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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 310432, 9 pages

http://dx.doi.org/10.1155/2013/310432

## Evaluation of Artificial Caudal Fin for Fish Robot with Two Joints by Using Three-Dimensional Fluid-Structure Simulation

Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka-shi, Osaka 558-8585, Japan

Received 18 January 2013; Accepted 1 March 2013

Academic Editor: Bo Yu

Copyright © 2013 Yogo Takada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A fish robot with image sensors is useful to research for underwater creatures such as fish. However, the propulsion velocity of a fish robot is very slow compared with live fish. It is necessary to swim at a speed several times faster than the speed of the current robots for various usages. Therefore, we are searching for the method of making the robot swim fast. The simulation before making the robot is important. We have made the computational simulation program of three-dimensional fluid-structure analysis. The flow around the caudal fin can be examined by analyzing the fin as an elastic body. We compared the results of numerical analysis with the results of PIV measurement. Both were agreed well. Because the performance of a fish robot with two joints is better than that of a fish robot with one joint, we searched for an excellent fin for the fish robot with two joints by using CFD. We confirmed that the swimming performance of a fish robot becomes very good when the caudal fin is rigid except for the root of the fin which is comparatively flexible.

#### 1. Introduction

A fin driving a fish robot which imitates a live fish never gives the caution to other underwater creatures. Therefore, the fish robot becomes useful for the investigation of the ecosystem in water, the usage of the environmental surveillance, and so on. Moreover, because the fish robot does not have the screw propeller which becomes tangled in water plants and fibrous floats, propulsive mechanism of caudal fin is suitable.

Many researchers have been studying propulsion mechanism about various caudal fins of the underwater creatures such as fishes in respect to the fluid phenomenon around the propulsive caudal fin for a long time [1–3]. However, the speed of fish robots manufactured by imaging live fish which swim with its caudal fin is very slow in comparison with the speed of an actual live fish. Live fishes can swim in the distance three times the fish size in length for one second. Besides, the propulsion efficiency of fish robot is much lower than live fishes. Though fish's swim mechanism is becoming clear step by step, those study results have not been tied into manufacturing of good-performance robots.

When practical use is tried for the ecosystem investigation, the same cruising speed as the fish is demanded. Therefore, it is necessary for the fish robot to swim at more high speed than the current state. Because the condition of an appropriate fin in each fish robot is different, repeated experiment is necessary in order to find the best fin in various fins made with a different condition whenever the fish robot is made for trial purposes. The best fin for the designed fish robot is hoped to be clarified before the fish robot is produced.

For a long time, we have conducted numerical flow analysis regarding fishes and fish robots [4–6]. The purpose of this study is to enable three-dimensional fluid-structure analysis to search for the good fin to use for a fish robot with two joints. For that purpose, three-dimensional elastic deformation analysis of the fin by finite element method (FEM) was combined with an existing three-dimensional fluid numerical analysis code (GTT code: generalized tank and tube code) [4]. Weak coupling method which solves alternately plural phenomena shown by separate governing equations has been employed. The propulsive performances of the fish robot that swam with carangiform locomotion of various fins were compared by using improved fluid-structure analysis code.

#### 2. Three-Dimensional Fluid-Structure Simulation

##### 2.1. Method of Flow Analysis

The equation of mass conservation and the equations of momentum conservation of flow are shown by the general conservation equation as follows: where is the induced variable, is the diffusion coefficient, and is the source term, respectively. Because the analytical object in this study is to flow around a moving object such as a fish or a robot, the fluid has been analyzed as a moving boundary problem where the computational grid moves. Equation (1) becomes (2) by considering the movement of the computational grid: where is the moving velocity of each computational grid. Table 1 shows the induced variable , the effective diffusion coefficient , and the source term in (2). In Table 1, is the effectiveness coefficient of viscosity. Because the Reynolds number of fluid in this study is very small, the analytical object can be considered to be laminar flow. The fluid around a fish robot is general water. Therefore, the incompressible fluid without the temperature change was added to the calculation condition.

In this research, we used the generalized tank and tube method (GTT method)which adopted the technique of conversion into generalized curvilinear coordinates based on the Tank and Tube method that Gosman et al. [7] and Spalding and Pun [8] developed.

In this method, first of all, the physical space shown by Cartesian coordinate system is divided into small hexahedral volume elements of arbitrary shape. Then, each control volume is converted into a cubic control volume in the calculation space by general curvilinear coordinate transformation. The allocation of independent variables such as pressure , density , and components of the velocity vector , , and is based on the regular grid (nonstaggered grid) method, and therefore, all variables are allocated at the center of each control volume. The pressure correction equation is calculated based on SIMPLER method [9], where pressure is calculated by satisfying the equation of continuity.

##### 2.2. Method of Elastic Deformation Analysis

In this research, the caudal fin of a fish robot is assumed to be a linear elastic body. The deformation of linear elastic body is modeled by equilibrium equation in consideration of inertia force. Newmark-*β* method [10] is used for time discretization, and FEM is used for space discretization:
where is the nabla differential operator, is the stress tensor, is the body force vector, and is the density.

To calculate by the FEM, the entire material to be analyzed is divided into 20-node hexahedral elements of some serendipity families. Figure 1 shows the example of one divided element. When the displacement vector in an arbitrary point in each element is defined by , is obtained as (4) by using displacement in each node: where is a form function of 20-node hexahedral element of serendipity family, is adisplacement along -axis on node , is adisplacement along -axis on node , and is adisplacement along -axis on node . Therefore, the displacement in each element is expressed as follows by the matrix form: where is expressed by the following matrix and :

When the virtual displacement is set as the weighting function and the fundamental equation (3) is integrated for a single element, the following (7) is obtained. In addition, when Green's theorem is applied to (7), (8) is obtained: where is the strain and is the virtual strain for the displacement .

By space discretization, (9)–(12) can be derived from the first term, second term, and third term in the left side of (8) and the right term of the same equation, respectively: where is a displacement-strain conversion matrix and is an elasticity coefficient matrix.

The dynamic equation in an element is shown as follows, as a result of arranging after substituting (9)–(12) for (8):

If (13) is overlapped over all elements, the dynamic equation of all elements becomes as follows: where is all elements mass matrix, is all elements stiffness matrix, and is the external force vector.

##### 2.3. Coupling of CFD and Deformation Calculation

The fluid analysis method and elasticity deformation method were described in the foregoing section. In this section, the coupling method of the fluid analysis and elastic deformation are described. There are a strong coupling method and a weak coupling method as a means to couple the fluid analysis and elastic deformation analysis.

The weak coupling method separately solves governing equations of the fluid and structure. Then, the weak coupling method conducts the calculation to satisfy dynamic equilibrium and a geometrically continuous condition on the fluid boundary surface and the structural boundary surface. It is inferior in respect of the accuracy of calculation and stability compared with the strong coupling method. However, different discretization techniques can be adopted to solve the governing equations of the fluid and structure.

In this study, three-dimensional CFD code (GTT code) of a finite volume method is adopted for the fluid analysis. We decided to use the weak coupling method because it is easy to combine elastic deformation analysis with the existing GTT code. Pressure on the fluid-structure boundary surface obtained by the fluid analysis is sent to the role part of the elastic deformation analysis as a dynamic boundary of beam. Moreover, displacement on the boundary surface obtained by the elastic deformation analysis is sent to the role part of the fluid analysis as a geometrical boundary. Dynamic equilibrium and a geometrical continuous condition on the fluid-structure boundary surface were satisfied by this method. We explain each step as follows.

*Step 1. *The time is increased by an interval .

*Step 2. *The computational grid for the fluid analysis is created based on the displacement of the fluid-structure boundary surface determined in the former time step. The Poisson equation with the Dirichlet condition is solved not to make the complex computational grid, and then the grid is smoothed. If the computational grid was not smoothed in this step, the grid lines would have crossed each other and CFD calculation would have stopped with errors.

*Step 3. *The fluid is analyzed by the GTT code by using the computational grid made with Step 2. The repetition calculation continues until a converged solution is obtained in consideration of the nonlinearity of the fluid.

*Step 4. *The converged solution of pressure obtained with Step 3 is given as a dynamic boundary condition of the structure, and the deformation analysis is conducted by the FEM.

The nonstationary fluid-structure analysis is conducted by repeating Steps 1 to 4.

#### 3. Experimental Fish Robot

It is necessary to compare the results of fluid-structure simulation and the results of the experiment to examine the validity of the program for the fluid-structure calculation used in this study. The fish robot for the experiment of the verification was manufactured. Then, the amount of deformation of caudal fin tip was examined after a flexible caudal fin was attached to the robot. Moreover, the flow behind the fish robot was measured by the PIV measurement.

Figure 2(a) shows the externals photograph of the fish robot produced for experimental use for the verification, and Figure 2(b) shows the structural drawing.

The manufactured fish robot was used to verify the result of the numerical analysis. Therefore, the driving system, the electric circuit, and so forth were put out on the surface of the water. Servomotors which can control the angle easily were used for driving the fish robot. The servomotor at the front joint is RB995b made. MiniStudio Inc; the rated voltage is 4.8 V, and the rated torque is 0.833 Nm. Then, the servo motor at the rear joint is DS385 made by Japan Remote Control Co.; the rated voltage is 4.8 V, and the rated torque is 0.196 Nm.

The total length of the fish robot is 190 mm, and the length of the body part is 120 mm. There are two rotation axles, and the rear of the body and the caudal fin are shaken from side to side through the shaft installed on the servomotors. The shafts restrain to move with housings so that the shafts do not move vertically and horizontally. All parts are rigid bodies from the head to the position (point A) which is at 20 mm behind joint R. The shape of the body part is NACA0012 which is one of the typical symmetrical airfoil. This shape is often used for the fluid analysis as fish's shape [11]. In addition, the part from the point A to the position of caudal fin edge (point B) is elastically deformable as a fin. In the experiment, a polystyrene board of 0.3 mm in thickness was used as a flexible caudal fin. The Young modulus of the polystyrene board was 2.74 GPa. the PIC18F2520 made by Microchip Technology Inc. was used for the control of the servo motor. Moreover, DC 5 V was supplied from the stabilized power supply PW18-1.3AT made by Kenwood Co., and it was connected to the control circuit and servo motors.

We examined the amount of the deformation of the caudal fin when the fish robot was moving. The fish robot was fixed to a circulating water channel, and the flow velocity in the circulating water channel was set to 0.2 m/s. The flow of water in the vicinity of the fin was taken of pictures at 200 frames per second from the lower side with the high-speed camera k-II which was made Katokoken Co. The amount of the fin's deformation was obtained from the pictures.

Particle image velocimetry (PIV) measurement was used to measure the flow velocity distribution of water around the caudal fin of the fish robot. with respect to the size of area for PIV measurement, the length in traveling direction is 150 mm, and the length in lateral direction is 184 mm. The area around the robot was illuminated with a laser sheet located at the side of the measurement chamber. A PIV laser G1000 (1 W), the above-mentioned CCD camera, and the software FlowExpert made by Katokoken were used to measure the flow velocity distribution.

#### 4. Flow Analysis around the Fish Robot by Three-Dimensional Fluid-Structure Simulation

In this section, the validity of fluid-structure simulation used in this study is examined by comparing the calculation results and the experimental results. Figure 3 shows the computational grid used in calculation. The computational grid was divided into , and the grid spacing in the vicinity of the caudal fin and its downstream region is narrower.

Table 2 shows the calculation condition of the fluid and the elastic body used in this calculation. As the upstream boundary condition in computation, the inlet velocity of water flow at the upstream end was uniformly set at 0.2 m/s. The density, Young's modulus, and the geometrical moment of inertia which have an affair with the flexibility of the fin are the same values as the polystyrene board of 0.3 mm in thickness which had been used to experiment in the foregoing section. Then, the computational grid used for the deformation analysis by the FEM is shown in Figure 3(d). The same computational grid as the fluid analysis is used in the deformation analysis. The range where is from 36 to 63 (: from 31 to 33, : from 19 to 33) shown in Figure 3(d) is calculated by the deformation analysis. In this study, 20-node serendipity element of the fin deformation described in section 2 was applied. The number of elements is 756, and the number of nodes is 4491. Three degrees of freedom of the displacement of each and direction, given at each node. Because the part of 20 mm in length where is from 36 to 40 is rigid body, the Young modulus of this part is set to GPa so as to avoid the elastic deformation. The center of rotation concerning the rear part of the robot body is the position where is 26, and that concerning the caudal fin is the position where is 36.

#### 5. Evaluation of the Propulsive Performance

We have examined various fins to discover good-performance caudal fin by using this fluid-structure simulation code. In this study, the five different caudal fins shown in Figure 4 were investigated in the simulation. The overall shape and area of each fin are the same. However they have different flexibilities. In Case 1, the entire fin is rigid and has a very high Young’s modulus. In Case 2, the entire fin is elastic sheet whose flexibility is the same as polystyrene, and the Young modulus is 2.74 GPa. In Case 3 and Case 4, the fins have a very soft part that bends easily. In Case 5, the fin has an elastic part and a rigid part. This fin is rigid except for the root of the fin which is flexible.

First of all, the distribution of the vorticity calculated based on the speed obtained by CFD and PIV measurement in case of each caudal fin was compared. This result is shown in Figure 5. It can be confirmed that the numerical result and the experimental result agreed well in any case. In Case 1, Case 2, and Case 5, vorticity is strong, and the vortices widely extend in Case 5. On the other hand, in Case 3 and Case 4 that have a soft part on each fin, vorticity is small, and the extension of the vortices is suppressed to small. Concerning the size of vortices, the big difference is not seen among Case 1, Case 2, and Case 5. The extension of vortices of Case 2 and Case 5 is a little narrower than Case 1 if they are seen in detail. In Case 3 and Case 4, the vorticity is smaller than the other cases, and the diffusion angle of the vortices is also narrower. The softness of the fin affects the diffusion angle. At s in Figure 5, each joint in the robot is controlled to become a defined angle by all cases. However, the position of the rear end of caudal fin is different in each case. The vortices extend widely if the rear end of the fin moves widely. Then, if the diffusion angle of Case 4 is compared with that of Case 3, the angle of Case 4 is narrower, and the vortices are smaller. The soft part in the fin of Case 4 is near the root of the fin. Therefore, the softness of the fin in the root will greatly influence the flexibility of the entire fin. In this study, reverse Karman vortex street was not generated in the numerical results and experimental results. It is guessed that this robot did not generate the reverse Karman vortex street because the shape of the fin is a rectangle.

Figure 6(a) shows the value of each static thrust obtained by the experiment that used a measurement device with a load cell. Then, Figure 6(b) shows the value of mean net propulsive force obtained by the simulation. Figure 6(c) shows the value of each side force. The propulsive force is qualitatively corresponding well, though a quantitative agreement cannot be confirmed because the physical quantity is different. The propulsive performance of Case 5 was the best in Figure 6, and the robot with the fin of Case 5 swam speediest in a water tank actually.

In Case 1, Case 2, and Case 5, because the water pressure on the fin surface is large, their net propulsive force and side force are strong. In contrast, in Case 3 and Case 4, the net propulsive force and the side force are weak because the water pressure on the fin surface is small.

Moreover, the fin of Case 5 was examined by using CFD analysis in detail. The relationship between the flexibility in the elastic part and the propulsive performance of the robot was investigated. The net propulsive force and the side force were calculated by CFD with respect to each Young's modulus and each length mm in Figure 7. Each net propulsive force and each side force are shown in Figure 8. The Young modulus of the elastic part is different from Case A to Case J. The material of Case A is polystyrene. In Case A, each propulsive force does not depend on length mm. All propulsive forces of Case A were almost the same value. The impellent becomes the maximum when is 6.5 mm (corresponding to the above-mentioned Case 5). On the other hand, when the length of becomes long, side force becomes small.

Young's modulus of the elastic part in each case is indicated in Table 3. According to Figure 8, if the Young modulus of fin is low, both the propulsive force and the side force are weak except for the case of mm. In Case C (the Young modulus is 0.913 GPa) and mm, the propulsive force is strongest. The propulsive performance improves theoretically, if the fin is made of a material which is three times softer than polystyrene.

The side force is weak if the fin is soft. However, when the material is soft like Case J, the side force does not depend on the length of . The rigid part in the fin edge may be twirled right and left because of the softness at the root of the fin. If the ratio of Young's modulus and length is the same, fin's overall flexibility and the propulsive performance becomes similar except for the case that the material is too soft. For instance, Case A (2.74 GPa), mm, and Case C (0.913 GPa), mm, have almost the same flexibility. Besides, each case's propulsive force and side force are almost the same. In a word, the suitable fin can be prepared by the adjustment of the length even if there is no material that has appropriate flexibility as a fin for the fish robot.

#### 6. Conclusions

In this study, three-dimensional fluid-structure analysis was conducted to examine the influence that the flexibility of the fin has on the propulsive performance on the simulation.

An improved three-dimensional flow numerical analysis code (GTT code) was used to simulate the fin deformation of a swimming fish robot. Three-dimensional elastic deformation analysis of the fin using FEM has been combined with the GTT code allowing the fluid-structure analysis to be carried out. To verify the simulation result, an experimental fish robot using a flexible fin was made for trial purposes. This robot has two degrees of freedom. The simulation result and experimental result for both the fin deformation and flow pattern behind the fin were compared. It has been confirmed that both are corresponding.

By using the three-dimensional fluid-structure interaction analysis, we confirmed good caudal fin for fish robot with two active joints is a rigid fin with a flexible material on the root. If robot manufacturer do not have most suitable material concerning the flexibility, good fin can be obtained by adjustment of the root length.

#### Conflict of Interests

There is no conflict of interests.

#### Acknowledgment

The authors would like to express their gratitude to the Japan Society for the Promotion of Science which subsidized them (Grant-in-Aid for Scientific Research (C), no. 24560301).

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