Yong-cheng LI （李永成）， Ding-yi PAN （潘定一）， Zheng MA （馬崢）

1. China Ship Scientific Research Center， Wuxi 214082， China， E-mail:liyongcheng702@163.com

2. Department of Engineering Mechanics， Zhejiang University， Hangzhou 310027， China

The mechanism of flapping propulsion of an underwater glider*

Yong-cheng LI （李永成）1， Ding-yi PAN （潘定一）2， Zheng MA （馬崢）1

1. China Ship Scientific Research Center， Wuxi 214082， China， E-mail:liyongcheng702@163.com

2. Department of Engineering Mechanics， Zhejiang University， Hangzhou 310027， China

To develop a bionic maneuverable propulsion system to be applied in a small underwater vehicle， a new conceptual design of the bionic propulsion is applied to the traditional underwater glider. The numerical simulation focuses on the autonomous underwater glider （AUG）'s flapping propulsion at Re=200 by solving the incompressible viscous Navier-Stokes equations coupled with the immersed boundary method. The systematic analysis of the effect of different motion parameters on the propulsive efficiency of the AUG is carried out， including the hydrofoil's heaving amplitude， the pitching amplitude， the phase lag between heaving and pitching and the flapping frequency. The results obtained in this study can provide some physical insights into the propulsive mechanisms in the flapping -based locomotion.

autonomous underwater glider， flapping propulsion， immersed boundary method

The autonomous underwater glider （AUG） is a new type of underwater vehicles and it is driven by its own buoyancy. Compared with the traditional underwater vehicle， it has the advantages of low noise， low energy consumption， and long range［1］.

Despite these advantages， some problems regarding the AUG should be given serious consideration. One of the most crucial problems is the “drift”. For collecting intense data， the gliding speed of the AUG has to be relatively low， which is only about 0.5 knot（0.25 m/s）. Under such a low speed， the movement of the AUG would be easily influenced by the ocean current， and it is not easy to continually follow the initially determined route.

In order to solve this problem， a conceptual design of the bionic propulsion method is adopted for the design of the AUG. In this paper， the bionic propulsion of a newly designed underwater glider is investigated by numerically solving the incompressible viscous Navier-Stokes equations coupled with the immersed boundary method to reveal the effect of hydrofoil's motion parameters on the propulsive efficiency， including the heaving amplitude， the pitching amplitude， the phase lag between heaving and pitching and the flapping frequency and to have an improved understanding of physical mechanisms of the flapping-based locomotion adopted by swimming animals.

As shown in Fig.1， the computational model is composed of the hull and the hydrofoils. The total length of the model is 1.200 m， where the middle part is a cylinder of 0.250 m in diameter and 0.625 m in length. The front part is a semi-ellipsoid of 0.175 m in semi-major axis， and the rear part is also a semi-ellipsoid of 0.400 m in semi-major axis. The hydrofoil is in the NACA0015 profile with a span length of 0.300 m and a chord length of 0.300 m， which is chosen as the characteristic length C.

Fig.1 Schematic diagram of the computational model

The bionic propulsion method is introduced into the design of the AUG， and and the hydrofoil's flapping is used to increase the AUG's advancing speed. The hydrofoil's motion is the combination of the heaving motion along the Y axis and the pitching motion around the Z axis， both directions of motion are sinusoidal， with a phase lag in the same motion cycle. The equations of the heave motion and the pitch motion are， respectively:

where0h is the heaving amplitude，0θ the pitching amplitude， f the flapping frequency and0ψ the phase lag. As a result of the hydrofoil's flapping， the underwater glider can move quickly. The schematic diagram of the movement is shown in Fig.2.

Fig.2 Schematic diagram of the motion process

The surrounding water around the AUG is considered as incompressible and viscous， and the Navier-Stokes equations of fluid motion is employed as［2，3］

where u is the velocity vector， p is the pressure，Re is the Reynolds number， which can be calculated as Re=U0L/ν with U0and L being the characteristic velocity and length scales， and f is the additional body force. To discretize the Navier-Stokes equations for numerical solutions， the Crank-Nicolson scheme is used for viscous terms and the Adams-Bashforth scheme is applied for other terms in Eq.（3）. In addition， the finite difference projection method is used to obtain the velocity and pressure fields. For simplification， the Reynlods number in the current study is chosen as 200， without any additional turbulent model to be applied.

The immersed boundary （IB） method is applied to capture the flapping motion of the hydrofoil.The additional body force f of the IB method near the moving boundary is modified according to the “direct forcing” approach［2］， in which the body force can be derived as

It is worth mentioning that unlike other bionic propulsion studies， this paper focuses on the practical application， to maintain a balance between the hull's average resistance and the hydrofoil's average thrust. Thus a glider can maintain a constant moving speed. The formula of balance is defined as

where D represents the drag experienced by the hull，F(xiàn) represents the thrust generated by the hydrofoils，and T is a motion period.

We here present some typical results on the bionic propulsion of the underwater glider. Based on the measurements and the modeling of the animal locomotion，the governing parameters used in this study are chosen as follows: the flapping frequency f=0.3Hz-1.0Hz， the phase lag between heaving and pitching ψ0=30o-110o， the heaving amplitude=0.05C-0.5C， the pitching amplitude θ=30oand the moving velocity V =0.5m/s-1.2m/s.

In order to characterize the propulsive efficiency of the underwater glider， the ratio of the kinetic energy of the body and the input work is usually employed［3，4］and defined as

where T is a movement period， and P the input power， which represents the energy required by the AUG to overcome the fluid force in the unit time and it consists of two parts， which arewhere1P is the power required by the hull to overcome the fluid resistance，2P is the power required by the hydrofoils to overcome the fluid dynamics， V is the average advancing speed， （）Lt is the vertical force acted on the hydrofoils and （）Mt is the torque around the Z axis.

Figure 3 shows the propulsive efficiency η versus the phase lag with the fixed pitching amplitude θ=30oand the flapping frequency f=0.6Hz .

Fig.3 Propulsive efficiency versus phase lag

It is seen from Fig.3 that the propulsive efficiency for each moving velocity increases to its maximum and then decreases with0ψ， the best phase lag increases constantly while the highest propulsive efficiency sees a slight change with the increase of the moving velocity. When the phase lag is aroundo90， the maximum value of the propulsive efficiency is obtained. So， in the following calculation， the phase lag is set aso90.

Figure 4 shows the curve of the propulsive efficiency versus the pitching amplitude and the moving velocity with the fixed flapping frequency f=0.6Hz and the phase lago90.

Fig.4 Propulsive efficiency versus pitching amplitude and moving velocity

As shown in Fig.4， similarly， the propulsive efficiency increases to its maximum and then gradually decreases with the increase of θ at several moving velocities. Furthermore， with the increase of the moving velocity， the highest propulsive efficiency experiences a sharp decline while the pitching amplitude corresponding to the maximum propulsive efficiency shows a slight change， abouto30. So it is recommended that the pitching amplitude is chosen aso30.

Figure 5 shows the propulsive efficiency versus the heaving amplitude and the moving velocity with the fixed pitching amplitude θ=30oand the phase lag ψ0=90o.

Fig.5 Propulsive efficiency versus heaving amplitude and moving velocity

As shown in Fig.5， there exists a certain heaving amplitude leading to the highest propulsive efficiency for a specified moving velocity and the best heaving amplitude increases constantly with the increase of the moving velocity while the corresponding propulsive efficiency shows a gradual decrease， which means that to obtain a high moving velocity means a sacrifice of the propulsive efficiency， and therefore the loss of the long range and the high duration.

Figure 6 shows the propulsive efficiency versus flapping frequency and the moving velocity with the fixed pitching angle 30oand the phase lag 90o.

Fig.6 Propulsive efficiency versus flapping frequency and moving velocity

As can be seen from Fig.6， at different moving velocities， the propulsive efficiency increases to itsmaximum and then gradually decreases with the increase of the flapping frequency. Besides that， the best flapping frequency increases constantly with the increase of the moving velocity while the maximum propulsive efficiency in the corresponding case decreases with the increase of the moving velocity.

Fig.7 （Color online） Instantaneous vortex structures for =f0.4 Hz， 0.6 Hz and 1.0 Hz

The propulsive behaviors of the flapping propulsion are closely associated with the vortex structures around the hydrofoils. In order to explain the above phenomenon， the vortex structures are obtained for three flapping frequencies =f0.4 Hz， 0.6 Hz and 1.0 Hz with V =1.2m/s ， θ=30oand ψ0=90o. The instantaneous vortex structures are shown in Fig.7.

As shown in Figs.7（a）-7（c） for f=0.6Hz ， the leading-edge vortex first moves along the upper surface of the hydrofoil to the trailing edge and falls off while a new leading-edge vortex emerges on the leading edge of the hydrofoil. The shedding leading-edge vortex is then connected with the tip vortices， lying in the two sides of the flapping hydrofoil， and is eventually closed with the trailing edge leading to vortex loops in the tail flow field. This phenomenon is consistent with the experimental observations of Von Ellenrieder［5，6］.

In the case of f=0.4Hz ， Figs.7（d）-（7f） show that the shedding vortices in the upper and lower surfaces of the hydrofoil separate from each other in the tail flow field， therefore， there is no vortex loop exists. In the case of a higher frequency f=1.0Hz ，F(xiàn)igs.7（g）-7（i） show that the vortices in the upper and lower surfaces of the hydrofoil separate earlier and they are overlapping with each other， so it is more difficult to form a vortex loop. Since the energy required for the propulsion is mainly derived from the vortex loop， so that may explain the results we have obtained above.

References

［1］CHEN Ya-jun， CHEN Hong-xun and Ma zheng Hydrodynamic analyses of typical underwater gliders［J］. Journal of Hydrodynamics， 2015， 27（4）: 556-561

［2］HUA R. N.， ZHU L. and LU X. Y. Locomotion of a flapping flexible plate［J］.Physics of Fluids， 2003， 25（12）: 121901.

［3］SHAO Xue-ming， PAN Ding-yi and DENG Jian et al. Numerical studies on the propulsive and wake structure of finite-span flapping hydrofoils with different aspect ratios［J］. Journal of Hydrodynamics， 2010， 22（2）: 147-154.

［4］PAN D.， DENG J. and SHAO X. et al. On the propulsive performance of tandem flapping hydrofoils with a modified immersed boundary method［J］. International Journal of Computational Method， 2016， 13: 1650025.

［5］Von ELLENRIEDER K.， PARKER K. and SORIA J. Flow structures behind a heaving and pitching finite-span wing［J］. Journal of Fluid Mechanics， 2003， 490: 129-138.

［6］TANG Chao， LU Xi-yun. Self-propulsion of a threedimensional flapping flexible plate［J］. Journal of Hydrodynamics， 2016， 28（1）: 1-9.

（August 18， 2016， Revised September 10， 2016）

* Project supported by the National Natural Science Foundation of China （Grant No. 51279184）.

Biography： Yong-cheng LI （1992-）， Male， Master Candidate

Ding-yi PAN，

E-mail: dpan@zju.edu.cn