Three-Dimensional Fluid–Structure Interaction Case Study on Elastic Beam

: A three-dimensional T-shaped ﬂexible beam deformation was investigated using model experiments and numerical simulations. In the experiment, a beam was placed in a recirculating water channel with a steady uniform ﬂow in the inlet. A high-speed camera system (HSC) was utilized to record the T-shaped ﬂexible beam deformation in the cross-ﬂow direction. In addition, a two-way ﬂuid-structure interaction (FSI) numerical method was employed to simulate the deformation of the T-shaped ﬂexible beam. A system coupling was used for conjoining the ﬂuid and solid domain. The dynamic mesh method was used for recreating the mesh. After the validation of the three-dimensional numerical T-shaped ﬂexible solid beam with the HSC results, deformation and stress were calculated for di ﬀ erent Reynolds numbers. This study exhibited that the deformation of the T-shaped ﬂexible beam increases by nearly 90% when the velocity is changed from 0.25 to 0.35 m / s, whereas deformation of the T-shaped ﬂexible beam decreases by nearly 63% when the velocity is varied from 0.25 to 0.15 m / s.


Introduction
Deformation of structures has been an area of interest for engineering. Because the coupling process between fluid and structure plays an important role in many engineering fields. Interaction of fluid and structures in underwater flexible beams is a complex problem in industrial applications. Experimental and numerical research has been conducted to examine fluid-structure interaction (FSI). A number of previous studies are summarized in this section. Important FSI problems were simulated numerically by Chimakurthi et al. [1]. The authors applied Ansys workbench system coupling and two-way fluid-structure interaction to various multiphysics coupled problems, such as FSI in an oscillating solid structure and sub-sea pipeline vibrations. This study is important, because it shows that Ansys workbench system coupling is highly suited to multiphysics coupled problems. Furthermore, the study results were also validated with other experimental and numerical studies. Gluck et al. [2] studied fluid-structure interaction numerically in different plate forms, such as vertical and L-shaped plates. The authors used two code samples for simulation flow and structure, one using the finite volume method for the flow side and the other using the finite element method for the solid side. The one-way fluid-structure interaction method was employed for the simulation effect of waves on a ship's hull by Dhavalikar et al. [3]. It was assumed that the ship's hull was a rigid body and wave loads were then simulated on a rigid body. Narayanan et al. [4] numerically investigated flow behavior past a cylinder with a flexible filament. The commercial software STAR-CCM+ was used for solving the governing equations.
the Lagrangian variables have been used in the immersed boundary (IB) method, particularly in biological fluid dynamics. A monolithic FEM/Multigrid method for the fluid-structure interaction of an incompressible elastic object in laminar incompressible viscous flow was presented by Hron and Turek [21]. Griffith and Luo [22] studied a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables. They used this method for solving FSI problems. The left ventricle of the heart was stimulated by this method. The arbitrary Lagrangian-Eulerian finite element method was applied by Nassiri [23]. They used this method for numerical simulation and experimental investigation of wavy interfacial morphology during high velocity impact welding. Tabatabaei et al. [24] studied the hydrodynamics behavior of an axisymmetric squid model numerically. They applied SST k-w turbulence model for simulation. Various fineness ratios, jet propulsion, and drag force were investigated for different swimming velocities. Squid's flow characteristics were studied numerically and experimentally by Olcay et al. [25]. Digital particle image velocimetry (DPIV) was used for obtaining velocity contours in the experimental region. Ansys Fluent commercial software was applied for solving governing equations in the flow field. They showed that numerical results were so close to the experimental data. Hydrodynamic forces of a moving cylinder and fixed cylinder were investigated numerically by Eren et al. [26]. They used the dynamic mesh method for recreating mesh and moving a cylinder in the incompressible flow.
To improve our understanding of fluid-structure interaction methodology for flow past a flexible beam, we studied the fluid-structure interaction experimentally and numerically for a single flexible beam. We use a high-speed camera system (HSC) for obtaining the deformation of the beam. The three-dimensional flexible beam model was then investigated numerically at three different velocities. Two-way FSI method was applied for numerical simulation, and also, a validation test was carried out. All in all, this study helps to improve our understanding of flexible beams deformations. The paper was organized as follows: Section 2 defines experimental setup, computational domain governing equations, and numerical methods. Section 3 presents validation of deformation T-shaped flexible beam, numerical results, and discussions. Section 4 contains the conclusions.

Experimental Arrangement and Analysis Methodology
Details of the experimental setup are explained in Sections 2.2 and 2.3. Section 2.2 presents the water channel, the high-speed camera, and the technique of high-speed imaging method. Section 2.3 describes the data analysis of the experiment results.

Experimental Setup
Experiments were carried out in a recirculating water channel of State Key Laboratory of Hydraulics and Mountain River Engineering (SKLHMRE) at Sichuan University (SCU), China. The water channel has a 12 m (length) × 0.5 m (width) × 0.6 m (height) test section with the mean flow velocity up to 0.25 m/s. The water level was maintained at 0.5 m during the experiments and a propeller velocity meter was used to measure the inflow velocity. Froude number is 0.11 (Fr = 0.11), so flow is subcritical in this study. Multiple measurement points in the same vertical plane of the channel at different depths are chosen to measure the velocity in a long period, and the mean flow velocity is calculated by time averaging and space averaging. A T-shaped flexible beam made of polyurethane was fixed in a vertical plane facing to the approaching flow. The dimensions of the T-shaped flexible beam are shown in Figure 1. Water was used for the fluid part and a polyurethane beam was applied for the solid part. The experiment was conducted indoors; physically essential properties of selected materials at room temperature are provided in Table 1.  For the beam motion and image recognition process, illumination was provided by an LED light sheet, and the beam was marked black. A non-intrusive technique of high-speed imaging method was employed to record the deformation of the flexible beam in this study. Images were captured at 1000 frames per second (fps) using a high-speed camera (Fastcam Mini UX100, Photron Inc., Chiyoda-Ku, Tokyo, Japan, maximum acquisition rate: 4000 fps with the resolution of 1280 × 1024 pixels). The resolution of the images is cut down to 616 pixels × 1024 pixels for saving camera memory space to only cover the area where the beam exists, as shown in Figure 2.   For the beam motion and image recognition process, illumination was provided by an LED light sheet, and the beam was marked black. A non-intrusive technique of high-speed imaging method was employed to record the deformation of the flexible beam in this study. Images were captured at 1000 frames per second (fps) using a high-speed camera (Fastcam Mini UX100, Photron Inc., Chiyoda-Ku, Tokyo, Japan, maximum acquisition rate: 4000 fps with the resolution of 1280 × 1024 pixels). The resolution of the images is cut down to 616 pixels × 1024 pixels for saving camera memory space to only cover the area where the beam exists, as shown in Figure 2.  For the beam motion and image recognition process, illumination was provided by an LED light sheet, and the beam was marked black. A non-intrusive technique of high-speed imaging method was employed to record the deformation of the flexible beam in this study. Images were captured at 1000 frames per second (fps) using a high-speed camera (Fastcam Mini UX100, Photron Inc., Chiyoda-Ku, Tokyo, Japan, maximum acquisition rate: 4000 fps with the resolution of 1280 × 1024 pixels). The resolution of the images is cut down to 616 pixels × 1024 pixels for saving camera memory space to only cover the area where the beam exists, as shown in Figure 2.  The standardization tests were carried out in still water in order to determine the actual distance per pixel ( Figure 3). The field of view was approximately 12.07 cm × 20.07 cm, leading to a spatial resolution of 0.0196 cm pixel −1 .

Data Analysis
To track the deformation of the T-shaped flexible beam, an image recognition Python code was developed to obtain quantitative data from images captured in the experiment ( Figure 4). A bilateral filter is a basic theory of image noise reduction, and it is better in edge-preserving than other filters, so we used a bilateral filter to reduce the noise of experimental images. Then, the Canny edge detector, which is an edge detection operator that uses a multi-stage algorithm, was used to detect edges of the beam in experimental images. Image binarization was set the grayscale value of the pixel on the image to 0 or 255, which is the process of presenting the whole image with an obvious black and white effect. The erosion, closing operation, and dilation are the morphological operations to enhance image features. We used erosion, closing operation, and dilation to make the edges detected by the Canny edge detector clearer, then the pixel coordinates of the edges were recorded. The pixel size of a picture can be detected. Thus, the scale S of actual distance and Pixel distance can be calculated through Figure 3. S is defined by S = actual distance/pixel distance. Therefore, the actual displacement D of the tracking point is calculated by D = S*L, where L is the pixel distance between the current and initial position of the tracking point. Details about image processing are illustrated in Howes [27]. The standardization tests were carried out in still water in order to determine the actual distance per pixel ( Figure 3). The field of view was approximately 12.07 cm × 20.07 cm, leading to a spatial resolution of 0.0196 cm pixel −1 . The standardization tests were carried out in still water in order to determine the actual distance per pixel ( Figure 3). The field of view was approximately 12.07 cm × 20.07 cm, leading to a spatial resolution of 0.0196 cm pixel −1 .

Data Analysis
To track the deformation of the T-shaped flexible beam, an image recognition Python code was developed to obtain quantitative data from images captured in the experiment (Figure 4). A bilateral filter is a basic theory of image noise reduction, and it is better in edge-preserving than other filters, so we used a bilateral filter to reduce the noise of experimental images. Then, the Canny edge detector, which is an edge detection operator that uses a multi-stage algorithm, was used to detect edges of the beam in experimental images. Image binarization was set the grayscale value of the pixel on the image to 0 or 255, which is the process of presenting the whole image with an obvious black and white effect. The erosion, closing operation, and dilation are the morphological operations to enhance image features. We used erosion, closing operation, and dilation to make the edges detected by the Canny edge detector clearer, then the pixel coordinates of the edges were recorded. The pixel size of a picture can be detected. Thus, the scale S of actual distance and Pixel distance can be calculated through Figure 3. S is defined by S = actual distance/pixel distance. Therefore, the actual displacement D of the tracking point is calculated by D = S*L, where L is the pixel distance between the current and initial position of the tracking point. Details about image processing are illustrated in Howes [27].

Data Analysis
To track the deformation of the T-shaped flexible beam, an image recognition Python code was developed to obtain quantitative data from images captured in the experiment (Figure 4). A bilateral filter is a basic theory of image noise reduction, and it is better in edge-preserving than other filters, so we used a bilateral filter to reduce the noise of experimental images. Then, the Canny edge detector, which is an edge detection operator that uses a multi-stage algorithm, was used to detect edges of the beam in experimental images. Image binarization was set the grayscale value of the pixel on the image to 0 or 255, which is the process of presenting the whole image with an obvious black and white effect. The erosion, closing operation, and dilation are the morphological operations to enhance image features. We used erosion, closing operation, and dilation to make the edges detected by the Canny edge detector clearer, then the pixel coordinates of the edges were recorded. The pixel size of a picture can be detected. Thus, the scale S of actual distance and Pixel distance can be calculated through Figure 3. S is defined by S = actual distance/pixel distance. Therefore, the actual displacement D of the tracking point is calculated by D = S*L, where L is the pixel distance between the current and initial position of the tracking point. Details about image processing are illustrated in Howes [27].

Numerical Methods
The two-way fluid-structure interaction (FSI) numerical method is explained in sections 2.5, 2.6, 2.7, and 2.8. The solution of the three-dimensional fluid domain is described in section 2.5. The structural dynamics of the T-shaped flexible beam is explained in section 2.6. Section 2.7 presents system coupling between fluid and solid domain. Details of the computational domain and boundary conditions are investigated in section 2.8.

Computational Fluid Dynamics (CFD)
The realizable k-ɛ turbulence model was applied for the turbulent flow simulation in the threedimensional fluid domain (Olcay et al. [28] and ANSYS Fluent Theory Guide [29]). The governing equations representing the continuity and momentum formulas as given below: where is the density and are the average velocity component of the fluid. P is pressure, is the source term for the momentum equation is the dynamic viscosity, is the eddy viscosity, and it is defined as = .
The transport equation for are for the realizable k−ɛ model given as,

Numerical Methods
The two-way fluid-structure interaction (FSI) numerical method is explained in Sections 2.5-2.8. The solution of the three-dimensional fluid domain is described in Section 2.5. The structural dynamics of the T-shaped flexible beam is explained in Section 2.6. Section 2.7 presents system coupling between fluid and solid domain. Details of the computational domain and boundary conditions are investigated in Section 2.8.

Computational Fluid Dynamics (CFD)
The realizable k-ε turbulence model was applied for the turbulent flow simulation in the three-dimensional fluid domain (Olcay et al. [28] and ANSYS Fluent Theory Guide [29]). The governing equations representing the continuity and momentum formulas as given below: where ρ is the density and u i and u j are the average velocity component of the fluid. P is pressure, S i is the source term for the momentum equation µ is the dynamic viscosity, µ t is the eddy viscosity, and it is defined as µ t = ρC µ k 2 ε . The transport equation for k and ε are for the realizable k − ε model given as, where k is the turbulent kinetic energy and ε is rate of dissipation. G k is turbulent kinetic energy generation because of the mean velocity gradients, G b is turbulent kinetic energy generation because of buoyancy, and Y M is fluctuating dilatation contribution to the overall dissipation rate. The model constants for realizable k-ε turbulence model are C 1ε = 1.44, C 2ε = 1.92, σ k = 1.0, C µ = 0.09, and σ ε = 1.3.

Computational Structural Dynamics (CSD)
Deformation of a three-dimensional flexible solid structure is described by the equation of motion, which can be expressed as follows: u is the nodal velocity vector, and {u} is the nodal displacement vector. Newmark time integration method with an improved algorithm (HHT) was used for the solution of Equation (5). The Newmark method and HHT method were applied for implicit transient analyses.
The Newmark method applies finite-difference expansions in the time interval ∆t. It is presumed that (Bathe [30]): where α and δ are Newmark integration parameters, ∆t is t n+1 − t n , {u n } is the nodal displacement vector at time t n , . u n is the nodal velocity vector at time t n , .. u n+1 is the nodal velocity vector at time t n+1 , and u n+1 is the nodal displacement vector at time t n+1 .
Since the primary aim is the calculation of displacement u n+1 , the governing Equation (5) can be computed at time t n+1 as: [M] ..
The solution of displacement at time t n+1 can be obtained by first rearranging Equations (6) and (7), such that: . u n+1 = . u n + a 6 .. where Once a solution is obtained for u n+1 , velocities and accelerations are computed as defined Equations (9) and (10). For the nodes where the velocity or the acceleration is obtained, a displacement constraint is computed from Equation (7). The HHT time integration method can help to have the desired property for the numerical damping in the full transient analysis (Chung and Hulbert [31]).
The basic form of the HHT method is defined as Equation (11) [M] ..
where α m and α f are two extra integration parameters for the interpolation of the acceleration and the displacement, velocity, and loads. It was also realized that the transient dynamic equilibrium equation considered in the HHT method is a linear combination of two successive time steps of n and n + 1 after comparing Equations (5) and (11).

CFD-CSD Coupling
We used Ansys Workbench-system coupling for simulation two-way fluid-structure interactions (Chimakurthi et al. [1]). Fluid Flow (Ansys Fluent) and the Transient Structural systems (Ansys Mechanical) are connected in system coupling. Reynolds-averaged Navier-Stokes (RANS) equations with the realizable k-ε turbulence model are solved in the computational domain by using the CFD solver (Fluent), and also, transient structural analysis is used to solve the T-shaped flexible beam deformation under the action of loads. Forces or stresses on the fluid side of the interface are transformed on to the solid side, and also, displacements or velocities on the solid side of the interface are transformed on to the fluid side in the system coupling method. Transferring in this system coupling includes the computation of weights and their subsequent use in the interpolation of data. It can happen between topologically similar and/or dissimilar element types, distributions, and dimensions such as surface to surface, volume to volume, point to volume, surface to volume, and vice-versa. Figure 5 shows the calculation procedure and detailed overview of the partitioned system coupling. The induced force on the beam was obtained after the flow field was calculated by using CFD solver from Ansys Fluent. Then, the displacement of the beam was solved by using a structure transient from Ansys Mechanical. This process gets continued until convergence is obtained in system coupling. [M] where are two extra integration parameters for the interpolation of the acceleration and the displacement, velocity, and loads. It was also realized that the transient dynamic equilibrium equation considered in the HHT method is a linear combination of two successive time steps of n and n + 1 after comparing Equations (5) and (11).

CFD-CSD Coupling
We used Ansys Workbench-system coupling for simulation two-way fluid-structure interactions (Chimakurthi et al. [1]). Fluid Flow (Ansys Fluent) and the Transient Structural systems (Ansys Mechanical) are connected in system coupling. Reynolds-averaged Navier-Stokes (RANS) equations with the realizable k-ɛ turbulence model are solved in the computational domain by using the CFD solver (Fluent), and also, transient structural analysis is used to solve the T-shaped flexible beam deformation under the action of loads. Forces or stresses on the fluid side of the interface are transformed on to the solid side, and also, displacements or velocities on the solid side of the interface are transformed on to the fluid side in the system coupling method. Transferring in this system coupling includes the computation of weights and their subsequent use in the interpolation of data. It can happen between topologically similar and/or dissimilar element types, distributions, and dimensions such as surface to surface, volume to volume, point to volume, surface to volume, and vice-versa. Figure 5 shows the calculation procedure and detailed overview of the partitioned system coupling. The induced force on the beam was obtained after the flow field was calculated by using CFD solver from Ansys Fluent. Then, the displacement of the beam was solved by using a structure transient from Ansys Mechanical. This process gets continued until convergence is obtained in system coupling.

Computational Model Geometry, Boundary Conditions, and Meshing
The T-shaped flexible beam was fixed on the channel bottom wall. Figure 6

Computational Model Geometry, Boundary Conditions, and Meshing
The T-shaped flexible beam was fixed on the channel bottom wall. Figure 6  and to the sides of the computational. Dimensions and material properties of T-shaped flexible beam and flow properties that were used in all simulations were the same with experiment case Table 1. The non-dimensional Reynolds number was applied for defining flow characteristics in this study. The Reynolds number was defined as Equation (12) = where is density, µ dynamic viscosity of the fluid, free stream velocity, and L is the characteristic length (i.e., height of beam). The highest water velocity was 0.25 m/s in the experimental setup, because the close-circuit water channel could supply 0.25 m/s as the highest water velocity throughout the channel. In the numerical study, three different velocities were used for the understanding of beam behaviors in various velocities. U = 0.15, 0.25, and 0.35 m/s were chosen for numerical simulations after validation. The Reynolds numbers were defined as 25,500, 42,500, and 59,500 for U = 0.15, 0.25, and 0.35 m/s Table 2. Commercial computational fluid dynamic (CFD) code Ansys Fluent and Ansys Mechanical programs were employed to solve the flow domain and solid part. A system coupling method was used to connect between flow domain and solid part. The coupled scheme was selected among five pressure-velocity coupling algorithms. The second-order upwind scheme was used for discretization of advective terms of the transport equations. Criteria of convergence were set to 10 -6 for the continuity and momentum equations. The solution of continuity and momentum equations were continued until criteria of convergence were achieved. Tetrahedron and prism with triangle base elements were set for meshing the fluid solution domain with high-density mesh near walls, and Tetrahedron mesh was used for the solid domain. The dynamic mesh method was applied to simulate the deformation of the T-shaped flexible beam. Totally 800,000-1,200,000 elements were employed to solve the fluid domain, and 21,210-73,000 elements were used to solve the solid domain, as illustrated in Figure 7. A mesh sensitivity study was also carried out for all models in the fluid domain. Table 3 shows the variety of total deformation along with the various number of elements at 0.25 m/s inlet velocity. It was identified that 1,200,000 elements for the fluid solution domain, and 72,425 elements for the solid domain were needed for obtaining good results at maximum velocity in our study. Three different coupling time steps were The non-dimensional Reynolds number was applied for defining flow characteristics in this study. The Reynolds number was defined as Equation (12) Re = ρUL µ where ρ is density, µ dynamic viscosity of the fluid, U free stream velocity, and L is the characteristic length (i.e., height of beam). The highest water velocity was 0.25 m/s in the experimental setup, because the close-circuit water channel could supply 0.25 m/s as the highest water velocity throughout the channel. In the numerical study, three different velocities were used for the understanding of beam behaviors in various velocities. U = 0.15, 0.25, and 0.35 m/s were chosen for numerical simulations after validation. The Reynolds numbers were defined as 25,500, 42,500, and 59,500 for U = 0.15, 0.25, and 0.35 m/s Table 2. Commercial computational fluid dynamic (CFD) code Ansys Fluent and Ansys Mechanical programs were employed to solve the flow domain and solid part. A system coupling method was used to connect between flow domain and solid part. The coupled scheme was selected among five pressure-velocity coupling algorithms. The second-order upwind scheme was used for discretization of advective terms of the transport equations. Criteria of convergence were set to 10 -6 for the continuity and momentum equations. The solution of continuity and momentum equations were continued until criteria of convergence were achieved. Tetrahedron and prism with triangle base elements were set for meshing the fluid solution domain with high-density mesh near walls, and Tetrahedron mesh was used for the solid domain. The dynamic mesh method was applied to simulate the deformation of the T-shaped flexible beam. Totally 800,000-1,200,000 elements were employed to solve the fluid domain, and 21,210-73,000 elements were used to solve the solid domain, as illustrated in Figure 7. A mesh sensitivity study was also carried out for all models in the fluid domain. Table 3   Mesh Resolution Deformation at t = 10 s 810,000 0.022400 890,000 0.01989 920,000 0.01934 1,020,000 0.019147 1,040,000 0.019144

Comparison between Experimental and Numerical Results
High-speed camera (HSC) measurements were carried out for the three-dimensional flexible beam model at one Reynolds number (42,500). The close-circuit water channel could supply 0.25 m/s as the highest velocity, so the highest Reynolds number for flexible beam was 42,500 in this experiment work. The numerical simulation was done at the same conditions, as used for the experiment. The results of numerical and experimental data were compared regarding the total deformation, Figure 8. The total deformation shape in the numerical model agrees well with the deformation obtained from HSC measurements for a Reynolds number of 42,500. A point (red point) was selected at the top of the T-shaped flexible beam for the tracking maximum displacement of the T-shaped flexible beam. When the T-shaped flexible beam has a vertical position, the red point is at position 1. After deformation, the red point changes from position 1 to position 2. The maximum displacement is the distance between position 1 and position 2. Table 4 shows the maximum deformation of the T-shaped flexible beam at t = 6 s and t = 10 s for numerical and experimental models.

Comparison between Experimental and Numerical Results
High-speed camera (HSC) measurements were carried out for the three-dimensional flexible beam model at one Reynolds number (42,500). The close-circuit water channel could supply 0.25 m/s as the highest velocity, so the highest Reynolds number for flexible beam was 42,500 in this experiment work. The numerical simulation was done at the same conditions, as used for the experiment. The results of numerical and experimental data were compared regarding the total deformation, Figure 8. The total deformation shape in the numerical model agrees well with the deformation obtained from HSC measurements for a Reynolds number of 42,500. A point (red point) was selected at the top of the T-shaped flexible beam for the tracking maximum displacement of the T-shaped flexible beam. When the T-shaped flexible beam has a vertical position, the red point is at position 1. After deformation, the red point changes from position 1 to position 2. The maximum displacement is the distance between position 1 and position 2. Table 4 shows the maximum deformation of the T-shaped flexible beam at t = 6 s and t = 10 s for numerical and experimental models.

Deformation and Stress Study
The T-shaped flexible beam was validated at a Reynolds number of 42500 (U = 0.25 m/s), and then, it was investigated at two different Reynolds numbers of 25500 (U = 0.15 m/s) and 59500 (U = 0.35 m/s). Totally, The T-shaped flexible beam was studied in three various Reynolds numbers

Deformation and Stress Study
The T-shaped flexible beam was validated at a Reynolds number of 42,500 (U = 0.25 m/s), and then, it was investigated at two different Reynolds numbers of 25,500 (U = 0.15 m/s) and 59,500 (U = 0.35 m/s). Totally, The T-shaped flexible beam was studied in three various Reynolds numbers numerically. Total deformation was calculated numerically in this study. Total deformation can be computed by using Equation (13) where U x is component deformation in the x direction, U y is component deformation in the y direction, U z is component deformation in the z direction. Figure 9 shows deformation of the T-shaped flexible beam increased in time, then it started to decrease and, finally, it had a constant value. The deformation of the T-shaped flexible beam increased with increasing velocity. The outer load also changed with velocity. The T-shaped flexible beam bent more and more when velocity increases. It seems that the maximum value of deformation happens early for minimum velocity. We can observe that the maximum value of deformation occurs at t = 0.47 s when velocity is U = 0.15 m/s, and also, it occurs at t = 0.511 s when velocity is U = 0.35 m/s. Maximum stress (Von Mises stress) of the T-shaped flexible beam happened at maximum inlet velocity between three different inlet velocities. Von Mises stress was calculated numerically in this study. Von Mises stress can be calculated by using Equation (14) where σ stress statestress in the x direction, σ is stress statestress in the y direction, and σ is stress statestress in the z direction. The maximum stress and maximum deformation of the beam have similar behavior, so we can figure out that stress changes like deformation in all velocities. Figure 10 shows maximum stress of beam that changes by time, and it occurs obviously on the bottom of the beam. Figures 11a-c show maximum principal stress, middle principal stress, and minimum principal stress of beam that changes by time.  Von Mises stress was calculated numerically in this study. Von Mises stress can be calculated by using Equation (14) where σ 1 stress statestress in the x direction, σ 2 is stress statestress in the y direction, and σ 3 is stress statestress in the z direction. The maximum stress and maximum deformation of the beam have similar behavior, so we can figure out that stress changes like deformation in all velocities. Figure 10 shows maximum stress of beam that changes by time, and it occurs obviously on the bottom of the beam. Figure 11a-c show maximum principal stress, middle principal stress, and minimum principal stress of beam that changes by time. Equivalent strain was computed numerically in this study. Equivalent strain can be calculated by using Equation (15)  Equivalent strain was computed numerically in this study. Equivalent strain can be calculated by using Equation (15) where ε 1 is principal strain in the x direction, ε 2 is principal strain in the y direction, ε 3 is principal strain in the z direction, and υ is effective Poisson's ratio. Figure 12 shows equivalent strain of beam that changes by time. where ε is principal strain in the x direction, ε is principal strain in the y direction, ε is principal strain in the z direction, and is effective Poisson's ratio. Figure 12 shows equivalent strain of beam that changes by time.

Contours Plots of Numerical Study
Deformation of the T-shaped flexible beam was shown for three different Reynolds numbers of 25,500 (U = 0.15 m/s), 42,500 (U = 0.25 m/s), and 59,500 (U = 0.35 m/s) at t = 10 s, Figure 13. The stress of the T-shaped flexible beam was illustrated for three different Reynolds numbers of 25,500, 42,500, and 59,500 at t = 10 s, Figure 14. It was realized that maximum deformation and stress occurred at the maximum Reynolds number, because the T-shaped flexible beam has a large pressure in the front surface.

Contours Plots of Numerical Study
Deformation of the T-shaped flexible beam was shown for three different Reynolds numbers of 25,500 (U = 0.15 m/s), 42,500 (U = 0.25 m/s), and 59,500 (U = 0.35 m/s) at t = 10 s, Figure 13. The stress of the T-shaped flexible beam was illustrated for three different Reynolds numbers of 25,500, 42,500, and 59,500 at t = 10 s, Figure 14. It was realized that maximum deformation and stress occurred at the maximum Reynolds number, because the T-shaped flexible beam has a large pressure in the front surface. where ε is principal strain in the x direction, ε is principal strain in the y direction, ε is principal strain in the z direction, and is effective Poisson's ratio. Figure 12 shows equivalent strain of beam that changes by time.

Contours Plots of Numerical Study
Deformation of the T-shaped flexible beam was shown for three different Reynolds numbers of 25,500 (U = 0.15 m/s), 42,500 (U = 0.25 m/s), and 59,500 (U = 0.35 m/s) at t = 10 s, Figure 13. The stress of the T-shaped flexible beam was illustrated for three different Reynolds numbers of 25,500, 42,500, and 59,500 at t = 10 s, Figure 14. It was realized that maximum deformation and stress occurred at the maximum Reynolds number, because the T-shaped flexible beam has a large pressure in the front surface. Pressure contours at the solution domain were plotted in Figure 15 for t = 10 s. When flow around a T-shaped flexible beam was studied, there was a large pressure on the front surface of T-shaped flexible beams. Figure 16 also showed lower pressure regions at the top region of the T-shaped flexible beam implying flow separation. It was noted that the pressure difference between the front and back surface of T-shaped flexible beams was large. In addition to pressure contours, streamline contours were plotted in Figure 16  Pressure contours at the solution domain were plotted in Figure 15 for t = 10 s. When flow around a T-shaped flexible beam was studied, there was a large pressure on the front surface of T-shaped flexible beams. Figure 16 also showed lower pressure regions at the top region of the T-shaped flexible beam implying flow separation. It was noted that the pressure difference between the front and back surface of T-shaped flexible beams was large. In addition to pressure contours, streamline contours were plotted in Figure 16  Pressure contours at the solution domain were plotted in Figure 15 for t = 10 s. When flow around a T-shaped flexible beam was studied, there was a large pressure on the front surface of T-shaped flexible beams. Figure 16 also showed lower pressure regions at the top region of the T-shaped flexible beam implying flow separation. It was noted that the pressure difference between the front and back surface of T-shaped flexible beams was large. In addition to pressure contours, streamline contours were plotted in Figure 16

Drag Coefficients Study
When the body locates in fluid flow, it can experience a certain amount of drag force (Olcay et al. [28], Batchelor [32], and Vasudev et al. [33]). Drag force was given by where F D_pressure and F D_viscous are drag forces in the x-direction due to the pressure and viscous effects. Here, p is the pressure on the T-shaped flexible beam, and τ w is the wall shear stress on the surface of T-shaped flexible beam. Drag force studies are shown in Table 5 for t = 6 s and t = 10 s. It was noted that the drag force increased with increased Reynolds numbers for T-shaped flexible beam. Table 5. Change in drag force with the Reynolds numbers for t = 6 s and t = 10 s. Once the drag force was obtained, the drag coefficient was studied using Equation (9).

Reynolds Umbers
where C d is drag coefficient, F Drag is Drag force, ρ is density of the fluid, U is velocity of fluid, and A is the reference area (the frontal area of the body). Drag coefficient was plotted in Figure 17. It was also realized that the drag coefficient decreased with increased Reynolds numbers for T-shaped flexible beam.

Drag Coefficients Study
When the body locates in fluid flow, it can experience a certain amount of drag force (Olcay et al. [28], Batchelor [32], and Vasudev et al. [33]). Drag force was given by where _ and _ are drag forces in the x-direction due to the pressure and viscous effects. Here, is the pressure on the T-shaped flexible beam, and is the wall shear stress on the surface of T-shaped flexible beam.
Drag force studies are shown in Table 5 for t = 6 s and t = 10 s. It was noted that the drag force increased with increased Reynolds numbers for T-shaped flexible beam. Table 5. Change in drag force with the Reynolds numbers for t = 6 s and t = 10 s. Once the drag force was obtained, the drag coefficient was studied using Equation (9).

Reynolds Umbers
where is drag coefficient, is Drag force, is density of the fluid, is velocity of fluid, and is the reference area (the frontal area of the body). Drag coefficient was plotted in Figure 17. It was also realized that the drag coefficient decreased with increased Reynolds numbers for T-shaped flexible beam.

Conclusions
In this study, the deformation of a T-shaped flexible beam was investigated at 0.25 m/s inlet velocity. A three-dimensional T-shaped flexible beam was placed into a close-circuit water channel for high-speed camera system (HSC) measurements. The results of a three-dimensional T-shaped flexible beam agreed well with the results of HSC measurements for 0.25 m/s inlet velocity. Then, two additional inlet velocities were noticed for a flexible beam, and those velocities were examined for the T-shaped flexible beam. A two-way FSI coupling method was employed for solving fluid and solid parts. The dynamic mesh method was used for grid, and mesh was updated in every time step in fluid and solid sides. Deformation, maximum stress, and minimum stress of the T-shaped flexible beam were calculated, and also, velocity distribution and pressure distribution of the flow around the T-shaped flexible beam were computed at various velocities in the numerical model. The results reveal that deformation and stress in the flexible beam has increased with increasing velocity. It was also found that a large pressure region was created on the front surface of the T-shaped flexible beam and flow separation happened in the head of the T-shaped flexible beam. It was concluded that high velocity caused the drag force to be larger when compared with low velocity, so high drag force caused a large deformation and high stress in the T-shaped flexible beam. We carried out a validation study. The results of experimental and numerical methods were compared in the present study. In this study, the percent error of maximum deformation between experimental and numerical methods is nearly 4%~5%. The study also revealed that the system coupling method can be used in fluid-structure interaction applications and a two-way FSI coupling method has high efficiency, so this method can be employed in various engineering fields such as mechanical, civil, and ocean engineering.