Numerical study on propulsive performance of fish-like swimming foils1

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Ser.B, 2006,18(6): 681-687

NUMERICAL STUDY ON PROPULSIVE PERFORMANCE OF FISH-LIKE SWIMMING FOILS* DENG Jian, SHAO Xue-ming, REN An-lu Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China, E-mail: [email protected] (Received October 19, 2005) ABSTRACT: Two-dimensional numerical simulations are performed to study the propulsive performance of fish-like swimming foils using the immersed-boundary method. A single fish as well as two fishes in tandem arrangement are studied. First, the effect of the phase speed on the propulsive performance of a single fish is analyzed. The wake structures and pressure distribution near the wavy fish are also examined. The results show good correlation with those by previous researchers. Second, two tandem fishes with the same phase speed and amplitude are studied. The results show that the fish situated directly behind another one endure a higher thrust than that of a single one. KEY WORDS: biomechanics, fish-like swimming, traveling wavy foil, immersed-boundary method

1. INTRODUCTION The study of the movement of fishes can be very informative in exploring mechanisms of unsteady flow control because movements in fish is a result of many millions of years of evolutionary optimization. Previous studies have shed light on the inviscid hydromechanics of fish-like propulsion, while predicting high propulsive efficiency. With the development of experimental equipments and novel numerical methods, there has been a better understanding of the principles of fish-like swimming. Some recent works have been reviewed in Refs. [1-3]. To better investigate the swimming ability of fishes, the wave-like swimming and flapping motions of the body are used as essential models to deal with the movement of fish. Previous researchers have shown the ability of caudal fin of a fish to produce a jet-like

wake similar to that of a flapping foil [4-7]. The numerical investigation on a fish-like traveling wavy plate and a smooth wavy wall undergoing transverse motion has also been performed [8,9]. Recently, a NACA0012 foil has also been used as the profile of the body at an equilibrium position of undulating motion [10]. In this study, the immersed-boundary method is used to investigate the viscous flow over the body of fishes. The NACA0012 foil, which undergoes a traveling wave motion, is employed to represent the profile of the body of the fishes. The flow structures and propulsive performance of the fish for different swimming parameters are discussed. Two fishes in tandem arrangement with the same parameters are also examined. 2.

PHYSICAL MODEL AND NUMERICAL METHOD As shown in Fig. 1, the NACA0012 foil with a traveling wavy motion is considered in this article. The length of the foil is L , and the free-stream velocity is U . The motion of the foil is described as

ys = Am ( x) cos[2πα ( x − ct )], 0 ≤ x ≤ 1

(1)

where α = L / λ , with λ being the wavelength of the traveling wave, Am and c are the amplitude and the phase speed of the traveling wave, respectively.

* Project supported by the National Natural Science Foundation of China (Grant No: 10472104) and National Laboratory of Hydrodynamics of China. Biography: DENG Jian (1981-), Male, Ph. D. Student

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and point s. It should be noted that the body-force is equal to zero for the grid points where the condition ds > dxy is satisfied. The nondimensional Navier–Stokes equations for incompressible viscous flow are written as follows,

y

x U

∇ ⋅V = 0,

L

Fig.1 Physical model and definition of the coordinates system

As suggested by previous researchers [8,9], to model the backbone undulation of fish swimming, the amplitude is approximated to be in the form of a quadratic polynomial as

Am ( x ) = C0 + C1 x + C2 x 2

(2)

In the simulations carried out in this study, the coefficients C2 and C0 are set equal to zero for simplification. An immersed-boundary method-based solver proposed by Zou et al.[11] and Deng et al.[12] is employed to simulate the fluid flow over the foil. The advantage of this method is that the complexity and cost of generating a body-conformal mesh at each time-step is eliminated, thereby reducing amount of the resources required to perform such simulations. Inside the obstacle body, a body forces added at the grid point, which is directly calculated using the expression given by

F n +1 = − RHS n +

U n +1 − u n Δt

(3)

where U n+1 is the velocity of solid boundary point at current time level t + Δt and u n is the corresponding fluid velocity at t time level. The term RHS n contains the convective, viscous, and pressure gradient terms in momentum equation at t time level. For the grid points outside the solid body, a certain interpolation procedure,

DV 1 2 = −∇p + ∇ V + Fadd Dt Re

(4)

where Fadd is the added force. The Euler-explicit time discretization scheme is applied for convective terms and the second-order-implicit Crank–Nicholson scheme is used for viscous terms. Spatial derivatives are discretized by second-order central finite difference. The pressure variable is solved using the pressure Poisson equation derived by applying the divergence operator to the momentum equations. A constant streamwise velocity has been used at the inlet and the lateral boundaries. The boundary condition at the outlet is set as ∂u ∂t + ua ∂u ∂n = 0 , where ua is the averaged streamwise velocity at the outlet. A pressure Neumann condition is applied to inflow, far field, and outflow boundaries. 3. METHOD VALIDATION To validate the numerical method, the flow around a circular cylinder is simulated. A rectangular domain is employed. The boundary conditions are imposed in such a way that the flow is from the left toward the right of the domain. A circular cylinder is placed inside the domain so that its center is 8D away from the inlet and 25D away from the outlet, where D denotes the cylinder diameter. The domain has a transverse dimension of 16D. These dimensions have been chosen to minimize the boundary effects on the flow development.

F ( x f , y f ) = (1- ds / dxy ) F ( xs , ys ) Fig.2

is needed to weaken the added body-force. Here dxy is the diagonal length of a grid element. The subscript f denotes the grid point in the fluid, and the subscript s denotes a corresponding virtual point on the boundary, and ds is the distance between point f

Nonuniform mesh used for the flow past a twodimensional circular cylinder. Only alternate grid lines are shown in both the directions

Figure 2 shows the 275 × 156 nonuniform mesh used in the present studies, and the grid near the Cylinder is uniform and has a constant spacing. A

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nonreflect boundary condition is used on the outlet boundary. A uniform constant velocity is specified at the domain entrance, as well as at the top boundary and bottom boundary. The simulations at are being performed at several Reynolds numbers in this study. The drag coefficients, which are obtained in studies by the authors, as well as the results from previous researches, are shown in Table 1. The comparison indicates a very good agreement. More detailed validation of this numerical method is given in Refs. [11-13]. Table 1 Comparison of mean drag coefficient (Cd) with those of other authors Lima Present Dennis et Park Re al. [14] et al. [15] et al. [16] work 10

2.78

2.81

2.98

20

2.05

2.01

2.04

2.06

40

1.52

1.51

1.54

1.52

1.46

1.42

1.35

1.40

1.32

1.33

1.39

1.30

50 80 100

1.06

Fd 1/ 2 ρU 2 L where Ff is the friction force, CD =

(5)

Fp the pressure

force, and the total drag force Fd = F f + Fp . Based on the definition [8,9], the total power (PT) required for the propulsive motion of the fish body consists of two components. One is the swimming power, required to produce the vertical oscillation of traveling wave motion and is defined as

Ps =

∫ Γ pividτ

(6)

where p is the pressure on the surface of the plate, and v denotes the y-component of velocity at the surface of the fish. The other power is needed to overcome the drag force and is represented as PD = −UFd . Here, the minus sign indicates that the velocity of swimming in fishes is reverse to that of the stream. Thus, the total power PT = PS + PD is obtained. 2

CL

CD

150

1.37

1.37

CL , CD

1 0 -1

4. A SINGLE FISH When considering a single fish, C1 is set to 0.2,

-2

and the phase speed c ranges from 0.5 to 2.5. The traveling wavelength λ is set equal to the length of the fish body. The Reynolds number is chosen as 500, which is according to a relevant study on an aquatic animal at intermediate Reynolds numbers (i.e. Re~ O (102)). In most previous studies, the Strouhal number is introduced to characterize the dynamics of the wake flow after a fish. The Strouhal number is defined by St = fA/U. In the current studies, it ranges from 0.2 to 1. 4.1 Forces and power consumption The drag force acting on the fish body and the power required for propelling are directly relevant to the study of locomotion in fishes. The total drag force on the wavy fish body consists of a friction drag and a pressure difference drag. The corresponding drag coefficients are defined as

2

CDF =

Ff 1/ 2 ρU L 2

,

CDP =

Fp 1/ 2 ρU 2 L

,

2

4

tU/D (a) c = 1.1

CL

6

8

6

8

CD

CL , CD

1 0 -1 -2

2

4

tU/D

(b) c = 1.8

Fig.3 Time history of lift and drag coefficients

The time history of drag and lift coefficients are shown in Fig.3, where Fig.3(a) and Fig.3(b) correspond to two different phase speeds, c = 1.1 and c = 1.8, respectively. The period of both the drag and lift coefficients accords well with the wavy motion, so more cycles can be observed for c = 1.8 in the same interval. The amplitude of the lift coefficient for c = 1.8 is greater than that for c = 1.1. When c = 1.1,

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the drag coefficient for most time is greater than zero, whereas when c = 1.8, the drag coefficient is mainly less than zero. This phenomenon is especially obvious if a time average of the drag coefficients or the powers is obtained, as described in the subsequent paragraphs.

, ,

1.0

the variation of

CD . When PD is negative, it

means that the fish body is propelled by the thrust, and PS increases for providing such thrust. The ratio of PD to PS can denote the propelling efficiency. In the relationship between the drag coefficient and the phase speed, the curve is always monotonic, as shown in Fig. 4. But when referring to the relationship between the efficiency and the phase speed, a peak value is determined at c = 2.0, as shown in Fig. 5, and the corresponding value of efficiency is η = 0.493.



0.0

-1.0 0.5

PD , monotonically decreases with c. It corresponds to

1.5 c (a) Drag force

2.5 0.50 0.46

PS , P D , P T

1.0 0.5

η

PS PD PT

0.38

1.5

2.0 c

2.5

0.0

Fig.5 Variation of Propulsive efficiency phase speeds

-0.5 1.5 c

0.5

2.5

(b) Power

Fig.4

0.42

Time-averaged drag force and power consumption versus c

The time-averaged drag force coefficients versus

c are shown in Fig. 4(a). With the increase of c , the time-averaged pressure difference drag coefficient CDP and total drag coefficient CD decrease, and the friction drag coefficient

CDF

a certain extent. It can be found that

increases to

CDP

becomes

negative and acts as thrust force when c > 1.1 approximately, and

CD

becomes negative when

c > 1.3 . This behavior is consistent with previous findings for viscous flow over a streamwise traveling wavy plate [8]. The variation of PS , PD , and PT with c are shown in Fig.4(b). With the increase of c, PS increases and becomes positive for c > 1 . When PS becomes negative, we say that the fish exhibits self-fluctuation, and no power input is required to fluctuate it. The power to overcome the drag force,

η

with various

4.2 Flow structures for different phase speeds To better understand the effect of c on the pressure difference drag force, Fig. 6 shows the pressure contours for different phase speeds. At c = 0.5 (in Fig. 6(a)), higher pressure distribution can be observed around the nose of the fish, and at the right side of the trough on the upper surface of the body of the fish as well. When c ≥ 1.1 , the pressure contours exhibit an obvious change, the higher pressure distribution can also be observed at the right side of the trough on the lower surface of the fish body, where only lower pressure distribution exists for the case of c < 1.1 . This phenomenon can also be observed when the pressure-difference drag time histories are examined, where the drag fluctuates around zero when c ≥ 1.1 , whereas remains positive when c < 1.1 . When c ≥ 1.1 , the averaged pressure-difference drag become negative, as shown in Fig. 4(a). Thus, it is reasonable to predict that the CDP time-average pressure difference drag decreases with the increase of c and becomes negative for c ≥ 1.1 . To understand the mechanism of the propulsion of the traveling wavy fish body, vortex structures in

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0.4

y

0.0

-0.4 -0.2

0.2

x

0.6

1.0

vorticity contours for c = 1.8. It is noted that the scale of shedding vortex and the lateral width of the vortices array are smaller than that in case c = 0.5. Furthermore, the structure of the wake has an essential change. It is a reversed Karman vortex street which has a reversed rotational direction when compared with classical Karman vortex street. The reversed Karman vortex street is of a thrust –type, therefore, it can be responsible for the production of thrust force shown in Fig. 4(a).

(a) c = 0.5

y

0.4

0.0

-0.4 -0.2

0.2

0.6

1.0

x (b) c = 0.8

y

0.4

5. TWO FISHES IN TANDEM ARRANGEMENT In this section, two foils being subjected to the traveling wave motion are employed to represent the swimming of two fishes. An attempt is made to investigate the difference between the flow by a single fish and that by two tandem fishes. In current studies, the downstream fish keeps pace with the upstream one, with c = 2.0 and C1 = 0.1. Several spacings between these two fishes are examined, ranging from S/L = 1.1 to 4.0.

0.0

-0.4 -0.2

0.2

0.6 x

1.0

(c) c = 1.1

(a) c = 0.5

y

0.4

0.0

-0.4 -0.2

0.2

0.6 x

(b) c = 1.8

1.0

(d) c = 1.5

Fig.6 Instantaneous pressure contours

the near wake of the fish body are investigated. Figure 7(a) shows the instantaneous vorticity contours for c = 0.5. The shear layer is generated along the surface of the fish body and rolls up to the downstream to form concentrated vortices behind the tail tip. It is noted that two vortices with opposite sign shed downstream with a staggered array during one cycle. Its structure is the well-documented drag-producing Karman vortex street [1], which is typically observed in the wake of bluff (nonstreamlined) objects for a specific range of Reynolds numbers (roughly 40 < Re < 2 ⋅105 ). Fig. 7(b) shows the instantaneous

Fig.7

Vorticity contours (the black denotes negative value and the gray denotes the positive value)

Figure 8 shows that the drag coefficients vary with the spacing. It can be seen that the mean drag coefficient on the downstream fish is reduced for all spacings. When the two fishes are sufficiently close together, for the spacing from S/L = 1.1 to 1.7, the flow in the gap region is considerably disturbed by both fishes, and even the drag on the upstream fish in reduced to a certain extent, especially for the simulation of S/L = 1.1. Till the spacing of S/L = 1.8, the value of the drag returns to that of a single fish. The variation of efficiency with the member spacing is shown in Fig. 9. The efficiency curves of both the fishes and also the total efficiency of the two-fish

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system are shown separately. The dashed line denotes the efficiency of a single fish, and this can act as the data comparable to the case of two tandem fishes, because both fishes in the tandem-fishes system are configured with the same parameters. It can be observed that the efficiency of the downstream fish is higher that that of the upstream fish, and also higher than the total efficiency of the two-fish system. As a common rule, the total efficiency is intermediate of the efficiency for both fishes. It can be concluded that the fish that was directly behind another fish endure a higher thrust and also a higher efficiency under a certain condition.

-▲- Efficiency of downstream fish 0.48

η

-○- Overall efficiency

0.36

-●- Efficiency of upstream fish

0.28 1.0

2.0

3.0

4.0

S/L

0.10 0.08 , ,

-■- Upstream -□- Downtream -0.06 -0.12

-▲- Upstream -△- Downtream

-0.18 -0.24

-●- Upstream -○- Downtream 1

3

2

4

S/L

Fig.8 Time-averaged drag coefficient versus member spacing

6. CONCLUSIONS The propulsive performance and vortex shedding structures of a single swimming fish and two fishes in tandem arrangement are numerically investigated by solving the two-dimensional incompressible Navier–Stokes equations using the immersed-boundary method. The NACA0012 foil with traveling wave motion is employed to represent the swimming fish. The effects of the phase speed c at a constant amplitude coefficient C1 = 0.2 on the propulsive performance are discussed. According to the variation of drag force and power consumption with c , a critical value of c =1.1 can be found. When c is larger than this critical value, the pressure difference drag force becomes negative and acts as thrust. The propulsive efficiency of the wavy fish also depends on the phase speed. The change of propulsive efficiency with phase speed is not monotonous. A peak value of the efficiency in the ranges between c = 0.5 and 2.5 can be found. The wake structures and pressure distribution near a single fish for different phase speeds are also examined, which is helpful to understand the physical mechanism of fish swimming.

Fig.9

Variation of efficiency with member spacing, dashed line denotes the efficiency of a single fish

Two fishes in tandem arrangement are also examined. It is found that under certain conditions, the efficiency of the downstream fish is higher than that of the upstream fish and also higher than the total efficiency of the two-fish system. This study is helpful, because it can be a first step toward the investigation of fish schooling behavior. REFERENCES [1] [2]

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