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Journal of Hydrodynamics Ser.B, 2006,18(2): 135-142

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A NUMERICAL STUDY ON A SIMPLIFIED TAIL MODEL FOR TURNING FISH IN C-START* YANG Yan The Laboratory for Biomechanics of Animal Locomotion, Graduate University of Chinese Academy of Sciences, Beijing 100049, China Department of Mechanics and Mechanical Engineering, University of Science and Technology of China, Hefei, 230026, China,E-mail: [email protected] TONG Bing-gang The Laboratory for Biomechanics of Animal Locomotion, Graduate University of Chinese Academy of Sciences, Beijing 100049, China (Received Apr. 19, 2005) ABSTRACT: Most freshwater fish are good at turning manoeuvres. A simulated fish tail model was numerically investigated and discussed in detail. This study deals with unsteady forces and moment exerted on the fish tail-fin in an initial sideways stroke and a subsequent return stroke motion, and visualizes the flow fields and vortex structures, in order to explore the flow control mechanism of the typical turning motion of fish. Further discussion on fluid dynamic consequences corresponding to two different bending forms of fish tail-fins in its C-start is given. The two-dimensional unsteady incompressible Navier-Stokes equations are solved with a developed pseudo-compressibility method to simulate the flow around the fish tail-fin. The computed results and the comparison with experiments indicate that (1) fish performs a turning motion of its body using the impulsive moment produced by the to-and-fro stroke, and each stage of the process exhibits its specific hydrodynamic characteristic, (2) fishes adopt two forms of tail-tip bend (single bend and double bend) to accomplish a C-start turning manoeuvre, in dependence of their physical situations and natural environments, (3) fish can control its turning motion by modulating some key kinematic parameters. KEY WORDS: fish swimming, turning manoeuvre, C-start, tail fin locomotion, 2D Navier-Stokes equations, CFD, physical mechanism, optimal control

1. INTRODUCTION Fishes have experienced a long evolution process

with natural selection, and exhibit a large variety of swimming gaits, relevant to their body shapes and surrounding environments[1,2]. Usually, pelagic fishes are experts in long-distance cruising (steady swimming), and freshwater fishes in maneuvering (transient swimming). The former movement of fish has traditionally been the focus of scientific attention among biomechanical researchers[3], including theoretical[4], experimental[5] and computational[6,7] studies. What they regard are the quasi-steady thrust and Froude efficiency. And in the last decade, engineers made out some biomimetic swimming robots mimic pelagic fishes, that had quite good capability of propulsion[8]. However, physical mechanism of fish maneuvering is little understood yet. And existing Autonomous Underwater Vehicles (AUV’s) are far insufficient at dextrous manipulation. Whereas, for freshwater fish, turning manoeuvre is one of the most important and ordinary skills. They change their direction of motion very often and sometimes rapidly, especially at some urgent moments like escaping and preying. There are many difficulties in research of fish manoeuvre, as experiments are hard to design and repeat, and the actual mechanical quantities hard to

* Project supported by the National Natural Science Fourndation of China(Grant No:10332040) and the Innovation Project of the Chinese Acadeny of Sciences (Grant No:KJCX-SW-L04). Biography: YANG Yan (1981-), Male, Ph.D. Student

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measure. Recently, some observational experiments from biological viewpoint were reported[9], most of which focused on behavioral and kinematic phenomena, while studies on hydrodynamic mechanism are few. Weihs[10] firstly estimated the axial and transverse forces exerting on fish with turning manoeuvres, using Lighthill’s slender body theory, which is an analytical inviscid potential flow theory, and the results could not be quite precise. To reveal the substaintial mechanism, researchers tend to deal with not complex real object but simplified mechanical model. Ahlborn et al.[11] visualized a model simulating fast-start fish tail and estimated the thrust with time. The model is a 2-D rigid flat plate flipping sideways to and fro about the hinged leading-edge in water. Hu et al.[12] analytically and numerically studied the similar model, and revealed the flow physics of large thrust production for fish fast-start. But the later two works were both concerned only with forward thrust, i.e. fast-start dynamics along a single direction.

Fig.1 Two bending forms of tail-fin tips in leftward C-start manoeuvre (data from Jing et al.[18, 19])

In general, a turn can be decomposed kinematically into two components, (1) body rotation about the Centre of Mass (CM) and (2) arc motion of CM. Among them, the former is the essential characteristic, as means changing the direction of motion, while some fishes can make a turn without or with little displacement of CM[13]. Experiments indicates that tail fin plays the most important role in

the turning motion[14], although pectoral fins and other fins are also controllable in some situations[15]. There are several modes of turning mainly using tail swing[16]. Herein, “C-start” is one typical turning manoeuvre used in escaping from enemies, which is characterized with two continuous motions through the process: (1) firstly a body rotation with some turning angle, (2) finally CM going forward (with a certain velocity Vf) in a new direction from rest at the beginning (Vo=0). C-start is named for fish contrasts their body into a “C” shape. Sakakibara et al.[17] used stereoscopic PIV to measure the velocity fields around a live fish with the above motion. Jing et al.[18,19] recorded C-start manoeuvres of freshwater fish, summarized the kinematic characteristics, and noted two kinds of C-start locomotion, distinguished by different bending forms of the tail-fin tips (single bend and double bend), as Fig.1 shows. The flow control mechanism of such typical turning motion, C-start, is what we have focused. We have as well devoted our attention to reveal the hydrodynamic characteristics of such motion. This study numerically solves the two-dimensional Navier-Stokes equations for unsteady incompressible flow, based on a simplified model simulating fish tail locomotion in C-start manoeuvres. We investigate the tail fin model extensively and reveal that the simplified model embodies the essential kinematic and dynamic features of turning motion. As a result, the study presents the flow physics and the unsteady hydrodynamic responses, i.e., how the moment and forces produced by the to-and-fro tail strokes contribute to body rotation and subsequent forward propulsion. And we try to reveal the secret of the two bending tail-tip forms from the hydrodynamic viewpoint. Furthermore, some optimal controls of this motion are discussed.

2. PHYSICAL MODEL Our physical model is inspired from Ahlborn et al.’s experiment[11] mentioned in Section 1. Here, we use a NACA0005 hydrofoil to model a fish tail. To mimic the two kinds of tail-fin bend during C-start manoeuvre more realistically, the model is designed to deform with time at the rear half. In spite of motion of fish body, the foil that represents fish tail-fin swings in water about the fixed leading-edge of tail peduncle, so as to produce forces and moment from fluid momentum changes. As shown in Fig.2(a) illustrates the sketch of the fish tail-fin model, and Fig.2 (b) the kinematic process of it. The foil depicted by thinner lines in Fig.2 (a) shows the locomotion of fish tail without deformation. The thicker lines present the locomotion of fish tail with a positive (single) bend and a negative (double) bend, as described for typicality by

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3.

1 1 A(x, t )= ± A1 (1-x)3 [ (1-cos(π t )] (0.5 ≤ x ≤ 1) 8 2

(1)

A1 is referenced amplitude for adjustment, determined by the actual maximum curvature according to measurements. Here, single bend also means the direction of bending deformation is the same with tail deflection, while double bend is opposite. Thus, the maximum of bend happens at the maximum stroke angle. Compared with Fig.1, in spite of forebody, the two bending forms of tail-fin are similar to that of real fish. Additionally, our model describes a turn to the right, while the experiment made left one, as are alike.

GOVERNING EQUATIONS AND NUMERICAL METHODS A robust CFD solver of two dimensional (2D) incompressible Navier-Stokes (N-S) equations is utilized in this study, which has been employed to many biofluiddynamic problems successfully [7,20,21]. In the frame fixed with the foil motion, employed the pseudo-compressibility method, the nondimensionalized N-S equations in the conservative form of x, y momentum and mass can be written as

∂Q ∂q + )dV + ∂τ

∫ ( ∂t

V (t )

∂F

∂G ∂Fv

∫ ( ∂x + ∂y + ∂x

V (t )

+

∂Gv )dV = 0 ∂y

(2)

where

⎡u2 + p⎤ ⎡u ⎤ ⎡u ⎤ ⎢ ⎥ Q = ⎢⎢v ⎥⎥ , q = ⎢⎢ v ⎥⎥ , F = ⎢ uv ⎥ , ⎢ βu ⎥ ⎢⎣0⎥⎦ ⎢⎣ p⎥⎦ ⎣ ⎦ ⎡ uv ⎤ ⎡ 2u x ⎤ ⎡ vs + u y ⎤ 1 ⎢ 1 ⎢ ⎢ ⎥ ⎥ 2 u y + vx ⎥ , Gv = 2v y ⎥⎥ G = ⎢v + p ⎥ , Fv = Re ⎢ Re ⎢ ⎢⎣ β v ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 0 ⎥⎦

β is the pseudo-compressibility coefficient, p is

Fig.2 Geometry and kinematics of fish tail model in C-start (“o” denotes no bend, “+” positive, “-”negative,sic passim)

Figure 2(b) indicates the kinematic parameters in a process. Referencing the experiment[11], which is similar to biological observation, we divide the process into 3 stages as follow: (1) forward stroke from initial position φi =0 to the maximum angle

φm =40º, through one accelerated and one decelerated phase (T1*=0.16s), (2) return stroke symmetrically with the reversed motion (T2*= T1*), (3) staying at the final position for a while (T3*=6 T1*).

pressure, u and v are velocity components in (x, y) frame, t is the physical time and τ is the pseudo time. Re is the Reynolds number. Physical quantities are nondimensionalized by three characteristic scales: length of foil L*=0.05m, time interval of Stage 1 T1*, and density of water ρ *=103kg/m3. Reynolds number is defined as Re=L*·L*/( ν · T1*)=1.04×104. Inflow velocity is given at inlet boundaries, and continuous outflow at outlet boundaries. On the foil wall, no-slip and no-penetration boundary is enforced, i.e. V=Vw, and for pressure ∂p/∂n=-an, where an is the normal acceleration of the wall. Hydrodynamic forces are computed by integrating the pressure and stresses on the body surface. And they are nondimensionalized as CT = −CD

Fx , 1/ 2 ρ *U *2 L*

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CL =

Fy 1/ 2 ρ *U *2 L*

CM =

,

M 1/ 2 ρ *U *2 L*2

(3)

Since the forces obtained by the tail from fluid act on fish body through the leading edge of peduncle, we calculate the moment about the leading edge of the foil. Above equations in Cartesian coordinate (x, y) are transformed into the generalized curvilinear coordinate ( ξ ,η ). And equations are discretized using the cell-centered Finite Volume Method (FVM)[20]. The Euler implicit scheme is used for discretization of time derivative, a third-order conservative difference scheme of MUSCL for convective term, and second-order central scheme for viscous term. ADI factorization method is implemented in evaluation of the equations. To treat the moving boundary that continuously deforms, we generate new body-fitted grids through a linear interpolation at every time step, while Space Conservation Law (SCL) is satisfied. A variety of validation tests were undertaken[7,20,21] for this CFD method, that prove out its applicability for unsteady and moving grid problems. And it was verified to reliable with theoretical modeling method on a problem analogical to this study[12]. After tests, we choose a physical time increment dt=0.001 for the computational stability condition, and a C-grid of 124 by 79. Figure 3 shows grids nearby the surface of a single-bending foil.

Fig.3 Grid near the surface of the foil (the case of single bend)

4. RESULTS AND DISCUSSION 4.1 Hydrodynamic analysis of common C-start We choose three typical cases to simulate, as a swinging tail without bend, with single bend and with

double bend. Fig.4 shows the hydrodynamic performance of the swinging foil in this study, i.e. the variation of thrust, side force and the moment coefficients. And we also examined the impulse of force/moment, which represents the increment of momentum/angular momentum, in a time interval, i.e., J i (t ) = ∫ Fi (t )dt = ∆ (mVi ), J m (t ) = ∫ M (t )dt = ∆ ( I ω )

(4)

Such impulses may lead to give time-averaged values of force/moment in this time interval, as shown in Table 1.

Fig.4

Forces and moment and relevant impulses with time

It is noted that the three cases are closer with each other and the case of rigid foil is the rough mean condition of the two bending forms. The result shows that they have common points. That is one reason why

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Table 1 The average values of forces and moment during every stage T1*(0-1) Accele-

Decele-

ration

ration

-

2.80

0.12

Accele-

Decele-

Ration

ration

1.41

-3.81

0.03

-

2.36

-0.44

0.93

-3.52

-

2.01

-1.04

0.46



-0.92

0.15



-0.51



Total

T3*(2-8)

-1.84

-0.21

-0.07

0.24

-1.60

-0.33

0.04

-2.96

0.52

-1.19

-0.36

0.04

-0.36

4.30

0.41

2.31

0.97

-0.01

0.70

0.11

3.21

0.23

1.69

0.89

0.00

-0.25

0.87

0.32

1.90

0.07

0.97

0.64

0.00



-4.35

0.15

-2.04

5.00

0.23

2.52

0.24

0.08



-3.85

0.67

-1.54

4.98

-0.28

2.28

0.36

-0.06



-3.40

1.53

-0.90

4.51

-0.89

1.76

0.43

-0.06

Time

Moment

Thrust

Side Force

T2*(1-2) Total

the two bending forms both belong to C-start. As described in Section 2, there are three distinctive stages kinematically. Thus, we analyze the common hydrodynamic characteristics of each stage in C-start in this section, using the case of rigid foil. Firstly, let us see the moment, which is the cause of rotation of fish body. According to Fig.4 and Table 1, the forward stroke produces a large negative (clockwise) moment in Stage 1. In the first accelerating phase, the absolute value of moment ascends quickly, as well as impulse of moment does. In the subsequent decelerating phase, the absolute value of moment descends quickly, but the negative value of angular impulse keeps quite large yet. The above dynamic process indicates that fish body will naturally rotate to the right quickly, and the rotation will last all through Stage 1. When it comes into Stage 2, the backward stroke brings a positive (anti-clockwise) moment, whose average value even exceeds that of first stroke. The impulse of moment does not change to positive immediately, but decrease to zero for a while. That determines fish body will continue rotating rightward with angular velocity reduced. Finally in the Stage 3, when the tail stops its motion, the moment and all forces almost disappear. And the impulse of moment stays on a value near to zero. Secondly, we investigate the thrust, which is related directly to the forward propulsion of fish body. The main trend and phases of thrust are similar to that of moment. Besides, it is noted that the thrust in Stage 2 is far larger than that in Stage 1. And at last, the impulse of thrust arrives to a high plateau without decline. It follows that fish gets propulsion chiefly by the return stroke, and in the end, fish body gets a large

Total

forward momentum. As a result, fish carries out a turn of C-start by swinging its tail towards the direction it wants to turn. The hydrodynamic characteristics of the whole process are described above. According to experiments of freshwater fish, Jing et al.[18,19] generalized the kinematic characteristics of the real motion of fish: (1) in the first stage, fish swings its tail quickly, while its body rotates quickly, and its CM has little displacement, (2) in the second stage, fish swings its tail back, while its body rotates deceleratively, and swims acceleratively ahead in the new direction. Such kinematic features fully match with our deduction. Besides these, we have found that some dynamic requirements implied in the whole process must be satisfied, i.e., (1) the mean moment in the whole process is close to zero, because fish must stop rotation after a certain rotated angle, (2) the side force is averagely little too, for fish do not have remarkable transverse movement in this case. 4.2 Flow mechanism and vortical patterns To reveal the reason why the forces and moment are produced in such a manner, we should explore the flow. As shown in Fig.5 and Fig.6, the time-changing flow fields and vortical patterns are illustrated. The numerically simulated flow visualizations coincide with those in the experimental results of rigid flat plate[11]. Moreover, a goldfish usually performs a C-start turn with single bend. Sakakibara et al.[17] measured the stereo flow field of a 45o turn of a goldfish, and sketched a diagram to explain its mechanism (Fig.7). The present numerical study confirms their schematic illustration. We found that: (1) the forward sideways stroke of fish tail produces a “side jet” shooting off along the tail tip, so the fish

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body is forced to turn, (2) in the succedent backward stroke, fish tail generates a “thrust jet” ejecting rearward, which pushes fish body ahead. So, fish can swing its tail to produce either “side jet” or “thrust jet”, depending on its need to rotate or swim ahead. For example, a fish can get a double turning, i.e., two continuous turnings through giving birth to another concurrent “side jet” after one, as Ref.[17] observed. A “jet” exhibit as a vortex pair two-dimensionally, and in three-dimensional space, it presents as two vortex rings.

Fig.6 Velocity vector and contours of vorticity in double-bend C-start

Fig.5 Velocity vector and contours of vorticity in single-bend C-start (solid lines present positive vorticity, dotted negative, and arrows velocity)

Fig.7

4.3 Special discussion on two bending tail-tip forms The two distinctive bending forms are discovered by Jing et al.[18, 19] (Fig. 1), just as Crucian Carp (Carassius auratus) performs a single-bend and yellow catfish (Pelteobagrus fulvidraco) a double-bend, while they both belong to C-start. After Fig.4 and Table 1, for all hydrodynamic quantities, the line of “o” is mostly between the lines of “+” and “-”, and all peak values and mean values of “+” are larger than that of “-”, that is to say, single-bend generates stronger fluid forces for its C-start motion than double-bend. The flow visualizations shown in Fig.5 and Fig.6 also exhibit some difference between these two situations, where the “side jet” in single-bend is stronger than that in double-bend. Moreover, we give the profile of the time-changing resultant force in single-bend and double-bend respectively in Fig.8, where the contrast is evident.

Fish tail comprises skeleton, muscles and tissues, which have certain strength and rigidity. The stronger the fluid forces acting on it, the harder fish tail becomes in taking the load. Therefore, with the same swing amplitude, the fish that performs single bend supports larger fluid loads than the fish performing a double bend. That is to say, double bend can be thought as a strategy to alleviate pressure from fluid, when its tail has not enough strength. Thus, we can speculate that this is the result of natural selection and adaptation. For instance, crucian have a strong physical condition, so they can accomplish a large turn and start, i.e., it has good manoeuvrability and agility. On the contrary, catfish are more infirm than crucian, and they cannot accomplish escaping as swift as crucian. Therefore, catfish need to dodge their predator more relying on shield around themselves, like aquatic weed and rugged ground. According to biological reports, yellow catfish are lentic (a

Schematic drawing of vortex ring while the fish makes a rightward turn (after Fig.8 of Ref. [17])

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biological term means living in still water) and get along almost near the bottom of water, while Crucian Carp live more widely, in not only deep but also shallow water, and not only still but also moving water. The different habitats reflect the difference in capability of subsistence and adaptation to circumstance.

gets fluid forces. However, the increase of input power is even larger than that of forces. It can be presumed there is a certain limit of φm for each fish. Such control may be named active control. Since we choose the duration of Stage 1 as the reference time, the duration of Stage 2, T2* is an obvious controlling variant. Changing of T2* means changing of the angular acceleration of the return stroke, so consequently the forces should change. As illustrated by Fig.10, the final impulses of thrust and moment have some change while T2* is altered. Sometimes, fish need to undertake different motion. Thereby, T2* is one convenient parameter to control for different hydrodynamic and further, kinematic effects. Considering an ideal condition of the C-start turning motion, a final zero impulse of moment is required, as well as a positive thrust impulse. Thus, according to Fig.10, we choose T2*≈1.5 as the optimal value for a normal C-start. Experimental data indicate the ratio T2*/ T1* is about 1-2[19]. This control may also conceptually named active control here. Ideally, in the end, CJM=0 and CJT>0 are required. Here, the larger T2* is, the less final values of all impulses become. So the best optimized situation happens when T2*=1.5 around.

Fig.8 The resultant force R in C-start with time

4.4 Discussion on several controlling factors Fish itself can control its turn actively and passively in some ways. The key controlling factors include swing amplitude and time interval of Stage 2. Besides, it may insert a resting stage just after Stage 1. We investigate the three situations to discuss the optimization of fish turning.

Fig.10

Fig.9

The coefficients of average thrust, moment and input power in Stage 1 versus swing amplitude φm

Fig.9 shows the effect of swing amplitude. Note that with the larger the swing angle φm , the larger fish

The effect of time interval of Stage 2 T2* (in the case of single bend)

Inserting a resting stage after Stage 1 is just like the situation T3* after T2*, the effect is clearly to retain the early impulse of thrust and moment, therefore naturally fish can hold the momentum and angular momentum of its body without expending any power. Such phenomenon was reported in some early work[22]. We called such control as passive control, for

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its ease.

5. CONCLUDING REMARKS This study aims at one typical turn manoeuvre as C-start. The hydrodynamic forces and moment acting on a to-and-fro stroking fish tail fin model is analyzed by numerically solving the two-dimensional incompressible Navier-Stokes equations. Visualizations of flow fields and vortex structures are presented, the flow physics is interpreted. And discussions on some controlling methods are explained. Through comparison with related experiments, the results are verified. To sum up, the following conclusions are drawn: Fish performs a C-start turning manoeuvre by a to-and-fro stroke of tail to the direction it wants to turn. (1) The forward stroke chiefly contributes to the body rotation, while propulsion minimal, (2) the return stroke mainly contributes to the body propulsion, and also decelerates the rotation. Finally fish gets a velocity towards a new direction from rest at the beginning. Fish tails operate C-start with two different ways, as single bend and double bend. Predicted results indicate that the single-bend fish bear the stronger fluid load, so it has better capability in manoeuvrability and agility than double-bend fish, as is adaptive to its relevant environment. Fish can control its turning motion with active or passive ways, through adjusting the amplitude and rhythm of swing. The study is helpful to bionics, especially for a new generation of AUV, which aims at improving its manoeuvrability and agility. Ideally, the numerical simulation for a turning whole fish in C-start is desirable and is the chief objective of our next work.

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