Three-dimensional Hydrodynamic Analysis of Propulsion of With Prosthetic Flippers

Xiaoqian Sun a*, Naomi Kato a Yasushi Matsuda b, Kazunori Kanda b, Yusuke Kosaka b Naoki Kamezaki c, Mari Taniguchi c

a Osaka University, Suita, Osaka, Japan b Kawamura Gishi Co. Ltd, Daito, Osaka, Japan c Sea Turtle Association of Japan, Hirakata, Osaka, Japan

Abstract—This study is to develop prosthetic flippers strokes are usually used by most freshwater turtles, for an injured sea turtle named “Yu” from the view- which have been documented in an extensive range of point of 3D (three-dimensional) hydrodynamic analysis previous studies [e.g. 1, 2, 3, 4, 5]. Flapping strokes are of sea turtles’ forelimb propulsion. Firstly template characterized by predominantly drosoventral forelimb matching method is used to compare the 3D movements movements, whereas rowing strokes are characterized of fore flippers in three cases respectively: those of a by predominantly anteroposterior forelimb movements healthy turtle, those of Yu with and without prosthetic combined with rotation of the foot (perpendicular to flippers. Secondly 3D hydrodynamic analyses for three flow during thrust and feathered during recovery) [6]. cases based on quasi-steady element theory are But specifically speaking, turtle species display carried out to investigate the hydrodynamic effects of considerable diversity in their styles of forelimb prosthetic flippers on the swimming performance of sea flapping or rowing. So quantifying the exact forelimb turtles. Finally the hydrodynamic effects are clarified kinematics and the corresponding thrust forces during and some remarks for designing new prosthetic flippers turtles’ swimming is a key, which is a significant in future are given. challenge because direct measurements of force generated by the free turtles’ swimming are not feasible. Index Terms—forelimb propulsion of sea turtle, Davenport et al. estimated the thrust force by prosthetic flipper, template matching method, wing attaching a force transducer to the shells of turtles [2]. element theory But this still puts some restriction on the free swimming of turtles. Walker et al. documented the changes in I. INTRODUCTION velocity and acceleration in by tracking the center of mass of an animal through an An injured female (Caretta artificial locomotor cycle [7]. Some other researchers caretta) named Yu was found and rescued by Sea Turtle tried to obtain the thrust force by examining the Association of Japan at Kiisuido in the summer of 2008. properties of the flow field around the aquatic animals. Her swimming speed was just 60% of that of healthy For example Drucker et al. employed digital particle adult sea turtle because only a half of the left forelimb image velocimetry (DPIV) to examine the vortex wake and two thirds of the right forelimb were left after being shed by freely swimming and then evaluated the attacked by a shark. Realizing that we could not put her thrust forces [8]. Our team concentrated on directly back into the sea under such a condition, “Yu Project” observing the forelimb movements of sea turtles and has begun since 2009 to develop prosthetic flippers in calculating the corresponding hydrodynamic forces. cooperation with veterinarians, a prosthetic company, Previously Isobe et al. [9] compared the 2D and 3D aquariums, universities and a public administration for motions of fore flippers between Yu itself, Yu equipped Yu. During this process studying the kinematics of sea with prosthetic flippers and “Sho” (a healthy sea turtle) turtles and evaluating the effects of prosthetic flippers in a pool of an aquarium, an artificial lagoon and a on the swimming performance of Yu become an water circulating tank. At the same time assuming the important job. flipper consist of a rigid wing and uniform flapping, Turtle species exhibit a diversity of kinematic rowing and feathering motion from root to tip, they patterns in their during swimming. Generally analyzed 2D hydrodynamic characteristics of different speaking, the flapping forelimb strokes are usually used flippers under uniform flow. But taking into account the by swimming marine turtles and the rowing forelimb 3D motion of fore flippers, we thought 2D analysis cannot accurately evaluate the swimming performance * Department of Naval Architecture and Ocean Engineering, School of sea turtles. In our following 3D analyses of forelimb of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, 565-0871, Japan. motions, the flipper is treated as flexible in spanwise E-mail: [email protected] direction and consists of several wing segments with

– 36 – JOURNAL OF AERO AQUA BIO-MECHANISMS, VOL.3, NO.1 different flapping, rowing and feathering motion from C. Background of Motion Analysis root to tip. A motion capturing software using template matching method was utilized to observe forelimb movements in II. MOVEMENTS ANALYSIS the form of time variation of the prescribed points on the forelimbs. As Sho is healthy and possesses A. Sea Turtles symmetric fore flippers, we will only analyze the Table.1 and Fig.1 show the details of the sea turtle movements of its left fore flipper during swimming. But “Yu” and “Sho”. as for the case of Yu, because its actual left flipper and actual right flipper show different shapes, we have to Table.1 Specifications of sea turtles for motion analysis analyze the movements of both flippers in the following Carapace Body Area (left Area (right context. Locations of the targeted points on Sho’s left Name length mass flipper) flipper) flipper, Yu’s flippers and the prosthetic flippers are Yu 0.797 [m] 100.5[kg] 0.039[m2] 0.026[m2] shown in Fig.4, Fig.5 and Fig.6 respectively. Two Sho 0.751 [m] 84.8[kg] 0.063[m2] 0.067[m2] targeted points located in the spanwise middle of the flipper are used to obtain the feathering motion of the flipper. The body fixed coordinate during motion analysis is defined as O-XYZ in Fig.17.

Fig.1 Photographs of “Yu” (left) and “Sho” (right)

B. Prosthetic Flippers The prosthetic flippers are made of a kind of copolymers by Kawamura Gishi Co. Ltd. The left and Fig.4 Motion captured points on Sho’s left flipper right flippers for “Yu” are shown in Fig.2. At first we (In the following context Lroot, LM and Ltip denote the let Yu put on the specially designed jacket (Fig.3). And root, the middle, and the tip of left flipper, respectively. then the prosthetic flippers are installed onto the Rroot, RM and Rtip denote the root, the middle and the corresponding forelimb of Yu. Finally Velcro tape is tip of right flipper respectively) used to tighten the prosthetic flippers around each sleeve of the jacket. Honestly speaking, the present shape of flippers and the procedure of installing prosthetic flippers onto the forelimb are the results of many trials and errors in the past. The specially designed jacket has the function of avoiding the prosthetic flippers coming off the forelimb.

Fig.5 Motion captured points on Yu’s fore flippers

Fig.2 Left and right prosthetic flippers for “Yu”

Fig.6 Motion captured points on the prosthetic flippers

Fig.3 Specially designed jacket for “Yu” equipped with D. Experimental Condition both prosthetic flippers

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Three videos were taken simultaneously from the left side, the right side and the upper side of Sho, Yu with and without prosthetic flippers at the water tank of Suma Aqualife Park KOBE.

E. Results of Movement Analysis Fig.7 shows the trajectories of the motion captured points on the left flipper of Sho in x-y plane of the body fixed coordinate. The period of motion is 3.0s. Fig.8 and Fig.9 show the trajectories of the motion captured Fig.8 Trajectories of motion captured points on the left points on the left and right flippers of Yu without flipper of Yu in x-y plane (period:2.4s) prosthetic flippers in x-y plane of the body fixed coordinate separately. Fig.10 and Fig.11 show the trajectories of the motion captured points on the left and right prosthetic flippers of Yu in x-y plane of the body fixed coordinate separately. First of all it is observed that Yu swims with a bigger frequency of flipper movements than Sho does, but if the prosthetic flippers are installed, the flipper movement frequency of Yu decreases. From the five figures we can see that the flippers of Sho and Yu describe a circular arc with large curvature in both the power stroke (from anterior Fig.9 Trajectories of motion captured points on the right position to posterior position) and the recovery stroke flipper of Yu in x-y plane (period:2.3s) (from posterior position to anterior position). On the other hand, in the case of Yu equipped with prosthetic flippers, the trajectory of the flippers in x-y plane is a circular arc with small curvature. Fig.12 shows the trajectories of the motion captured points on the left flipper of Sho in x-z plane of the body fixed coordinate. Fig.13 and Fig.14 show the trajectories of the motion captured points on the left and right flippers of Yu in x-z plane of the body fixed coordinate. Fig.15 and Fig.16 show the trajectories of Fig.10 Trajectories of motion captured points on the left the motion captured points on the left and right prosthetic flipper of Yu in x-y plane (period: 2.93s) prosthetic flippers of Yu in x-z plane of the body fixed coordinate. The trajectories of the flippers of Sho and Yu are ovals but in the case of Yu equipped with prosthetic flippers the trajectory of the flippers is similar to an oval with twist at the posterior position for the left prosthetic flipper and at the middle position for the right prosthetic flipper.

Fig.11 Trajectories of motion captured points on the right prosthetic flipper of Yu in x-y plane (period:2.96s)

Fig.7 Trajectories of motion captured points on the left flipper of Sho in x-y plane (period: 3s) (The arrows in Fig.7-16 denote trajectory direction with time variation)

Fig.12 Trajectories of motion captured points on the left flipper of Sho in x-z plane

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o x  y  z  : The origin is located at the leading point of the same cross-section of the forelimb. can be obtained by rotating around axis oz with  . o x  y  z  : The origin is also located at the leading point of the same cross-section. Plane o x  z  coincides with the cross-section. can be got by rotating around ox with angle .

Fig.13 Trajectories of motion captured points on the left xX    Xo      flipper of Yu in x-z plane y   Y   Yo …(Eq.1)      zZ    Zo x   x  cos sin 0  x         y  RZ ( )  y     sin  cos  0  y  …(Eq.2)        z   z  0 0 1  z  x   x   1 0 0  x            y  RX ( )  y   0 cos  sin   y  …(Eq.3) z   z   0 sin cos  z          Fig.14 Trajectories of motion captured points on the right flipper of Yu in x-z plane

Fig.15 Trajectories of motion captured points on the left Fig.17 Coordinates definition during 3D analysis prosthetic flipper of Yu in x-z plane

Fig.16 Trajectories of motion captured points on the right prosthetic flipper of Yu in x-z plane Fig.18 Divisions of Sho’s left forelimb during analysis

III. METHOD FOR 3D ANALYSIS B. Determination of and During flipper movement, the shape of forelimb

changes instantly as a combination of flapping, rowing A. Coordinates Definition and feathering motions and its flexibility. In order to Altogether four coordinates are defined during hydro- reflect the direction variation as accurately as possible, dynamic analysis (Fig.17). we divide each forelimb into four sections. Every O XYZ : We set the front point of the carapace as the moment inside each section all the body fixed origin and the centerline of the carapace as the X-axis. coordinates can be assumed to possess the same o xyz : The origin is located at the leading point of direction properties, namely at each moment inside each one cross-section of the forelimb. The three axes are section the values of  and for different cross-sections parallel to the axes of respectively.

– 39 – JOURNAL OF AERO AQUA BIO-MECHANISMS, VOL.3, NO.1 are uniquely determined separately. Let us take the we modify 2D wing theory by using lifting line theory. division of Sho’s left flipper as example (Fig.18). The vortex wake consists of streamwise vortices due to The values of  and  of Section Lroot-LI2-LI1 can the spanwise circulation gradient and transverse vortices be calculated by considering the direction of Triangle due to the variation of the circulation with time. Given Lroot-LI2-LI1, which can be assumed to be included in the large aspect ratio of flippers and the low frequency the representative surface of this section. For the other of flipper motion, we assume the transverse vortices are three sections the same method is adopted. very small, the flow within each chord-wise cross- Next we concentrate ourselves on only one section to section of the flipper can be dealt with 2D wing theory get the values and for that section (Fig.17). Line and 3D effect is only considered by the difference of segment AB denotes the chord of one cross-section induced velocities and induced angles of attack. As for (LI1-LI2, LM1-LM2 or LO1-LO2 in Fig.18) that lies in the calculation of added mass, given the density of the plane o x  z  , and that C denotes one point (Lroot, water is much larger than that of air, we chose to obtain LIM, LOM or Ltip in Fig.18) of the triangles above. the corresponding value of each segment along the In coordinate O XYZ , the values of AXYZ(,,) , spanwise direction without any assumption on the flow AAA because of its simplicity. By this way we simplified the BXYZ(,,)BBB and CXYZ(,,)CCC can be obtained by hydrodynamic calculation of sea turtles’ flippers. movement analysis. Then the expression of surface Because the chord length of cross-sections vary ABC can be given as follows: greatly in different position of the forelimbs, in order to XXYYZZAAA   x y z reflect this variation we subdivide each section of the …(Eq.4) XBABABABBB X Y  Y Z  Z  x y z  0 flipper in the previous part of this article into three XXYYZZ   x y z segments separately, which is shown in (Fig.18). CACACA CCC Considering each subsection as a regular 3-D wing, This determinant can be expanded as follows: each subsection can be represented with the middle  yzBCCBBCCBBCCB yzx  xz  xzy  xy  xyz  0 chord length of each subsection as the representative …(Eq.5) chord length of the 3-D wing, with the width of each Therefore m , a normal vector to surface ABC should subsection as the span length of the 3-D wing. have the following expression: For each segment the two-point hinge oscillating wing theory is adopted to simulate crosssection motion, m yzBCCBCBBCBCCB  yz,, xz  xzxy  xy  (Eq.6) which is coupled with heaving and pitching motions. On the other hand, the unit vector of axis oy  is: Then we can calculate the corresponding thrust in each no x  y  z   0 1 0 …(Eq.7) segment. Finally the resultant force can be obtained by The expression of this unit vector in coordinate adding all the thrust forces in each subsection. o xyz can be got by using (Eq.2) and (Eq.3). In one cross-section of the fore flipper, schematic compositions of velocities are given in Fig.19. 0   sin cos  11    …(Eq.8) no xyz  RZX R  1   cos  cos       0   sin  If we assume is in the plane ABC, we have that m is perpendicular to n : mn0 …(Eq.9) Because AB is also perpendicular to , we also have AB n 0 …(Eq.10) Substituting (Eq.7) and (Eq.8) into (Eq.9) and (Eq.10), finally we can obtain that y()() x y x y  z x z  x z Fig.19 Velocity compositions in one cross-section tan  BBCCBBCBBC …(Eq.11) xBBCCBBBCCB()() x y x y  z y z  y z Here, the origin o is actually the leading point of tan (x sin  y cos  ) / z …(Eq.12)   BBB each cross-section. U andW are the incoming flow Up to here, the values of and can be obtained. velocity components in the ox  and oy  direction. Here we use the motion of one quarter point on the C. Hydrodynamic Force Calculation chord of wing to represent wing movement. Assuming In the water the fore flippers serve as . The that xz,  is the coordinate of the leading point movement of forelimbs is unsteady, but here we use 11 two-point hinge oscillating wing theory, which is a in o x  z  , the coordinate of the one-quarter point on quasi-steady theory developed by Nagai [10]. the chord of wing can be given as: There are altogether three important forces during cc x x cos z   z   sin …(Eq.13) hydrodynamic analysis: lift force, drag force and added 1/4 144 1/4 1 mass force. During the calculation of lift and drag force Then the velocity at the one-quarter point can be got:

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dx c   1/4  TLDF  sinv   cos  v  n sin  u1/4  u 1  dcos / dt 1/4 1/4 …(Eq.21) dt 4 …(Eq.14) ZLDF cos   sin    cos  dz  v1/4 v 1/4 n 1/4  c w1/4   w 1  dsin / dt dt 4 According to (Eq.1) and (Eq.4), the expressions of lift force and vertical force in coordinate o xyz and u  and w  are the velocity components of the leading 1 1 O XYZ can be obtained: point A of the cross-section in o x  y  z  , which can be TTZZZcos   sin  sin  ,   cos  …(Eq.22) calculated by movement analysis. T is thrust force along O-X direction and Z is vertical At the one-quarter point of the chord, force along O-Z direction. 2 2 U1/4 U   u 1/4  () W   w 1/4  …(Eq.15) 1 Ww 1/4  (v )1/4  tan Uu 1/4  The geometric angle of attack is calculated as:

1/4() v 1/4  …(Eq.16)

The geometric lift coefficient CL and the geometric drag coefficient CD can be obtained from the curves in Fig.20, which is referred from the paper by Okamoto and Jinba [11]. Taking into account the downwash phenomenon of 3D wing, the effective angle of attack can be calculated according to [12]: Fig.21 Schematic relationship between  and ,    ,/,/  C   AR AR  b2 S ...(Eq.17) ff1/4 L D. The Determination of  Where ε is induced angle of attack, AR is the aspect During previous part, parameters except were ratio of wing, b is the span of wing and S is the projecting area of wing. Next the effective lift introduced. So in this part the process of how to get the value of  will be explained. coefficient CL  can be got from Fig.20. The eff In Fig.21, we assume that AB is the chord of one effective drag coefficient is the sum of geometric drag cross-section. BE and BF are perpendicular to plane coefficient CD and induced drag coefficient CDi. o xy and o xz respectively. G is the intersecting point C C  C,/ C  C  C2  AR …(Eq.18)  Deff D Di Di L L   between plane BEF and line ox . We assume Finally actual Lift and drag acting one the wing are: FAG  and EAG  . and are defined as

1122the upper view projecting value and the side view LCUSDCUS   ,    …(Eq.19) 22LDeff1/4 eff 1/4 projecting value separately of the angle between the line here  denotes fluid density and S is the projecting area of the flipper located at the spanwise middle (line of each segment. segment AB in Fig.4) and the carapace line (line segment OX in Fig.4). The values of and can be obtained by movement analysis. Assuming that the coordinates of point B

is xBBB,, y z  , we can get the relations below:

yBBBB xtan , z x tan Then the vector parallel to OB can be written as

co xyz  (1,tan ,tan ) According to (Eq.2) and (Eq.3), the expression of c at Fig.20 Curves between CL (CD) and angle of attacks coordinate o x  y  z  can be calculated: The added mass can be approximated by the added cos sin  tan  cx mass of the fluid of the column of which diameter is  cco x  y  z  sin cos   cos  cos  tan   sin  tan    y  equal to the chord length of the wing c, and of which  sin sin  cos  sin  tan  cos  tan  c length is the same as the wing span b. Then Added mass z force can be obtain as below: …(Eq.23)  dU( sin ) can be considered as the angle between co x  y  z  F   bc2 1/4 1/4 …(Eq.20) n 4 dt and ox  . And then the value of can be obtained: Up to now, the expressions of thrust force and lateral  arccosc / c2  c 2  c 2 …(Eq.24) force in coordinate o x  y  z  are given below:  x x  y  z  

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4 IV. RESULTS AND DISCUSSIONS force(N) body drag thrust 2 A. Hydrodynamic Analysis of Sho’s left flipper 0 Using the theoretical analysis mentioned above, we discuss here the relationship between drag force acting -2 on the body of sea turtle and thrust force produced by the left forelimb of Sho. -4 Fig.22 shows time variation of swimming velocity -6 components of Sho. Fig.23 gives the time variation of t(sec) th -8 angle of attacks of the 7 segment of Sho’s left flipper 0 0.5 1 1.5 2 2.5 3 in Fig.18. On the other hand, hydrodynamic drag Fig.24 Time variation of total thrust force generated by coefficients on turtle body were referred from the paper left flipper and total body drag force of Sho by Watson and Granger [13]. We plotted time variation of total thrust force generated by left flipper and body drag force in Fig.24. The average thrust force generated by the left flipper is -0.9873N, the total thrust force can be got as twice of the thrust generated by left flipper and the average body drag force is 2.0114N. Consequently the total thrust force and body drag force can be balanced. In order to verify the validity of our method in carrying out the hydrodynamic analysis of sea turtle, some further calculations were performed. The total Fig.25 Total thrust force and body drag of Sho in thrust force (twice of the thrust force generated by the different swimming velocities (Force is set plus) left flipper) and the body drag force in several swimming velocities were calculated and then the B. Hydrodynamic Analysis of Yu’s Forelimbs Without tendency of both forces were compared (Fig.25). The Prosthetic Flippers velocity of the intersecting point in Fig.25 is 0.269m/s Fig.26 shows the time variation of swimming velocity and the actual swimming velocity during Sho’s components of Yu with actual flippers. Fig.27 and calculation is 0.278m/s. So we can say our method can Fig.28 give the time variation of angle of attacks of the almost predict the hydrodynamic analysis of sea turtles’ 8th segment for Yu’s actual left flipper and Yu’s actual forelimb movements. right flipper. Fig.29 is the time variation of total thrust force generated by both actual flippers and body drag 0.1 velocity(m/s) U force. The average thrust force generated by left flipper V W is -0.3789N, the average thrust force generated by right 0 flipper is -0.5381, and the average body drag force is

-0.1 0.9228N. Consequently the total thrust force and body drag force can be almost balanced.

-0.2 The verification calculations in different swimming velocities were also done (Fig.30). The total thrust force

-0.3 is the sum of the thrust force generated by the left and right actual flippers. The velocity of the intersecting t(sec) -0.4 point in Fig.30 is 0.203m/s and the actual swimming 0 0.5 1 1.5 2 2.5 3 Fig.22 Time variation of swimming velocity velocity during Yu’s calculation is 0.187m/s. Although components of Sho there exist some discrepancies, our method can predict the hydrodynamic analysis of Yu’ forelimb movements. 100 angle of attack 0.1 angle of attack(degree) velocity(m/s) 80

60 0 40

20 -0.1 0

-20 -0.2 -40 U -60 -0.3 V W -80 t(sec) t(sec) -100 -0.4 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 th Fig.23 Time variation of angle of attacks of the 7 Fig.26 Time variation of swimming velocity segment for Sho’s left flipper components of Yu with actual flippers

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100 angle of attack(degree) average thrust force generated by right prosthetic flipper angle of attack 80 is -1.1661N, and the average body drag is 2.573N. The 60 average thrust and body drag can be balanced. 40 The verification in different swimming velocities is 20

0 shown in Fig.35. The velocity of the intersecting point

-20 in Fig.35 is 0.268m/s and the actual swimming velocity

-40 is 0.293m/s. Although the discrepancy is a little bigger, -60 almost we can say that if installing prosthetic flippers -80 our method can also predict well the hydrodynamic t(sec) -100 0 0.5 1 1.5 2 2.5 analysis of Yu’s forelimb movements. th Fig.27 Time variation of angle of attack of the 8 0.2 velocity(m/s) segment for Yu’s actual left flipper 0.1

100 angle of attack(degree) 0 angle of attack 80 -0.1 60 U -0.2 V 40 W 20 -0.3

0 -0.4 -20 -0.5 -40 t(sec) -0.6 -60 0 0.5 1 1.5 2 2.5 3 -80 Fig.31 Time variation of swimming velocity t(sec) -100 0 0.5 1 1.5 2 2.5 components of Yu with prosthetic flippers th Fig.28 Time variation of angle of attack of the 8 100 angle of attack(degree) angle of attack segment for Yu’s actual right flipper 80 60 6 body drag force(N) 40 thrust 4 20

0 2 -20

-40 0 -60

-2 -80 t(sec) -100 0 0.5 1 1.5 2 2.5 3 -4 Fig.32 Time variation of angle of attack of the 7th t(sec) -6 0 0.5 1 1.5 2 2.5 segment of Yu’s left forelimb with prosthetic flippers Fig.29 Time variation of total thrust force generated by 100 angle of attack(degree) angle of attack Yu’s both actual flippers and total body drag of Yu 80 60

40

20

0

-20

-40

-60

-80 t(sec) -100 0 0.5 1 1.5 2 2.5 3 Fig.33 Time variation of angle of attack of the 7th

Fig.30 Total thrust force and body drag force of Yu in segment of Yu’s right forelimb with prosthetic flippers different swimming velocities 10 force(N) body drag 8 thrust C. Hydrodynamic Analysis of Yu’s Forelimbs With 6 Prosthetic Flippers 4 Fig.31 shows the time variation of swimming velocity 2 components of Yu with prosthetic flippers. Fig.32 and 0 Fig.33 give the time variation of angle of attack of the -2 th 7 segment of Yu’s left and right prosthetic flippers. -4 Fig.34 gives the time variation of total thrust force -6 t(sec) -8 generated by Yu’s both flippers and body drag force 0 0.5 1 1.5 2 2.5 3 with prosthetic flippers. The average thrust force Fig.34 Time variation of total thrust generated and total generated by left prosthetic flipper is -1.0399N, the body drag of Yu with prosthetic flippers

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larger than that generated by the left one. So in the future we think it is better to develop new prosthetic flippers or only install the left prosthetic flipper, which can make both flippers generate equal thrust and therefore Yu’s swimming motion will become smooth.

REFERENCE [1] Warren F. Walker, Jr., “Swimming in sea turtles of Fig.35 Time variation of total thrust force and total the family Cheloniidae”, Copeia, 2, pp.229-233 (1971). body drag force of Yu with prosthetic flippers [2] J. Davenport, Sarah A. Munks and P.J. Oxford, “A comparison of the swimming of marine and freshwater D. Comparison Among Three Cases turtles”, Proceedings of The Royal Society of London 1) Our method can predict well the hydrodynamic B, vol.220, No.1221, pp.447-475 (1984). characteristics in the case of Sho’s flippers, Yu’s actual [3] S. Renous and V. Bels, “Comparison between flippers and Yu’s prosthetic flippers. aquatic and terrestrial locomotions of the leatherback 2) Prosthetic flippers play a positive role in generating sea turtle (Dermochelys coriacea)”, Journal of Zoology, thrust force. But the thrust generated by right prosthetic Vol.230, Issue 3, pp.357-378 (1993). flipper is larger than that generated by the left one. [4] J. Wyneken, “Sea turtle locomotion: mechanisms, 3) The angle of attack of Sho’s left flipper stays in the behavior, and energetics”, The Biology of Sea Turtles large lift coefficient area for a long time in one period. (P. Lutz and J. Musick eds), Boca Raton Florida: CRC But the angle of attack of Yu’s actual left flipper varies Press, pp.165-198 (1997). largely, and much of it is not in the large lift coefficients [5] J.A. Walker and M.W. Westneat, “Mechanical area. After installing the prosthetic flippers, the performance of aquatic rowing and flying”, Proceedings situation of angle of attack was improved. of The Royal Society of London B: Biological Science, 4) From validation results, we can see for Sho’s case it vol.267, No.1455, pp.1875-1881 (2000). has good agreement but for Yu’s cases with and [6] C.M. Pace, R.W. Blob and M.W. Westneat, especially without prosthetic flippers, there exist some “Comparative kinematics of the forelimb during discrepancies. The first reason is from the validation swimming in red-eared slider (Trachemys scripta) and method itself: sea turtles’ swimming velocity is not spiny softshell (Apalone spinifera) turtles”, The Journal constant, but during our validation we adopted the of Experimental Biology 204, pp.3261-3271 (2001). relationship of average velocity and average thrust to [7] J.A. Walker and M.W. Westneat, “Labriform inspect the validity of our predicting method. And the propulsion in : kinematics of flapping aquatic asymmetry of Yu’s real and prosthetic flippers flight in the bird wrasse Gomphosus varius (labridae)”, aggravates this inconstancy of swimming velocities. The The Journal of Experimental Biology 200, pp.1549- second reason may come from that when Yu swims with 1569 (1997). prosthetic flippers, it cannot suit itself with the [8] E.G. Drucker and G.V. Lauder, “Locomotor forces and therefore cannot swim regularly, which on a swimming fish: three-dimensional vortex wake increase the difficulty of taking good videos. dynamics quantified using digital particle image velocimetry”, The Journal of Experimental Biology V. CONCLUSION 202, pp.2393-2412 (1999). To contribute to design and development of [9] Y. Isobe, N. Kato, H. Suzuki and others, “Motion prosthetic flippers, we contradistinguished the 3D analysis of sea turtle with prosthetic flippers”, Proc. of kinematics and hydrodynamics of three cases and The International Society of Offshore and Polar validated that our proposed method can predict well the Engineers 2010 Conference, 2, pp.335-342 (2010). hydrodynamics of sea turtles’ forelimb motions. [10] M. Nagai, I. Teruya, K. Uechi and T. Miyazato, We also tried finding some bijective correspondence “Study on an oscillating wing propulsion mechanism”, between kinematics and propulsive force generation. Transactions of the Japan Society of Mechanical One is that flipper flexibility plays a great role during Engineers. B, 62(593): 200-206, (1996). the generation of thrust force. Flexible Sho’s flippers [11] M. Okamoto and Y. Jinba, “Experimental study on and Yu’s actual flippers behave curvilinear motions and aerodynamic characteristics of wing planforms at low can bend actively in both chordwise and spanwise Reynolds number”, Research Reports of Akita National directions, which correspondingly produce effective College of Technology 44, pp.42-50, (2009). thrust force. But Yu’s prosthetic flippers behave nearly [12] K. Kato, A. Ooya, and K. Karasawa, “Introduction linear motions and can only passively utilize the to Aerodyanmics”, University of Tokyo Press, ISBN: flexibility, which to some extent affect the effective 978-4-13-061043-0, pp.41-42, (1982). generation of thrust. [13] K. P. Watson, R. A. Granger, “Hydrodynamic From the viewpoint of making prosthetic flippers, we effect of a satellite transmitter on juvenile green turtle”, can see the thrust generated by right prosthetic flipper is The Journal of Experimental Biology, Vol 201, Issue 17, pp. 2497-2505, (1998).

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