CHIN.PHYS.LETT.

Vol. 25, No. 2 (2008) 597

Two-Dimensional Self-Propelled Fish Motion in Medium: An Integrated Method for Deforming Body Dynamics and Unsteady Fluid Dynamics ∗ YANG Yan( ö)1,2 , WU Guan-Hao(Ç)Í)3 , YU Yong-Liang({[ )1 , TONG Bing-Gang(Ö[j)1∗∗ 1

The Laboratory for Biomechanics of Animal Locomotion, Graduate University of Chinese Academy of Sciences, Beijing 100049 2 Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026 3 State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084

(Received 31 October 2007) We present (1) the dynamical equations of deforming body and (2) an integrated method for deforming body dynamics and unsteady fluid dynamics, to investigate a modelled freely self-propelled fish. The theoretical model and practical method is applicable for studies on the general mechanics of animal locomotion such as flying in air and swimming in water, particularly of free self-propulsion. The present results behave more credibly than the previous numerical studies and are close to the experimental results, and the aligned vortices pattern is discovered in cruising swimming.

PACS: 45. 50. −j, 47. 11. Df, 47. 63. Mc, 45. 20. D−, 02. 60. Jh As we know, fishes swim in water by propelling themselves, with their body deforming actively and fins moving.[1,10] Two basic questions arise: (1) How can we describe the kinematics of a deforming body, and furthermore write the governing equations of its dynamics? (2) The mechanism of self-propelled freely swimming of deforming body in medium is determined by the interaction of body dynamics and unsteady fluid dynamics, so how can we obtain the solution of the coupled processes simultaneously? Firstly, for a deforming body, or named deformable medium, whose time-dependent shape is known beforehand, the more concerned issue is its global kinematics under the action of external fluid forces and torque, rather than the kinematics of each mass point or segment. The theory of classical mechanics gives the dynamical equations of mass point, particle system, and rigid body. However, little further has been described aimed at the deforming body yet,[2] and several previous works have used different approaches. Carling et al.[6] and Kern and Koumoutsakos[7] numerically studied self-propelled anguilliform swimming. The former[6] used a model consisting of a number of discrete segments, each of which was dealt with as a mass point; the latter[7] mentioned that a certain local coordinate system should be set up, but they did not present the theoretical formula of its conditions. In another related work, Eldredge[8] used a three-linkage-rigid-bodies model to substitute for a continuous fish body. Secondly, previous relevant studies implemented and simplified somewhat unreasonable models with some artificial constraints. In the earliest simulat∗ Supported

ing work[3] and the subsequent ones,[4,5] researchers usually introduced a hypothetical free upstream flow condition,[3,4] or gave the prescribed whole trajectory of fish motion.[5] All of the above hypotheses are inconsistent with real conditions and could not reflect the feedback of fluid to fish body, i.e. the fish is not freely or self-propelled at all, so that such works mainly discussed the flow or vorticity characteristics. Recently, Wang[9] assumed that the fish’s centre of mass stays on the fixed point of the fish body, which is inconsistent with the fact of the body deformation, and leads to remarkable unreasonable lateral force and yawing motion in steady cruise swimming. In this Letter, we deduce the dynamical equations of a general continuous deforming body systematically, and give an integrated method for deforming body dynamics and unsteady fluid dynamics. The results are close to experiments and verify that the present method is reasonable. Fish swimming is a coupled interaction process of deforming body dynamics (BD) and fluid dynamics (FD). Firstly, the body performs certain timechanging deformation through its neural control and muscular locomotion. Secondly, the deforming fish body arouses the ambient fluid medium, generating fluid forces (the process of FD). Thirdly, the fluid forces cause the motion of fish body (the process of BD). Again, the motion and deformation of fish body produce fluid forces (the process of FD) simultaneously, as turns to the second point. As an actively deforming continuous body, its whole-body motion[10] attracts much more attention than the complex motions of all particles or parts in a discrete system.

by National Natural Foundation of China under Grant No 10332040. whom correspondence should be addressed. Email: [email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd °

∗∗ To

598

YANG Yan et al.

Considering a two-dimensional problem, it can be proved that the whole-body motion has three degrees of freedom (DOFs) when the deformation information is known beforehand, two of translation and one of rotation, similar with a rigid body. We set up the centre-of-mass system Cx0 y 0 , whose three-DOF motion in the inertial coordinate system, is considered as the ‘whole-body motion’ of the deforming body. The origin of Cx0 y 0 frame is placed on the centre of mass (c.m.) of the deforming body, and the body has no rotational impulse in this frame, as there exists no external forces. Thus, the ‘whole-body motion’, i.e. the translation and the rotation, is only determined by the external actions, while the deforming motion is only determined by the internal actions; both motions are independent of each other, given by the mechanics of particle system.[2] Write the theorem of linear and angular momentum of the particle system, the deforming body here, in the inertial coordinate system, ∫ d u(s, t)ρ(s)w(s)ds = F (e) (t), (1a) dt ∫ d [r(s, t) − r C (t)] dt (e)

×u(s, t)ρ(s)w(s)ds = M C (t),

(1b)

where the body model is one-dimensional, and s is the curve coordinate along the centreline, as shown in Fig. 1. Along s, w(s) is the section width, ρ(s) the fish body’s density, which is equal ∫ to water’s, so that the mass of the body is m = ρ(s)w(s)ds. In addition, r(s, t) is the position on the centreline, in the inertial coordinate system Oxy, and u(s, t) is its velocity. The position of c.m., as well as the origin of 1 ∫ r(s, t)ρ(s)w(s)ds, so the Cx0 y 0 frame, is r C = m 0 r (s, t) = r(s, t) − r C (t) is the position vector from (e) c.m. F (e) (t) is the total external force and M C (t) the total external torque with respect to c.m.

Fig. 1. Inertial coordinate system Oxy, and the c.m. coordinate system Cx0 y 0 , of the sketched fish model.

Decompose the motion u(s, t) as u(s, t) = uC (t) + ω(t) × r 0 (s, t) + u0 (s, t).

(2)

Vol. 25

1 ∫ u(s, t)ρ(s)w(s)ds is the velocity m of c.m., as well as time derivative of r C (t), and ω C (t) is the angular velocity of the Cx0 y 0 frame; is the velocity of deforming motion in the Cx0 y 0 frame. The translation and rotation, first two terms on the righthand side of Eq. (2), compose the whole-body motion, which is determined by the external actions. Consider the whole-body motion only, i.e. take the last term of Eq. (2) off, and substitute u(s, t) into Eqs. (1a) and (1b). We obtain the equations of the whole-body motion, ∫ d uC (t)ρ(s)w(s)ds = F (e) (t), (3a) dt ∫ d r 0 (s, t) × [ω C (t) dt where uC (t) =

×r 0 (s, t)]ρ(s)w(s)ds = M C (t), (e)

(3b)

Define the instantaneous moment of ∫inertia about c.m. ∫ 0 0 as IC (t) = (r · r )ρ(s)w(s)ds = |r 0 |2 ρ(s)w(s)ds, which changes with time. Therefore we have the simple form of Eqs. (3a) and (3b) as d [muC (t)] = F (e) (t), dt d (e) [IC (t)ω C (t)] = M C (t). dt

(4a) (4b)

Eliminating Eqs. (4a) and (4b) respectively from Eqs. (1a) and (1b), we find the governing equations of the deforming motion in the Cx0 y 0 frame, equivalent to the conservation laws of linear and angular momentum: ∫ d u0 (s, t)ρ(s)w(s)ds = 0, (5a) dt ∫ d r 0 (s, t) × u0 (s, t)ρ(s)w(s)ds = 0. dt (5b) Practically, the time-changing deformation of the deforming body may be given in a convenient body coordinate like oξη, so u0 (s, t) in Eqs. (5a) and (5b) is unknown. The transformation between oξη and Cx0 y 0 at every instant is given by solving Eqs. (5a) and (5b), therefore the deforming motion u0 (s, t) is attained. The whole-body motion, u0 C (t) and ω c (t), are solved by Eqs. (4a) and (4b). So far, the complete dynamical equations of the deforming body are set up by Eqs. (4a), (4b), (5a), and (5b). For example of fish swimming in water, the external action is the unsteady fluid force (including torque) through the interaction of body and water. The governing equations of the fluid dynamics (FD) here is the well-known Navier–Stokes equations for incompressible viscous flow, which is solved with finite volume method (FVM) in this work.[11] The numerical

No. 2

YANG Yan et al.

method for FD in this work has been well validated and verified.[11] In order to solve dynamical equations (4a) and (4b), Euler’s numerical integrating method is implemented. Equations (5a) and (5b) are a set of one-variable root finding problems at each time step, which can be easily solved with basic numerical methods. For the coupled process of FD and BD, computation is advanced step by step with a loose coupling, where the time step of FD and of BD are determined by tests to ensure the numerical stability and precision. In this case, computational grid number is 206×81, and nondimensional time step of FD and BD both are 2.5 × 10−5 , when the residual value do not decrease visibly anymore.

599

body from rest (t = 0) to constant periodic undulation (t > t0 = 1.0T ). Therefore, the Reynolds number Re in the nondimensionalized Navier–Stokes equations is defined as ReL = L2 ρ/µT = 5 × 105 , where ρ = 0.998 × 103 kg·m−3 and µ = 1.002 × 10−3 N·s·m−2 are the density and the dynamic viscosity of water at 20◦ C.

Fig. 3. Lateral force CL vs time, compared with the result calculated by the previous method.[13]

Fig. 2. Net thrust and lateral force (nondimensionalized) vs time in saithe’s swimming from initial rest to cruising stably.

Here we present one typical case of fish’s selfpropelled swimming simulations, where the integrated method is applied, that is, a saithe (pollachius virens) swimming from rest to cruise, which is a typical type of subcarangiform body and/or caudal fin propulsion in terms of biomechanics of fish swimming.[12] An NACA0012 foil is used to simulate the twodimensional fish body. The deforming of a saithe is described as the undulation of its centreline, as approximatively fitted from experiments[12] in the natural curve coordinate s along the centreline, h(s, t) = hmax a1 (s)a2 (t) sin[2π(s/λ − t/T )],

Fig. 4. Resultant kinematics, translational and angular velocities (vx , vy and ωC ) of the cruising saithe versus time.

(6)

where

 s ≤ 0.2L  0, a1 (s) = ( s/L − 0.2 )2  , 0.2L < s ≤ L 1 − 0.2 ( ) { t 1 − sin 2πt t0 , 0 ≤ t ≤ t0 , 2π a2 (t) = t0 1, t > t0 ,

Fig. 5. Vorticity cloud, velocity profiles and the jets sketch, in the wake of cruising saithe (all the quantities are nondimensionalized).

(7)

and L(= 0.37 m) is the body length, T (= 0.278 s) and λ(= 1.04L) are the cycle and the wavelength of undulation respectively, hmax = 0.083L the maximum half-amplitude of the deflection at tail tip. As Eq. (7) presents, the saithe undulates its posterior part of the

We find that our results are more credible than previous numerical studies,[9,13] and close to the experimental data.[12] Firstly, Fig. 2 gives the net thrust history and the lateral force history, the two components tend to a stable periodic equilibrium in cruising, and their magnitudes are within the same scale. In several

600

YANG Yan et al.

previous studies, which did not completely calculate the self-propelled whole-body motion, the lateral force is used to display unreasonable large value, one scale larger than its thrust.[9,13] In fact, as shown in Fig. 3, with the artificial constraints released, the lateral force is in present work is much smaller and more reasonable. Secondly, the resultant whole-body kinematics of saithe is shown in Fig. 4, including the stable and periodic translation and rotation, which are consistent with common sense and simulating results,[6,7] though in experiments it is difficult to measure such little oscillations. Furthermore, for quantitative comparing, the final mean cruising velocity v¯x , 0.65 body length per cycle, as shown in Fig. 4, is close to the experimental value 0.85,[12] considering difference with model and experiments, such as the two-dimensional model, undulation fitting function and experimental errors. Meanwhile, the reverse von K´ arm´ an vortex street is found in the wake, similar to that observed in many kinds of fish swimming.[1] Moreover we discover the vortices with alternating sense approximately aligns all exactly on a straight line uniformly, which confirms Wu’s theoretical prediction,[14] different from common two row zigzag patterns.[1] In addition, the jets between every two adjacent vortices have no backward components but primarily lateral ones, and vortices shedding from the tail just stay where they are and diminish gradually, instead of moving away, because the saithe cruises uniformly with ‘momentumless wake’.[14] In summary, we have proposed the dynamical

Vol. 25

equations of a deforming body and set up an integrated method to solve the coupled problem of deforming body dynamics and unsteady fluid dynamics numerically, to overcome the shortcoming of artificial constrains in the previous works. Practice reveals that the present method gives credible results, which is helpful to understand the nature and hydrodynamic characteristics of fish freely and self-propelled swimming and will surely be useful for the design of powerful biology inspired underwater vehicles.

References [1] Fish F E and Lauder G V 2006 Ann. Rev. Fluid Mech. 38 193 [2] Greiner W 2003 Classical Mechanics: Systems of Particles and Hamiltonian Dynamics (New York: Springer) [3] Liu H, Wassersug R and Kawachi K 1996 J. Exp. Biol. 199 1245 [4] Lu X Y and Yin X Z 2005 Acta Mech. 175 197 [5] Wolfgang M J et al 1999 J. Exp. Biol. 202 2303 [6] Carling J, Williams T L and Bowtell G 1998 J. Exp. Biol. 201 3143 [7] Kern S and Koumoutsakos P 2006 J. Exp. Biol. 209 4841 [8] Eldredge J D 2006 Bioinspiration and Biomimetics 1 S19 [9] Wang L 2007 PhD Thesis (Nanjing: Hohai University) (in Chinese) [10] Webb P W 2004 IEEE J. Oceanic Eng. 29 547 [11] Yang Y and Tong B G 2006 J. Hydrodynamics B 18 135 [12] Videler J J 1993 Fish Swimming (London: Chapman and Hall) [13] Hu W R, Tong B G and Liu H 2007 J. Hydroynamics B 19 135 [14] Wu T Y 2001 Adv. Appl. Mech. 38 291