NUMERICAL SIMULATION OF FISH SWIMMING WITH RIGID PECTORAL FINS*

XU Yi-gang, WAN De-cheng

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: xuyigang2007@163.com

NUMERICAL SIMULATION OF FISH SWIMMING WITH RIGID PECTORAL FINS*

XU Yi-gang, WAN De-cheng

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: xuyigang2007@163.com

The numerical simulation of the self-propelled motion of a fish with a pair of rigid pectoral fins is presented. A Navier-Stokes equation solver incorporating with the multi-block and overset grid method is developed to deal with the multi-body and moving body problems. The lift-based swimming mode is selected for the fin motion. In the lift-based swimming mode, the fin can generate great thrust and at the same time have no generation of lift force. It can be found when a pair of rigid pectoral fins generates the hydrodynamic moment, it may also generate a lateral force opposite to the centripetal direction, which has adverse effect on the turn motion of the fish. Furthermore, the periodic vortex structure generation and shedding, and their effects on the generation of hydrodynamic force are also demonstrated in this article.

pectoral fins, lift-based swimming mode, self-propelled motion, overset grid method

Introduction

Presently, there are more pervasive applications of underwater vehicles including seabed oil and gas explorations, scientific deep ocean survey and so on[1]. In the underwater vehicles field for the offshore industry, delicate motion of underwater vehicles is needed to survey the complicated seabed terrain. However, it is difficult for the present underwater vehicles, for which motion is controlled by thrusters, to perform hovering and quick turning both in the vertical plane and in the horizontal plane over seabed terrain.

Pectoral fins are used by fish and aquatic animals to execute a wide variety of swimming manoeuvres. They can be used as rudders to control pitch during high speed swimming and to enhance low speed maneuverability, and they can also be employed as flapping like high degree of freedom propulsors to propel and manoeuvre the animal at low speed[2]. Because of this, the pectoral fin has served as a source of inspiration for a range of bio-robotic devices[3]. These robotic devices have been developed as alternatives for propellers and traditional control surfaces for underwater vehicles and have been used as experimental tools for investigating of the hydrodynamics and control of fish swimming.

Many investigations of the kinematic of fish pectoral fin have been carried out. The kinematic of fish pectoral fins was experimentally measured, and the flow field around the pectoral fin was visible by using Digital Particle Image Velocimetry (DPIV) technology[4]. There also exist many studies of the numerical simulation of fish swimming. Ramamurti et al.[5]developed an incompressible flow solver combined with unstructured mesh generation and adaptive re-meshing to investigate the fluid dynamics of flapping aquatic flight in the bird wrasse. Walker and Westmeat[6]used a quasi-steady blade-element model that accounted for unsteady phenomena to estimate the mean thrust and the mechanical efficiency for the oscillating appendage. Kato and Liu[7,8]developed an underwater vehicles equipped with mechanical pectoral fin to experimentally and numerically study the hydrodynamic performance of the mechanical pectoral fin, as well as its control performance in water, wavesand currents. Suzuki and Kato[9]applied the overset grid system in an unsteady Navier-Stokes equations solver to numerically study the unsteady flow around a mechanical pectoral fin. Wang et al.[10]established a three Degree Of Freedom (DOF) kinematic model of a rigid pectoral fin and numerically studied the hydrodynamic performance of the pectoral fin with variable parameters of kinematic model. Chen et al.[11]investigated the hydrodynamics of batoid swimming motions using the three-dimensional simulation of a self-propelled body in still water and the kinematics of batoid swimming is characterized by large amplitude undulations of the pectoral fins. Xu and Wan[12]investigated the hydrodynamics of tuna swimming through numerically solving three-dimensional unsteady incompressible Navier-Stokes equations. Lu et al.[13]presented some typical works, which include measurement on kinematics of free-swimming fish and prediction of dynamics acting on an arbitrarily deformable body, numerical and experimental simulations of flow over flapping and traveling wavy bodies. Zhang et al.[14]presented a comprehensive study of the effects of the caudal fin shape on the propulsion performance of a candal fin in harmonic heaving and pitching through a series of numerical simulation and hydrodynamic experiments. Zhang et al.[15]discussed the drag reduction mechanism of shark skin based on the direct numerical simulation of the turbulent flow over a real shark skin.

In this article, an unsteady, multi-block, overset grid and Navier-Stokes equation coupled solver is developed to numerically simulate the self-propelled motion of a fish with a pair of rigid pectoral fins. The coefficient of force, velocity and the 3-D vortex structures are presented to analyze the performance of fish swimming with rigid pectoral fins.

Fig.1 Body-fin model

Fig.2 Pectoral fin model

1. Methods

1.1 Body-fin model

Figure 1 shows the body-fin model, which includes a fish body and a pair of pectoral fins. Geometric models of the pectoral fin and the fish body are constructed based on a real fish. Figure 2 shows the outline of the pectoral fin. The span and chord of the pectoral fin are 0.1 m, and 0.097 m, respectively. The cross-sections of the pectoral fin along the span and the cross-sections of the fish body along the body length are approximated to be of elliptical shapes.

1.2 Kinematic model of the pectoral fin

Regardless of the flexible deformation of the pectoral fin, the movement of the pectoral fin consists of three basic motions, i.e., rowing backwards and forwards, flapping up and down, and feathering, as shown in Fig.3.

Fig.3 Illustration of three basic motions of a rigid pectoral fin

Fig.4(a) Definition of body-fixed coordinate system(X,Y,Z) and space-fixed coordinate system

Fig.4(b) Definition of angles of fin motion in body-fixed coordinate system (X0,Y0,Z0)

The body-fixed coordinate system has its origin O at the base of fin, with the X-axis from tail to head,Z-axis from dorsal to venter, and Y-axis normal to X-Z plane, as shown in Fig.4(a). The rowing angle, the feathering angle and flapping angle are all defined in Fig.4(b), which vary sinusoidally using the following definitions

where ωfinand t denote the angular velocity of the pectoral fin and time, respectively,φR0and φFE0denote the average value of the rowing angle and feathering angle, respevctively, φRA, φFEAand φFLAdenote the amplitude of the rowing angle, feathering angle and flapping angle, respevctively, ΔφFEand ΔφFLthe phase of the feathering angle and flapping angle, respevctively.

1.3 Overset grid method

Since it is difficult to generate a unique grid system around a fish body and a pair of moving pectoral fins in the entire solution domain by using a structured grid system, an overset grid method is employed to solve this problem.

By applying the overset grid method[16], the computational domain consists of 3 domain meshes, i.e., the fish body domain and the two subdomains of the two pectoral fins. In each domain, an O-O type grid system surrounding the pectoral fin or the fish body is established independently. Figure 5 shows the overset grid system for body-fin model.

Fig.5 Overset grid system for body-fin model

In the overset grid method, the grid points out of the main domain is got rid of and marked as hole points. In the overlapping region, some grid points of the main domain are marked as the inside boundary points of the main domain, and some grid points of the subdomains are marked as the outside boundary points of the subdomains. In order to express the fin motion in accordance with the progression of time, all the grid points are shifted using the following relations[9]:

here, the superscripts 0 and n denote the 0th and n-th time step, and the Superscripts 1, 2 and 3 represent the steps of feathering, flapping and rowing motions, respectively. The subscripts i,j andk express respectively the grid point, and the subscript“pivot” denotes the pivot point of the origin at the fin. 1.4 Solution of Navier-Stokes equation

The governing equations are the incompressible, unsteady Navier-Stokes equations, written in strongthe governing equations, because the laminar flow isassumed.

In the above equations, β is the pseudo-compressibility coefficient, pis pressure, u, v and w are three components of velocity, V(t) is the volume of the cell, S(t) is the surface of the cell, ugis the velocity of the moving grid, n=(nx,ny,nz) are components of the unit outward normal vector corresponding to all the faces of a polyhedron cell,t denotes physical time and τ is pseudo time, andRe is the Reynolds number. Note that the term q associated with pseudo time is designed for an inner-iteration at each physical time step, and will vanish when the divergence of velocity is driven to zero in order to satisfy the equation of continuity. All the fluid variables are made dimensionless with respect to the chord length of the fin L, the characteristic velocity U and the fluid densityρ. The dimensionless parameter of the Reynolds number is defined as

where ν is the kinetic viscosity.

Time-dependent solutions of the incompressible Navier-Stokes equations are formulated in the Arbitrary Lagrangian-Eulerian (ALE) manner using the finite volume method and are performed in a timemarching manner using the pseudo-compressibility method, with special treatment of conservation of mass and momentum both in time and in space. A third-order upwind differencing scheme, using the flux-splitting method, is implemented for convection terms in the Monotone Upstream-centered Schemes for Conservation Law (MUSCL) fashion. The viscousity terms are evaluated using a second-order central differencing method based on Gaussian integration in a manner of finite volume method. An implicit approximation-factorization method, based on the Euler implicit scheme, is employed for time integration[18]. For each independent domain, the Navier-Stokes equation is numerically solved independently except some special treatment to the hole points.

The outside boundary of the main domain consists of upstream and downstream boundaries. The inflow condition for the upstream velocity components is (u,v,w)=(ubound,vbound,wbound) while pressure is set at zero. At the downstream zero-gradient condition is taken for both velocity and pressure. On the fin surfaces and fish body surface, the no-slip condition is applied for the velocity components. To incorporate the dynamic effect due to the acceleration of the oscillating body, pressure divergence at the surface stencils is derived from the momentum equation, such that (u,v,w)=(ubody,vbody,wbody), ?p/?n=-a0n[18].

The velocity and pressure of the inside boundary of the main domain are interpolated from thesubdomains. And the velocity and pressure of the outside boundary of the subdomains are interpolated from the main domain. The interpolation is implemented using a tri-linear interpolation technique. Because the relative location of the pectoral fins and fish body changes with time, the flow information exchange between the grids is performed at each time step. The numerical method in this article is reliable to deal with the unsteady flow and moving body problems, which has been validated in previous articles[12,19].

1.5 Equation of fish motion

To realize the fish self-propelled at each time step, the motion of the self-propell ed fish is calculated using Newton’s laws of motion.

where (Fx,Fy,Fz)isthe sum of hydrodynamic force acting on the surfaceof the body and on the surface of the pectoral fins in three directions, (Mx, My,Mz)is the sum of hydrodynamic moment acting on the surface of the body and on the surfaceof the pectoral fins, r is the position of the center of the fish, and φ is the azimuth of the fish.

1.6 Solution procedure of Navier-Stokes solver for fish motion

The overset grid method and Newton’s laws of motion are incorporated into the Navier-Stokes equation solver. Thus an unsteady, multi-block, overset grid Navier-Stokes equation solver is developed. Consequently, the overall solution procedure for this Navier-Stokes solver is summarized as follows[9]:

(1) Shift the grid of the subdomains in accordance with the defined rowing, flapping and feathering angles at time t. Then update the grid of the three domains using the location and azimuth of the fish and pectoral fins.

(2) Determine the hole grid points in the subdomains, which are out of the computation domain, such as the grid points located in the fish body, then get rid of these hole grid points.

(3) Determine the grid points within the overlapping region, which need interpolating the flow variables from other grid. Calculate the parameters for the interpolation and store them.

(4) Set the outer boundary condition of the two subdomains by interpolating the flow variables from the main solution domain. Then solve the flow variables in the two subdomains, respectively.

(5) Set the inner boundary condition of the main domain by interpolating flow variables from the subdomains. Then solve the flow variables of the main solution domain.

(6) Proceed to Step 7 if steady solutions are computed in the pseudo time step, if not, return to Step 4.

(7) Calculate the hydrodynamic force of the fish and pectoral fins in three directions and the hydrodynamic moment in X-Y plane. Then the location and azimuth of the fish are obtained using Newton’s Laws of Motion.

(8) Return to Step 1, the time step is increased.

2.Computational results

2.1Free s wimming straightly forward

2.1.1 Computational conditi on

The fish is assumed to swim freely in still water, and the inflow velocity is setas zero.The frequency of the pectoral fin motion ωfinis set as 12 (the period T=π/6s). The characteristic velocity is defined as vL, where v is the mean angular velocity of the fin andis twice the amplitude of flapping motion multiplied by fin-beat frequency, L is the chord length of the fin. This results in reference velocity calculated as U=φFLAωfinL /π. Then the Reynolds number is equal to 1.63×104. Here the laminar flow is assumed and turbulence quantities are not considered. The time step (Δt)is chosen as 0.004 s (about 131 time steps in one period).

In the current calculation, only the motion of the pectoral fin is considered, and the fish body is assumed to be rigid. The lift-based swimming mode, which consists only flapping and feathering motions, is selected for fin motion. The parameters of the fin motion for the lift-based swimming mode is optimized so that the motion of the pectoral fin generates the maximum average thrust and the minimum average lift force in the case of still water[20]. Based on the optimization results, both of the pectoral fins are set with the parameters as follows

2.1.2 Hydrodynamic forces and swimming process

The h ydrodynamic coefficients Cx, Cy, Cz, Cmare defined as follows

where Fx,FyandFzrepresent the hydrodynamic forces inX, YandZ directions, respectively, where Cmdenotes the hydrodynamic moment in the X-Y plane.Here L,U and S are the chord length of the fin, the characteristic velocity and the surfacearea of a pectoral fin, respectively.

Fig.6(a) Coefficient of hydrodynamic force in Xdirection and Z direction

Figure 6(a) shows the coefficient of hydrodynamic force in the X direction andZ direction. It canbe seen that the hydrod ynamic forceCxandCzfluctuate with time, which shows the unsteady flowacco mpanying the fin motion. In one period, the maximum Czand the min imum Czappear once respectively, but the maximum Cxand the minim umCxappear twice respectively.

Fig.6(b) Velocity of the fish in X direction andZ direction

Fig.6(c) Position of the fish in X direction and Z direction

Figure 6(b) shows the velocity of the fish in X direction and Zdirection. The velocity Uxand Uzalso fluctuate with time, but the fluctuation amplitude is smaller compared with that of the hydrodynam ic force as the velocity is calculated by the integration of hydrodynamic force along the time. The velocity of the fish reaches steady 0.35 m/s since 6 s in the X direction and the velocity keeps almost 0 m/s all the time in theZ direction, which implies the mean force generated is mainly in the X direction and the mean force in the Z direction is zero. Thus the fish swims straightly forward along the X direction.

The hydrodynamic force in each direction consists of friction component and pressure component, such as the hydrodynamic force in the X direction

here Fx_frepresents the friction component of the hydrodynamic force in the X direction and Fx_p represents the pressure component of the hydrodynamic force in theX direction. Figure 7 shows the each component of the hydrodynamic force in X direction from10-th period to11-th period. Itcanbe seen the friction component keeps almost unchanged with time and the pressure component fluctuates with time, so it is supposed the fluctuation of the thrust is mainly due to the pressure component.

Fig.7 Component of hydrodynamic force in X direction

The maximum positive force appears twice in oneperiod, respectively at t=10.04T, t=10.54T. And the maximum negative force also appears twice in one period, respectively at t=10.24T,t= 10.74T.

2.1.3 Vortex structures

To visualize the three-dimensional vortex structures in the flow field, Jeong and Hussion proposed that the vorticity-induced pressure is sectionally minimumin a vortex. They showed that this amounts to a “λ2-criterion” and λ2is calculated as follows:

where D andΩare the strain rate tensorand vorticitytensor, respectively,λ2is the second eige-nvalue of the symmetric tensor G andλ2＜0. Figure 8 shows iso-vorticity surface distributions for the vortex structures by λ2=-1 att=10.04T, t= 10.24T,t=10.54T andt=10.74T.

Fig.8 Iso-surface of λ2=-1

At t=10.04T, the fin flaps down and the feathering angle is positive, the fin pushes considerable fluidinto the downstream region, so the maximum positive force appears, as Fig.7 shows. At thesame time, itcan be observed from Fig.8(a) that avortex structure is generated near the fin edge. At t= 10.24T, the fin flaps up and the feathering angle is still positive, the fin pushes considerable fluid into the upstream region, so the maximum negativeforce appears, as Fig.7 shows. At the same time, itcan be observed from Fig.8(b) that a vortex structuresheds fromthe fin edge. Att=10.54T, the fin still flaps up but the feathering angle is negative, so the maximum positive force appears, and a vortex structure generates near the fin edge. At t=10.74T, the fin flaps down and the feathering angle is still negative, so the maximum negative force appears, and a vortex structure sheds from the fin edge. All these demonstrate that if the fin motion result in pushing the fluid into the downstream, the positive force (thrust) appears, if the fin motion result in pushing the fluid into the upstream, the negative force (resistance) appears. Furthermore, there exists a close relationship between the force generation and the vortex structure. The generation of a vortex structure near the fin edge accompanies the generation of maximum thrust, and a vortex structure shedding from the fin edge accompanies the generation of maximum resistance.

2.2 Fish turning performance in X-Y plane

The computation condition is the same as above except the motion parameters of the fin. To realize the turning motion, one way is to set the two pectoral fins with different values of ΔφFL. Thus the motions of the two fins generate inverse hydrodynamic force and a hydrodynamic moment in the X-Y plane. The motion of the right pectoral fin is set with the parameters as follows

The motion of the left pectoral fin is set with the parameters as follows

It can be seen the only difference for the motion parameters with the case of fish swimming straightly forward is that the parameter ΔφFLof the left fin is set to-80o.

Fig.9(a) Coefficient of hydrodynamic moment in X-Y plane

Fig.9(b) Angular velocity of the fish in X-Y plane

Figure 9 shows the coefficient of hydrodynamic moment and the angular velocity of the fishin the X-Yplane. The angular velocity of the fish reaches steady7.5o/s since 9 s. Thus the fish swim per-

fo rms turning motion in the X-Y plane.

Fig.9(c) Azimuth of the fish in X-Y plane

Fig. 10 Course of the parameter ΔφFLof the left pectoral fin varying with time

Fig.11(a) Coefficient of hydrodynamic force in X direction and Y direction

Fig.11(b) Velocity of the fish in X direction and Y direction

Fig.11(c) Position of the fish in X direction and Y direction

2.3 Fish swimming combining straight motion and turning motion

Themotion of the right pectoral fin is set with the parameters as follows

The motion of the left pectoral fin is set with the parameters as the same as the right one except ΔφFL. The parameter ΔφFLof the left pectoral fin changes withtime. Figure 10 shows the course of the parameter ΔφFLof the left pectoral fin varying with time. Figure 11 shows the coefficient of hydrodynamic force, the velocity and the position of the fishin X and Ydirections inthe whole course ofthe fish swimming. It can be seen the course of fish motion can be divided to three stages:

Stage 1: From 0 s to 1.05 s (2T), the fish swims straightly in the X direction.

Stage 2: From 1.05 s (2T) to 3.14 s (6T), the fish perform turn motion in the X-Y plane.

Stage 3: After 3.14 s (6T), the turn motionends and the fish swims straightly in the new direction.

Fig.12 Lateral velocity and the positio n of the fish

In particular, Fig.12 shows the lateral velocity and position of the fish during Stage 2. It is seen that the lateral velocity and position is negative, which implies the lateral force is opposite to the centripetal force needed to perform turn motion. Therefore, when apair of rigid pectoral fin generate the hydrodynamic moment, it may also generate a lateral force opposite tothe centripetal direction, which is supposed tohave adverse effect on the turn motion of the fish.

Figure 13 shows iso-vorticity surfacedistributions of the vortex structures by λ2=-1 during Stage 2. It shows no periodic vortex structure shedding from the left fin. It is supposed the direction of fishmotion affects the vortex shedding. Figure 14 shows iso-vorticity surface distributions of the vortex structures by λ2=-1 during Stage 3. It shows the vortex structure restart periodically shedding from theleft fin, and at the same time, the turn motion of the fish ends.

Fig.13 Iso-surface of λ2=-1

Fig.14 Iso-surface of λ2=-1

3. Conclusions

An unsteady, multi-block, overset grid and Navier-Stokes equation coupled solver has been developed to numerically simulate the self-propelled motion of a fish with a pair of rigid pectoral fins in this article. The overset grid method is proved to be effective in dealing with the multi-block moving objects problems of fish swimming with paired rigid pectoral fins. For lift-based swimming mode combining flapping and feathering motions, there exists a set of parameters of fin motion, which can generate great thrust and at the same time keep lift force almost zero. Adjusting the parameterΔφFLcan change the forces generation of the paired pectoral fin and the fish motion. The periodic vortex structure generation and shedding is closely related to the hydrodynamic force generation of the pectoral fin. The generation of a vortex structure near the fin edge accompanies the generation of maximum thrust, and a vortex structure shedding from the fin edge accompanies the generation of maximum resistance. In this article, it is concluded that when a pair of rigid pectoral fin generate the hydrodynamic moment, it may also generate a lateral force opposite to the centripetal direction, which has adverse effect on the turn motion of the fish. Therefore, the motion of paired pectoral fin and the fluctuation of the fish body are both considered to perform the turn motion in future work.

Acknowledgements

This work was supported by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (Grant No. 2008007). The authors gratefully acknowledge the kindness of Prof. Liu Hao of Chiba University, Japan for providing the basic Navier-Stokes code of fish swimming.

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November 23, 2011, Revised February 19, 2012)

* Project supported by the National Natural Science Foundation of China (Grant Nos. 50739004, 11072154), the Foundation of State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University (Grant No. GKZD 010053-11).

Biography: XU Yi-gang (1987-), Male, Master

WAN De-cheng,

E-mail: dcwan@sjtu.edu.cn