed shape. The objective is to judge the flap positions and flow regimes at varying computational time and to observe transient deformation response under different flow velocities. Likewise, the drag coefficients are also predicted for various flap alignments. Henceforward, the numerical model is suitable to accurately predict the response of thick and relatively thinner structures.  
The flap bending in the current work is observed for low Reynolds number; however, the bending and periodic nature of flap motion is schedule by another 3D FSI model for higher Reynolds number in case of water as a working fluid. For this purpose, experiments will be performed for thinner flaps and the numerical model will be helpful in understanding the flow physics around the flaps. This will be further helpful in designing the flexible structures in hemodynamic, biomedical and structural dynamics.  
Conceptualization, T.B.; Methodology, T.B. and F.H.; Software, T.B.; Validation T.B.; Formal Analysis, T.B. and F.H.; Investigation, T.B.; Writing--original draft preparation, T.B.; writing--review and editing, T.B., F.H., M.H. und R.S.; supervision, M.H. und R.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded to T.B. by a scholarship from the funds of the Freistaat Sachsen Germany, grant number Landesstipendium and to F.H. by Deutsche Forschungsgemeinschaft (DFG) Germany, grant number DFG 1494/25-1.  
The authors declare no conflict of interest.  
The following symbols are used in the manuscript:  
\begin{tabular}{l l} L & Flap length & (m) \\ w & Flap width & (m) \\ b & Flap thickness & (m) \\ Re & Reynolds number & \\ \(\Omega\) & Domain & \\ \(\Gamma\) & Fluid-structure interface & \\ t & Time & (sec) \\ v & Velocity & (m s\({}^{-1}\)) \\ \(\sigma\) & Stress & (N m\({}^{-2}\)) \\ n & Unit vector & \\ \(\varepsilon\) & Strain defined in Equation (4) & (m m\({}^{-1}\)) \\ H, \(\lambda\) & Lam\(\mathrm{\acute{e}}\) parameters & \\ \(\delta\) & Identity tensor defined in Equations (4) and (7) & \\ \(\rho\) & Density & (kg m\({}^{-3}\)) \\ u & Displacement & (m) \\ E & Young's Modulus & (N m\({}^{-2}\)) \\ v & Poisson ratio & \\ P & Static pressure & \\ \(\tau\) & Viscous stress tensor defined in Equation (7) & \\ e & Velocity gradient defined in Equation (7) & \\ U & Fluid mesh velocity & (m s\({}^{-1}\)) \\ K & Stiffness matrix & \\ \(\mu\) & Dynamic viscosity & (kg m\({}^{-1}\) s\({}^{-1}\)) \\ v & Kinematic viscosity & (m\({}^{2}\) s\({}^{-1}\)) \\ G & Shear Modulus & (N m\({}^{-2}\)) \\ x, y & Bending coordinates & (m) \\ \end{tabular}