EFFECTS OF TWO TYPES OF CONTROLLABLE DEFORMATION ON ENERGY EXTRACTION OF A FLEXIBLE HYDROFOIL

Energy extraction capacity of controllably flexible hydrofoil was studied under two identified deformation modes. Deformation modes, flexure parameters (flexure amplitude  and flexure coefficient ) and motion parameters (reduced frequency f* and pitching amplitude 0) were investigated to understand the effects of controllably flexible deformation on energy extraction. The results reveal that deformation modes affect the effective angle of attack and vortex structure, which influence hydrodynamic performance. The energy extraction capacity improves from the deformation mode 2 to the rigid hydrofoil and then to the deformation mode 1. Under the deformation mode 1, lift, moment and power coefficients are increased obviously with the increase of , while they increase slightly with . Power coefficients and efficiency are sensitive to , which influences the development of leadingedge vortices. The flexible coefficient  affects the wake structure, which has less impact on variation of force coefficient. As the increase in f*, averaged power coefficients firstly increase and then decrease. Further, the optimal f* is subjected to 0. Interestingly, a critical reduced frequency f*s, which is generally increase with increasing 0, was found under three modes. The condition that f* > f*s. is a prerequisite for subsequent adjustments of flexure modes and parameters according to different requirement of power coefficient under different tidal currents. The range of high efficiency () is: deformation mode 1 (36.1%<<54.3%) > rigid hydrofoils (34.2%<<41%) > deformation mode 2 (26.9%<<30.3%).


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be improved by a flexible tail, thus improving the power extraction efficiency [32,33]. The leading and trailing edge deformation increase the camber of deforming foil surfaces, which could improve the energy harvesting efficiency [34,35].
As mentioned in the literature [31], the chordwise flexibility of hydrofoils affects the pressure distribution, vortices formation and shedding time scales, which directly affects the hydrodynamic characteristics of hydrofoils. Tidal current conditions are complex and changeable. By controlling the flexible wing to deform reasonably respect to flow velocity conditions, the power extraction efficiency could be improved, which could expand the application range of tidal current energy. In other words, the optimal power and efficiency could be obtained by reasonable deformation of hydrofoils in places with high or low flow rate. To the knowledge of authors, there are few works about flexible hydrofoils and little research referring to the influences of flexure parameters on power extraction performance. In addition, systematic analysis for effects of flexibility under different motion parameters on power extraction has not been well developed yet and need to be further investigated. Zhu et al. found that the energy extraction efficiency could be improved by the active synchronous arc deformation and in their study both leading and trailing edges were deformable [36]. However, in this study, the part from leading edge to pitching axis is rigid, and the rest is flexible. By tuning the adjustable parameters of the flexible amplitude  and flexible coefficient , the shapes of upper and lower surfaces of hydrofoils can gradually change corresponding to the change of chord line. In addition, deformation modes, flexure parameters and a wide range of kinematic parameters are studied. A comprehensive study is conducted in terms of force coefficients, effective angle of attack, vortices structures, and power extraction capabilities. Furthermore, the appropriate range of parameters is presented and the evolutions of pressure distribution are also discussed.

Numerical method
In this study, unsteady incompressible viscous flows around an oscillating hydrofoil are governed by equations as follows: where , and ⍴ are the pressure, viscosity and density, respectively; and F denote the velocity and force in direction; and F denote the velocity and force in y direction.
Spalart-Allmaras turbulence model was selected for simulations [5,37]. Pressure Implicit with Splitting of Operator algorithm was employed. In addition, the movement and deformation of hydrofoils was controlled by User defined function (UDF). The pressure discretization is second order scheme. Momentum and modified turbulent viscosity discretization are second order upwind schemes. Absolute convergence criteria was 10 -5 in the present study. Power coefficients and force coefficients are determined after six cycles in order to ensure the stable convergence computations.
A mesh model is established for simulation, as shown in Fig. 1. The square fluid Yubing Zhang, Yong Wang Effects of two types of controllable deformation Yudong Xie, Guang Sun, Jiazhen Han on energy extraction of a flexible hydrofoil 112 computational domain is 75c  75c and the center is the pitching axis location of hydrofoils. The mesh is composed of three sub-domains. The external-domain (red) and middle-domain (blue) are connected with a sliding interface. The interface ensures that the external-domain only has a heaving movement. In addition, the middle-domain experiences a synchronous heaving/pitching movement. It must be noted that the grid of middle-domain is only moved as a whole grid, which means that it is not re-meshed. However, the grid of internal-domain (green) is re-meshed due to the deformation of hydrofoils. Quadrilateral cells are used for boundary layers. In order to ensure the accuracy of deformation motion, a fine mesh of the internal-domain is created. The internal-domain experiences a synchronous heaving/pitching/deforming motion, which avoids grid distortion and ensures minimum effects of grid deformation on simulations. show the down-stroke of the rigid hydrofoil, deformation mode 1 and deformation mode 2, respectively. And the up-stroke is the reversed process, thus no display here. The research focus in this work is the effect of flexure of hydrofoils on power extraction performance, thus, the traditional periodic motions of hydrofoils are adopted. And following equations express the heaving and pitching motion pattern: where ḣ(t) and ) t (   are the heaving velocity and pitching angular velocity, respectively; h0 and 0 are the heaving and pitching amplitude respectively; is the frequency;  is the phase difference between heaving and pitching motion. The chord length c is 0.25m; the heaving amplitude h0/c is 1; the pitching axis location is c/3 from the leading edge. The phase difference  is kept at 90; the flow velocity U is 2m/s; the reduced frequency determined as  On the basis of Bose's chordwise deformation theory [38], a modified flexure model is proposed and the chordwise flexure deformation (x, t) can be described by the following function: where c is the chord length; is the coordinate value of cells of hydrofoil thread in relative coordinate system. It must be noted that the value of is determined in relative coordinate system, in which the hydrofoil only occurs deformation motion. The relative coordinate system performs the combined heaving/pitching motion in global coordinate system. The part from trailing edge to pitching axis is flexible, and the rest is rigid.  is the flexure amplitude, denoting the maximum deformation value of trailing edge tips. A dimensionless parameter , which is determined as  =/c, is used.  is the flexure coefficient, denoting the curvature of chord line. These various meaning is discussed in the following paragraph.  is the phase difference of flexure deformation direction, and 0 refers to the deformation mode 1 and  refers to the deformation 2.
Le and Ko [31] demonstrated that power extraction efficiency improved 30% when the chordwise flexure is 20%. Different from their research, this study focuses on the different  under the variation of . In other words, the different  denotes the different curvature shapes under the same . Fig. 3(a) displays the deformation mode 1 during down-stroke at t/T=0.25, which is consistent with Fig. 2(b). Fig. 3(b) displays the deformation mode 2 during downstroke at t/T=0.25, which is consistent with Fig. 2

Power extraction parameters
The lift coefficients (C ), drag coefficients (C ) and moment coefficient (C ) were obtained by the following equations: where F is the lift force; F is the drag force; F is the moment; and b is the span length, which is equal to one unit length for 2D model. A vertical linear velocity (ḣ(t)), which is caused by the heaving motion of hydrofoils, results in that the instantaneous equivalent inflow velocity is not the flow velocity (U), which can be seen in Fig. 2. The equivalent inflow velocity (U(t)) can be determined as: The effective angle of attack is the interior angle between the line from the leading edge to the trailing edge tips [34] and U(t), as shown in Fig. 2. In other words, the angle is the combination of the velocity induced, pitching angles and flexible deformation. The angle induced by the flexure of hydrofoils cannot be ignored. Therefore, the effective angle of attack of rigid hydrofoils (e), deformation mode 1 (e1) and deformation mode 2 (e2) can be expressed as: where (t) is the flexible deformation of trailing edge tips.
The instantaneous power coefficient Cp(t), averaged power coefficient Cp and efficiency () can be can be written as: where P(t) is the instantaneous power. It must be noted that the instantaneous power consists of two components: heaving motion power ( ) t ( h F y  ) and pitching motion power ( ) t ( F m   ) [31].The user defined function (UDF) was used to calculate the force and velocity [4,15,19]. The force of the hydrofoil thread unit is extracted through UDF, and the force of the grid unit is added up to be the total force. H is the rigid foil's swept distance. H1 and H2 are the swept distance of deformation mode 1 and deformation mode 2, respectively, as shown in Fig. 2.

The independence validation of grid number and time step
In this section the tests were done under deformation mode 1 in h0/c=0.5, 0=80, f*=0.1, =8% and =1. Fig. 4 shows Cy for the different cases during one cycle. It can be seen that the results are gradually converged with grid number increase and time step size decrease. Table. 1 shows Cp and its error in one cycle. The errors are determined based on the case 7.
As shown in table. 1, the errors of case 6 is 1.3%, which within 3%. It can be concluded that at this level the influences of grid number and time step are very limited and can be reasonably ignored. Therefore, in this study, a number of 1.2110 5 cells and 1600 ts/cycle are chosen for all simulations.   [4]. Their simulation results were compared with prototype experiments. Fig. 5 exhibits that Cy and Cx in present study show good agreement with that of Kinsey and Dumas [4]. Therefore, it can be revealed that the code parameter settings and iteration scheme in this study are correctness. To quantify the influence of  and  on hydrodynamic of an oscillating hydrofoil, the averaged power coefficient (Cp) and efficiency () are computed and listed in Table. 2.Cp is the sum of averaged lift power coefficients (Cpl) and averaged moment power coefficients (Cpm). Cx is averaged drag coefficients, which are used to understand the negative influence of and  on performance of oscillating hydrofoils.
In Table. 2  of flexure hydrofoils at =5% and =2.5 is 0.317, which is increase of about 17.8% compared to the rigid hydrofoil. As  increases, Cp and  is increased slightly. However, the increases are obviously with the increase of  from 0 to 15% at =0.7 and further the increase is slightly as  from 15% to 20%. Furthermore, Cp and  are decreased gradually as increasing  from 20% to 30%. Similar tendency has been reported by Le et al [31], where efficiency increases from 38% (at A=0.15) to 39.2% (at A=0.2), while further increase A from 0.25 to 0.3 causes the decrease of efficiency from 39.8% to 39.6%.
It can be found in Table. 2 that the improvements of Cp are mainly attributed to the increase of Cpl. It indicated that the power extraction depends on the heaving motion, which is consistent with Xie et al. [13]. However, it must be noted that with the increases of  and , Cpm is increased, which means that the consumption of power is gradually increasing. In addition, Cx also increases with  and  increasing. Therefore, =15% and =1 is selected for further researches. Variations of instantaneous lift and power coefficient with  are presented in Fig. 6. Lift coefficients (Cy) increase slightly with  increasing from 0.7 to 2.5 ( Fig. 6(a)). However, Cy of all flexible cases (=0.7-2.5) are larger than that of rigid hydrofoil (=0). Fig. 6(b) shows the variation of Cp, which has the similar trend with Fig. 6(a).  It can be seen that any increase in  causes to the increase of lift coefficients. The maximum lift coefficient is 3.21 at =20% and t/T=0.2, which are approximately 40.2% higher than that of rigid hydrofoil (2.29). It can be seen in Fig. 7(b) and (c) that with the increase of , Cx and Cm coefficients are increased, which increases the consumption of energy. Fig. 7(d) is provided to show the instantaneous power coefficient variation with  at fixed =0.7. The instantaneous power coefficient increases obviously with  increasing from 5% to 20%. At t/T=0.23, maximum power coefficient of 2.04 is achieved, which is increase of about 40.6% as compared to the rigid hydrofoil (1.45). According to Eq. (18), the power (P(t)) is positively related to force (Fy). What's more, there is a great correlation between force coefficients and the effective angle of attack (AOA). Therefore, the study of AOA can better understand the essence of the above phenomenon. Fig.   8(a) shows the AOA of  varying from 0-20%. As the increase of , the AOA is gradually increased. Compared with , the AOA cannot be changed obviously by the variation of  from 0.7 to 2.5 ( Fig. 8(b)). It can be revealed that  can be used to alter the AOA of hydrofoils. Due to the AOA is less than the dynamic stall angle, the increase of AOA is beneficial for the improvement of Cy. Because h0, 0 and f* is constant, the heaving velocity and angular velocity are changeless. According to the equation (18), the increased Cy will improve the instantaneous power. Therefore, it can be determined that Cp and  are directly affected by the lift of hydrofoils, which depends on AOA. Fig. 8 the AOA of (a)  =5%-20% at =0.7, and (b) =0.7-2.5 at  =5%

Evolutions of pressure distribution and vortex field
From Eq. (10)- (12), it can be seen that Cy, Cx and Cm are subjected to the variation of forces, which are related to the pressure distribution of the surfaces on hydrofoils. Pressure distribution made a major contribution to force coefficients [10]. Therefore, observation of pressure distribution is necessary to analyze the effects of flexure parameters ( and ) on hydrodynamic performance.
It can be found in Fig. 9(a) that pressure difference mainly locates near the leading edge. In addition, the pressure difference of flexure hydrofoils (=0.7-2.5) is very similar and greater than that of rigid hydrofoils (=0).
The parameter  (5%, 10%, 15% and 20%) is equivalent to the maximum chordwise flexures of trailing edge tips respect to the chord length. It can be observed in Fig. 9(b) that the pressure difference of rigid hydrofoils (=0) is smaller compared with flexible hydrofoils (=5%-20%). With the increase of , the pressure difference is increased gradually. It must be Effects of two types of controllable deformation Yubing Zhang, Yong Wang on energy extraction of a flexible hydrofoil Yudong Xie, Guang Sun, Jiazhen Han 119 noted that with  increasing, the projection distance on the x axis gradually approaches the origin, as marked by red arrows.
It can be seen in Fig. 9(a) and (b) that the pressure difference caused by larger  at fixed  is larger than that caused by varying  at fixed . In other words, compared with adjusting , adjusting  is more conducive to the increase of pressure difference. This indicated that more power can be extracted with a higher .  during down-stroke phase. It must explain that up-stroke phase is the converse process of Fig.   10, and thus no exhibition in this section. At t/T=0, with  increase vortices are gradually generating. At t/T=0.15 the vortex is moving along the surface of hydrofoils. The pitch angle reaches peak values at t/T=0.25. However, when  increasing from 0 to 10%, boundary layer thickens gradually, which can be illustrated at t/T=0.25 and 0.35. The increase of  leads to reverse pressure gradient increasing greatly, which in return increases obviously boundary layer thickness. When  is greater than 10%, the boundary layer evolves into vortex at t/T=0.5. In addition, a second leading edge vortex can be found when  is greater than 15%, causing the fluctuation of lift coefficient. As shown in Fig. 10, no vortex is shedding away from near wall of hydrofoil, which means that no dynamic stall occurs.

Influences of reduced frequency and pitching amplitude
The Cp for three modes (rigid hydrofoils, deformation mode 1 and mode 2) are shown in Fig. 11 at 0.025f*0.325, 40080, =15% and =1. As shown in Fig. 11, the trend of Cp Yubing Zhang, Yong Wang Effects of two types of controllable deformation Yudong Xie, Guang Sun, Jiazhen Han on energy extraction of a flexible hydrofoil 120 is increased firstly and then decreased under three modes. It indicates that a peak value ofCp exists in the range of f* and 0. The peak value is gradually increased as f* and 0 increases. The Cp of deformation mode 1 improved by approximately 55% compared with the rigid hydrofoil at f*=0.175 and 0=70. Interestingly, Cp of deformation mode 2 is lower than that of other types. Therefore, the deformation mode 2 can be used as adjustment measures in high velocity zone in order to protect turbines. This adjustment measure is carried out without stopping the machine, and the pressure on surface can be greatly reduced by simply changing the deformation mode.
In addition, for the different 0, there always exists a critical reduced frequency f*s. When f*< f*s, Cp for three modes is very similar. However, when f*> f*s, as f* increase, the sensitivity of Cp is significantly increased. The f*s is generally increase with increasing 0. To understand the influences of f* on Cp, f* of 0.075, 0.175 and 0=80 are selected for further research. Fig. 12 exhibits Cy over one cycle for three modes. As shown in Fig. 12 (a), Cy of deformation mode 1 is slightly large at 0-0.2t/T and 0.5-0.7t/T, however, the value is decrease seriously at 0.2-0.4t/T and 0.7-0.9t/T, resulting the limited difference of Cy in three modes. Therefore, Cp for three modes is very similar when f*< f*s. Fig. 12(b) shows that when f*> f*s, Cy of deformation mode 1 is the highest of the three in a period, resulting in a large difference of Cp. In addition, the lift coefficient curve is relatively smooth, indicating that no stall has occurred.  Fig. 13 shows AOA over one cycle for three modes under f*=0.075 and f*=0.175 at θ0=80. From Fig. 13(a) and (b), AOA of deformation mode 1 is larger than that of rigid hydrofoils, which is consistent with Fig. 2(b). However, compared with rigid hydrofoils, deformation mode 2 decreases the AOA of hydrofoils.
At a low frequency ( Fig. 13(a)), the corresponding heaving velocity is lower due to the same heaving amplitude, which leads to a lower induced velocity. According to the Eq. (6,(14)(15)(16), the decrease of an induced velocity increases the actual AOA. When the AOA is greater than the critical stall angle, hydrofoils will get into stall states, thus reducing Cy (Fig. 12(a)). Conversely, at a high frequency (Fig. 13(b)), AOA is decreased due to the increase of induced velocities. Therefore, Cy (Fig. 12(b)) keeps a large value for a long time, which is beneficial for the improvement of Cp.   Fig. 14(a) that from t/T=0 to t1, boundary layer separation does not occur, and the lift coefficient increases steadily, but vortex separation occurs at t/T=t2 stage, resulting in a serious drop in lift coefficient. This corresponds to the lift curve in Fig. 12(a) above. At t/T=t3 stage, the vortex is shedding away from the surface of hydrofoils and interacting with the wake vortex.
As shown in Fig.14(b), no leading-edge vortex sheds in the whole cycle, which means that there is no stall occurring. The lift coefficient curve transitions smoothly, which corresponds to the lift coefficient curve ( Fig. 12(b)). By comparing the three modes, it can be seen that deformation mode 1 has some degree of stall delay, which can be illustrated at t/T=0-t3. In addition, deformation mode affects the motion state of the fluid near the lower surfaces at down-stroke. Under the deformation mode 1, because the flexure direction is the same as the pitching direction, an anticlockwise direction vortex near the trailing edge (t/T=t2), which caused by pitching motion, is strengthened by deformation movement. The vortex will lead to the separation of the boundary layer. For comparison, due to the flexure direction is opposite to the pitching direction under deformation mode 2, the deformation motion weakens the vortex caused by pitching motion, which leads to the fluid being closer to the lower surfaces at down-stroke. It can be seen in Fig. 15(a) that at lower f* the pressure difference in three modes is very similar, which leads to a limited difference in lift coefficient ( Fig. 12(a)). However, the pressure difference of deformation mode 1 is larger than that of rigid hydrofoil and deformation mode 2 at higher f* ( Fig. 15(b)), resulting the higher lift coefficient ( Fig. 12(b)). As shown in Fig. 16(a), the peak  of 41% and the zone of high  (34.2%<<41%) were achieved at 0.1<f*<0.175 and 65<0<80. The highest  of deformation mode 1 was up to 54.3% and the zone of high  (36.1%<<54.3%), which is at 0.1<f*<0.25 and 55<0<80, is wider than that of rigid hydrofoils. However, compared with the rigid hydrofoil and deformation mode 1, the deformation mode 2 have a lower peak  of 30.3% and a narrower zone of high  (26.9%<<30.3%) at 0.075<f*<0.15 and 70<0<80. It can be inferred that the deformation mode 2 has obvious effects on reducing power-extraction efficiency without

Conclusions
In this study, the effects of ,  , f*, 0 and deformation modes on energy extraction capacity of a flexible hydrofoil were investigated using the commercially available software Fluent. The results are as follows: (1) Energy extraction capacity is affected seriously by the chordwise flexibility. Power coefficients and efficiency are sensitive to , while the influences of flexible coefficient  are very limited. In addition, Cp and  are increased obviously with  increasing from 0 to 15%, while the increase of them is very limited as  further increase from 15% to 20%. As  increase from 0 to 10%, boundary layer thickens, while a leading-edge vortex shedding along the near wall to the trailing edge when  further increase from 10% to 20%.
(2) Flexible deformation modes affect AOA and vortex structure, which alter the pressure difference, and thus affect the power extraction capacity. The order of power extraction capacity is: deformation mode 1 > rigid hydrofoils > deformation mode 2.
(3) Interestingly, there is a critical reduced frequency f*s under three modes, when f*< f*s, Cp for three modes is very similar; However, when f*> f*s, as f* increase, the sensitivity of Cp is significantly increased. In addition, the f*s is generally increase with increasing θ0.
Therefore, the adjustment of flexure modes and parameters, which is made according to different requirement of power coefficient under different tidal currents, should be carried out under the condition that f* is larger than f*s.
The interaction between the flexible hydrofoil and the fluid flow in the real tidal currents is complicated. The fluid-structure interaction can be calculated by a fully two-way coupling approach [40][41][42]. In the future, this approach should be used for the simulation of the flexible hydrofoil.