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Hydrodynamic performance of an autonomous underwater glider with a pair of bioinspired hydro wings–A numerical investigation
Ocean Engineering 163 (2018) 51–57
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Hydrodynamic performance of an autonomous underwater glider with a pair of bioinspired hydro wings–A numerical investigation
T
Yongcheng Lia, Dingyi Panb,∗, Qiaosheng Zhaoa, Zheng Maa, Xijian Wanga a b
State Key Laboratory of Hydrodynamics, China Ship Scientific Research Centre, Wuxi, 214082, China Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Autonomous underwater glider (AUG) Bio-inspiration Fluid structure interaction Travelling wave
A conceptual design of autonomous underwater glider (AUG) is introduced, with a pair of bio-inspired hydro wings undergoing travelling wave motion. Hydrodynamic performance of current AUG is investigated by numerically solving the incompressible viscous Navier–Stokes equations of its surrounding flow coupling with the immersed boundary method to capture the moving boundaries. The force balance along the stream direction is guaranteed by adjusting the kinematic parameters of the travelling wave motion, thus the cruise state of the AUG with a constant velocity can be achieved. The propulsive efficiency under cruise state is therefore directly and well defined. The effects of kinematic parameters on propulsive performance are investigated, and the nondimensional Strouhal number, as a combination of these kinematic parameters, turns to be a dominant parameter of the propulsive efficiency. There is an optimum range of Strouhal number, in which the AUG achieves its maximum efficiency. The results obtained in current study can provide certain technical supports for the design and development of small submersible vehicle.
1. Introduction Autonomous underwater glider (AUG) is a novel autonomous underwater vehicle (AUV) without external active propeller. Since Stommel (1989) first put forward the concept of underwater glider, several kinds of AUGs have been built in succession, which are used both in military and civil applications. Compared with the traditional AUV, AUG has several advantages, such as low noise, low energy consumption, long operational range and endurance and great operational flexibility (Davis et al., 2002; Claustre and Beguery, 2014; Chen et al., 2015). AUG is a buoyancy-driven device, and it periodically changes its net buoyancy by a hydraulic pump to glide upwards and downwards alternately. As AUG dives and ascends, its internal ballast changes its position to control attitude, and its body and wings provide longitudinal hydrodynamic force to drive itself moving forward (Asakawa et al., 2012; Mitchell et al., 2013; Javaid et al., 2014; Zhao et al., 2014; Ye and Qi, 2013). However, apart from aforementioned advantages, some problems concerning AUG should be given due attention. One of the most crucial problems is ‘drifting with the current’. For achieving intense data collection, the gliding speed of AUG has to be relatively low, which is only about 0.5 knot (0.25 m/s). Under such a low speed, movement of AUG would be easily influenced by ocean current, and it
∗
Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (D. Pan).
https://doi.org/10.1016/j.oceaneng.2018.05.052 Received 18 December 2017; Received in revised form 14 April 2018; Accepted 27 May 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
is not able to be continually maintained following the initially desired route, thus failing to complete specific task. Hence, AUG may encounter a risk of mission failure when operated in a strong-current (Li et al., 2016; Liu et al., 2017). Therefore, additional control strategies are required to make AUG's trajectory as desired, and meanwhile without significant energy consumption. Aiming at this, extensive studies have been reported recently. Niu et al. (2016) proposed a prototype design of hybrid vehicle to improve AUG's gliding speed by using of the additional propeller positioned at the rear part of AUG and the trial result shows that the hybrid underwater glider can achieve the maximum gliding speed of 3 knots. Similarly, Claus & Bachmayer (2016) adopted an energy optimal depth controller design methodology for a long range AUV, presented with applications to a propeller driven hybrid AUG. Jeong et al. (2016) designed a novel underwater glider having a high horizontal speed of the maximum 2.5 knots with the help of a controllable buoyancy engine to regulate the amount of buoyancy drastically. Zhang et al. (2017) utilized supercavitation to reduce drag and increase their underwater speed. Most of the research mentioned above make effort on utilizing external propeller to increase AUGs' gliding speed. Consequently, the external propeller shall import additional drag which will cause negative impact to its endurance and gliding range. Besides, problems of vibration and noise will destroy AUG's concealment with the
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the model (fuselage) is 1.2 m, where the middle part is a cylinder, scaled with 0.25 m in diameter and 0.625 m in length. The front part is a semi-spheroid, with 0.175 m in semi-major axis, and the rear part is also a semi-spheroid, with 0.4 m in semi-major axis. The hydro wings are designed with rectangular wing by using the NACA0015 profile. It has a span length of 0.3 m and a chord length of 0.3m, which is chosen as the characteristic length C. In the following, all the variables are nondimensionised, e.g. length of chamber line, hydrodynamic forces, following the approach presented in Xu and Xu (2017) and Xu et al. (2017). The bioinspired travelling wave motion of the two hydro wings has been as the propulsion method of current AUG. Each point of the hydro wing surface along its chord length direction makes a transverse oscillation and its lateral displacement can be described by the following sinusoidal function,
introduction of propeller. Owing to these, alternative solutions to solve this problem from other perspectives are required. It is worth noting that, in nature, many creatures have acquired extraordinary abilities of locomotion, such as high efficiency, fast manoeuvrability and low noise etc., through natural selection and evolution. This inspires scientists and engineers to explore the nature of their excellent athletic ability. Among them, fish swimming in lake and sea are typical kind of creatures with extraordinary athletic ability (Sfakiotakis et al., 1999). In the past several decades, great endeavours have been devoted in the study on hydrodynamic mechanism of fish swimming (Triantafyllou et al., 2000, 2004) as well as the design of robotic fish (Triantafyllou and Triantafyllou, 1995; Roper et al., 2011, Raj and Thakur, 2016). On the other hand, a complementary design of bioinspired AUG also attracts attentions recently (Fish et al., 2003; Georgiades et al., 2009; Li et al., 2016), which can achieve high efficient propulsion by mimicking fish swimming, and meanwhile fulfil the requirement of enough space for payload and avoid the deformation fuselage like robotic fish. In detail, the bioinspired AUG consists of two major parts, i.e. a ‘biomimetic propeller’ and a rigid cantered fuselage. With the help of ‘biomimetic propeller’, e.g. flapping wing or wavy plate, AUG can maintain its initial gliding route, and at the same time, resistance experienced by AUG will not increase significantly causing minimal impact to gliding range and endurance. Instead of using hybrid driven mode, the newly designed bioinspired AUG is supposed to implant a driven-mode of hydrofoil, which also means, under complex underwater environment, AUG would be driven by its ‘biomimetic propeller’, similar with fish in nature, whereas in rest time, when the environment is relatively stable, the AUG may undergo usual gliding. In this paper, a conceptual design of bioinspired AUG is introduced, with a pair of bioinspired hydro wings undergoing travelling wave motion. Hydrodynamic performance of current AUG is investigated by numerically solving the incompressible viscous Navier–Stokes equations of surrounding flow. We would like to figure out the effect of hydro wing parameters on the propulsive efficiency of AUG, to achieve an improved understanding of physical mechanisms relevant to the biomimetic locomotion adopted by current AUG. The immersed boundary method is employed to capture the moving boundary of the hydro wings, which has been successfully applied in previous studies on fish-like flapping wing (Shao et al., 2010b; Pan et al., 2016), undulatory foil (Shao et al., 2010a), flexible plate (Pan et al., 2010, 2014), etc. An evaluation process which is first proposed in our previous work (Li et al., 2016) is also applied to investigate the hydrodynamic propulsive efficiency of current AUG in its cruise state. In the following, the physical model of current AUG is presented in Sec. 2, following by the introduction of numerical method in Sec. 3. The simulating results and relevant discussion are given in Sec. 4. Conclusion is drawn in the last section.
2π y (x , t ) = A (x )sin ⎛ x − 2πft ⎞, ⎝ λ ⎠
(1)
A (x ) = a0 + a1·x + a2 ·x 2,
(2)
where λ is the wave length, f is the frequency, x the distance from a specific point along the chord length of the hydro wing to the leading edge of the hydro wing, and A(x) is a function used to describe the transverse magnitude of the hydro wing and its expression is shown as Eq. (2), where a0, a1 and a2 are all constants. Here, we let a0 = −0.02 and a2 = −2a1, thus it makes sure the transverse magnitude lies in the maximum depth of the hydro wing's trailing edge. As examples, Fig. 2 shows the camber line deformation of the hydro wing in one period under different parameters. In order to give an intuitive description of the hydro wing's travelling wave motion, Fig. 3 shows the hydro wing deformation in a whole motion period. With the undulation of the hydro wings, thrust force is produced to overcome the drag of ocean current. In our preliminary design of the AUG prototype, the hydro wing surface is made of flexible rubber, while inside the wing, several rigid fin rays are uniformly distributed along the chord length. Each fin ray is connected with a separate electric motor by a crank train. Therefore, the travelling wave of the wing surface is activated with this mechanism. A similar design was adopted by Clark and Smits (2006). 3. Numerical method and validation The surrounding water around AUG is considered as incompressible and viscous, and the non-dimensional Navier–Stokes equations of fluid motion is employed as
2. Physical model
∂u 1 2 + (u ·∇) u = −∇P + ∇ u + f, ∂t Re
(3)
∇ ·u = 0,
(4)
where u is the velocity vector, P the pressure, Re the non-dimensional Reynolds number which can be calculated as Re]U0C/ν with U0 and C the characteristic velocity and length scale, and ν the dynamic viscosity, and f 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 the other terms in Eq. (3). In addition, the finite difference projection method is used to obtain the velocity and pressure fields. For simplification, the Reynolds number in current study is around 200, taking the characteristic velocity to be one, without any additional turbulent model to be applied. The immersed boundary (IB) method is applied to capture the travelling wave motion of the hydro wings. The additional body force f of IB method near the moving boundary is modified according to the ‘direct forcing’ approach, in which the body force can be derived as
Fig. 1 shows the sketch of current physical model, which is made up with one centred fuselage and two side hydro wings. The total length of
fin + 1 =
Fig. 1. Sketch of current physical model. 52
V n + 1 − uin + RHSin, Δt
(5)
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Fig. 2. Schematic diagram of the camber line deformation of the hydro wing in one period under different wavelength (λ) and amplitude (a1), (a) λ = 1.0, a1 = −0.08, (b) λ = 2.5, a1 = −0.08, (c)λ = 1.0, a1 = −0.20.
whereVn+1 is the boundary velocity in the next time step, and RHSin represents all the other terms in Eq. (3). The main strategy of this method can be concluded as follow: the whole domain including the solid immersed body is treated as fluid; an added force term, which is equal to zero except at the nearby points of the immersed boundary, is needed in the government equations to enforce the solid body movement. In this method, the entire simulations could be carried out with the fixed Cartesian grid, which improves the efficiency of this simulation approach. A detailed description of the numerical method and the relevant validation can be found in our previous papers (Shao et al., 2010b; Shao and Pan, 2011; Pan et al., 2014, 2016). In current simulation, velocity boundary condition is imposed at inlet and no back-flow condition is applied at outlet. The other surrounding boundaries are all set as the uniform velocity condition. Based on our convergence tests for the present problem, the computational domain for fluid flow is chosen as [-5,10] × [-4,4] × [-4,4] in x, y and z directions and the mesh in each direction has a finest spacing of 0.03. The time step is chosen as 0.001.
cruise velocity, and input power is that consumed for the hydro wings to oscillate with the travelling wave motion. In view of this, both the output and input powers can be derived as right left Pout = Pout + Pout = U0·T left + U0·T right,
Pin = Pinleft + Pinright =
Γ
4.1. Hydrodynamic performance evaluation of current AUG Unlike previous studies on one single biomimetic wing/foil that there is a non-zero fluid force acting on the wing/foil, in current study, we only focus on the cruise state of the AUG with a constant velocity. Therefore, the net fluid force acting on the whole AUG is nearly zero. Concerning current AUG model, the oscillating hydro wings produce thrust to overcome the fluid drag force acting on the fuselage and the force balance along stream direction in one period can be represented as
1 Γ
∫ D·dτ = Γ1 ∫ T·dτ , Γ
Γ
(8)
Here, Pout and Pin are output and input powers of the system, U0 is cruise velocity, Tleft and Tright are thrust forces generated by the left and right hydro wings respectively, pleft and pright are the pressure distributions along the wing surfaces, and vleft and vright are the velocities along the y direction on wing surfaces. With above definitions, the propulsive efficiency can be obtained directly as η = Pout/Pin. In order to achieve the zero net fluid force state of the AUG, the kinematic parameters for the travelling wave motion are adjusted accordingly. For simplicity, only the average fluid force in each period is kept to zero, the parameters are fixed when the adjustment is achieved. A typical result is shown here. In this case, the parameter a2 = −2a1 in Eq. (2) is used as an adjustable parameter, while the rest kinematic parameters are fixed as a0 = −0.02, λ = 1.5 and f = 1.5Hz. When a1 = −0.16, the net fluid force on the AUG is nearly zero, and therefore, a cruise state is achieved. The vertical and horizontal forces of all the components (left and right wings and fuselage) are shown in Fig. 4. Overall, all the forces oscillate periodically with a constant frequency correlated to the travelling wave frequency. The force curves of left and right hydro wings collapse to each other, as a result of the same geometry shape and symmetrically arranged on either side of the fuselage. The mean forces in one single period can also be obtained directly from Fig. 4. The mean vertical forces (FL) of all the components are very small, O (10−3), as shown in Table 1, and the net force along vertical direction is nearly zero neglecting its gravity and buoyancy forces. For the force components along horizontal direction, each hydro wing is owing a negative mean force (FD), while the FD of the fuselage is positive, indicating that the twin hydro wings produce thrust forces to overcome the drag force acting on the fuselage. The resultant force of the whole AUG along horizontal direction is also very close to zero. Therefore, Eq. (6) is fulfilled and the AUG is in its cruise state. In the following, the effects of the kinematic parameters on the propulsive efficiency of the AUG are investigated. Motivated by the
4. Results and discussion
D =T ⇒
∮ (pleft ·v left + pright ·v right) dτ .
(7)
(6)
where D represents the drag force acting on the fuselage, T represents the thrust force generated by the twin hydro wings, and Γis an oscillating period. Meanwhile, the output and input power of the AUG system can be defined directly. The output power can be considered as the power generated by the oscillating hydro wings to maintain the
Fig. 3. Schematic diagram of the travelling wave motion of current AUG in one period, (a) t = 0 T, (b) t = 1/4 T, (c) t = 1/2 T, (d) t = 3/4 T. 53
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Fig. 4. Vertical and horizontal force histories of left and right hydro wings and fuselage.
frequency. There is an optimum travelling frequency for each moving velocity, leading to the highest propulsive efficiency. Besides, it can be also found that the highest propulsive efficiency shows a little decrease when the moving velocity increases, which means if we want to gain high moving velocity we have to pay price for cutting down the endurance. The magnitudes of corresponding wave amplitudes, a1, is shown in Fig. 5 (b). The magnitudes of wave amplitude of all the moving velocity have the same trend: the smaller wave frequency the larger wave amplitude, and also the small moving velocity corresponding to a small wave amplitude, which is consistent with the results of propulsive efficiency. As mentioned above, the effects the wave frequency, wave amplitude and moving velocity can be combined as the effect of the dimensionless number St, and the corresponding result of Fig. 5 (a) and (b) is shown in Fig. 6. The results of all moving velocity almost collapse, and the optimum St for maximum propulsive efficiency is around 0.4–0.6. Therefore, for the study of current AUG model, Strouhal number is still a key parameter for the evolution of propulsive efficiency of biomimetic locomotion.
Table 1 Average vertical and horizontal forces of twin hydro wings and fuselage.
mean FL mean FD
left hydro wing
right hydro wing
fuselage
−0.0066 −0.1946
−0.0066 −0.1946
−0.0030 0.3901
biomimetic of animal locomotion, the parameters used in current study are chosen in proper ranges as: the travelling wave frequency f = 0.8–3.0, the travelling wave length λ = 0.8–2.5 and the cruise velocity V = 0.5–1.2, and the relevant Reynolds number varies from 100 to 240, accordingly. 4.2. Effects of travelling wave frequency on propulsive efficiency In the following two subsections, we are going to investigate the effects of travelling wave parameters on the propulsive efficiency of current AUG. The effect of frequency is first studied in this subsection. The oscillating or fluctuating frequency might be one of the most important parameters in biomimetic locomotion (Triantafyllou et al., 1991; Shao et al., 2010b). The dimensionless Strouhal number (St) combing the effect of travelling wave frequency and amplitude is introduced as
2h max⋅f St = , U0
4.3. Effects of the travelling wave amplitude on propulsive efficiency The effects of travelling wave amplitude are investigated in this subsection. The travelling wave frequency is fixed as f = 1.5Hz, the wave amplitude is varied in the range of a1 = 0–0.5 and the wave length λ is adjusted to obtain a state of zero net force. Fig. 7 (a) shows the result of propulsive efficiency versus travelling wave amplitude. As shown in Fig. 7 (a), the propulsive efficiency increases monotonous with a1 to the peak and then decrease with further increase of a1. The magnitude of maximum efficiency of each moving velocity decreases with the increase of moving velocity, which is similar with the trend showing in Fig. 5. The magnitude of wave length λ to the wave amplitude a1 is shown in Fig. 7 (b). Obviously, the wave length has a linear dependence on the magnitude of wave amplitude. Linear fittings are also made in Fig. 7 (b), the fitting lines under different moving velocity
(9)
where f and U0 are the hydrofoils' travelling frequency and AUG's advancing speed, hmax represents hydrofoils' maximum travelling amplitude along the Y direction and it can be calculated from Eqs. (1) and (2) easily. In current simulation, the travelling wave length is fixed as λ = 2.5, and the travelling wave amplitude, a1, is adjusted to achieve a cruise state. The result of frequency effect is shown in Fig. 5 (a), and it is seen from the result that the propulsive efficiency for each moving velocity increases to its maximum and then decreases with the increasing of
Fig. 5. Propulsive efficiency versus travelling frequency under different moving velocities and the corresponding magnitudes of travelling wave amplitudes a1. 54
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Fig. 8. Propulsive efficiency versus travelling wave length under different moving velocities.
Fig. 6. Propulsive efficiency versus dimensionless Strouhal number.
show the similar slope, around −1.8. In contrast, the curves of propulsive efficiency versus travelling wave length is also plot in Fig. 8, according to the same set of data in Fig. 7. The curves show similar trend with those in Fig. 7 (a), whereas the difference is that the corresponding wave lengths of maximum efficiency under different moving velocity are in a relative small range, around λ = 2.4, and the curves under different moving velocity are close to each other. Therefore, the effect of travelling wave length shows an equilibrium to the combined effects of wave amplitude and moving velocity. On the other hand, similar to Fig. 6, the effect of Strouhal number on propulsive efficiency is plot in Fig. 9. The maximum propulsive efficiency is achieved around St = 0.6–1.0, which is a little larger than the range in Fig. 6. 4.4. Discussions
Fig. 9. Propulsive efficiency of AUG versus different Strouhal numbers under different moving velocity.
4.4.1. Flow structures around the AUG The flow structures around and in the wake of the AUG is closely related to the fluid force acting on the AUG, therefore, here we present some typical vortex structures under different conditions to investigate the coherence between vortex topology and hydrodynamic performance of the AUG. The vortex iso-surfaces with λ2 definition (Jeong and Hussain, 1995) in the wake are illustrated in Fig. 10. For comparison, three typical cases under the same advancing velocity is V = 0.5 and travelling wave length λ = 2.5, and different travelling wave frequencies, i.e. f = 1.0, 1.2 and 2.0. The corresponding Strouhal number are St = 0.52, 0.43 and 0.24, and the resulted propulsive efficiencies are η = 36%, 42% and 31.5%, respectively. As can be seen from Fig. 10, the vortex rings/tubes emerge in the wakes of hydro wings under different travelling frequencies. This phenomenon is similar with our previous work (Li et al., 2016). It is worth mentioning that the expanding angle (or obliquity) of the shedding vortex rings, i.e. the angle between the centre line of vortex rings and horizontal line, varies one by one. For case f = 1.0 as shown in Fig. 10 (a), the oscillating amplitude a1 is larger than the one of case
f = 1.2 [Fig. 10 (b)], and therefore it generates a larger expanding angle of shedding vortex wake. Corresponding pressure contours and velocity distributions are shown in Fig. 11. Small low-pressure area is distributed only around the tail of the wing as shown in Fig. 11 (a) and (b). Since the tail is bending downward, a low-pressure area around the bottom surface of the tail plays a positive role in the propulsion of the wing and the lower pressure the larger thrust force. Whereas for case f = 2.0 [Fig. 10 (c)], a larger expanding angle is also resulted even with a smaller oscillating amplitude, comparing with the case for f = 1.2 of higher propulsive efficiency. It indicates that, with a large travelling wave frequency, strong momentum exchanging happens during the shedding of the vortex ring. It is confirmed by the corresponding pressure contours as shown in Fig. 11 (c), a large low-pressure area is distributed around the whole bottom surface of the wing. In a sum, the vortex shedding from the trailing edge of hydro wing is accompanied by the release of energy. The shedding vortex ring is inclined to the inflow direction, the corresponding released energy is decomposed into
Fig. 7. Propulsive efficiency versus travelling amplitude and travelling wave length under different moving velocities. 55
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Fig. 10. Instantaneous vortex structures for cases under different conditions at t = 3.0 T.
Fig. 11. Pressure contours and velocity distributions in cross-section plane of AUG hydro wing, corresponding to Fig. 10.
differences among them lie in the width and thickness of the vortex rings. Similar conclusions were also stated by Blondeaux et al. (2005) and Dong et al. (2006). Second, in spite of the Reynolds number effect on thrust and efficiency as reported in Ashraf et al. (2011) and Deng et al. (2016), the major concerns in current study is the effects of kinematic parameters with fixed Reynolds number. The results of aforementioned studies indicate that the thrust/efficiency tendencies with the foil thickness at different Reynolds number are similar (Ashraf et al., 2011) and also the efficiency tendency with the frequency (Deng et al., 2016). In other words, the Reynolds number does not have significant effect on the thrust/efficiency tendencies with those kinematic parameters. Therefore, the efficiency tendency obtained in our study at Re = 200 may qualitatively similar with the tendency at higher Reynolds number. Last but not least, self-propelled micro swimmers are of great interests for in vivo biomedical applications (Zhang et al., 2009; Kosa et al., 2012). Those micro swimmers can be used to diagnose diseases, delivery drug, etc. From a fluid mechanic point of view, it is a smallscale vehicle moves in a viscous fluid with a low velocity, and the relevant Reynolds number is normally in the order of 102. The improvement of propulsive efficiency is also an important issue for the design of these micro swimmers. Current study on the bioinspired AUG may also provide insights in the field of micro swimmer design.
horizontal and vertical components, and the horizontal part is correlated to the propulsive performance. Therefore, it can be conjectured that the less expanding angle, the more horizontal energy is released and the better the propulsive efficiency. 4.4.2. Reynolds number effects In current study, the Reynolds number of the flow is fixed around Re = 200, which might be much lower than the traditional flow condition at which the Reynolds number is around 105 to 106. The reason that the flow with a relative low Reynolds number in current study can be explained into the following three aspects: First, our intention of simulation at Re = 200 is basically from the pioneer simulation work of biomimetic propulsion studies by Blondeaux et al. (2005) and Dong et al. (2006), in which propulsive performance of a single biomimetic wing was numerical investigated with Reynolds number in the range from 100 to 400. Our current study can be considered as an extension of these pioneer works. Instead to the single biomimetic wing, the whole AUG model with wing–fuselage interaction is simulated. Meanwhile, a corresponding new definition of propulsion efficiency based on the friction of fuselage and thrust of hydro wing is proposed. In view of this, a close Reynolds number with previous work is chosen. On the other hand, concerning the Reynolds number effects, our preliminary results indicate that the vortex topologies are similar for cases with different Reynolds number. The 56
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5. Conclusions A conceptual design of AUG with a pair of bioinspired hydro wings undergoing travelling wave motion is introduced in this paper and the flow field around the AUG is simulated by solving the incompressible viscous Navier–Stokes equations coupled with the optimized immersed boundary method to capture the moving boundaries. The hydrodynamic performance of AUG as well as the vortex structures in the wake of the AUG are investigated and discussed. The force balance along the stream direction is guaranteed by adjusting the kinematic parameters of the travelling wave motion, thus the cruise state of the AUG with a constant velocity can be achieved. Thereafter, the propulsive efficiency under cruise state is well defined. Here we briefly summarize the results obtained in current work. 1. The main kinematic parameters of the hydro wing, e.g. the travelling wave amplitude, travelling wave frequency and the wave length has the similar effects on AUG's propulsive efficiency. Concretely, propulsive efficiency increases to its maximum and then decreases with the increasing of above parameters. To estimate the combined effect of these parameters, the non-dimensional Strouhal number is proposed, and there is an optimum range of Strouhal number in which the propulsive efficiency reaches its maximum, regardless of any single parameter. 2. To achieve a cruise state, the adjusted wave length has a linear dependence on the magnitude of wave amplitude and the linear fitting slopes under different moving velocities close to each other, around −1.8, which can provide a much convenient and support for the control of such bioinspired AUG. 3. The vortex structures in the wake of current AUG have the close connection with the propulsive efficiency and the high propulsive efficiency is usually correlated to a small expanding angle of vortex rings or vortex ring obliquity. Acknowledgments The authors would like to acknowledge the support provided by Zhejiang Provincial Natural Science Foundations (Nos. LY18A020002, LQ18E090010) and the National Natural Science Foundations of China (No. 91634103, 51279184). References Asakawa, K., Kobayashi, T., Nakamura, M., Watanabe, Y., Hyakudome, T., Itoh, Y., Kojima, J., 2012. Results of the first sea-test of Tsukuyomi: a prototype of underwater gliders for virtual mooring. In: 2012 Oceans, 14-19 Oct. 2012, pp. 1–5. Ashraf, M., Young, J., Lai, J., 2011. Reynolds number, thickness and camber effects on flapping airfoil propulsion. J. Fluid Struct. 27, 145–160. Blondeaux, P., Fornarelli, F., Guglielmini, L., Triantafyllou, M.S., Verzicco, R., 2005. Numerical experiments on flapping foils mimicking fish-like locomotion. Phys. Fluids 17, 113601. Chen, Y.J., Chen, H.X., Ma, Z., 2015. Hydrodynamic analyses of typical underwater gliders. J. Hydrodyn. 27, 556–561. Clark, R.P., Smits, A.J., 2006. Thrust production and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 562, 415–429. Claus, B., Bachmayer, R., 2016. Energy optimal depth control for long range underwater vehicles with applications to a hybrid underwater glider. Aut. Robots 40, 1307–1320. Claustre, H., Beguery, L., 2014. SeaExplorer glider breaks two world records. Sea Technol. 55, 19–21. Davis, R.E., Eriksen, C.C., Jones, C.P., 2002. Autonomous Buoyancy-driven Underwater Gliders. Taylor and Francis, London. Deng, J., Sun, L., Teng, L., Pan, D., Shao, X., 2016. The correlation between wake
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Hydrodynamic performance of an autonomous underwater glider with a pair of bioinspired hydro wings–A numerical investigation
T
Yongcheng Lia, Dingyi Panb,∗, Qiaosheng Zhaoa, Zheng Maa, Xijian Wanga a b
State Key Laboratory of Hydrodynamics, China Ship Scientific Research Centre, Wuxi, 214082, China Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Autonomous underwater glider (AUG) Bio-inspiration Fluid structure interaction Travelling wave
A conceptual design of autonomous underwater glider (AUG) is introduced, with a pair of bio-inspired hydro wings undergoing travelling wave motion. Hydrodynamic performance of current AUG is investigated by numerically solving the incompressible viscous Navier–Stokes equations of its surrounding flow coupling with the immersed boundary method to capture the moving boundaries. The force balance along the stream direction is guaranteed by adjusting the kinematic parameters of the travelling wave motion, thus the cruise state of the AUG with a constant velocity can be achieved. The propulsive efficiency under cruise state is therefore directly and well defined. The effects of kinematic parameters on propulsive performance are investigated, and the nondimensional Strouhal number, as a combination of these kinematic parameters, turns to be a dominant parameter of the propulsive efficiency. There is an optimum range of Strouhal number, in which the AUG achieves its maximum efficiency. The results obtained in current study can provide certain technical supports for the design and development of small submersible vehicle.
1. Introduction Autonomous underwater glider (AUG) is a novel autonomous underwater vehicle (AUV) without external active propeller. Since Stommel (1989) first put forward the concept of underwater glider, several kinds of AUGs have been built in succession, which are used both in military and civil applications. Compared with the traditional AUV, AUG has several advantages, such as low noise, low energy consumption, long operational range and endurance and great operational flexibility (Davis et al., 2002; Claustre and Beguery, 2014; Chen et al., 2015). AUG is a buoyancy-driven device, and it periodically changes its net buoyancy by a hydraulic pump to glide upwards and downwards alternately. As AUG dives and ascends, its internal ballast changes its position to control attitude, and its body and wings provide longitudinal hydrodynamic force to drive itself moving forward (Asakawa et al., 2012; Mitchell et al., 2013; Javaid et al., 2014; Zhao et al., 2014; Ye and Qi, 2013). However, apart from aforementioned advantages, some problems concerning AUG should be given due attention. One of the most crucial problems is ‘drifting with the current’. For achieving intense data collection, the gliding speed of AUG has to be relatively low, which is only about 0.5 knot (0.25 m/s). Under such a low speed, movement of AUG would be easily influenced by ocean current, and it
∗
Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (D. Pan).
https://doi.org/10.1016/j.oceaneng.2018.05.052 Received 18 December 2017; Received in revised form 14 April 2018; Accepted 27 May 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.
is not able to be continually maintained following the initially desired route, thus failing to complete specific task. Hence, AUG may encounter a risk of mission failure when operated in a strong-current (Li et al., 2016; Liu et al., 2017). Therefore, additional control strategies are required to make AUG's trajectory as desired, and meanwhile without significant energy consumption. Aiming at this, extensive studies have been reported recently. Niu et al. (2016) proposed a prototype design of hybrid vehicle to improve AUG's gliding speed by using of the additional propeller positioned at the rear part of AUG and the trial result shows that the hybrid underwater glider can achieve the maximum gliding speed of 3 knots. Similarly, Claus & Bachmayer (2016) adopted an energy optimal depth controller design methodology for a long range AUV, presented with applications to a propeller driven hybrid AUG. Jeong et al. (2016) designed a novel underwater glider having a high horizontal speed of the maximum 2.5 knots with the help of a controllable buoyancy engine to regulate the amount of buoyancy drastically. Zhang et al. (2017) utilized supercavitation to reduce drag and increase their underwater speed. Most of the research mentioned above make effort on utilizing external propeller to increase AUGs' gliding speed. Consequently, the external propeller shall import additional drag which will cause negative impact to its endurance and gliding range. Besides, problems of vibration and noise will destroy AUG's concealment with the
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the model (fuselage) is 1.2 m, where the middle part is a cylinder, scaled with 0.25 m in diameter and 0.625 m in length. The front part is a semi-spheroid, with 0.175 m in semi-major axis, and the rear part is also a semi-spheroid, with 0.4 m in semi-major axis. The hydro wings are designed with rectangular wing by using the NACA0015 profile. It has a span length of 0.3 m and a chord length of 0.3m, which is chosen as the characteristic length C. In the following, all the variables are nondimensionised, e.g. length of chamber line, hydrodynamic forces, following the approach presented in Xu and Xu (2017) and Xu et al. (2017). The bioinspired travelling wave motion of the two hydro wings has been as the propulsion method of current AUG. Each point of the hydro wing surface along its chord length direction makes a transverse oscillation and its lateral displacement can be described by the following sinusoidal function,
introduction of propeller. Owing to these, alternative solutions to solve this problem from other perspectives are required. It is worth noting that, in nature, many creatures have acquired extraordinary abilities of locomotion, such as high efficiency, fast manoeuvrability and low noise etc., through natural selection and evolution. This inspires scientists and engineers to explore the nature of their excellent athletic ability. Among them, fish swimming in lake and sea are typical kind of creatures with extraordinary athletic ability (Sfakiotakis et al., 1999). In the past several decades, great endeavours have been devoted in the study on hydrodynamic mechanism of fish swimming (Triantafyllou et al., 2000, 2004) as well as the design of robotic fish (Triantafyllou and Triantafyllou, 1995; Roper et al., 2011, Raj and Thakur, 2016). On the other hand, a complementary design of bioinspired AUG also attracts attentions recently (Fish et al., 2003; Georgiades et al., 2009; Li et al., 2016), which can achieve high efficient propulsion by mimicking fish swimming, and meanwhile fulfil the requirement of enough space for payload and avoid the deformation fuselage like robotic fish. In detail, the bioinspired AUG consists of two major parts, i.e. a ‘biomimetic propeller’ and a rigid cantered fuselage. With the help of ‘biomimetic propeller’, e.g. flapping wing or wavy plate, AUG can maintain its initial gliding route, and at the same time, resistance experienced by AUG will not increase significantly causing minimal impact to gliding range and endurance. Instead of using hybrid driven mode, the newly designed bioinspired AUG is supposed to implant a driven-mode of hydrofoil, which also means, under complex underwater environment, AUG would be driven by its ‘biomimetic propeller’, similar with fish in nature, whereas in rest time, when the environment is relatively stable, the AUG may undergo usual gliding. In this paper, a conceptual design of bioinspired AUG is introduced, with a pair of bioinspired hydro wings undergoing travelling wave motion. Hydrodynamic performance of current AUG is investigated by numerically solving the incompressible viscous Navier–Stokes equations of surrounding flow. We would like to figure out the effect of hydro wing parameters on the propulsive efficiency of AUG, to achieve an improved understanding of physical mechanisms relevant to the biomimetic locomotion adopted by current AUG. The immersed boundary method is employed to capture the moving boundary of the hydro wings, which has been successfully applied in previous studies on fish-like flapping wing (Shao et al., 2010b; Pan et al., 2016), undulatory foil (Shao et al., 2010a), flexible plate (Pan et al., 2010, 2014), etc. An evaluation process which is first proposed in our previous work (Li et al., 2016) is also applied to investigate the hydrodynamic propulsive efficiency of current AUG in its cruise state. In the following, the physical model of current AUG is presented in Sec. 2, following by the introduction of numerical method in Sec. 3. The simulating results and relevant discussion are given in Sec. 4. Conclusion is drawn in the last section.
2π y (x , t ) = A (x )sin ⎛ x − 2πft ⎞, ⎝ λ ⎠
(1)
A (x ) = a0 + a1·x + a2 ·x 2,
(2)
where λ is the wave length, f is the frequency, x the distance from a specific point along the chord length of the hydro wing to the leading edge of the hydro wing, and A(x) is a function used to describe the transverse magnitude of the hydro wing and its expression is shown as Eq. (2), where a0, a1 and a2 are all constants. Here, we let a0 = −0.02 and a2 = −2a1, thus it makes sure the transverse magnitude lies in the maximum depth of the hydro wing's trailing edge. As examples, Fig. 2 shows the camber line deformation of the hydro wing in one period under different parameters. In order to give an intuitive description of the hydro wing's travelling wave motion, Fig. 3 shows the hydro wing deformation in a whole motion period. With the undulation of the hydro wings, thrust force is produced to overcome the drag of ocean current. In our preliminary design of the AUG prototype, the hydro wing surface is made of flexible rubber, while inside the wing, several rigid fin rays are uniformly distributed along the chord length. Each fin ray is connected with a separate electric motor by a crank train. Therefore, the travelling wave of the wing surface is activated with this mechanism. A similar design was adopted by Clark and Smits (2006). 3. Numerical method and validation The surrounding water around AUG is considered as incompressible and viscous, and the non-dimensional Navier–Stokes equations of fluid motion is employed as
2. Physical model
∂u 1 2 + (u ·∇) u = −∇P + ∇ u + f, ∂t Re
(3)
∇ ·u = 0,
(4)
where u is the velocity vector, P the pressure, Re the non-dimensional Reynolds number which can be calculated as Re]U0C/ν with U0 and C the characteristic velocity and length scale, and ν the dynamic viscosity, and f 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 the other terms in Eq. (3). In addition, the finite difference projection method is used to obtain the velocity and pressure fields. For simplification, the Reynolds number in current study is around 200, taking the characteristic velocity to be one, without any additional turbulent model to be applied. The immersed boundary (IB) method is applied to capture the travelling wave motion of the hydro wings. The additional body force f of IB method near the moving boundary is modified according to the ‘direct forcing’ approach, in which the body force can be derived as
Fig. 1 shows the sketch of current physical model, which is made up with one centred fuselage and two side hydro wings. The total length of
fin + 1 =
Fig. 1. Sketch of current physical model. 52
V n + 1 − uin + RHSin, Δt
(5)
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Fig. 2. Schematic diagram of the camber line deformation of the hydro wing in one period under different wavelength (λ) and amplitude (a1), (a) λ = 1.0, a1 = −0.08, (b) λ = 2.5, a1 = −0.08, (c)λ = 1.0, a1 = −0.20.
whereVn+1 is the boundary velocity in the next time step, and RHSin represents all the other terms in Eq. (3). The main strategy of this method can be concluded as follow: the whole domain including the solid immersed body is treated as fluid; an added force term, which is equal to zero except at the nearby points of the immersed boundary, is needed in the government equations to enforce the solid body movement. In this method, the entire simulations could be carried out with the fixed Cartesian grid, which improves the efficiency of this simulation approach. A detailed description of the numerical method and the relevant validation can be found in our previous papers (Shao et al., 2010b; Shao and Pan, 2011; Pan et al., 2014, 2016). In current simulation, velocity boundary condition is imposed at inlet and no back-flow condition is applied at outlet. The other surrounding boundaries are all set as the uniform velocity condition. Based on our convergence tests for the present problem, the computational domain for fluid flow is chosen as [-5,10] × [-4,4] × [-4,4] in x, y and z directions and the mesh in each direction has a finest spacing of 0.03. The time step is chosen as 0.001.
cruise velocity, and input power is that consumed for the hydro wings to oscillate with the travelling wave motion. In view of this, both the output and input powers can be derived as right left Pout = Pout + Pout = U0·T left + U0·T right,
Pin = Pinleft + Pinright =
Γ
4.1. Hydrodynamic performance evaluation of current AUG Unlike previous studies on one single biomimetic wing/foil that there is a non-zero fluid force acting on the wing/foil, in current study, we only focus on the cruise state of the AUG with a constant velocity. Therefore, the net fluid force acting on the whole AUG is nearly zero. Concerning current AUG model, the oscillating hydro wings produce thrust to overcome the fluid drag force acting on the fuselage and the force balance along stream direction in one period can be represented as
1 Γ
∫ D·dτ = Γ1 ∫ T·dτ , Γ
Γ
(8)
Here, Pout and Pin are output and input powers of the system, U0 is cruise velocity, Tleft and Tright are thrust forces generated by the left and right hydro wings respectively, pleft and pright are the pressure distributions along the wing surfaces, and vleft and vright are the velocities along the y direction on wing surfaces. With above definitions, the propulsive efficiency can be obtained directly as η = Pout/Pin. In order to achieve the zero net fluid force state of the AUG, the kinematic parameters for the travelling wave motion are adjusted accordingly. For simplicity, only the average fluid force in each period is kept to zero, the parameters are fixed when the adjustment is achieved. A typical result is shown here. In this case, the parameter a2 = −2a1 in Eq. (2) is used as an adjustable parameter, while the rest kinematic parameters are fixed as a0 = −0.02, λ = 1.5 and f = 1.5Hz. When a1 = −0.16, the net fluid force on the AUG is nearly zero, and therefore, a cruise state is achieved. The vertical and horizontal forces of all the components (left and right wings and fuselage) are shown in Fig. 4. Overall, all the forces oscillate periodically with a constant frequency correlated to the travelling wave frequency. The force curves of left and right hydro wings collapse to each other, as a result of the same geometry shape and symmetrically arranged on either side of the fuselage. The mean forces in one single period can also be obtained directly from Fig. 4. The mean vertical forces (FL) of all the components are very small, O (10−3), as shown in Table 1, and the net force along vertical direction is nearly zero neglecting its gravity and buoyancy forces. For the force components along horizontal direction, each hydro wing is owing a negative mean force (FD), while the FD of the fuselage is positive, indicating that the twin hydro wings produce thrust forces to overcome the drag force acting on the fuselage. The resultant force of the whole AUG along horizontal direction is also very close to zero. Therefore, Eq. (6) is fulfilled and the AUG is in its cruise state. In the following, the effects of the kinematic parameters on the propulsive efficiency of the AUG are investigated. Motivated by the
4. Results and discussion
D =T ⇒
∮ (pleft ·v left + pright ·v right) dτ .
(7)
(6)
where D represents the drag force acting on the fuselage, T represents the thrust force generated by the twin hydro wings, and Γis an oscillating period. Meanwhile, the output and input power of the AUG system can be defined directly. The output power can be considered as the power generated by the oscillating hydro wings to maintain the
Fig. 3. Schematic diagram of the travelling wave motion of current AUG in one period, (a) t = 0 T, (b) t = 1/4 T, (c) t = 1/2 T, (d) t = 3/4 T. 53
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Fig. 4. Vertical and horizontal force histories of left and right hydro wings and fuselage.
frequency. There is an optimum travelling frequency for each moving velocity, leading to the highest propulsive efficiency. Besides, it can be also found that the highest propulsive efficiency shows a little decrease when the moving velocity increases, which means if we want to gain high moving velocity we have to pay price for cutting down the endurance. The magnitudes of corresponding wave amplitudes, a1, is shown in Fig. 5 (b). The magnitudes of wave amplitude of all the moving velocity have the same trend: the smaller wave frequency the larger wave amplitude, and also the small moving velocity corresponding to a small wave amplitude, which is consistent with the results of propulsive efficiency. As mentioned above, the effects the wave frequency, wave amplitude and moving velocity can be combined as the effect of the dimensionless number St, and the corresponding result of Fig. 5 (a) and (b) is shown in Fig. 6. The results of all moving velocity almost collapse, and the optimum St for maximum propulsive efficiency is around 0.4–0.6. Therefore, for the study of current AUG model, Strouhal number is still a key parameter for the evolution of propulsive efficiency of biomimetic locomotion.
Table 1 Average vertical and horizontal forces of twin hydro wings and fuselage.
mean FL mean FD
left hydro wing
right hydro wing
fuselage
−0.0066 −0.1946
−0.0066 −0.1946
−0.0030 0.3901
biomimetic of animal locomotion, the parameters used in current study are chosen in proper ranges as: the travelling wave frequency f = 0.8–3.0, the travelling wave length λ = 0.8–2.5 and the cruise velocity V = 0.5–1.2, and the relevant Reynolds number varies from 100 to 240, accordingly. 4.2. Effects of travelling wave frequency on propulsive efficiency In the following two subsections, we are going to investigate the effects of travelling wave parameters on the propulsive efficiency of current AUG. The effect of frequency is first studied in this subsection. The oscillating or fluctuating frequency might be one of the most important parameters in biomimetic locomotion (Triantafyllou et al., 1991; Shao et al., 2010b). The dimensionless Strouhal number (St) combing the effect of travelling wave frequency and amplitude is introduced as
2h max⋅f St = , U0
4.3. Effects of the travelling wave amplitude on propulsive efficiency The effects of travelling wave amplitude are investigated in this subsection. The travelling wave frequency is fixed as f = 1.5Hz, the wave amplitude is varied in the range of a1 = 0–0.5 and the wave length λ is adjusted to obtain a state of zero net force. Fig. 7 (a) shows the result of propulsive efficiency versus travelling wave amplitude. As shown in Fig. 7 (a), the propulsive efficiency increases monotonous with a1 to the peak and then decrease with further increase of a1. The magnitude of maximum efficiency of each moving velocity decreases with the increase of moving velocity, which is similar with the trend showing in Fig. 5. The magnitude of wave length λ to the wave amplitude a1 is shown in Fig. 7 (b). Obviously, the wave length has a linear dependence on the magnitude of wave amplitude. Linear fittings are also made in Fig. 7 (b), the fitting lines under different moving velocity
(9)
where f and U0 are the hydrofoils' travelling frequency and AUG's advancing speed, hmax represents hydrofoils' maximum travelling amplitude along the Y direction and it can be calculated from Eqs. (1) and (2) easily. In current simulation, the travelling wave length is fixed as λ = 2.5, and the travelling wave amplitude, a1, is adjusted to achieve a cruise state. The result of frequency effect is shown in Fig. 5 (a), and it is seen from the result that the propulsive efficiency for each moving velocity increases to its maximum and then decreases with the increasing of
Fig. 5. Propulsive efficiency versus travelling frequency under different moving velocities and the corresponding magnitudes of travelling wave amplitudes a1. 54
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Fig. 8. Propulsive efficiency versus travelling wave length under different moving velocities.
Fig. 6. Propulsive efficiency versus dimensionless Strouhal number.
show the similar slope, around −1.8. In contrast, the curves of propulsive efficiency versus travelling wave length is also plot in Fig. 8, according to the same set of data in Fig. 7. The curves show similar trend with those in Fig. 7 (a), whereas the difference is that the corresponding wave lengths of maximum efficiency under different moving velocity are in a relative small range, around λ = 2.4, and the curves under different moving velocity are close to each other. Therefore, the effect of travelling wave length shows an equilibrium to the combined effects of wave amplitude and moving velocity. On the other hand, similar to Fig. 6, the effect of Strouhal number on propulsive efficiency is plot in Fig. 9. The maximum propulsive efficiency is achieved around St = 0.6–1.0, which is a little larger than the range in Fig. 6. 4.4. Discussions
Fig. 9. Propulsive efficiency of AUG versus different Strouhal numbers under different moving velocity.
4.4.1. Flow structures around the AUG The flow structures around and in the wake of the AUG is closely related to the fluid force acting on the AUG, therefore, here we present some typical vortex structures under different conditions to investigate the coherence between vortex topology and hydrodynamic performance of the AUG. The vortex iso-surfaces with λ2 definition (Jeong and Hussain, 1995) in the wake are illustrated in Fig. 10. For comparison, three typical cases under the same advancing velocity is V = 0.5 and travelling wave length λ = 2.5, and different travelling wave frequencies, i.e. f = 1.0, 1.2 and 2.0. The corresponding Strouhal number are St = 0.52, 0.43 and 0.24, and the resulted propulsive efficiencies are η = 36%, 42% and 31.5%, respectively. As can be seen from Fig. 10, the vortex rings/tubes emerge in the wakes of hydro wings under different travelling frequencies. This phenomenon is similar with our previous work (Li et al., 2016). It is worth mentioning that the expanding angle (or obliquity) of the shedding vortex rings, i.e. the angle between the centre line of vortex rings and horizontal line, varies one by one. For case f = 1.0 as shown in Fig. 10 (a), the oscillating amplitude a1 is larger than the one of case
f = 1.2 [Fig. 10 (b)], and therefore it generates a larger expanding angle of shedding vortex wake. Corresponding pressure contours and velocity distributions are shown in Fig. 11. Small low-pressure area is distributed only around the tail of the wing as shown in Fig. 11 (a) and (b). Since the tail is bending downward, a low-pressure area around the bottom surface of the tail plays a positive role in the propulsion of the wing and the lower pressure the larger thrust force. Whereas for case f = 2.0 [Fig. 10 (c)], a larger expanding angle is also resulted even with a smaller oscillating amplitude, comparing with the case for f = 1.2 of higher propulsive efficiency. It indicates that, with a large travelling wave frequency, strong momentum exchanging happens during the shedding of the vortex ring. It is confirmed by the corresponding pressure contours as shown in Fig. 11 (c), a large low-pressure area is distributed around the whole bottom surface of the wing. In a sum, the vortex shedding from the trailing edge of hydro wing is accompanied by the release of energy. The shedding vortex ring is inclined to the inflow direction, the corresponding released energy is decomposed into
Fig. 7. Propulsive efficiency versus travelling amplitude and travelling wave length under different moving velocities. 55
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Fig. 10. Instantaneous vortex structures for cases under different conditions at t = 3.0 T.
Fig. 11. Pressure contours and velocity distributions in cross-section plane of AUG hydro wing, corresponding to Fig. 10.
differences among them lie in the width and thickness of the vortex rings. Similar conclusions were also stated by Blondeaux et al. (2005) and Dong et al. (2006). Second, in spite of the Reynolds number effect on thrust and efficiency as reported in Ashraf et al. (2011) and Deng et al. (2016), the major concerns in current study is the effects of kinematic parameters with fixed Reynolds number. The results of aforementioned studies indicate that the thrust/efficiency tendencies with the foil thickness at different Reynolds number are similar (Ashraf et al., 2011) and also the efficiency tendency with the frequency (Deng et al., 2016). In other words, the Reynolds number does not have significant effect on the thrust/efficiency tendencies with those kinematic parameters. Therefore, the efficiency tendency obtained in our study at Re = 200 may qualitatively similar with the tendency at higher Reynolds number. Last but not least, self-propelled micro swimmers are of great interests for in vivo biomedical applications (Zhang et al., 2009; Kosa et al., 2012). Those micro swimmers can be used to diagnose diseases, delivery drug, etc. From a fluid mechanic point of view, it is a smallscale vehicle moves in a viscous fluid with a low velocity, and the relevant Reynolds number is normally in the order of 102. The improvement of propulsive efficiency is also an important issue for the design of these micro swimmers. Current study on the bioinspired AUG may also provide insights in the field of micro swimmer design.
horizontal and vertical components, and the horizontal part is correlated to the propulsive performance. Therefore, it can be conjectured that the less expanding angle, the more horizontal energy is released and the better the propulsive efficiency. 4.4.2. Reynolds number effects In current study, the Reynolds number of the flow is fixed around Re = 200, which might be much lower than the traditional flow condition at which the Reynolds number is around 105 to 106. The reason that the flow with a relative low Reynolds number in current study can be explained into the following three aspects: First, our intention of simulation at Re = 200 is basically from the pioneer simulation work of biomimetic propulsion studies by Blondeaux et al. (2005) and Dong et al. (2006), in which propulsive performance of a single biomimetic wing was numerical investigated with Reynolds number in the range from 100 to 400. Our current study can be considered as an extension of these pioneer works. Instead to the single biomimetic wing, the whole AUG model with wing–fuselage interaction is simulated. Meanwhile, a corresponding new definition of propulsion efficiency based on the friction of fuselage and thrust of hydro wing is proposed. In view of this, a close Reynolds number with previous work is chosen. On the other hand, concerning the Reynolds number effects, our preliminary results indicate that the vortex topologies are similar for cases with different Reynolds number. The 56
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5. Conclusions A conceptual design of AUG with a pair of bioinspired hydro wings undergoing travelling wave motion is introduced in this paper and the flow field around the AUG is simulated by solving the incompressible viscous Navier–Stokes equations coupled with the optimized immersed boundary method to capture the moving boundaries. The hydrodynamic performance of AUG as well as the vortex structures in the wake of the AUG are investigated and discussed. The force balance along the stream direction is guaranteed by adjusting the kinematic parameters of the travelling wave motion, thus the cruise state of the AUG with a constant velocity can be achieved. Thereafter, the propulsive efficiency under cruise state is well defined. Here we briefly summarize the results obtained in current work. 1. The main kinematic parameters of the hydro wing, e.g. the travelling wave amplitude, travelling wave frequency and the wave length has the similar effects on AUG's propulsive efficiency. Concretely, propulsive efficiency increases to its maximum and then decreases with the increasing of above parameters. To estimate the combined effect of these parameters, the non-dimensional Strouhal number is proposed, and there is an optimum range of Strouhal number in which the propulsive efficiency reaches its maximum, regardless of any single parameter. 2. To achieve a cruise state, the adjusted wave length has a linear dependence on the magnitude of wave amplitude and the linear fitting slopes under different moving velocities close to each other, around −1.8, which can provide a much convenient and support for the control of such bioinspired AUG. 3. The vortex structures in the wake of current AUG have the close connection with the propulsive efficiency and the high propulsive efficiency is usually correlated to a small expanding angle of vortex rings or vortex ring obliquity. Acknowledgments The authors would like to acknowledge the support provided by Zhejiang Provincial Natural Science Foundations (Nos. LY18A020002, LQ18E090010) and the National Natural Science Foundations of China (No. 91634103, 51279184). References Asakawa, K., Kobayashi, T., Nakamura, M., Watanabe, Y., Hyakudome, T., Itoh, Y., Kojima, J., 2012. Results of the first sea-test of Tsukuyomi: a prototype of underwater gliders for virtual mooring. In: 2012 Oceans, 14-19 Oct. 2012, pp. 1–5. Ashraf, M., Young, J., Lai, J., 2011. Reynolds number, thickness and camber effects on flapping airfoil propulsion. J. Fluid Struct. 27, 145–160. Blondeaux, P., Fornarelli, F., Guglielmini, L., Triantafyllou, M.S., Verzicco, R., 2005. Numerical experiments on flapping foils mimicking fish-like locomotion. Phys. Fluids 17, 113601. Chen, Y.J., Chen, H.X., Ma, Z., 2015. Hydrodynamic analyses of typical underwater gliders. J. Hydrodyn. 27, 556–561. Clark, R.P., Smits, A.J., 2006. Thrust production and wake structure of a batoid-inspired oscillating fin. J. Fluid Mech. 562, 415–429. Claus, B., Bachmayer, R., 2016. Energy optimal depth control for long range underwater vehicles with applications to a hybrid underwater glider. Aut. Robots 40, 1307–1320. Claustre, H., Beguery, L., 2014. SeaExplorer glider breaks two world records. Sea Technol. 55, 19–21. Davis, R.E., Eriksen, C.C., Jones, C.P., 2002. Autonomous Buoyancy-driven Underwater Gliders. Taylor and Francis, London. Deng, J., Sun, L., Teng, L., Pan, D., Shao, X., 2016. The correlation between wake
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