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Influence of lateral distance and phase difference on energy absorption performance of undulating airfoils in a side-by-side arrangement
European Journal of Mechanics / B Fluids 68 (2018) 193–200
Contents lists available at ScienceDirect
European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Influence of lateral distance and phase difference on energy absorption performance of undulating airfoils in a side-by-side arrangement Li Ding, Diangui Huang * School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
highlights • • • •
We simulate effects on efficiency of two side-by-side undulating airfoils. Within a certain range, efficiency of two airfoils is higher than a single one. Efficiency increases first and then decreases as lateral distance increases. As phase difference varies, curve of efficiency takes on an antisymmetric form.
article
info
Article history: Received 12 October 2017 Received in revised form 28 November 2017 Accepted 19 December 2017 Available online 29 December 2017 Keywords: Traveling wave motion Airfoils in a side-by-side arrangement Energy absorption efficiency Lateral distance Phase difference
a b s t r a c t It is recognized that traveling wave motion can extract energy from moving water or wind under certain motion parameters. The NACA0012 airfoil section is used as a two-dimensional simplified model of the fishlike body in this paper. We numerically investigated the influence of lateral distance and phase difference on energy absorption performance of two undulating airfoils in a side-by-side arrangement. The findings show that: within a certain range, the energy absorption efficiency of the side-by-side undulating airfoils is higher than the data of a single airfoil. When the phase difference is 0◦ , the energy absorption efficiency of undulating airfoils increases first and then decreases with the increasing lateral distance, existing a maximum efficiency. When the lateral distance is constant, as the phase difference varies, the curve of absorption efficiency almost takes on an antisymmetric form with the period of 360◦ . As the phase difference increases, one of the airfoil’s efficiency decreases first, then increases and finally decreases, while the other one’s efficiency increases first, then decreases and finally increases. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction In the nature selection and billions of years of evolutionary process, fish and other aquatic organisms gradually have a high swimming efficiency and mobility [1,2]. As a common phenomenon, fish school swimming has attracted the attention of many researchers. Fish school swimming can reduce the probability of arrest, improve foraging efficiency, and enhance the communication and reproduction between fish. According to the recent study [3], it is found that the duration of swimming of fish school can be increased by 2–6 times compared with the single fish, which indicates that fish school swimming can increase the thrust and reduce the energy consumption. Weihs [4] used the non-viscous flow model theory to analyze the three kinds of favorable interaction between the individuals in fish school, which are the interference of the flow in the same horizontal layer, lateral interference and the interference
* Corresponding author.
E-mail address: [email protected] (D. Huang).
https://doi.org/10.1016/j.euromechflu.2017.12.007 0997-7546/© 2017 Elsevier Masson SAS. All rights reserved.
between the horizontal layers. Dong et al. [5] have investigated flow over traveling wavy foils in a side-by-side arrangement and analyzed the hydrodynamics and flow structure for the flow over the in-phase and anti-phase traveling wavy foils. They concluded that when the flow is over the in-phase traveling wavy foils, the wavy foils save the swimming power while the forces acting on the anti-phase traveling wavy foils are enhanced. Based on the two-dimensional model, Chang [6] analyzed the mutual interference between fishes in inline, ‘triangular’ and ‘diamond’ shaped arrays in his doctoral thesis. The results showed that in the twodimensional model, the fish at the end of the array could be subjected to beneficial flow disturbances. In order to reveal the energy saving mechanism in fish school swimming, Wang et al. [7] used an adaptive mesh projection method and immersion boundary method to numerically simulate the basic unit of fish school swimming with three fishes. They found that when the middle fish falls behind about 0.4 times of body length, its swimming frequency is only 54% of that of the other two fishes, to demonstrate that the ‘channeling effect’ is remarkable.
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L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
Nomenclature L
λ A(x) U d T u, v x, yw cm c y ff y fp p
vy µ
Psr Ps t E ˙ m ds
ρ ηE θ
Amax h Cdd
Φd
Length of camber line (body length) Wavelength The space-varying amplitude The incoming flow speed Lateral difference The undulation period The velocity components in the x and y directions Cartesian coordinates The speed of a wave Dimensionless wave speed Skin friction Pressure drag The pressure acting on the top surface The y component of velocity The dynamic viscosity of a fluid The output power A dimensionless power output Time The kinetic energy The mass flow rate of air (or water) The infinitesimal element of surface A function of air density The energy utilization efficiency Phase difference The maximum amplitude of the undulation wave The maximum swept height of the traveling wave The drag coefficient The motion phase delay
Since fish can generate thrust through the traveling wave, does the traveling wave motion harvest energy from the wind or water? Huang et al. [8] found that at a certain range of dimensionless velocity c, the undulating foil could absorb energy from the water. It is a new way to harvest energy, which provides a new model for absorption of wind and water energy. Li [9] studied the effect of the wavelength and amplitude on the power coefficient and energy extraction efficiency at Re = 4×105 and found that there is an optimal case that making the energy extraction efficiency to the maximum. Ji [10] systematically investigated the effects of Reynolds number on energy extraction performance, pointing out that when the wavelength is constant, as the Reynolds number increases, the energy extraction efficiency first increases and then deceases. This is the new type of harvesting energy which is different from the traditional wind turbine and flapping wing It is worthy of further study. In order to study the law of energy absorption of undulating airfoil, this paper numerically simulates the side-by-side undulating airfoils. By changing the lateral distance and phase difference of the upper airfoil, we studied the effects of these two parameters on harvesting energy of the side-by-side undulating airfoils. 2. Physical and numerical method
Fig. 1. Physical model.
Fig. 2. Traveling wave motion diagram.
In the course of traveling wave motion, the type of motion is unique when given the wavelength and maximum amplitude and kept the length of camber line (body length) unchanged. Thus undulatory motion equation [11,12] can be expressed as: yw (x, t) = A(x) cos(
2π
λ
x−
2π T
t)
(1)
where T is the period. We only talked about the case of constant amplitude here, so the amplitude equation is expressed as: A(x) = Amax = c0 L.
(2)
The following is the speed of a wave cm determined by its wavelength and wave period. cm =
λ T
.
(3)
Herein the speed of a wave is non-dimensionalized with the incoming flow speed U and gets dimensionless wave speed as presented as follows: c=
λ TU
.
(4)
2.1. Physical model In this paper, the NACA0012 airfoil shape is used as a twodimensional simplified model of the airfoil, which is traveling in the water. The physical model is shown in Fig. 1 and the traveling wave motion diagram is shown in Fig. 2. As shown in these two figures, L is the length of camber line (body length), d is the lateral distance of two airfoils, U is the velocity of the flow, λ is the wavelength of the traveling wave, A is the amplitude.
2.2. Definitions of the captured energy of an undulatory swimming body and its energy absorption efficiency The friction force and the pressure difference force [11–13] affect the model of the traveling wave motion, where the friction force comes from viscous shear force of the model surface, the pressure difference force comes from the pressure difference between the model surface flows. In the present study, the y
L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
(a) Computational domain diagram.
195
(b) Local mesh around the airfoil.
Fig. 3. Computational mesh.
components of both types of drag forces on an infinitesimal strip f p of the airfoil with undulatory motion are represented by fy and fy which have the following expressions respectively:
⎧ ⎨f f = µ[2 ∂v − dyw ( ∂v + ∂ u )] y on the upper surface ∂y dx ∂ x ∂y ⎩ p fy = −p
(5)
⎧ ⎨f f = µ[−2 ∂v + dyw ( ∂v + ∂ u )] on the lower surface y ∂y dx ∂ x ∂y ⎩ p fy = p
(6)
where u and v are the velocity components in the x and y directions.
µ is the dynamic viscosity of a fluid. p is the pressure acting on the top surface. The expression for the y component of velocity of the infinitesimal body is: y [t + dt] − y [t]
vy =
dt
Psr =
(fyf
+
v
fyp ) y ds
.
E=
2
˙ 2= mU
1 2
.
(8)
(ρ hU)U
2
(9)
˙ is the mass flow rate of air (or water) through a reference where m area in kg/s and h is the maximum swept height of the traveling wave. Therefore, the energy absorption efficiency of the undulating airfoil can be defined as follows:
ηE =
Psr E
.
The computational mesh was generated in Pointwise. As shown in Fig. 3(a), the entire computational domain is a circle with a radius of 40L which is divided into A and B regions. Region A was discretized into structured mesh while region B was used an unstructured grid for computing unsteady flow around the airfoils. The spring smoothing and remeshing techniques of FLUENT were employed in the present dynamic meshing. Local mesh is shown in Fig. 3(b). All simulations were performed using a commercial CFD software FLUENT15.0. The two-dimensional incompressible unsteady Navier–Stokes equations are employed as governing equations as follows: →
(7)
If Psr is divided by the value of ρ U3 L, a dimensionless power output from the modeled airfoil Ps is then achieved. In addition, E is the kinetic energy per unit time of the flow that passes through the airfoil can be calculated as: 1
3.1. Computational mesh and numerical schemes
∇·u =0
So the output power of the undulatory swimming airfoil can be calculated as follows:
∮
3. Numerical simulation
(10)
(11)
→
ρ(
→ ∂u → → + u · ∇ u) = ρ f + ∇ · σ ∂t
(12) →
where ∇ is the spatial gradient operator, f is the body force on → micro-unit. And the velocity u is defined as: →
u = (u, v)T .
(13)
The stress tensor of fluid σ is given as: →
→
→T
σ = −p I + µ(∇ u + ∇ u ). →
(14)
Here I is unit tensor, µ is the dynamic viscosity. Through the comparison of several turbulence models, the Transition-SST model was selected in this paper eventually. Furthermore, the Y+ value in the near-wall region are clarified. As the value of Reynolds number of this simulation is 195,000, the Y+ value range is from 0 to 7. The inlet and outlet boundary conditions were set into speed inlet and pressure outlet respectively, and the surface of the airfoils was set into solid wall. The working fluid is water. A pressure based solver was used for the solution.
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Fig. 4. Experimental set up of hydrodynamic force measurement [15].
Fig. 5. Comparison between numerically simulation and experiment.
Table 1 Results for mesh independency and convergence of time step. Cells
∆t
Lower efficiency
Upper efficiency
134,834 281,091 613,715 281,091 281,091
1000 1000 1000 500 2000
0.30091 0.27246 0.27298 0.27477 0.27612
0.30081 0.28093 0.27993 0.28099 0.28084
3.2. Mesh independency and convergence of time step validation To confirm the mesh independency and convergence of time step (∆t), we have performed extensive validation of them. The motion parameters were c = 0.5, λ = 0.12 m, Amax = 0.015 m, d = 0.5L, θ = 0◦ . Three different time steps and grids were employed. The results are shown in Table 1. It can be seen that as the number of mesh grows, the values of efficiency are close to each other. Moreover, the value of efficiency is almost unchanged while the time step increases. For saving the computing resources, we finally chose the mesh of 281,091 cells and 1000 time-steps. 3.3. Numerical model validation In order to verify the accuracy and reliability of numerical simulation, we refers to the experiment that Nishio et al. [14,15] completed about fishlike propulsive motion. In this experiment, they estimated the thrust and drag forces at different phase lag angle Φd . As shown in Fig. 4, the experimental model is a three-dimensional model with a NACA0018 airfoil section, which is divided into four joints that controlled by the PC. During the experiment, the model was placed in a circulating water tank and fixed at its 1/4 chord length. Dragging the model horizontally by the motor, the thrust in this direction was measured by the applied load. When the total horizontal force is zero, the applied load force is the drag force. Here, drag force was reflected by the drag coefficient Cdd and phase lag angle Φd was the movement phase difference between the model head and tail. The model’s chord length is 0.15 m, the aspect ratio is 0.18, the wave period is two seconds and the cruising speed is 0.35205 m/s. Fig. 5 shows the result about the comparison of numerical simulation and the experiment. The Nishio’s experimental condition parameters and motion modes were selected to simulate the twodimensional flexible body with NACA0018 airfoil propelling in water for finding the relationship between the drag coefficient Cdd and the different phase lag angle Φd .
Fig. 6. The curve of energy absorption varying with lateral distance.
As shown in Fig. 5, the change trade of the drag coefficient with phase lag angle by simulated is very close to the experimental result. Compared with it, the computed drag coefficient is a little bit bigger than the experimental result at small values of Φ d . However, at large values of Φ d , the drag coefficient by simulated is similar with the data by experimented. In general, the aforementioned numerical model is accurate and reliable. 4. Numerical results and discussions According to the relevant research, in this study, the incoming flow velocity U is 1 m/s, body length L is 0.2 m, the wavelength λ is 0.12 m, the dimensionless velocity c is 0.5 and the maximum amplitude Amax is 0.015 m. Based on the above parameters, we investigated the influence of lateral distance and phase difference on energy absorption performance of undulating airfoils of NACA0012 airfoil in a side-by-side arrangement. 4.1. Influence of lateral distance The energy absorption performance of the side-by-side undulating airfoils was studied when the phase difference was 0◦ , which meant that the two airfoils were in the same traveling wavy motion. The curve of energy absorption varying with lateral distance is presented in Fig. 6. It can be seen that the energy absorption efficiency of two undulating airfoils increases first and then decreases
L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
197
Fig. 7. Total pressure contour of single airfoil.
Fig. 9. Vorticity contour of single airfoil.
with the increase of lateral distance d. Furthermore, the energy absorption efficiencies of two airfoils are both higher than the data of single one in a certain range. When d = 0.5L, the efficiency of airfoils achieve the maximum values with the upper one got 28.093% and the lower one got 27.246%. As the lateral distance continues to increase, the energy absorption efficiency of sideby-side undulating airfoils decreases and is eventually consistent with the single one’s value. Because of in-phase traveling wave, the energy absorption efficiency of the upper and the lower airfoils is substantially the same at each distance. Fig. 7 displays the total pressure contour of single undulating airfoil. When the fluid flows over the airfoil, a large area of low pressure in the wake of the airfoil is observed clearly which suggesting that the undulating airfoil extracts energy from moving water. Fig. 8 shows the total pressure contours of the side-by-side undulating airfoils in different lateral distances in the same phase wave. When the distance is small, the interaction between the two
airfoils is strong, the low pressure area is bigger and the color of the area is darker. Compared with the single one, the side-byside undulating airfoils absorb more energy from water, as shown in Fig. 8(a), (b). As the lateral distance increases, the interaction becomes weak, the scale of low pressure area gradually decreases and the color of it becomes more shallow as shown in Fig. 8(c). So the energy extracted from moving water is less. As the distance increases to 3L in Fig. 8(d), the total pressure distribution of the two airfoils is almost the same, indicating that when the lateral distance is enough large, the interaction between them disappears finally. In this case, the energy absorption efficiency is close to that of single airfoil. Fig. 9 presents the vorticity contour of single undulating airfoil. It can be seen that the vortex in the wake of the airfoil is in a staggered formation. Fig. 10 shows the vorticity contours of the side-by-side undulating airfoils in different lateral distances in the same phase wave. As shown in Fig. 10(a), when the lateral distance
(a) d = 0.35L.
(b) d = 0.5L.
(c) d = 1L.
(d) d = 3L.
Fig. 8. Total pressure contours of the side-by-side undulating airfoils in different lateral distances.
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(a) d = 0.35L.
(b) d = 0.5L.
(c) d = 1L.
(d) d = 3L.
Fig. 10. Vorticity contours of the side-by-side undulating airfoils in different lateral distances.
between the airfoils is small, the interference of the vortices in the wake of the two airfoils is strong. The negative and positive vortices are respectively integrated but dissipate quickly. When the distance is 0.5L, the interaction of the two airfoils is strong. The vortex structure disorders and dissipates slowly that are shown in Fig. 10(b). With the distance increasing, the vortices shedding is slower and the vortex street is longer. The interference gradually decreases until it disappears, as shown in Fig. 10(c), (d). 4.2. Influence of phase difference In this section, we studied the influence of phase difference θ on the energy absorption performance of the side-by-side undulating airfoils Fixing the distance (d = 0.5L), the different phase difference is simulated numerically. Fig. 11 shows the trend of energy absorption efficiency with varying phase difference. We can conclude that the curve of absorption efficiency is almost in an antisymmetric form with the period of 360◦ with the changing phase difference. Moreover, the energy absorption efficiency of both is higher than the data of single one. When θ is 0◦ , the two airfoils are in the same phase so that the energy absorption efficiency is almost consistent. In the range of 0◦ to 180◦ , the efficiency of the upper airfoil decreases first and then increases with the increase of the phase difference. At θ = 60◦ , the efficiency is the lowest which is 24.797%. However, the efficiency of the lower airfoil increases first and then decreases with the increasing phase difference. At θ = 120◦ , the efficiency is the highest, reaching to 30.387%. When θ is 180◦ , the two airfoils are in the reverse wave so that the difference of two energy absorption efficiency is very small. In the range of 180◦ to 360◦ , the efficiency of the upper airfoil increases first and then decreases with the increase of the phase difference. At θ = 300◦ , the efficiency is the
Fig. 11. The relationship of absorption and phase difference in constant distance (d
= 0.5L).
highest, reaching to 30.415%. However, the efficiency of the lower airfoil decreases first and then increases with the increase of the phase difference. At θ = 300◦ , the efficiency is the lowest which is 24.387%. When θ is 360◦ , the two airfoils return to the in-phase wave. Fig. 12 shows the total pressure contours of different phase differences in the same time. When there is a phase difference in the wave, the total pressure distribution changes significantly. As shown in Fig. 12, the total pressure distribution of two airfoils is very different from that of single airfoil (Fig. 7), indicating that
L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
(a) θ = 0◦ .
(b) θ = 60◦ .
(c) θ = 180◦ .
(d) θ = 300◦ .
199
Fig. 12. Total pressure contours of different phase differences.
(a) θ = 0◦ .
(b) θ = 60◦ .
(c) θ = 180◦ .
(d) θ = 300◦ .
Fig. 13. Vorticity contours of different phase differences.
the interference between the side-by-side airfoils is strong and they can absorb more energy from moving water. When θ is 0◦ in Fig. 12(a), the upper and lower airfoils are in an in-phase wave, the energy absorption efficiency is the same. When θ is 60◦ in Fig. 12(b), the area of low-pressure changes and the low pressure of the upper airfoil moves to the lower airfoil, which indicates that the lower airfoil extracts more energy compared to the upper one. When θ is 180◦ in Fig. 12(c), the upper and lower airfoils are in anti-phase wave and the total pressure distribution is symmetrical so that the efficiency of the two airfoils are the same. When θ is 300◦ in Fig. 12(d), the low pressure of the lower airfoil moves to the upper airfoil, which indicates that the upper airfoil extracts more energy compared to the lower one. Vorticity contours of different phase differences in the same time are shown in Fig. 13. As shown in Fig. 13, the phase difference between the side-by-side airfoils makes the vortex structure more complex, which is completely different from the vorticity contour
of the single airfoil (Fig. 9). When θ is 0◦ in Fig. 13(a), the vortex structures in the wake of the in-phase airfoils are the same. When θ is 60◦ in Fig. 13(b), because of the existence of phase difference, the positive vorticity of the upper airfoil moves to the lower airfoil and creates a strong disturbance in the wake of the lower airfoil. When θ is 180◦ in Fig. 13(c), the vortex structures in the wake of the antiphase airfoils are the opposite. When θ is 300◦ in Fig. 13(d), the situation is opposite to θ = 60◦ , the negative vorticity of the lower airfoil moves to the upper airfoil and creates a strong disturbance in the wake of the upper airfoil. 5. Conclusion The NACA0012 airfoil section is used as a two-dimensional simplified model of the fishlike body in this paper, which is traveling in the water. We investigated the influence of lateral distance and phase difference on energy absorption performance of undulating
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airfoils of NACA0012 airfoil shape in a side-by-side arrangement within a certain range. Here, we briefly summary the results as follows: (1) When the phase difference between the side-by-side undulating airfoils is 0◦ , with the increase of the lateral distance d, the energy absorption efficiency of undulating airfoils increases first and then decreases. In addition, the energy absorption efficiency of both is almost higher than the data of single one in a certain range. When d = 0.5L, the efficiency of airfoils reaches the maximum value with the upper one got 28.093% and the lower one got 27.246%. As the lateral distance is enough far, the interference of the two airfoils disappears and return to the single wave. (2) When the lateral distance is constant, with the change of phase difference, the curve of absorption efficiency is nearly in an antisymmetric form with the period of 360◦ . Moreover, the energy absorption efficiency of both is higher than the data of single one. As the increase of the phase difference, the efficiency of the upper airfoil decreases first, then increases and finally decreases. At θ = 60◦ , the efficiency of it is the lowest which is 24.797%, while at θ = 300◦ , the efficiency of it is the highest, reaching to 30.415%. However, the efficiency of the lower airfoil increases first, then decreases and finally increases. At θ = 120◦ , the efficiency of it is the highest, reaching to 30.387%, while at θ = 300◦ , the efficiency of it is the lowest which is 24.387%. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 51536006, 51576133), and the program of the Shanghai Science and Technology Commission (No. 17060502300).
References [1] J. Gray, Studies in animal locomotion VI. The propulsive powers of the dolphin, J. Exp. Biol. 13 (2) (1936) 192–199. [2] B.G. Tong, X.Y. Lu, A review on biomechanics of animal flight and swimming, Adv. Mech. 34 (1) (2004) 1–8. [3] L.P. Zhang, X.H. Chang, X.P. Duan, Numerical simulation of undulated swimming of two in-line fishes based on dynamic hybrid mesh, in: Beijing: Unconventional Aerodynamics Symposium. 2007, pp. 28-34. [4] D. Weihs, Hydromechanics of fish schooling, Nature 241 (5387) (1973) 290– 291. [5] G.J. Dong, X.Y. Lu, Characteristics of flow over traveling wavy foils in a side-byside arrangement, Phys. Fluids 19 (5) (2007) 99. [6] X.H. Chang, Unsteady Incompressible Flow Solver on Moving Hybrid Grids and Numerical Study of Flow Mechanism for Fish Swimming. Mianyang: China Aerodynamics Research and Development Center, 2012. [7] L. Wang, C.J. Wu, Energy saving mechanism of ‘‘channeling effect’’ in fish school swimming, Chin. J. Theor. Appl. Mech. 43 (1) (2011) 18–23. [8] D. Huang, J. Zhang, X. Sun, New method for harvesting energy from fluid flow based on undulatory motion, Int. J. Green Energy 14 (6) (2017) 540–547. [9] H.Z. Li, A kind of new wind energy extraction device–traveling wavy plate, J. Eng. Therm. Energy Power (9) (2017) 74–79. [10] F. Ji, D.G. Huang, Effects of Reynolds number on energy extraction performance of a two dimensional undulatory flexible body, Ocean Eng. 142 (2017) 185– 193. [11] G.J. Dong, Numerical Investigation on the Viscous Flow Around a Flexible Body with Lateral Motion, University of Science and Technology of China, Hefei, 2006. [12] J. Deng, Simulation of Three-Dimensional Flow with Moving Bodies and Analysis of Vortical Structures in the Wake, Zhejiang University, Hangzhou, 2007. [13] X.H. Chang, Unsteady numerical method based on dynamic hybrid mesh and simulation of bio-fluid flow. Mianyang: China Aerodynamics Research and Development Center, 2007. [14] S. Nishio, Z. Chen, Y. Doi, et al., Analysis of fish-like propulsion, J. Visual. Soc. Jpn 23 (2003) 41–44. [15] S. Nishio, K. Nakamura, A study on the propulsive performance of fish-like motion using waving wing model, J. Soc. Naval Archit. Jpn. (2002) 17–24.
Contents lists available at ScienceDirect
European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu
Influence of lateral distance and phase difference on energy absorption performance of undulating airfoils in a side-by-side arrangement Li Ding, Diangui Huang * School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
highlights • • • •
We simulate effects on efficiency of two side-by-side undulating airfoils. Within a certain range, efficiency of two airfoils is higher than a single one. Efficiency increases first and then decreases as lateral distance increases. As phase difference varies, curve of efficiency takes on an antisymmetric form.
article
info
Article history: Received 12 October 2017 Received in revised form 28 November 2017 Accepted 19 December 2017 Available online 29 December 2017 Keywords: Traveling wave motion Airfoils in a side-by-side arrangement Energy absorption efficiency Lateral distance Phase difference
a b s t r a c t It is recognized that traveling wave motion can extract energy from moving water or wind under certain motion parameters. The NACA0012 airfoil section is used as a two-dimensional simplified model of the fishlike body in this paper. We numerically investigated the influence of lateral distance and phase difference on energy absorption performance of two undulating airfoils in a side-by-side arrangement. The findings show that: within a certain range, the energy absorption efficiency of the side-by-side undulating airfoils is higher than the data of a single airfoil. When the phase difference is 0◦ , the energy absorption efficiency of undulating airfoils increases first and then decreases with the increasing lateral distance, existing a maximum efficiency. When the lateral distance is constant, as the phase difference varies, the curve of absorption efficiency almost takes on an antisymmetric form with the period of 360◦ . As the phase difference increases, one of the airfoil’s efficiency decreases first, then increases and finally decreases, while the other one’s efficiency increases first, then decreases and finally increases. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction In the nature selection and billions of years of evolutionary process, fish and other aquatic organisms gradually have a high swimming efficiency and mobility [1,2]. As a common phenomenon, fish school swimming has attracted the attention of many researchers. Fish school swimming can reduce the probability of arrest, improve foraging efficiency, and enhance the communication and reproduction between fish. According to the recent study [3], it is found that the duration of swimming of fish school can be increased by 2–6 times compared with the single fish, which indicates that fish school swimming can increase the thrust and reduce the energy consumption. Weihs [4] used the non-viscous flow model theory to analyze the three kinds of favorable interaction between the individuals in fish school, which are the interference of the flow in the same horizontal layer, lateral interference and the interference
* Corresponding author.
E-mail address: [email protected] (D. Huang).
https://doi.org/10.1016/j.euromechflu.2017.12.007 0997-7546/© 2017 Elsevier Masson SAS. All rights reserved.
between the horizontal layers. Dong et al. [5] have investigated flow over traveling wavy foils in a side-by-side arrangement and analyzed the hydrodynamics and flow structure for the flow over the in-phase and anti-phase traveling wavy foils. They concluded that when the flow is over the in-phase traveling wavy foils, the wavy foils save the swimming power while the forces acting on the anti-phase traveling wavy foils are enhanced. Based on the two-dimensional model, Chang [6] analyzed the mutual interference between fishes in inline, ‘triangular’ and ‘diamond’ shaped arrays in his doctoral thesis. The results showed that in the twodimensional model, the fish at the end of the array could be subjected to beneficial flow disturbances. In order to reveal the energy saving mechanism in fish school swimming, Wang et al. [7] used an adaptive mesh projection method and immersion boundary method to numerically simulate the basic unit of fish school swimming with three fishes. They found that when the middle fish falls behind about 0.4 times of body length, its swimming frequency is only 54% of that of the other two fishes, to demonstrate that the ‘channeling effect’ is remarkable.
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L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
Nomenclature L
λ A(x) U d T u, v x, yw cm c y ff y fp p
vy µ
Psr Ps t E ˙ m ds
ρ ηE θ
Amax h Cdd
Φd
Length of camber line (body length) Wavelength The space-varying amplitude The incoming flow speed Lateral difference The undulation period The velocity components in the x and y directions Cartesian coordinates The speed of a wave Dimensionless wave speed Skin friction Pressure drag The pressure acting on the top surface The y component of velocity The dynamic viscosity of a fluid The output power A dimensionless power output Time The kinetic energy The mass flow rate of air (or water) The infinitesimal element of surface A function of air density The energy utilization efficiency Phase difference The maximum amplitude of the undulation wave The maximum swept height of the traveling wave The drag coefficient The motion phase delay
Since fish can generate thrust through the traveling wave, does the traveling wave motion harvest energy from the wind or water? Huang et al. [8] found that at a certain range of dimensionless velocity c, the undulating foil could absorb energy from the water. It is a new way to harvest energy, which provides a new model for absorption of wind and water energy. Li [9] studied the effect of the wavelength and amplitude on the power coefficient and energy extraction efficiency at Re = 4×105 and found that there is an optimal case that making the energy extraction efficiency to the maximum. Ji [10] systematically investigated the effects of Reynolds number on energy extraction performance, pointing out that when the wavelength is constant, as the Reynolds number increases, the energy extraction efficiency first increases and then deceases. This is the new type of harvesting energy which is different from the traditional wind turbine and flapping wing It is worthy of further study. In order to study the law of energy absorption of undulating airfoil, this paper numerically simulates the side-by-side undulating airfoils. By changing the lateral distance and phase difference of the upper airfoil, we studied the effects of these two parameters on harvesting energy of the side-by-side undulating airfoils. 2. Physical and numerical method
Fig. 1. Physical model.
Fig. 2. Traveling wave motion diagram.
In the course of traveling wave motion, the type of motion is unique when given the wavelength and maximum amplitude and kept the length of camber line (body length) unchanged. Thus undulatory motion equation [11,12] can be expressed as: yw (x, t) = A(x) cos(
2π
λ
x−
2π T
t)
(1)
where T is the period. We only talked about the case of constant amplitude here, so the amplitude equation is expressed as: A(x) = Amax = c0 L.
(2)
The following is the speed of a wave cm determined by its wavelength and wave period. cm =
λ T
.
(3)
Herein the speed of a wave is non-dimensionalized with the incoming flow speed U and gets dimensionless wave speed as presented as follows: c=
λ TU
.
(4)
2.1. Physical model In this paper, the NACA0012 airfoil shape is used as a twodimensional simplified model of the airfoil, which is traveling in the water. The physical model is shown in Fig. 1 and the traveling wave motion diagram is shown in Fig. 2. As shown in these two figures, L is the length of camber line (body length), d is the lateral distance of two airfoils, U is the velocity of the flow, λ is the wavelength of the traveling wave, A is the amplitude.
2.2. Definitions of the captured energy of an undulatory swimming body and its energy absorption efficiency The friction force and the pressure difference force [11–13] affect the model of the traveling wave motion, where the friction force comes from viscous shear force of the model surface, the pressure difference force comes from the pressure difference between the model surface flows. In the present study, the y
L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
(a) Computational domain diagram.
195
(b) Local mesh around the airfoil.
Fig. 3. Computational mesh.
components of both types of drag forces on an infinitesimal strip f p of the airfoil with undulatory motion are represented by fy and fy which have the following expressions respectively:
⎧ ⎨f f = µ[2 ∂v − dyw ( ∂v + ∂ u )] y on the upper surface ∂y dx ∂ x ∂y ⎩ p fy = −p
(5)
⎧ ⎨f f = µ[−2 ∂v + dyw ( ∂v + ∂ u )] on the lower surface y ∂y dx ∂ x ∂y ⎩ p fy = p
(6)
where u and v are the velocity components in the x and y directions.
µ is the dynamic viscosity of a fluid. p is the pressure acting on the top surface. The expression for the y component of velocity of the infinitesimal body is: y [t + dt] − y [t]
vy =
dt
Psr =
(fyf
+
v
fyp ) y ds
.
E=
2
˙ 2= mU
1 2
.
(8)
(ρ hU)U
2
(9)
˙ is the mass flow rate of air (or water) through a reference where m area in kg/s and h is the maximum swept height of the traveling wave. Therefore, the energy absorption efficiency of the undulating airfoil can be defined as follows:
ηE =
Psr E
.
The computational mesh was generated in Pointwise. As shown in Fig. 3(a), the entire computational domain is a circle with a radius of 40L which is divided into A and B regions. Region A was discretized into structured mesh while region B was used an unstructured grid for computing unsteady flow around the airfoils. The spring smoothing and remeshing techniques of FLUENT were employed in the present dynamic meshing. Local mesh is shown in Fig. 3(b). All simulations were performed using a commercial CFD software FLUENT15.0. The two-dimensional incompressible unsteady Navier–Stokes equations are employed as governing equations as follows: →
(7)
If Psr is divided by the value of ρ U3 L, a dimensionless power output from the modeled airfoil Ps is then achieved. In addition, E is the kinetic energy per unit time of the flow that passes through the airfoil can be calculated as: 1
3.1. Computational mesh and numerical schemes
∇·u =0
So the output power of the undulatory swimming airfoil can be calculated as follows:
∮
3. Numerical simulation
(10)
(11)
→
ρ(
→ ∂u → → + u · ∇ u) = ρ f + ∇ · σ ∂t
(12) →
where ∇ is the spatial gradient operator, f is the body force on → micro-unit. And the velocity u is defined as: →
u = (u, v)T .
(13)
The stress tensor of fluid σ is given as: →
→
→T
σ = −p I + µ(∇ u + ∇ u ). →
(14)
Here I is unit tensor, µ is the dynamic viscosity. Through the comparison of several turbulence models, the Transition-SST model was selected in this paper eventually. Furthermore, the Y+ value in the near-wall region are clarified. As the value of Reynolds number of this simulation is 195,000, the Y+ value range is from 0 to 7. The inlet and outlet boundary conditions were set into speed inlet and pressure outlet respectively, and the surface of the airfoils was set into solid wall. The working fluid is water. A pressure based solver was used for the solution.
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Fig. 4. Experimental set up of hydrodynamic force measurement [15].
Fig. 5. Comparison between numerically simulation and experiment.
Table 1 Results for mesh independency and convergence of time step. Cells
∆t
Lower efficiency
Upper efficiency
134,834 281,091 613,715 281,091 281,091
1000 1000 1000 500 2000
0.30091 0.27246 0.27298 0.27477 0.27612
0.30081 0.28093 0.27993 0.28099 0.28084
3.2. Mesh independency and convergence of time step validation To confirm the mesh independency and convergence of time step (∆t), we have performed extensive validation of them. The motion parameters were c = 0.5, λ = 0.12 m, Amax = 0.015 m, d = 0.5L, θ = 0◦ . Three different time steps and grids were employed. The results are shown in Table 1. It can be seen that as the number of mesh grows, the values of efficiency are close to each other. Moreover, the value of efficiency is almost unchanged while the time step increases. For saving the computing resources, we finally chose the mesh of 281,091 cells and 1000 time-steps. 3.3. Numerical model validation In order to verify the accuracy and reliability of numerical simulation, we refers to the experiment that Nishio et al. [14,15] completed about fishlike propulsive motion. In this experiment, they estimated the thrust and drag forces at different phase lag angle Φd . As shown in Fig. 4, the experimental model is a three-dimensional model with a NACA0018 airfoil section, which is divided into four joints that controlled by the PC. During the experiment, the model was placed in a circulating water tank and fixed at its 1/4 chord length. Dragging the model horizontally by the motor, the thrust in this direction was measured by the applied load. When the total horizontal force is zero, the applied load force is the drag force. Here, drag force was reflected by the drag coefficient Cdd and phase lag angle Φd was the movement phase difference between the model head and tail. The model’s chord length is 0.15 m, the aspect ratio is 0.18, the wave period is two seconds and the cruising speed is 0.35205 m/s. Fig. 5 shows the result about the comparison of numerical simulation and the experiment. The Nishio’s experimental condition parameters and motion modes were selected to simulate the twodimensional flexible body with NACA0018 airfoil propelling in water for finding the relationship between the drag coefficient Cdd and the different phase lag angle Φd .
Fig. 6. The curve of energy absorption varying with lateral distance.
As shown in Fig. 5, the change trade of the drag coefficient with phase lag angle by simulated is very close to the experimental result. Compared with it, the computed drag coefficient is a little bit bigger than the experimental result at small values of Φ d . However, at large values of Φ d , the drag coefficient by simulated is similar with the data by experimented. In general, the aforementioned numerical model is accurate and reliable. 4. Numerical results and discussions According to the relevant research, in this study, the incoming flow velocity U is 1 m/s, body length L is 0.2 m, the wavelength λ is 0.12 m, the dimensionless velocity c is 0.5 and the maximum amplitude Amax is 0.015 m. Based on the above parameters, we investigated the influence of lateral distance and phase difference on energy absorption performance of undulating airfoils of NACA0012 airfoil in a side-by-side arrangement. 4.1. Influence of lateral distance The energy absorption performance of the side-by-side undulating airfoils was studied when the phase difference was 0◦ , which meant that the two airfoils were in the same traveling wavy motion. The curve of energy absorption varying with lateral distance is presented in Fig. 6. It can be seen that the energy absorption efficiency of two undulating airfoils increases first and then decreases
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Fig. 7. Total pressure contour of single airfoil.
Fig. 9. Vorticity contour of single airfoil.
with the increase of lateral distance d. Furthermore, the energy absorption efficiencies of two airfoils are both higher than the data of single one in a certain range. When d = 0.5L, the efficiency of airfoils achieve the maximum values with the upper one got 28.093% and the lower one got 27.246%. As the lateral distance continues to increase, the energy absorption efficiency of sideby-side undulating airfoils decreases and is eventually consistent with the single one’s value. Because of in-phase traveling wave, the energy absorption efficiency of the upper and the lower airfoils is substantially the same at each distance. Fig. 7 displays the total pressure contour of single undulating airfoil. When the fluid flows over the airfoil, a large area of low pressure in the wake of the airfoil is observed clearly which suggesting that the undulating airfoil extracts energy from moving water. Fig. 8 shows the total pressure contours of the side-by-side undulating airfoils in different lateral distances in the same phase wave. When the distance is small, the interaction between the two
airfoils is strong, the low pressure area is bigger and the color of the area is darker. Compared with the single one, the side-byside undulating airfoils absorb more energy from water, as shown in Fig. 8(a), (b). As the lateral distance increases, the interaction becomes weak, the scale of low pressure area gradually decreases and the color of it becomes more shallow as shown in Fig. 8(c). So the energy extracted from moving water is less. As the distance increases to 3L in Fig. 8(d), the total pressure distribution of the two airfoils is almost the same, indicating that when the lateral distance is enough large, the interaction between them disappears finally. In this case, the energy absorption efficiency is close to that of single airfoil. Fig. 9 presents the vorticity contour of single undulating airfoil. It can be seen that the vortex in the wake of the airfoil is in a staggered formation. Fig. 10 shows the vorticity contours of the side-by-side undulating airfoils in different lateral distances in the same phase wave. As shown in Fig. 10(a), when the lateral distance
(a) d = 0.35L.
(b) d = 0.5L.
(c) d = 1L.
(d) d = 3L.
Fig. 8. Total pressure contours of the side-by-side undulating airfoils in different lateral distances.
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(a) d = 0.35L.
(b) d = 0.5L.
(c) d = 1L.
(d) d = 3L.
Fig. 10. Vorticity contours of the side-by-side undulating airfoils in different lateral distances.
between the airfoils is small, the interference of the vortices in the wake of the two airfoils is strong. The negative and positive vortices are respectively integrated but dissipate quickly. When the distance is 0.5L, the interaction of the two airfoils is strong. The vortex structure disorders and dissipates slowly that are shown in Fig. 10(b). With the distance increasing, the vortices shedding is slower and the vortex street is longer. The interference gradually decreases until it disappears, as shown in Fig. 10(c), (d). 4.2. Influence of phase difference In this section, we studied the influence of phase difference θ on the energy absorption performance of the side-by-side undulating airfoils Fixing the distance (d = 0.5L), the different phase difference is simulated numerically. Fig. 11 shows the trend of energy absorption efficiency with varying phase difference. We can conclude that the curve of absorption efficiency is almost in an antisymmetric form with the period of 360◦ with the changing phase difference. Moreover, the energy absorption efficiency of both is higher than the data of single one. When θ is 0◦ , the two airfoils are in the same phase so that the energy absorption efficiency is almost consistent. In the range of 0◦ to 180◦ , the efficiency of the upper airfoil decreases first and then increases with the increase of the phase difference. At θ = 60◦ , the efficiency is the lowest which is 24.797%. However, the efficiency of the lower airfoil increases first and then decreases with the increasing phase difference. At θ = 120◦ , the efficiency is the highest, reaching to 30.387%. When θ is 180◦ , the two airfoils are in the reverse wave so that the difference of two energy absorption efficiency is very small. In the range of 180◦ to 360◦ , the efficiency of the upper airfoil increases first and then decreases with the increase of the phase difference. At θ = 300◦ , the efficiency is the
Fig. 11. The relationship of absorption and phase difference in constant distance (d
= 0.5L).
highest, reaching to 30.415%. However, the efficiency of the lower airfoil decreases first and then increases with the increase of the phase difference. At θ = 300◦ , the efficiency is the lowest which is 24.387%. When θ is 360◦ , the two airfoils return to the in-phase wave. Fig. 12 shows the total pressure contours of different phase differences in the same time. When there is a phase difference in the wave, the total pressure distribution changes significantly. As shown in Fig. 12, the total pressure distribution of two airfoils is very different from that of single airfoil (Fig. 7), indicating that
L. Ding, D. Huang / European Journal of Mechanics / B Fluids 68 (2018) 193–200
(a) θ = 0◦ .
(b) θ = 60◦ .
(c) θ = 180◦ .
(d) θ = 300◦ .
199
Fig. 12. Total pressure contours of different phase differences.
(a) θ = 0◦ .
(b) θ = 60◦ .
(c) θ = 180◦ .
(d) θ = 300◦ .
Fig. 13. Vorticity contours of different phase differences.
the interference between the side-by-side airfoils is strong and they can absorb more energy from moving water. When θ is 0◦ in Fig. 12(a), the upper and lower airfoils are in an in-phase wave, the energy absorption efficiency is the same. When θ is 60◦ in Fig. 12(b), the area of low-pressure changes and the low pressure of the upper airfoil moves to the lower airfoil, which indicates that the lower airfoil extracts more energy compared to the upper one. When θ is 180◦ in Fig. 12(c), the upper and lower airfoils are in anti-phase wave and the total pressure distribution is symmetrical so that the efficiency of the two airfoils are the same. When θ is 300◦ in Fig. 12(d), the low pressure of the lower airfoil moves to the upper airfoil, which indicates that the upper airfoil extracts more energy compared to the lower one. Vorticity contours of different phase differences in the same time are shown in Fig. 13. As shown in Fig. 13, the phase difference between the side-by-side airfoils makes the vortex structure more complex, which is completely different from the vorticity contour
of the single airfoil (Fig. 9). When θ is 0◦ in Fig. 13(a), the vortex structures in the wake of the in-phase airfoils are the same. When θ is 60◦ in Fig. 13(b), because of the existence of phase difference, the positive vorticity of the upper airfoil moves to the lower airfoil and creates a strong disturbance in the wake of the lower airfoil. When θ is 180◦ in Fig. 13(c), the vortex structures in the wake of the antiphase airfoils are the opposite. When θ is 300◦ in Fig. 13(d), the situation is opposite to θ = 60◦ , the negative vorticity of the lower airfoil moves to the upper airfoil and creates a strong disturbance in the wake of the upper airfoil. 5. Conclusion The NACA0012 airfoil section is used as a two-dimensional simplified model of the fishlike body in this paper, which is traveling in the water. We investigated the influence of lateral distance and phase difference on energy absorption performance of undulating
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airfoils of NACA0012 airfoil shape in a side-by-side arrangement within a certain range. Here, we briefly summary the results as follows: (1) When the phase difference between the side-by-side undulating airfoils is 0◦ , with the increase of the lateral distance d, the energy absorption efficiency of undulating airfoils increases first and then decreases. In addition, the energy absorption efficiency of both is almost higher than the data of single one in a certain range. When d = 0.5L, the efficiency of airfoils reaches the maximum value with the upper one got 28.093% and the lower one got 27.246%. As the lateral distance is enough far, the interference of the two airfoils disappears and return to the single wave. (2) When the lateral distance is constant, with the change of phase difference, the curve of absorption efficiency is nearly in an antisymmetric form with the period of 360◦ . Moreover, the energy absorption efficiency of both is higher than the data of single one. As the increase of the phase difference, the efficiency of the upper airfoil decreases first, then increases and finally decreases. At θ = 60◦ , the efficiency of it is the lowest which is 24.797%, while at θ = 300◦ , the efficiency of it is the highest, reaching to 30.415%. However, the efficiency of the lower airfoil increases first, then decreases and finally increases. At θ = 120◦ , the efficiency of it is the highest, reaching to 30.387%, while at θ = 300◦ , the efficiency of it is the lowest which is 24.387%. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 51536006, 51576133), and the program of the Shanghai Science and Technology Commission (No. 17060502300).
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