Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow

Journal of Bionic Engineering 14 (2017) 99–110

Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow Jiapu Zhan1, Bing Xu1, Jie Wu1,2, Jing Wu1 1. Department of Aerodynamics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2. Jiangsu Key Laboratory of Hi-Tech Research for Wind Turbine Design, Nanjing 210016, China

Abstract A numerical investigation on the power extraction performance of a semi-activated flapping foil in gusty flow is conducted by using the commercial software FLUENT. The foil is forced to pitch around the axis at one-third chord and heave in the vertical direction due to the period lift force. Different from previous work with uniform flow, an unsteady flow with cosinusoidal velocity profile is considered in this work. At a Reynolds number of 1100, the influences of the mechanical parameters (spring constant and damping coefficient), the amplitude and frequency of the pitching motion, the amplitude of the gust fluctuation and the phase difference between the pitching motion and the gusty flow on the power extraction performance are systematically investigated. Compared with the case of uniform flow, the capability energy harvesting of the system is enhanced by the introduction of the gusty flow. For a given pitching amplitude and frequency, the power extraction efficiency increases with the gust fluctuation amplitude. Moreover, with an optimal phase difference between pitch and gust (φ = 180˚), the efficiency can be further enhanced due to the generation of high lift force. Keywords: semi-activated, flapping foil, gusty flow, power extraction performance Copyright © 2017, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(16)60381-5

1 Introduction With the consumption of fossil energy, the development of newly and environmentally friendly sustainable energy is looming ahead. As a replacement of coal, oil and gas in the generation of electric power, the wind power industry has developed rapidly in the past decades. Inspired by the motion of wings/fins of insects/fish, a new type of energy harvesters through the oscillation of the foils has been studied experimentally and numerically. Compared to the traditional horizontal-axis or vertical-axis rotating turbine blade based energy converters, the main advantages of oscillating foil based energy harvesters are the alleviated impact upon the environment and wildlife as well as the reduced high noise output due to their relatively low blade tip speed[1,2]. In fact, the concept of using an oscillating foil, which has been widely studied[3–6], to harvest energy from surrounding flow through a gravity wave in water is first put forward by Wu[7], who indicated that the greatest possible rate of energy extraction was obtained Corresponding author: Jie Wu E-mail: [email protected]

by the optimum mode of heaving and pitching of the foil. Later on, Mckinney and Delaurier[8] found that energy extraction is also possible in a uniform flow through the harmonic oscillation of the foil. Based on the concept of oscillating foil, Jones and Platzer[9] proved that a foil undergoing pitching/heaving could be used as propellers as well as energy harvesters, depending on the range of kinematic parameters. Recent biomechanics studies illustrated that swimming fish can reduce the cost of locomotion[10] or even move forwards without energy consumption at a relatively low Strouhal number when they are in the Karman vortex street[11]. The wake behind a flapping foil is the thrust type for the purpose of locomotion while that is the drag type for energy extraction. Based on the activating mechanism of the device, so far, the studies on the flapping foil based energy converters can be roughly classified into three categories[12–19]. The first type is the systems with prescribed pitching and plunging motions. Obviously, without taking into account the activating mechanism, the power extracted by these systems equals to the whole

100

Journal of Bionic Engineering (2017) Vol.14 No.1

hydrodynamic energy. The second type is the semiactivated systems in which the pitching motion is imposed and then the plunging motion is induced. For these systems, energy input is necessary to perform the pitching motion, whereas energy extraction is achieved through the heaving motion of the foil induced by the dynamic lifting forces. Positive net power extraction is possible if the energy extracted from the heaving motion exceeds the energy consumed to activate the pitching motion. As a matter of fact, the existing flapping foil based energy harvesters applied in industry often employ this design. The last type is the self-sustained systems. For these systems, the motions of the foil in the heaving and pitching directions rely on the flow induced instability. This not only simplifies the mechanical design, but also guarantees the positive energy extraction. Specifically, depending on the factor of the pitch axis location as well as the torsional spring, four different dynamic behaviors have been identified[20], i.e., stationary response, periodic pitching and heaving motions, chaotic response, and flip over. Among these four patterns, only the periodic response is suitable for power extraction for the predictable motion of the foil. To explore the performance of a semi-activated system, deferent combinations of related parameters have been considered. A foil mounted on a vertical spring-damper base and imposing a prescribed pitch motion was investigated by Zhu et al.[21] with the two-dimensional analytical methodology. They found that under the condition of the best frequency, pitch angle and damper strength parameter combinations, an efficiency of 25% with small amplitudes of motion can be obtained. Experiments of a NACA0015 foil elastically supported in plunge and driven in pitch have been conducted by Abiru and Yoshitake[22] at a Reynolds number Re = 5.0×104. They found that a high wing mass was needed to motivate the hydroelastic responses. Wu et al.[23] found that performance of the power extraction can be improved by placing one or two solid walls near the foil which is known as ground effect. Moreover, non-sinusoidal oscillating motions have also been introduced to enhance the energy harvesting performance of the system. Ashraf et al.[24] reported a 17% increase in power generation and 15% increase in efficiency by the adopting of non-sinusoidal pitch-plunge motion. Similarly, Teng et al.[25] reported that when the pitching amplitude is small, non-sinusoidal pitching motions can

indeed improve the performance of the system. To the best of our knowledge, the effect of the incoming gusty flow has never been taken into account in the semi-activated system. Therefore, the power extraction performance of a semi-activated flapping foil in gusty flow is numerically investigated in this work. To conduct numerical simulations, the commercial software FLUENT is employed. The motion of the foil is implemented through the dynamic mesh strategy and the adoption of sliding interface. A NACA0015 airfoil with forced pitching and induced plunging motions at a Reynolds number of 1100 is considered in this work. In addition, the pitching axis is fixed at the one-third chord of the foil. First, the damping coefficient and the strength of the spring are systematically studied to find the optimal choice. After that, the effects of the strength of the gust and the phase angle between the pitching and the gust on the force behaviors and power extraction performance are investigated in details. Based on the numerical results obtained, it is found that forces and power extraction performance are greatly changed due to the gusty flow.

2 Problem description and methodology 2.1 Problem description A two dimensional NACA0015 airfoil with chord length of c and mass of m is considered in this work, as sketched in Fig. 1a. A damper with damping coefficient b and spring constant k is attached to the foil to mimic the power extractor. To drive the pitching motion of the foil, external energy input is needed, and positive net power extraction is possible if the extracted power is larger than the energy input. Similar to most of flapping foil cases, a sinusoidal pitching motion mode is employed. Thus, the governing equations for the pitching motion and the resulting plunging motion can be expressed as[20,23]:

θ (t ) = θm cos(wt ),

(1)

mh + bh + kh = Fy ,

(2)

where θ(t) is the instantaneous pitching angle, θm is the pitching amplitude and w is the angular flapping frequency w = 2πf. The reduced frequency f* is employed in this work and its definition is f* = fc/Uave, where Uave is the mean freestream velocity that will be described in the following section. The Reynolds number based on the

Zhan et al.: Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow

U (t ) = U ave (1 + A cos(2πf g t + ϕ )),

101 (3)

U (t)/Uave

where, Uave is the mean freestream velocity, and A is the gust fluctuation amplitude. fg is the gust frequency and φ is the phase difference between the pitching motion of the foil and the incoming flow. Based on the previous study[26], it is found that the ratio of the flapping frequency and the gust frequency has little effect on the power extraction efficiency. Then the gust frequency fg is set to be the flapping frequency f. In addition, since we do not focus on the effect of kinematics of the flapping foil, the reduced flapping frequency is fixed at f* = 0.2 according to previous work[20].

Fig. 1 Sketch of (a) semi-activated flapping foil and (b) velocity profile of wind gust.

mean freestream velocity and the chord length is fixed at Re = 1100, which is similar to previous work[20,23]. Fy is the lifting force exerted on the foil, and h is the displacement of the foil in the transverse direction. As indicated by Zhu and Peng[20], the flapping foil can achieve high power extraction when the distance between the leading edge of the foil and the foil pitching axis is in the range of 0.3c and 0.4c. Following this suggestion, therefore, the foil pitching axis is located at the one-third chord in the current study. The inertial of the foil is neglected in the work of Zhu and Peng[20], but it is known that the foil inertial has an important impact on the power extraction performance. However, we attempt to concentrate on the effects of other parameters of the system (i.e., b and k). Thus, m = 1 is used in this work. 2.2 Flow field The gust may be complicated, but in this work, a simple wind gust profile with a single frequency harmonic oscillation is adopted, as plotted in Fig. 1b, which is the same as that in the work of Prater and Lian[26]. Thus, the wind gust can be described as:

2.3 Governing equations and performance of the system For two dimensional unsteady and incompressible flows, the governing equations can be expressed as[12]: JG (4) ∇ ⋅ V = 0, JG JG DV (5) ρ = −∇p + μΔV , Dt JG where V is the velocity vector, ρ and p are the fluid density and pressure, respectively, and μ is the dynamic viscosity. A Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm is utilized to deal with the pressure-velocity coupling problem. A second order accurate scheme is used for pressure discretization and the gradient is calculated based on the Least-Squares Cell Based method. The discretization of the transverse displacement of the foil is realized through the explicit four step Runge-Kutta method[12].

1 htn+1 = htn + ( K1 + K 2 + K3 + K 4 )Δt , 6

(6)

1 htn+1 = htn + ( L1 + L2 + L3 + L4 )Δt , 6

(7)

where htn and htn are the instantaneous displacement of the foil and its first order derivative, and Ki and Li (i = 1,2,3,4) can be expressed as Eq. (8). The motion of the foil is achieved through the dynamic mesh strategies. As can be seen from Fig. 2, three non-conformal sliding interfaces are introduced. One is a circle around the foil that is related to the pitching

Journal of Bionic Engineering (2017) Vol.14 No.1

102

⎧  ⎪ K1 = htn , ⎪ ⎪ Δt  ⎪⎪ K 2 = htn + L1 , 2 ⎨ ⎪ K = h + Δt L , 2 tn ⎪ 3 2 ⎪ ⎪ K = h + Δt ⋅ L , 3 tn ⎪⎩ 4

Fy

b  k htn − htn m m m Fy b  k Δt Δt L2 = − (htn + L1 ) − (htn + K1 ) 2 2 m m m . Fy b  Δt k Δt L3 = − (ht + L2 ) − (htn + K 2 ) 2 m m n 2 m Fy b  k − (ht + Δt ⋅ L3 ) − (htn +Δt ⋅ K 3 ) L4 = m m n m

L1 =

−

(8)

Then the dimensionless coefficients for the net power are defined as: Cop =

P 1 3 c ρU ave 2

= Cph − Cpθ = Cl

h Ωc − Cm , (10) U ave U ave

where Cl and Cm are the lift and torque coefficients that can be expressed as: Cl =

Fy 1 2 ρU ave c 2

Cm =

,

M . 1 2 2 ρU ave c 2

(11)

Thus, the time averaged net power extraction coefficient over one period of the combined pitching and plunging motions T can be computed by: Cop =

1 T Cop dt , T ∫0

(12)

The total power available in the incoming flow passing through the swept area can be defined as: 3 Pa = ( ρU ave d ) / 2 , where d is the overall vertical extent of the foil motion. As a result, the net power extraction efficiency can be calculated as[12]: Fig. 2 Grid details with zoom levels showing the non-conformal sliding interface.

motion, and others are two lines in the vertical direction that is related to the plunging motion. When the pitching motion is activated, the power consumed is Pθ = MΩ, where M is the instantaneous torque about the pitching axis, and Ω is the angular velocity of the foil. Meanwhile, the power extracted from the flow can be expressed as: Ph = FyVy, where Vy is the heaving velocity. So the net power extraction from the system is[20]: P = Ph − Pθ = Fy

dh dθ −M , dt dt

(9)

η=

P c = Cop , Pa d

(13)

2.4 Numerical validation

To validate the dynamic mesh strategy for the study on the flapping foil based power extraction, the case of a NACA0015 airfoil with an imposed pitching motion and an induced plunging motion is simulated. Two sets of parameters, which are the same as the parameters used by Wu et al.[23] are chosen: θm = 20˚ and f* = 0.1; θm = 40˚ and f* = 0.2. The mean power extraction coefficient from heaving motion Cph , mean power con-

Zhan et al.: Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow

sumption coefficient due to pitching motion Cpθ , and the net power extraction efficiency η are presented in Table 1. Compared to the results of Wu et al.[23], good agreement is achieved. Table 1 Parametric study of flow over a flapping NACA0015 foil at Re = 1100 θm = 20˚, f* = 0.1

θm = 40˚, f* = 0.2

Cph

Cpθ

η(%)

Cph

Cpθ

η(%)

Present

0.058

0.0059

8.29

0.184

0.064

12.7

Wu et al.[23]

0.061

0.0061

8.51

0.184

0.065

12.5

Table 2 Results for different time steps (f* = 0.2 and θm = 20˚) Time step

Cph

Cpθ

η(%)

T/500

0.0573

0.00582

8.25

T/1000

0.0576

0.00589

8.29

T/2000

0.0577

0.00590

8.31

103

To validate the independence of the force calculations with respect to spacial discretization, three different meshes are simulated. The energy harvesting efficiency calculated by the coarse mesh (58971 cells), medium mesh (80536 cells) and fine mesh (90886 cells) are 8.21%, 8.29% and 8.32%, respectively. The difference between the medium mesh and the fine mesh is less than 0.4%. Hence, the medium mesh with the moderate time step is adopted in the following simulations. To further validate the time-discretization independence, three different time steps, T/500 (T is the period of the flapping motion), T/1000 and T/2000 are adopted. The results at f* = 0.2 and θm = 20˚ are presented in Table 2. From the results, it is found that the largest difference of η between three different time steps is less than 0.7%. The force and moment coefficients over a period of time plotted in Fig. 3 show little variation among the three different time steps. Thus, the moderate time step of T/1000 is small enough to guarantee temporal accuracy.

3 Results and discussions

0.6 ∆t = T/500 0.4

After validating the numerical method, the power extraction performance of a pitching-motion-activated flapping foil is systematically investigated. The mainly focus of this work are the influence of mechanical parameters of the foil (i.e., b and k), the amplitude of the gust flow and the phase angle between the pitching and the incoming flow.

∆t = T/1000 ∆t = T/2000

0.2 0.0 −0.2 −0.4 −0.6 0.0

0.2

0.4

t/T

0.08

0.6

0.8

1.0

0.8

1.0

∆t = T/500 ∆t = T/1000 ∆t = T/2000

0.04

0.00

−0.04

−0.08 0.0

0.2

0.4

t/T

0.6

Fig. 3 Time evolution of lift and torque coefficients with different time steps.

3.1 Uniform flow Before studying the gust flow, the cases of the foil in uniform flow are first simulated to obtain the optimal mechanical parameters of the foil. Since the motion is symmetric, the mean forces are always zero. Thus, the Root Mean Square (RMS) values of the lift and torque coefficients, i.e., (Cl)RMS and (Cm)RMS, are plotted in Fig. 4 against the reduced frequency f* with different k at θm = 45˚ and b* = 2π. As can be seen from Fig. 4a, for all k* considered in this work, the RMS lift coefficient (Cl)RMS first decreases and then increases with f*. On the other hand, it is noted that (Cl)RMS always has a maximum value at k* = 100 for a given flapping frequency. Otherwise, when f* ≤ 0.2, (Cl)RMS increases with k*. And when f* > 0.2, the variation trend is reverse. It is clear from Fig. 4b that the RMS torque coefficient (Cm)RMS increases with f* for all k* considered. When the spring constant is small

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Journal of Bionic Engineering (2017) Vol.14 No.1 (a)

k*= 0 k*= 1 k*= 2 k*= 5 k*= 7.5 k*= 10 k*= 100

0.4

− Cph

(Cl)RMS

0.3

0.2

0.1

0.0 0.10 (b)

0.15

0.20 f*

0.25

0.30

0.3 0.2

− Cop

(Cm)RMS

0.1 0.0 k*= 0 k*= 1 k*= 2 k*= 5 k*= 7.5 k*= 10 k*= 100

−0.1 −0.2 −0.3 −0.4 0.10

0.15

(c)

0.20 f*

0.25

0.20 f*

0.25

0.30

0.2

Fig. 4 Comparison of (a) RMS lift coefficient ((Cl)RMS) and (b) RMS torque coefficient ((Cm)RMS) for a semi-activated flapping foil in the uniform flow at θm = 45˚ and b* = 2π.

To deeply understand the contribution of pitching and plunging motions to the energy extraction performance, the mean values of power extraction coefficient Cph , the net power extraction coefficient Cop as well as the net power extraction efficiency η are plotted in Fig. 5. Although Cph is related to the lift coefficient, the influence of k* on Cph shows a difference compared to (Cl)RMS. As can be seen from Figs. 5a and 5b, Cph roughly increases with f*, whilst Cop decreases with f*. This is mainly because that with the increase of the flapping frequency, the power needed to activate the pitching motion exceeds the power extracted from the plunging motion. When f* ≤ 0.2, as plotted in Fig. 5a, it seems that k* = 1 is the best choice to harvest energy through the plunging motion. In addition, as f* increases, the

η

0.0

(k* ≤ 10), (Cm)RMS generally decreases with k* for a given flapping frequency.

k*= 0 k*= 1 k*= 2 k*= 5 k*= 7.5 k*= 10 k*= 100

−0.2

−0.4 0.10

0.15

0.30

Fig. 5 Comparison of (a) mean power extraction coefficient ( Cph ), (b) mean net power extraction coefficient ( Cop ), and (c) power extraction coefficient efficiency η for a semi-activated flapping foil in the uniform flow at θm = 45˚ and b* = 2π.

difference of Cph between different k* becomes small except for k* = 100. It is noted that Cph with k* = 100 is nearly close to zero and changes slightly with the increase of f*. On the other hand, as plotted in Fig. 5b, the maximum value of Cop is always at k* = 1 when f* ≤ 0.2. As f* further increases, Cop reaches its

Zhan et al.: Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow

Fig. 6, Cph and Cop first increase and then decrease with θm when b* > 0.5π, as plotted in Figs. 7a and 7b, respectively. Whilst at b* = 0.5π, they monotonically increase with θm, and then stay almost unchanged when θm exceeds 75˚. Due to the irregular variation of force behavior and

− Cph

maximum value at different k*. Although Cop shows irregular variation with respect to k*, the maximum η at a given f* is fixed at k* = 5, as shown in Fig. 5c. This may be because that k* = 5 can provide good combination of force behavior and displacement. After checking the effect of k*, the parametric study on b* is conducted. Fig. 6 plots the variation of (Cl)RMS and (Cm)RMS with pitching amplitude at different b*. The dimensionless coefficient k* is fixed at 5. As seen from Fig. 6a, (Cl)RMS first increases and then decreases with θm when b* > 0.5π. But when b* > 0.5π, (Cl)RMS monotonically increases with θm. At a given θm, a high b* always generates large (Cl)RMS and (Cm)RMS. Fig. 6b shows that (Cm)RMS monotonically increases with θm for all b*.

105

Generally, the energy harvesting performance of the semi-activated flapping foil system is determined by the force behaviors. Fig. 7 correspondingly shows Cph , Cop and η varying with θm. Similar to the results in (a) 1.8 b*= 0.5π b*= π b*= 1.5π b*= 2π

(Cl)RMS

− Cop

1.5

1.2

0.9 45.0

52.5

60.0

(b) 0.35

θm

67.5

75.0

82.5

67.5

75.0

82.5

b*= 0.5π b*= π b*= 1.5π b*= 2π

(Cm)RMS

η

0.30

0.25

0.20

0.15 45.0

52.5

60.0

θm

Fig. 6 Comparison of (a) RMS lift coefficient (Cl)RMS) and (b) RMS torque coefficient (Cm)RMS) for a semi-activated flapping foil in the uniform flow at k* = 5.

Fig. 7 Comparison of (a) mean power extraction coefficient Cph , (b) mean net power extraction coefficient Cop , and (c) power extraction efficiency η for a semi-activated flapping foil in the uniform flow at k* = 5.

Journal of Bionic Engineering (2017) Vol.14 No.1

power extraction performance, the power extraction efficiency also shows complex variation due to different combination of mechanical parameters. As shown in Fig. 7c, η first increases and then decreases with θm. When b* > 0.5π, the peak of η occurs at θm = 67.5˚. But when b* = 0.5π, the peak value appears at θm = 75˚. From the figure, it is clear that when θm ≤ 67.5˚, b* = π can produce the best energy harvesting performance (η = 30% at b* = π and θm = 67.5˚), while b* = 0.5π is superior when θm > 67.5˚ (η = 32% at b* = 0.5π and θm = 67.5˚). Based on the results above, it is known that the mechanical parameters of the foil affect not only the force behavior but also the energy harvesting performance of the system. For the mass considered in this work (m = 1), k* = 5 is recommended and will be applied in the following studies to achieve the high net power extraction efficiency. Meanwhile, when the pitching amplitude θm ≤ 67.5˚, it is suggested to use b* = π to obtain high net energy harvesting efficiency. But when θm > 67.5˚, b* = 0.5π is an optimal choice. To systematically investigate the influence of the gust parameters, two representative sets of mechanical parameters, i.e., θm = 60˚, b* = π and θm = 75˚, b* = 0.5π, are considered in the following sections. 3.2 Gust flow Since the gust frequency is set to be the same as the flapping frequency of the foil, the parameters concerned are the gust fluctuation amplitude (A) and the phase difference between the gusty flow and the pitching motion of the foil (φ).

3.2.1 Effect of gust amplitude To investigate the influence of the gust amplitude on the power extraction efficiency, the phase difference is fixed at φ = 45˚ and 180˚. Fig. 8 displays the energy harvesting efficiency η varying with the gust amplitude, in which A = 0 means the uniform flow cases. It is known from the figure that the gusty flow can enhance the capability of energy harvesting in general. The larger the amplitude is, the higher the energy harvesting efficiency is. In particular, the higher pitching amplitude can extract more power at the same phase difference. At a given θm, φ = 180˚ is better for power extraction than φ = 45˚, especially when the gust amplitude is large (A ≥ 0.2).

η

106

Fig. 8 Variation of energy extraction efficiency η with respect to the gust fluctuation amplitude.

To further check the influence of the gust amplitude on the power extraction process in detail, the time evolutions of lift coefficient Cl, non-dimensional heaving velocity Vy/Uave, contribution from pitch Cpθ, and net power extraction coefficient Cop within one gust cycle are plotted in Fig. 9. The cases with θm = 60˚ are considered. As a reference, the results of uniform flow are also involved. In the figure, three sub-regions with good synchronization between Cl and Vy/Uave are denoted, which can lead to positive contribution from heave Cph. At A = 0.1 as shown in Fig. 9b, two negative peak values of Cl (sub-regions 1 and 3) are larger than that in the uniform flow, shown in Fig. 9a. Since Vy/Uave is hardly modified by the gusty flow and the contribution Cpθ is small, the resultant positive peak values of Cop (sub-regions 1 and 3) are also higher. As a consequence, larger Cop (increases slightly from 0.33 to 0.336) as well as η (increases from 27.4% to 28.1%) are achieved. As A increases up to 0.2 (shown in Fig. 9c), the negative peak values of Cl continue to decrease. It means that more power is extracted from the heaving motion. As a result, Cop and η also increase (up to 0.361 and 30.2%, respectively). On the other hand, when the phase difference φ varies from 45˚ to 180˚ (shown in Fig. 9d), the positive peak value of Cl (sub-region 2) is greatly enhanced. Consequently, the corresponding positive peak value of Cop is also increased obviously, which then results in the improvement of Cop and η (reaches 0.392 and 31.9%, respectively). The results in Fig. 9 explain well the increase of η in Fig. 8.

Zhan et al.: Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow Cop Vy/Uave Cpθ Cl

3 2

2 1

0

0

−1

−1 − Cop = 0.330 η = 27.4%

−2 1

3

2

1

−3

0.0

0.2

0.4

t/T (a)

0.6

0.8

1.0

Cop Vy/Uave Cpθ Cl

3 2

0.0

0.2

1

0

0

−1

−1

2

1

− Cop = 0.361 η = 30.2%

3

0.6

0.8

1.0

0.8

− Cop = 0.392 η = 31.9%

2

1

−3 t/T (c)

0.6

t/T (b)

−2

−3 0.4

0.4

3 1.0

Cop Vy/Uave Cpθ Cl

2

1

0.2

2

3

−2

− Cop = 0.336 η = 28.1%

−2

−3

0.0

Cop Vy/Uave Cpθ Cl

3

1

107

0.0

0.2

0.4

t/T (d)

0.6

3

0.8

1.0

Fig. 9 Time evolution of Cl, Vy/Uave, Cpθ and Cop at θm = 60˚. (a) Uniform flow; (b) A = 0.1, φ = 45˚; (c) A = 0.2, φ = 45˚; (d) A = 0.2, φ = 180˚.

3.2.2 Effect of phase difference between gust and pitch Besides the gust amplitude, another parameter concerned in this work is the phase difference φ. To examine the influence of φ on the power extraction performance, two typical gust amplitudes, i.e., A = 0.2 and 0.4, are chosen. In addition, φ varies from 0˚ to 270˚ with an increment of 45˚. As can be seen from Fig. 10, for a given pitching amplitude (for example, θm = 60˚), η is sensitive to φ. With the increase of φ, η first decreases and reaches the minimum value at φ = 90˚ (η = 28.5% for A = 0.2 and η = 31.5% for A = 0.4). After that, it starts to increase and arrives the maximum value at φ = 180˚ (η = 32.6% for A = 0.2 and η = 44.6% for A = 0.4). Thus, it is known that φ = 0˚ and 180˚ are the nice choices for the power extraction of semi-activated flapping foil system in the gusty flow. Similar to Fig. 9, the time evolutions of Cl, Vy/Uave, Cpθ and Cop within one gust cycle at some typical φ are presented in Fig. 11. The cases with θm = 75˚ are

θm = 60˚ A = 0.2 θm = 60˚ A = 0.4 θm = 75˚ A = 0.2 θm = 75˚ A = 0.4

0.5

0.4 θm = 75˚ uniform flow

0.3

θm = 60˚ uniform flow 0

45

90

135 Φ

180

225

270

Fig. 10 Variation of power extraction efficiency η with respect to the phase difference between gust and pitch.

considered, and gust amplitude is fixed at A = 0.2. Again, the results of uniform flow are included. At φ = 90˚ as shown in Fig. 11b, the positive peak value of Cl in sub-region 2 and negative peak value of Cl in sub-region 3 are reduced and increased, respectively,

Journal of Bionic Engineering (2017) Vol.14 No.1

108 Cop Vy/Uave Cpθ Cl

3 2

2

1

1

0

0

−1

−1 − = 0.525 C op η = 32.0%

−2 −3

2

1

0.0

0.2

0.4

t/T (a)

0.6

2

3

0.8

1.0 0.0

1

0

0

−1

−1

−3 0.0

− Cop = 0.557 η = 33.7%

2

1 0.2

0.2

0.4

t/T (c)

0.6

0.4

t/T (b)

0.6

0.8

0.8

− = 0.562 C op η = 34.5%

−2 3

3 1.0

Cop Vy/Uave Cpθ Cl

2

1

2

1

−3

3

−2

− = 0.526 C op η = 32.1%

−2

Cop Vy/Uave Cpθ Cl

3

Cop Vy/Uave Cpθ Cl

3

2

1

−3 1.0 0.0

0.2

0.4

t/T (d)

0.6

0.8

3 1.0

Fig. 11 Time evolution of Cl, Vy/Uave, Cpθ and Cop at θm = 75˚ and A = 0.2. (a) Uniform flow; (b) φ = 90˚; (c) φ = 135˚; (d) φ = 180˚.

as compared to the uniform flow as plotted in Fig. 11a. This then makes Cop and η change a little bit. When φ increases up to 135˚ as plotted in Fig. 11c, it is noted that the negative peak value of Cl in sub-region 1 is slightly decreased, whilst the positive peak value of Cl in sub-region 2 is greatly enhanced. As a consequence, η increases from 32.1% to 33.7%. For the case of φ = 180˚ (Fig. 11d), the positive peak value of Cop in sub-region 2 is further increased, whilst the values of Cop in sub-regions 1 and 3 are nearly unchanged. It then results in the further increase of η (increases up to 34.5%). Again, the results in Fig. 11 explain well the change of η in Fig. 10. Based on the results above, it is known that the lift force plays an important role in the power extraction process. In order to explore the variation of the force behavior at different φ in detail, the flow field around the foil is closely checked. Fig. 12 shows four instantaneous vorticity contours near the foil over one flapping period. At the instant of t = 0T/4, the foil is at the positive

peak of pitching angle, the clockwise vortex formed at the upper surface of the foil moves from leading edge to trailing edge. With the gradual increase of φ (from 90˚ in Fig. 12b to 180˚ in Fig. 12d), the strength of incoming flow is weakened, and it is always smaller than that of uniform flow. As a result, the strength of the clockwise vortex is reduced. At the instant of t = 1T/4, the foil is traveling to the lowest location of the downstroke. At this moment, the foil is rotating in the clockwise direction with large angular velocity. Two obvious counterclockwise vortices are formed around the foil. The stronger one near the leading edge travels downstream along the lower surface, and the weaker one at the trailing edge is going to interact with the clockwise vortex on upper surface. The counterclockwise vortices reduce the pressure along the lower surface. As a consequence, the dynamic stall occurs, which leads to a sharp decrease of the lift force as shown in Fig. 11. With the increase of φ, the strength of leading edge counterclockwise vortex is slightly

Zhan et al.: Power Extraction Performance of a Semi-activated Flapping Foil in Gusty Flow

109

Fig. 12 Instantaneous vorticity contours at θm = 75˚ and A = 0.2. (a) Uniform flow; (b) φ = 90˚; (c) φ = 135˚; (d) φ = 180˚

weakened, whilst the strength of trailing edge counterclockwise vortex is enhanced obviously. At the instant of t = 2T/4, the foil reaches the negative peak of pitching angle. Since the flow is attached to the most parts of foil surface, the positive lift force can hold at a high level, as shown in Fig. 11. With the increase of φ, the strength of incoming flow is enhanced, and it is always larger than that of uniform flow. For the case of φ = 90˚, the counterclockwise vortex still exists and is moving downstream to interact with the trailing edge. But for the case of φ = 180˚, the counterclockwise vortex has departed from the foil, and a new clockwise vortex is being formed on the upper surface of leading edge. This can explain well the smooth increase of Cl from Fig. 11b to Fig. 11d. At the instant of t = 3T/4, the foil is approaching the highest position of the upstroke and is going to rotate in the counterclockwise direction. As a consequence, a strong clockwise vortex around the leading edge has been formed and is moving to the downstream. Due to the clear flow separation, the lift force drops rapidly (Fig. 11). However, with the increase of φ, the strength of clockwise vortex is increased, which can delay the drop of lift force. The vortex development at different φ indicates that the flow around the flapping foil is complicated, and it is related to the unsteady hydrodynamics. By adjusting the phase difference between the pitching motion and the

gusty flow, good performance of power extraction can be achieved, which is due to the modification of lift force. For a semi-activated flapping foil in the gust flow, the recommended phase difference is 180˚.

4 Conclusion In this work, the power extraction behavior of a semi-activated flapping foil in the gusty flow is numerically examined. The foil pitching axis is located at the one-third chord, the cosinusoidal velocity profile is employed to model the gusty flow, and the Reynolds number is chosen as 1100. First, the flapping foil is placed in the uniform flow. Under the condition of fixed foil mass, the optimal spring constant and damping coefficient for achieving high power extraction efficiency are found. When the gusty flow is introduced, a stronger gust fluctuation amplitude generates the higher power extraction efficiency at a given pitching amplitude and frequency. By adjusting the phase difference between pitch and gust, the energy harvesting efficiency can be further increased. The present work can be used to design the wind turbine in the industry.

Acknowledgment Jie Wu acknowledges the support of the Fundamental Research Funds for the Central Universities (Grant No. NS2016017). This work is also supported by

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Journal of Bionic Engineering (2017) Vol.14 No.1

the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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