Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins

Journal of Bionic Engineering 8 (2011) 25–32

Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins Md. Mahbubar Rahman, Yasuyuki Toda, Hiroshi Miki Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita City, Osaka 565-0871, Japan

Abstract The undulating fin propulsion system is an instance of the bio-inspired propulsion systems. In the current study, the swimming motion of a squid-like robot with two undulating side fins, mimicking those of a Stingray or a Cuttlefish, was investigated through flow computation around the body. We used the finite analytic method for space discretization and Euler implicit scheme for time discretization along with the PISO algorithm for velocity pressure coupling. A body-fitted moving grid was generated using the Poisson equation at each time step. Based on the computed results, we discussed the features of the flow field and hydrodynamic forces acting on the body and fin. A simple relationship among the fin’s principal dimensions was established. Numerical computation was done for various aspect ratios, fin angles and frequencies in order to validate the proposed relationship among principal dimensions. Subsequently, the relationship was examined base on the distribution of pressure difference between upper and lower surfaces and the distribution of the thrust force. In efficiency calculations, the undulating fins showed promising results. Finally, for the fin, the open characteristics from computed data showed satisfactory conformity with the experimental results. Keywords: biomimetics, squid robot, undulating fin, propulsion efficiency, hydrodynamics, CFD Copyright © 2011, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60003-6

1 Introduction Natural selection is sought to produce the best possible designs for specific biological functions as each species has to evolve through a unique and optimum interaction with its environment, where only the best designs can sustain the evolutionary selection process. Robotic engineers borrow the sense and structures from the animals for similar functions in their designs to optimize the creations. The fish swimming modes were classified into two types[1] based on the propulsive structure used: Body and/or Caudal Fin (BCF) and Median and/or Paired Fin (MPF). The BCF movement can achieve greater thrust and acceleration while the MPF is generally employed at slow speed, offering greater maneuverability and better propulsive efficiency. The present study was based on a robot characterized to the latter group that used the undulating side fin for propulsion. This study was on a squid-like robot that uses the undulating side fin for its propulsion akin to that of a Corresponding author: Md. Mahbubar Rahman E-mail: [email protected]

squid. In the field of underwater robotic research, undulating-fin robots offer exceptional advantages over propeller in preserving the undisturbed condition of its surroundings for data acquisition. Though the movement of this biomimetic type of robot is slow, it has compelling areas of applications such as in covert operations, in experimentations where minimally disturbed surrounding is of prime importance. The demand for high-efficiency underwater vehicle is increasing and many researchers are getting interested in studying this challenging but interesting field. Many new concepts of biologically inspired underwater propulsion system have so far been developed[2–4], including several undulating-finned robots[5–7]. Unfortunately, the development of efficient and environmentally friendly underwater vehicle with undulating side fins has not gained the desired momentum, which however offers us adequate opportunity to contribute to this field. In our laboratory, a squid-like underwater robot with undulating side fins similar to that of a Stingray or a

Journal of Bionic Engineering (2011) Vol.8 No.1

Cuttlefish has been under active investigation for many years (Fig. 1). Our investigation aimed at the development of an efficient and environmentally friendly underwater vehicle. Our work initiated with the computation of the flow field around the simple model with undulating fins and resistance body[8]. The subsequent work elucidated the features of the flow field and the hydrodynamic forces acting on the body and fins, which has augmented the understanding of the complex fluid mechanics that fishes use to propel themselves. We have reported our first experimental model having a strut operated at very low frequency[9] followed by the second model[10] that consisted of 16 servomotors for both sides, each of which was connected to heavy power cables. The strut of the first model and the bulky cables of the second model hindered the free movements of these robots. Consequently, we have constructed the third model by excluding the strut and heavy cables[11]. The third model had 17 servo motors for both sides to produce independent motion individually for any fin with the servo controller installed into the microcomputer housed inside the model. The third model demonstrated motion with 6 degrees of freedom and was amenable to greater control. However, it had some limitations in adjustment of vertical gravity center and in communicating over a large distance. In the fourth model, these limitations were addressed by using wireless Local Area Network (LAN) communication device and the gravity center adjustment system in three directions inside the model. The fin width was kept the same as the third model but the fin length was increased a little resulting in a change in aspect ratio. Recently the preliminary experimental results demonstrating the improvement in the performance of the new model has been reported[12].

The experimental and numerical studies are complementary to each other. Experiments give the total force characteristics while the detail analysis of force can only be obtained through the numerical study. As such we conducted the numerical study for the fourth model to recognize the flow physics around the undulating fin and to investigate the mechanism of thrust generation which was limited to one fin angle and few specific frequencies[13]. The present work focuses on the similar phenomenon but for two fin angles and a large number of frequencies. The objective of this study is to establish a simple relationship among the undulating side fin’s principal particulars and hydrodynamic forces through numerical analysis of the flow physics around the fin.

2 Methodology 2.1 Grid generation The model sketch and coordinate system are shown in Fig. 2a. The body is symmetric with respect to x-axis, so only half of the body (shaded region) was simulated to save computational cost. The widths of the resistance body and the fin are denoted by b and bm respectively. Also u, v and w are the velocity components in x, y and z directions respectively. Computation for the fin geometries similar to that of experiment was conducted for comparisons. However, in the computation, the model was approximated as rectangular flat plate for simplicity in grid generation. At first, the numerical grid was constructed around a rectangular flat plate of same area of the robot (including fin). Then two fins were produced at the lateral sides of the flat plate by gradually increasing the vertical amplitude using Eq. (1) (Fig. 2b).

I=

zm z m ⎛t⎞ − cos ⎜ ⎟ π, 2 2 ⎝T ⎠

(1)

where, zm is the vertical amplitude of fin edge and T is nondimensional period. All variables were nondimensionalized using resistance body length (L), advanced Y bm

Z

(a)

Fig. 1 Photo of (a) squid and (b) squid-like robot.

Y

e

b/2

Body motion

Inside of fin Outside of fin

i lin ge

Le ad in g

ed

ge

v

dg

Z

w

u X

X

T ra

26

(b)

Fig. 2 (a) Simple model sketch and coordinate system; (b) computational model including undulating fin.

Rahman et al.: Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins

θ = θ m sin(2πnt − 2πx),

{

(2)

}

θ m = arcsin ⎡ 1 − 0.905 ( x − 0.5 ) sin Θ ⎤ , ⎢⎣

2

⎥⎦

(3)

On inlet: u = 1, v = w = p = 0; On upper and lower sides (i.e., z = ±3): u = 1, v = 0; On lateral side (i.e., y = +3): u = 1, w = 0; ∂u ∂v ∂w ∂p = = = = 0; On outer boundary: ∂x ∂x ∂x ∂x ∂y ∂z On body surface: v = b , w = b ; ∂τ ∂τ ∂u ∂w ∂p =v= = = 0. On the symmetry plane (i.e., y = 0): ∂y ∂y ∂y

3 Result and discussion 3.1 Forces The computation was first conducted on the flat plate having similar area of the model in order to validate the codes used for numerical study. In the simulation, the velocity of the flow was gradually accelerated from 0 to 1 and then kept at steady state. The computational frictional force of flat plate agreed well with the well-known Blasius solution (B/L × 0.01328) in the steady state region (Fig. 4). We analyzed the hydrodynamic forces produced by different fins for which computation was conducted at different aspect ratios

F

where, θ is the deflection angle from flat plate (0 < x < 1), θm is angle amplitude (0 < x < 1), n is frequency, t is time, x is the distance along fin from leading edge, and Θ is the maximum fin angle from the flat position. As mentioned above n, t and x are all dimensionless. The dimensional value of n is N and n=NL/U. A moving grid was constructed around the body to handle the unsteady motion. The numerical grid was generated around half of the body at each time step using Poisson equation. At the first period, undulating side fins were produced at the lateral sides of the flat plate by gradually increasing its amplitude, while the other periods were kept symmetric. The grid independency was confirmed by checking different number of grids. The enlarged view near the fin of the grid is shown in Fig. 3. The body was covered by 41 and 16 grid points in x and y directions respectively. The minimum grid spacing was 0.0015 in the y and z direction. The computational domain and the number of grids for half domain are shown in Table 1.

2.2 Method of computation The governing equations and the method of computation were briefly discussed in the initial study[8]. The Navier-Stokes equations and continuity equations were discretized by the 12-point finite-analytic method[14–15] in space. Euler implicit scheme was used for time discretization along with the PISO algorithm for velocity pressure coupling. The laminar flow computation was carried out at Re = 10,000. The boundary conditions of the computation were as follows

U

speed (U), density (ρ) and their combinations. So, the nondimensional body length and the advanced speed were 1. The time history of fin angle in x-direction θ(x,t) was expressed as

27

Fig. 3 Computational grid near the fin. Table 1 Computational domain and grid points for half domain Axis

Computational domain

Number of grid points

x

−1 ~ 4

90

y

0~3

40

z

−3 ~ 3

51

Fig. 4 Frictional force for a flat plate in acceleration stage and constant velocity stage.

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(AR = width/length) and fin angles for different frequencies. Sufficient convergence was ensured by taking adequate iteration in all the cases. The time history of x-directional forces at aspect ratio (AR) of 0.1, frequency (n) of 4 and the maximum fin angle of 30˚ was drawn (Fig. 5). In Fig. 5, XF and XP are the force due to frictional stress and pressure respectively, and XT is the total force. Note that, the force direction is positive x; that means the positive and negative values show the resistance and thrust respectively. The forces are nondimensionalized by ρU2L2. The robot we used to mimic the flat fishes with undulating side fins, which usually use the progressive wave based swimming mode. In this mode, to propel forward, the progressive wave must be faster than the uniform flow. In this kind of lift-based swimming mode, no recovery stroke is necessary because lift force is generated during the upstroke and downstroke of the fin. In this fashion, x, y and z direction forces are produced during the movement of the undulating side fins (Fig. 6). The z-direction force is large but the body does not go up or down because of its flat shape. Also the mean of z-component force is zero. The y-component

is also cancelled out due to the symmetrical movements of two side fins in opposite directions. So, only the x-direction force acts on the body. This figure also indicates two peaks in one period for the x-component force, which confirmed that the fin produces force in both upstroke and downstroke movements. The total forces for one period of aspect ratio 0.1 at different frequencies (n) and maximum fin angles (Θ), as shown in Fig. 7, reveals that, for Θ = 30˚ and n = 3, the total force is very close to zero. This is in agreement with the experiment as the self propulsion frequency of the model is similar to this value. The amount of thrust and the propulsive efficiency depend mostly upon the aspect ratio and maximum fin angle. High aspect ratio fins are more efficient as they induce less drag per unit of lift or thrust produced. The effect of changes in aspect ratio and fin angle on the x-direction force were calculated for the same frequency (n = 4) (Fig. 8). The thrust force dominated the frictional resistance of the body for AR = 0.1 in case of Θ = 30˚ and the similar value was found from AR = 0.075 in case of Θ = 45˚.

XF XT XP

0.00 0.0

0.5

1.0

1.5

2.0

Force

Force

0.01

−0.01

−0.02

t

Fig. 5 Time history of x-directional forces for AR = 0.1, n = 4 and Θ = 30˚.

Fig. 6 The x, y and z direction pressure forces produced by undulating side fin. 0.005

0.005

−0.005

Flat plate n = 2, Θ = 30˚ n = 2, Θ = 45˚ n = 3, Θ = 30˚ 1.0 n = 4, Θ = 30˚ n = 3, Θ = 45˚

−0.010

n = 5, Θ = 30˚

Force

0.000 0.0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

−0.005

n = 4, Θ = 45˚ n = 6, Θ = 30˚

−0.020 t/T

Fig. 7 Time history (one period) of the x-directional total force for AR = 0.1 at different frequencies and maximum fin angles.

0.8

1.0

AR = 0.05, Θ = 30˚ AR = 0.075, Θ = 30˚ AR = 0.1, Θ = 30˚ AR = 0.05, Θ = 45˚ AR = 0.075, Θ = 45˚ AR = 0.1, Θ = 45˚

−0.010 −0.015

−0.015

−0.025

0.000 0.0

−0.020 −0.025

t/T

Fig. 8 Time history (one period) of the x-directional total force for n = 4 at different aspect ratios and maximum fin angles.

Rahman et al.: Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins

3.2 Contours Different contours were drawn to visualize the surface distribution of forces and velocities. The contour of x-direction velocity distribution on a flat plate, as shown in Fig. 9a, shows the boundary layer very clearly around the flat plate. Figs. 9b to 9e also show the x-direction velocity distribution around the undulating side fins at different time steps in one period. From the figures, the velocity gradients at different positions of the fin are clearly visible. When the fin pushed the surrounding water during its movement, the velocity increased in that region. The undulations of the fins reduced the boundary layer, resulting in increase in velocity gradient and, hence, shear stress. The pressure difference between upper and lower surfaces during the undulations was also computed, as shown in Fig. 10. The prediction of pressure difference between two surfaces is important because it is the main reason behind the thrust force generation. The thrust was computed by the product of pressure difference between upper and lower surfaces with the normal vectors in x-direction (Fig. 11). The pressure difference distribution and the thrust distribution on the body and the fin at four time steps (Fig. 12) reveal two important phenomena. Firstly, both for the negative and the positive pressure differences, the x-direction forces y U: 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3

Legend & coordinate

29

have maximum values. This confirms previous discussion on Fig. 6, which shows that the fin produces thrust during both of its upstroke and downstroke movements. Secondly, the pressure difference also exists on the body part, but the body could not produce thrust force because the normal vectors are zero here.

P:−0.5 −0.3−0.1 0.1 0.3 0.5

z x y

Fig. 10 Pressure distribution on the upper and lower surfaces of the fin at y = 0.275. Z PD: −1.0−0.8 −0.6−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

X

Pressure difference (Upper surface – Lower surface)

Y

Normal vectors on x-direction

x XF: 0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

z

Thrust force distribution on fin surface

(a) Flat plate

Fig. 11 The method of thrust calculation.

(b)

T/4 (c)

T/2 (d)

3T/4 (e)

T

Fig. 9 The u-velocity distribution on a flat plate and around a fin (AR = 0.1, Θ = 30˚) for n = 4 at four time steps in one period.

Fig. 12 (a) Distribution of pressure difference (pupper – plower); (b) Distribution of x-directional thrust force on the full body of AR = 0.1, n = 4 and Θ = 30˚ at four time steps in one period.

Journal of Bionic Engineering (2011) Vol.8 No.1

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4.1 Computed result The hydrodynamic force produced by the undulating side fins was computed to examine the relation between the thrust coefficient (Kx) and advance coefficient (J), similarly as in propeller chart. The thrust force (Tx) was estimated by subtracting the flat plate resistance from the total x-directional force in the similar manner as the experiment. In the calculation an average value was taken from a conversed period. The thrust coefficient Kx in x-direction and advanced coefficient were computed as

Kx = J=

T T U2 = × = Tx × J 2 , ρ N 2 L4 ρU 2 L2 N 2 L2

(4)

1 ⎛ U ⎞ ⎜= ⎟. n ⎝ NL ⎠

(5)

The fin open characteristics or the fin performance chart for maximum fin angle of 30˚ and 45˚ for aspect ratios 0.05, 0.075 and 0.1 are shown in Fig. 13 and Fig. 14 respectively. Each graph apposed in the expected tendency with the change of aspect ratio and fin angle. It was also found that the thrust coefficient produced by fin at maximum angle of 45˚ is nearly twice compared to that of 30˚. 4.2 Efficiency The efficiency of propulsion (η) is defined as the ratio of the useful power obtained, PE, to the power actually delivered to the propeller, PD, i.e., η = PE / PD. In case of the model we studied, the delivered power or work (PD) by the undulating side fins was the power yielded in y and z direction. PD was calculated by summing the products of the total force with the corresponding velocities. The obtained power (PE) was the x-direction thrust force, as the velocity was one unit in this direction. Both the pressure force and the thrust, in accordance with our definition (total force-flat plate resistance, i.e. the TF-FPR) were taken into consideration to analyze the trend of efficiency of our model (Fig. 15). When only the pressure force was considered as the obtained power (PE), the efficiency increases with the advance coefficient (J), but when it accounted the frictional resistance, the efficiency drops down after certain value. This can be explained by the fact that with the increase in frequency, the frictional resistance also increases, which decreases the efficiency.

Kx

4 Fin open characteristics

Fig. 13 Fin open characteristics, Θ = 30˚. 0.0016 AR = 0.1 0.0014 0.0012 0.0010 0.0008 0.0006 AR = 0.075 0.0004 0.0002 AR = 0.05 0.0000 0.2

0.3

0.4

−0.0002

0.5

0.6

0.7

J

Fig. 14 Fin open characteristics, Θ = 45˚. 0.60 0.50 0.40

PE as PD (Θ = 30˚, AR = 0.1) TF-FPR as (Θ = 30˚, AR = 0.1) PE as PD (Θ = 45˚, AR = 0.1) TF-FPR as PD (Θ = 45˚, AR = 0.1) PE as PD (Θ = 30˚, AR = 0.075) TF-FPR as PD (Θ = 30˚, AR = 0.075)

0.30 0.20 0.10 0.00 0.1

0.2

0.3

0.4

0.5

0.6

J

Fig. 15 Efficiency of the undulating side fin at different conditions.

Rahman et al.: Computational Study on a Squid-Like Underwater Robot with Two Undulating Side Fins

The effect of changes in aspect ratio and maximum fin angle on the efficiency of our squid-like robot was also estimated. Efficiency was not affected by the changes in aspect ratio and the maximum fin angle under the assumption that pressure force was the obtained power. In contrary, when the frictional resistance was included in the obtained power, the efficiency was affected both by changes in the aspect ratio and the maximum fin angle. 4.3 Verification The relation among thrust coefficient, aspect ratio and fin angle was derived as. 3

2

U ⎞ 1 1 ⎛b ⎞ ⎛ K x = Θ 2 ⋅ ⎜ m ⎟ ⋅ ⎜1 − ⎟ ⋅ ∫ ∫ f ( s, b) dsdb, (6) ⎝ L ⎠ ⎝ nλ ⎠ 0 0 in which f ( s, b) = 4π 2 Cpb 2 cos 2 ( 2πk ( s − ct ) ) , c is the phase velocity of progressive wave and kc = n. This equation indicates that the thrust coefficient is proportional to the cube of aspect ratio and square of maximum fin angle. These two relationships were established by the computation (Fig. 16). We know, (0.1/0.075)3 = 2.37; so in Fig. 16, if we divide the result of AR = 0.1 by 2.37, we can obtain a curve very close to that AR = 0.075. In the similar manner, as (45/30)2 = 2.25, the thrust generated by fin at maximum fin angle Θ = 45˚ is about 2.25 times of that at 30˚ irrespective of fin’s aspect ratio.

31

respectively in Fig. 17. The dot line and the solid line respectively show the computed results for AR = 0.075 and 0.1. As the aspect ratios of the computed and experimental models were different, for more comparison the results for AR similar to the experiments were calculated using the previous formula, as shown by dash and dash-dot lines. The calculated results show good agreement with the experimental results of corresponding aspect ratios. A little disparity was seen between the experiment and computed results at first sight. But, this was due to the dissimilarities of the models. The fin open characteristics for Θ = 45˚ were also shown in Fig. 18. This result shows better conformity than that for Θ =30˚. This was because the same area of the fin generated more pressure force when the maximum fin angle was increased. This is also consistent with the previous discussion. Over all, the computed results approached the expected trend with the changes of the aspect ratio and maximum fin angle. 0.0002

Exp.model-4 (AR = 0.09) Exp.model-3 (AR = 0.136) Computed (AR = 0.075) Computed (AR = 0.1) Calculated (AR = 0.09) Calculated (AR = 0.136)

0.0015

0.0010

0.0005

0.0000 0.10 0.15

0.20

0.25

0.30

0.0005

0.35

0.40 0.45 0.50

0.55

J

Kx

Fig. 17 Comparison among the computed, calculated and experimental results, Θ = 30˚. 0.006 Exp.model-4 (AR = 0.09) Exp.model-3 (AR = 0.136) Computed (AR = 0.075) Computed (AR = 0.1) Calculated (AR = 0.09) Calculated (AR = 0.136)

0.005 0.004 0.003

Fig. 16 The relation between Kx & AR and Kx & Θ.

4.4 Comparison The computed and calculated results were compared with the experimental ones. The experimental results for the third and the fourth models with aspect ratios 0.136 and 0.09 are shown by dots and triangles,

0.002 0.001 0.000 0.1

0.2

0.3

0.5

0.4

0.6

0.7

0.8

J

Fig. 18 Comparison among the computed, calculated and experimental results, Θ = 45˚.

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Journal of Bionic Engineering (2011) Vol.8 No.1

5 Conclusion The swimming motion of a squid-like body with two undulating side fins was investigated through flow computation around the body. The relationship between the relative velocity of fluid and the dimension of the fin was investigated based on the distribution of pressure difference between upper and lower surfaces and thrust force distribution. The computational model exhibited a considerable agreement with the experiment observations. A simple relationship among the fin’s principal dimensions and hydrodynamics was established using the computed results. The comparison between the force measurement and integrated force using CFD result for simple model showed that the CFD techniques can predict the hydrodynamic force produced by the undulating side fin.

Acknowledgement This research has been partly supported by the Grant-in-Aid for Scientific Research, Japan (2009-2011, Project number 21246129).

References [1] [2]

[3]

[5]

Low K H. Modeling and parametric study of modular undulating fin rays for fish robot. Mechanism and Machine Theory, 2009, 44, 615–632.

[7]

Shang L, Wang S, Tan M and Dong X. Motion control for an underwater robotic fish with two undulating long-fins. The Proceedings of 48th IEEE Conference on Decision and Control, Shanghai, China, 2009, 6478–6483.

[8]

Toda Y, Hieda S, Sugiguchi T. Laminar flow computation around a plate with two undulating side fins. Journal of Kansai Society of Naval Architects, 2002, 237, 71–77.

[9]

Toda Y, Fukui K, Sugiguchi T. Fundamental study on propulsion of a fish-like body with two undulating side fins. The Proceedings of the First Asia Pacific Workshop on Marine Hydrodynamics, Kobe, Japan, 2002, 227–232.

[10] Toda Y, Suzuki T, Uto S, Tanaka N. Fundamental study of a fishlike body with two undulating side-fins. In: Kato N, Kamimura S (Eds.). Bio-Mechanisms of Swimming and Flying, Springer, Tokyo, 2004, 93–110. [11] Toda Y, Ikeda H, Sogihara N. The motion of a fish-like under-water vehicle with two undulating side fins. The Third International Symposium on Aero Aqua Biomechanisms, Ginowan, Okinawa, Japan, 2006, P27. [12] Toda Y, Danno M, Sasajima K, Miki H. Model experiments on the squid-like under-water vehicle with two undulating

Breder C. The locomotion of fishes. Zoologica, 1926, 4,

side fins. The 4th International Symposium on Aero Aqua

159–256.

Biomechanisms, Shanghai, China, 2009.

Sfakiotakis M, Lane D M, Davies J B C. Review of fish

[13] Rahman M M, Toda Y, Miki H. Study on the performance of

swimming modes for aquatic locomotion. IEEE Journal of

the undulating side fins with various aspect ratios using

Oceanic Engineering, 1999, 24, 237–252.

computed flow, pressure field and hydrodynamic forces.

Kato N, Ayers J, Morikawa H. Bio-mechanism of Swimming

Proceedings of the 5th Asia-Pasific Workshop on Marine

and Flying, Springer-Verlag, Tokyo, Japan, 2004. [4]

[6]

Hydrodynamics, Osaka, Japan, 2010, 333–338.

Riggs P, Bowyer A, Vincent J. Advantages of a biomimetic

[14] Chen C J, Chen H C. Finite analytic numerical method for

stiffness profile in pitching flexible fin propulsion. Journal

unsteady two dimensional Navier-Stokes equation. Journal

of Bionic Engineering, 2010, 7, 113–119.

of Computational Physics, 1984, 53, 209–226.

Zhang Y H, Jia L B, Zhang S W, Yang J, Low K H. Com-

[15] Tahara Y. Computation of viscous flow around series 60

putational research on modular undulating fin for biorobotic

model and comparison with experiments. Journal of Kansai

underwater propulsor. Journal of Bionic Engineering, 2007,

Society of Naval Architects, 1993, 220, 29–47.

4, 25–32.