# Principal Parameters Analysis of the Double-Elastic-Constrained Flapping Hydrofoil for Tidal Current Energy Extraction

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## Abstract

**:**

## 1. Introduction

^{4}, Duarte [35,36] experimentally studied the same type of system and found that the pivot axis should be located at least 0.29 chord length from the leading edge to achieve continuous operation. They also revealed that the system can perform much better by increasing the heaving natural frequency. More recently, in 2020, a follow-up numerical study was then conducted by them on this system, which obtained a much higher efficiency of 53.8% [26]. In 2022, R. Fernandez-Feria innovatively introduced the flexible foil into the double-elastic constraint system and studied its flutter instability [37].

## 2. Motion Description

_{h}) models the energy conversion into electricity by an electric generator. The total pitch damping coefficient (D

_{θ}) is corresponding to the sum of some undesired viscous pitch damping, as well as some coulomb friction (dry friction) in pitch, due to the inherent presence of friction in a real experimental setup. The foil is free to pitch (z direction, right-handed helix rule) about a pitch axis located at a distance c

_{EA}from the leading edge and to heave in the y direction as schematically shown. The motion is not possible in any other direction nor about any other axis. In addition, gravity acts in the span direction, or z direction, hence playing no role in the blade dynamics.

_{∞}. In case there are small force disturbances, such as uneven incoming flow or pulsating force caused by turbulence, the foil will leave the equilibrium position and oscillate in the two directions of heave and pitch. This oscillation is caused by the flutter instability, i.e., the coalescence flutter phenomenon. The energy to stimulate the coalescence flutter comes from the kinetic energy of the tidal current, so the tidal current energy can be obtained through this phenomenon. Generally, we obtain energy in the heave direction, because according to the existing research, the energy harvested by the pitch motion is negligible compared with the heaving motion [36].

## 3. Methodology

_{θ}x

_{θ}. However, from Equation (1), there is no necessary connection between S and m

_{θ}. In order not to introduce new non-dimensional parameters, m

_{h}is used instead of m

_{θ}in the dimensionless parameter construction of S. S* is expressed as the mathematically static imbalance arm length. As the key parameters affecting the energy acquisition efficiency of this system are very rich, including not only the parameters listed in Table 2 but also airfoil, three-dimensional effect, Re number, non-uniform incoming flow, oblique flow, etc., a comprehensive parametric study is thus beyond the scope of the present work. Considering the focus of this paper, the parameter setting of this problem is simplified as follows.

_{∞}c/γ) of 2 × 10

^{5}based on the freestream velocity and the chord length. Such a large Reynolds number also ensures that the boundary layers are turbulent, hence making the use of a turbulence model in the fully turbulent mode adequate, as stated in Section 4.1. According to the previous research, the tidal current energy obtained by the flapping hydrofoil mainly comes from the heave motion. Therefore, in order to simplify the analysis, this paper does not consider the influence of pitch damping and sets the pitch damping to zero. Meanwhile, from the research of Boudreau et al. [25] and Goyaniuk et al. [38], s* is very small, so this paper sets s* as zero. That is, the mass center of the foil coincides with the position of the rotating axis. The position of the rotating axis of the head heave motion is selected at 35% chord length from the leading edge [35]. Since the turbine dynamics is analyzed via 2D numerical simulations, forces per unit span are obtained, and the span length is therefore considered to be equal to one (b = 1). Through the above simplification, there are only five main parameters affecting the energy acquisition efficiency of the double-elastic-constrained flapping hydrofoil, including k

_{h}*, k

_{θ}*, ω

_{h}*, ω

_{θ}*, and ζ

_{h}. The final selected baseline case and the parameters describing this baseline are shown in Table 3. Considering that the damping ratio directly affects the amount of power obtained, there should be an optimal damping. So, it is necessary to analyze the influence of the damping ratio (0.0~6.0) for each parameter configuration.

_{max}and pitch amplitude Δθ

_{max}are shown in Equation (2). Taking heaving motion as an example, a stroke refers to the process that the heave coordinate starts from the zero point and then passes through the zero point twice; heave amplitude in one stroke Δ Y

_{i}refers to the distance (in y direction) from the lowest edge to the uppermost edge in a single movement.

_{i}is the heave amplitude of the pitch axis location in one stroke, m; Δθ

_{i}is the pitch amplitude in one stroke, radian; N represents the number of cycles and is a set of positive integers.

_{y}v

_{y}in Equation (3). Therefore, two metrics characterizing the energy extraction, namely the hydraulic efficiency (η) and the average power coefficient (c

_{P}), both based on the cycle-averaged power dissipated in the eddy-current brake which models the energy extraction, are also defined in Equations (4) and (5). The hydraulic efficiency (η) of the turbine expresses the ratio between the total power harvested and the hydraulic power available in the cross section of the flow swept by the foil, whereas the average power coefficient (c

_{P}) of total power harvested by the turbine is normalized in terms of the projected surface of the foil instead, where the unknown maximum heave amplitude (ΔY

_{max}) is replaced by the known chord length (c).

_{0}and t

_{1}are the start time and end time of total N cycles, respectively; A is defined as swept area ($A=2b\Delta {Y}_{\mathrm{max}}$), the product between the span length b and the maximum heave amplitude ΔY

_{max}during its motion; V

_{∞}is freestream velocity.

## 4. Numerical Method

#### 4.1. Governing Equations

#### 4.2. Mesh and Method

^{8}of the chord length. In addition, the refined near-wall mesh could ensure the y plus value is about 1.0.

#### 4.3. Validation

_{h}* = 2.0, ω

_{h}* = 0.707, k

_{θ}* = 0.08, and ω

_{θ}* = 0.894, and the comparison results of foil performance with different meshes are shown in Figure 5. The incoming flow velocity is defined as 1 m/s, and the Re number is about 2 × 10

^{5}. The time step Δt = 0.001 s is selected in the simulation. The position of the rotation axis coincides with the center of gravity and is set at 0.35c.

_{max}), pitch angle (Δθ

_{max}), and reduced frequency (f*) are compared, respectively, as shown in Figure 6. It can be seen that although there are still some errors between the two results, the overall trend is consistent, and the coincidence degree of some working conditions is relatively high. It is speculated that there are two main reasons for this problem. On the one hand, the flutter of this system is always accompanied by a large angle of attack and complex separation, and the accurate simulation of the separation has always been the difficulty of this kind of problem. On the other hand, the attitude of the oscillating foil changes rapidly and violently at any time in the experiment, and it is difficult to accurately evaluate the influence of the water attached to the oscillating foil in the numerical method. In addition, this is obviously not a simple problem. It requires a lot of research, which is not what this paper can carry out. Follow-up research can be carried out in the future.

## 5. Results and Analysis

_{h}are analyzed. Secondly, the flow field and force at the working point with high efficiency and power coefficient are analyzed; then, the effects of the dimensionless spring stiffness coefficient (k

_{h}*, k

_{θ}*) and the frequency ratio (ω

_{h}*, ω

_{θ}*) on the performance of the flapping hydrofoil are analyzed.

#### 5.1. Influence of Damping Coefficient on Energy Acquisition Efficiency

_{p}, heave amplitude ΔY

_{max}, and pitch amplitude Δθ

_{max}) with different damping coefficients are given in Figure 7. As can be seen from Figure 7, on the whole, this flapping-hydrofoil system can start and work regularly within a wide range of damping coefficients, but the hydraulic efficiency and power coefficient are sensitive to the change of the damping coefficient. With the increase in the damping coefficient, the hydraulic efficiency increases rapidly at first and then remains almost unchanged. The highest efficiency can reach about 50%, which is very close to the highest efficiency of 53.8% achieved by Matthieu Boudreau in 2020 [26]; a noticeable feature of the power coefficient is that it does not change monotonically with the change of the damping coefficient, but there is a peak. Under the condition of a small damping coefficient, the power coefficient increases rapidly with the increase of the damping coefficient, but with the further increase of the damping coefficient, the heave amplitude decreases suddenly, and the power coefficient decreases slowly. Under the baseline case working condition, the maximum power coefficient of 0.75 is reached with the damping coefficient of 0.15.

_{p}is due to the different definitions of the expression. That is, η is the total harvested power normalized by the power available in the flow window harvested by the device, while c

_{p}is the total harvested power normalized by a characteristic power based on the chord length of the foil. The hydraulic efficiency η defined in Equation (4) in this paper takes the heave amplitude as the characteristic length. Because the heave amplitude decreases with the increase of the damping coefficient, the efficiency does not show a decreasing trend under a large damping coefficient but continues to increase, which is inconsistent with the actual energy obtained by the flapping hydrofoil; as shown in Equation (5), the power coefficient c

_{p}takes the fixed chord length c as the characteristic length, which represents the actual energy-extraction performance. When the damping coefficient is too small or too large, its energy-extraction performance is not good. Therefore, it is more reasonable to use the power coefficient c

_{p}to evaluate the energy-extraction performance of this system. For example, a very high value of hydraulic efficiency η combined with small amplitudes of motion would result in a low value of c

_{p}. In this scenario, the device would be very efficient, but very little power would be harvested from the flow. However, considering that the hydraulic efficiency η is also of certain significance to the working characteristics analysis, and both of the two metrics need to be optimized in order to obtain an interesting turbine, the hydraulic efficiency η will still be listed and discussed in this paper.

_{max}in Figure 7.

#### 5.2. Flow Field, Motion, and Force Analysis

_{p}=0.7 in the baseline case. In accordance with the result in Figure 9, there could be a great difference between the motion of the double-elastic-constrained system and the sinusoidal motion of the conventional fully constrained flapping hydrofoil. In an effort to display the foil wake vortex diagrams of different attitudes and positions, the times selected in Figure 10 are not divided equally but according to the time when different pitching angles (θ = 0, θ = θ

_{max}/2, θ = θ

_{max}) are reached. It can be seen from Figure 10 that, during every half cycle, a large leading-edge vortex is produced (Figure 10c,g), developed backward (Figure 10d,h), and shed off at the lowest pitch position (Figure 10a,e). It shows an obvious well-defined 2S wake pattern, which has been discussed by Zhuo Wang et al. [40]. Furthermore, it should be noted that when the flapping hydrofoil approaches the equilibrium position, that is, when the pitch angle is close to 0°, such as in Figure 10a,e, a topical separation vortex will be generated on the leading edge of the foil. The topical separation vortex is marked by the arrow in Figure 10a,b,e,f. However, due to the pitching motion of the hydrofoil, the separated vortex turns to the upstream side of the hydrofoil. Then, the topical separation vortex could be impacted by the incoming flow, and it does not have the conditions for continuous development. This separation vortex gradually disappears in the subsequent development, which could be confirmed in Figure 10c,g. In addition, it seems to not affect the subsequent vortex’s street development. According to the time evolution of vortex development in Figure 10, it is speculated that this topical separation vortex is caused by the excessively high pitching speed of the foil when passing through the 0° position.

_{y}v

_{y}curve. Figure 12 shows the time traces of F

_{y}, v

_{y}, and F

_{y}v

_{y}with damping coefficient ζ = 0.177 in the baseline case. As known from the v

_{y}curve, the motion of the flapping hydrofoil in the y direction is close to sinusoidal, which is similar to that of many fully constrained foil systems, so we could make a further comparison in future work. Nevertheless, this is not the case for F

_{y}. From the F

_{y}curve in Figure 12, there is a sudden jump near the moment of v

_{y}= 0, which corresponds to the stage of rapid pitch movement of the flapping hydrofoil. Combined with the wake vortex diagrams in Figure 8 and Figure 10, it can be analyzed that the rapid pitching process of the flapping hydrofoil is corresponding to the shedding of the large separation vortex on the leading edge of the foil. Due to a large shedding vortex and a small separation vortex successively forming on the foil, which have been described in the analysis of Figure 10, F

_{y}has two peaks, one large and one small. The large shedding vortex has completely fallen off at the end of the rapid pitching stage; hence, the hydrodynamic effect is stronger, and the F

_{y}shows a significantly larger peak. However, at this time, there is a certain phase difference between F

_{y}and v

_{y}, so the average mechanical power at this stage is negative on the F

_{y}v

_{y}curve, which has an adverse impact on energy acquisition. As such, the power F

_{y}v

_{y}is positive in the pure heaving phases but negative in the stroke reversal phases. However, the F

_{y}v

_{y}curve shows that the positive work is still large in the whole stroke. Zhuo Wang [40] has also given the time trace curve of C

_{fy}in the research of the double-elastic flapping system but has not discussed the F

_{y}v

_{y}curve, and their C

_{fy}curve characteristics are quite similar to the F

_{y}curve in this paper.

#### 5.3. Effects of Pitch Motions

_{h}*, k

_{θ}*, ω

_{h}*, ω

_{θ}*, and ζ

_{h}. Based on the previous discussion of damping coefficient ζ

_{h}, in this part, we further study the influence of pitch spring stiffness coefficient k

_{θ}* and frequency ratio in the pitch direction ω

_{θ}* on the performance of the flapping hydrofoil.

_{θ}*. In order to ensure that the frequency ratio of the pitching motion ω

_{θ}* remains unchanged, while changing the spring stiffness ratio, the moment of inertia of the flapping-hydrofoil system is also adjusted, and other settings are consistent with the baseline case in Table 3. The performances as a function of the damping coefficient ζ for different pitch spring stiffness coefficient k

_{θ}* are prescribed in Figure 13.

_{p}of the flapping-hydrofoil system are significantly improved as a whole. The power coefficient c

_{p}reaches as high as 0.8, the efficiency η reaches 38% when the spring stiffness is 0.16, and the damping coefficient is about 0.177. However, the change of the pitch spring stiffness coefficient has little effects on the heave and pitch amplitudes. The heave amplitude only increases slightly with the increase of the spring stiffness coefficient, and the pitch amplitude has little change, which is between 75~90°. Moreover, in addition to the four torsion spring stiffness ratios listed in Figure 13, we also analyzed another two cases where the spring stiffness ratio is 0.25 and 0.5. Unfortunately, with these two settings, the double-elastic-constrained foil cannot obtain sustained motions, which is consistent with the conclusion that the system cannot be stable after the spring stiffness ratio is increased to a certain range obtained by Duarte in 2019 [35]. For the consideration of reliability and energy efficiency of the system, pitch spring stiffness coefficient k

_{θ}* = 0.08 is selected in the later analysis in this paper.

_{θ}*. With the baseline case, we keep the stiffness ratio of the pitch spring at 0.08 and increase the frequency ratio in the pitch direction ω

_{θ}* from 0.632 to 1.789 by changing the inertia moment I

_{θ}of the flapping hydrofoil. Figure 14 lists the performance curves of the flapping hydrofoil with four different ω

_{θ}*, which show that the frequency ratio in the pitch direction also has a great impact on the system performance. Noticeable, at a small frequency ratio, such as 0.632, the efficiency and power coefficient are both particularly small. In addition, in this working condition, the heave amplitude is very small, and the pitch angle is more than 100°. This phenomenon can be explained; due to the moment of inertia I

_{θ}being large, the pitch motion of the flapping hydrofoil has poor responsiveness to the flow, resulting in a large deviation from the usual working state. Hence, the energy acquisition performance of system is poor. With the other three frequency ratios (0.894, 1.265, and 1.789), the performance variation trend shown in Figure 14 is basically consistent. Firstly, from Figure 14a, when the frequency ratio ω

_{θ}* increased from 0.894 to 1.789, the overall trend of the efficiency η is almost the same. The efficiency with these three frequency ratios reaches more than 50% when the damping ratio is larger than 0.5, which basically reaches the highest efficiency achieved by Boudreau in 2020 [26], whereas the peak point of power coefficient c

_{p}shown in Figure 14b is very sensitive to the frequency ratio ω

_{θ}*. On one hand, the value of the peak point varies from 0.6 to 0.8 with the increase of the frequency ratio ω

_{θ}*, but the law of change is uncertain; on the other hand, the damping ratio of the power coefficient peak point is different. With the increase of frequency ratio ω

_{θ}*, the damping ratio of the power coefficient peak increases gradually, which can be seen from the three dotted lines representing the peak position in Figure 14b. Then, as seen from Figure 14c, the heave amplitude ΔY

_{max}/c of these three frequency ratios ω

_{θ}* does not change obviously and only increases slightly; lastly, in Figure 14d, the pitch amplitude Δθ

_{max}decreases significantly with the increase of frequency ratio ω

_{θ}

^{*}, which decreases from 80°~90° to 60°~70° as a whole.

_{θ}*, pitch spring stiffness coefficient k

_{θ}* has a more obvious impact on the performance of the system, such as the efficiency and power coefficient.

#### 5.4. Effects of Heave Motions

_{h}* and frequency ratio in the heave direction ω

_{h}* on the performance of the flapping hydrofoil is discussed. The double-elastic-constrained flapping-hydrofoil system obtains energy in the heaving direction, that is, the damping is applied to the heaving motion. Thus, the damping coefficients ζ will be greatly affected when the spring stiffness and the mass of the heave system are adjusted and then affect the performance of the whole system.

_{h}* remains unchanged. Then, the spring stiffness ratio is increased from 0.2 to 2.0, and a series of performance curves are obtained, as shown in Figure 15. Firstly, from the efficiency curve shown in Figure 15a, it can be seen that the system with large spring stiffness ratio coefficient k

_{h}* has higher efficiency η with the same damping coefficient, and the maximum efficiency even reaches 67.7%, which exceeds the maximum value of 59.26% by Bates’s theory. This is because the definition of hydraulic efficiency η in Equation (4) used maximum heave amplitude at the pitching axis position ΔY

_{max}as the sweep width. When the pitch angle of the flapping hydrofoil is large, the actual swept width of the leading edge or trailing edge of the foil is much larger than that of the pitch axis position. In order to compare these two kinds of efficiencies, with k

_{h}* = 2.0, the hydraulic efficiency η

_{2}calculated by the larger value of the maximum heaving amplitude in the leading-edge point or trailing-edge point is also listed in Figure 15e. The curve of hydraulic efficiency η

_{2}with the damping coefficient shows a characteristic of a single peak, which is very close to that of the power coefficient c

_{p}. In addition, the peak value of efficiency η

_{2}is about 30%. It can be further concluded that as metrics characterizing the energy extraction of the double-elastic-constrained flapping-hydrofoil system, power coefficient c

_{p}should be more reasonable than hydrodynamic efficiency η. As shown in Figure 15b, the power coefficient c

_{p}obviously decreases with the increase of the heave spring stiffness coefficient k

_{h}*, which is opposite to that of efficiency η. When the heave spring stiffness ratio k

_{h}* is 0.2, the system does not form stable oscillation under a small damping coefficient. Hence, the highest power coefficient c occurs when the heave spring stiffness ratio k

_{h}* is 0.5, reaching 1.05, and its corresponding efficiency η is 37.8%. Although it is possible to further optimize the power coefficient c

_{p}in the range of k

_{h}* = 0.2 ~0.5, considering the risk that the flapping hydrofoil cannot work normally, the working condition of k

_{h}* = 0.5 is selected in the later analysis. In Figure 15c,d, the variation trend of the heaving amplitude and pitching amplitude with the heave spring stiffness ratio k

_{h}* is consistent, and both of them increase significantly with the decrease of the spring stiffness ratio.

_{h}* = 0.5, this part further discusses the influence of frequency ratio in the heave direction ω

_{h}* on the performance of the flapping hydrofoil. As shown in Figure 16, this paper gives the foil performance and motion characteristic curves with four different frequency ratios ω

_{h}* (0.577–1.414). In Figure 16a, the influence of frequency ratio ω

_{h}* on hydraulic efficiency η is not very clear. On the whole, the efficiency curves first increase rapidly with the increase of the damping coefficient and then gradually tend to increase slowly. Only under different frequency ratio ω

_{h}* conditions, the rate of hydraulic efficiency η increasing with the damping coefficient is slightly different. As seen in Figure 16b, the effect of frequency ratio ω

_{h}* on power coefficient c

_{p}is obvious. The peak values of the power coefficient are different with different frequency ratio ω

_{h}*, and the damping coefficients ζ of the peak point are also different. On the whole, the damping coefficients of the peak points increase gradually with the increase of frequency ratio ω

_{h}*. It is noticeable that, when the frequency ratio in the heave direction ω

_{h}* is one, the peak of the power factor is the lowest. With the increase or decrease of frequency ratio ω

_{h}*, the peak of the power factor increases for both. The power factor reaches the peak value of 1.05 when the frequency ratio ω

_{h}* is 0.707 and 1.414. It is conjectured that when the frequency ratio in the heave direction ω

_{h}* = 1, it is close to pitch motion frequency ratio ω

_{θ}* = 0.894. Hence, the system may tend toward resonance, which may result in a decrease of the system power factor. The particularity of this operating point (ω

_{h}* = 1) can also be seen in the other three figures in Figure 16. Compared with the other three working conditions of frequency ratio ω

_{h}*, the performance curve and motion curve of this working condition (ω

_{h}* = 1) fluctuate more irregularly. Therefore, the influence of frequency ratio ω

_{h}* on the power factor of the flapping hydrofoil needs more and further research. The heave amplitude and the pitch amplitude, as shown in Figure 16c,d, are also obviously affected by frequency ratio ω

_{θ}*. On the whole, with the increase of the frequency ratio in heave motion ω

_{θ}*, the heave amplitude and the pitch amplitude both increase, except for some working points.

_{p}of the double-elastic-constrained system is usually near the damping ratio of 1.0–1.5. From the range in the red circle in Figure 17, the reduced frequency of 0.15 is a more moderate choice, which can provide a suggestion for the mechanical design of the double-elastic-constrained system.

## 6. Conclusions

^{5}. Through the analysis performance of the turbine blade dynamics, flow field, motion, and force of the flapping hydrofoil, the following conclusions are obtained:

- (1)
- In the range of parameters studied in this paper, the power coefficient c
_{p}of the double-elastic-constrained flapping-foil turbine exceeding 1.0 has been achieved, the corresponding efficiency is up to 37.8%, and it is even comparable with the most efficient modern turbines, which capture around 35~45% of available flow energy [26]. The optimal system dynamic parameters are k_{h}* = 0.5, ω_{h}* = 0.707, k_{θ}* = 0.08, ω_{θ}* = 1.265, and ζ_{h}= 0.53, respectively; in addition, compared with hydrodynamic efficiency η, power coefficient c_{p}is more reasonable to measure the energy-extraction ability of the double-elastic-constrained flapping-hydrofoil system. - (2)
- On the whole, this double-elastic-constrained flapping-foil system can start and work regularly within a wide range of damping coefficients. However, the hydraulic efficiency and power coefficient are sensitive to the change of the damping coefficient. When other parameters remain unchanged, with the increase of the damping coefficient, the hydraulic efficiency increases rapidly at first and then gradually tends toward a maximum. The highest efficiency can reach about 50%, whereas the power coefficient does not change monotonically with the increase of the damping coefficient; there is a peak. Noticeably, for different working conditions, the optimal damping coefficient changes greatly. For example, the optimal damping coefficient of the basic case in this paper is 0.17, and the optimal damping coefficient obtained after optimizing the parameters is 0.53. So, it is very necessary to optimize the damping coefficient for different working conditions.
- (3)
- The motion parameters of the double-elastic-constrained flapping-foil system are also affected by the variation of the damping coefficient. The heave amplitudes of the foil always decrease gradually with the increase of the damping coefficient. Although the pitching amplitudes also show a downward trend, the pitch amplitude of the foil keeps a large value under all damping coefficients (about 70~90°). The large pitch amplitude will inevitably make the foil work at a large angle of attack. In addition, the damping coefficient will also affect the periodic characteristics of the flapping hydrofoil’s motion. At a higher damping coefficient, the motion of the foil presents obvious periodic characteristics. The heave displacement is close to the sinusoidal curve, while the pitch angle curve is much fuller; at a small damping coefficient, the periodicity of the foil motion is not obvious, and the randomness is enhanced. This randomness characteristic is more obvious when there is no damping.
- (4)
- In the study of the four main parameters of heave spring stiffness coefficient k
_{h}*, pitch spring stiffness coefficient k_{θ}*, frequency ratio in the heave direction ω_{h}*, and frequency ratio in the pitch direction ω_{θ}*, the most obvious parameter affecting the energy acquisition performance of the system is the spring stiffness coefficients. With the increase of pitch spring stiffness coefficient k_{θ}*, the efficiency and power coefficient of the flapping hydrofoil increase significantly; the influence trend of heave spring stiffness coefficient k_{h}* on the efficiency and power factor is different. With the increase of the stiffness ratio of the heave spring stiffness coefficient k_{h}*, the efficiency increases obviously, while the power coefficient decreases. However, frequency ratios in the heave direction and in the pitch direction (ω_{h}*, ω_{θ}*) both have little influence on the peak value of the efficiency and power coefficient, but they will cause the change of damping coefficients of the peak point, and the law is not very clear, which needs further research. - (5)
- Under all working conditions involved in this paper, the reduced frequency of the double-elastic-constrained system is roughly in the range of 0.11~0.16. If the range of maximum power factor is considered, the optimal reduced frequency is about 0.15, which can provide a suggestion for the mechanism design of the double-elastic-constrained system.

_{h}*, ω

_{θ}*) both have little influence on the peak value of the efficiency and power coefficient, but they will cause the change of the damping coefficients of the peak point, and the law is not very clear. Therefore, further analysis of various parameters can be made in this regard; on the other hand, all the conclusions of this paper are based on the analysis results of numerical calculations, which can be verified by a series of experiments in future work, enrich the test data of the flapping foil, and further demonstrate the correctness of this paper. Furthermore, the 3D effects are inevitable when performing such simulations. This task can be imagined to be very arduous. However, it is of great significance for foil-based flow structure, aerodynamic/hydrodynamic performance, and energy collection efficiency.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Simplified schematic of the symmetrical, rigid, double-elastic-constrained foil with symbolic representation of key parameters.

**Figure 4.**Comparison of four sets of grids with different mesh refinement schemes. (

**a**) Each grid layer’s refinement is 2 layers; (

**b**) each grid layer’s refinement is 4 layers; (

**c**) each grid layer’s refinement is 8 layers; (

**d**) each grid layer’s refinement is 12 layers.

**Figure 5.**Hydrodynamic performance and motion metrics of flapping hydrofoil as functions of the damping coefficient with four different mesh refinement schemes: (

**a**) hydraulic efficiency; (

**b**) power coefficient; (

**c**) heave amplitude; (

**d**) pitch amplitude.

**Figure 6.**Comparisons of various computational performance metrics with previous experimental results. (

**a**) Efficiency (η); (

**b**) heave amplitude (ΔY

_{max}); (

**c**) pitch angle (Δθ

_{max}); (

**d**) reduced frequency (f*) (where ${f}^{*}=\frac{c}{T{V}_{\infty}}=\frac{Nc}{\left({t}_{1}-{t}_{0}\right){V}_{\infty}}$).

**Figure 7.**Variation of flapping-hydrofoil performance with different damping coefficients under baseline case.

**Figure 8.**Wake vortex diagrams of the flapping hydrofoil with four damping coefficients in baseline case: (

**a**) ζ = 0.0; (

**b**) ζ = 0.177; (

**c**) ζ = 0.354; (

**d**) ζ = 0.530 (the time selected is when the pitch angle is zero).

**Figure 9.**Heavy displacement time history curves with different damping ratios under baseline case conditions.: (

**a**) ζ = 0.177; (

**b**) ζ = 0.

**Figure 10.**Time evolution of wake vortex diagrams being formed during one cycle with damping coefficient ζ = 0.177 in baseline case: (

**a**) 0T, θ = 0; (

**b**) 0.06T, θ = θ

_{max}/2; (

**c**) 0.31T, θ = θ

_{max}; (

**d**) 0.44T, θ = θ

_{max}/2; (

**e**) 0.49T, θ = 0; (

**f**) 0.54T, θ = −θ

_{max}/2; (

**g**) 0.84T, θ = −θ

_{max}; (

**h**) 0.95T, θ = −θ

_{max}/2.

**Figure 11.**Time traces of the flapping motions with damping coefficient ζ = 0.177 in the baseline case. (The three phases represented by Roman numerals in the figure represent fast pitching regime (region I), slow pitching regime (region II), and transitional pitching regime (region III)).

**Figure 12.**Time traces of the F

_{y}, v

_{y}, and F

_{y}v

_{y}with damping coefficient ζ = 0.177 in the baseline case.

**Figure 13.**Foil performance parameters as a function of the damping coefficient ζ for various values of pitching spring stiffness coefficient k

_{θ}*. (

**a**) Efficiency η; (

**b**) power coefficient c

_{p}; (

**c**) heave amplitude ΔY

_{max}/c; (

**d**) pitch amplitude Δθ

_{max}.

**Figure 14.**Foil performance parameters as a function of the damping coefficient (ζ) for various values of the pitching motion frequency ratio (ω

_{θ}*). (

**a**) Efficiency η; (

**b**) power coefficient c

_{p}; (

**c**) heave amplitude ΔY

_{max}/c; (

**d**) pitch amplitude Δθ

_{max}. (The three dotted lines in (

**b**) representing the peak position of power coefficient c

_{p}with different frequency ratio).

**Figure 15.**Foil performance as a function of the damping coefficient ζ for different heave spring stiffness coefficients (k

_{h}*). (

**a**) Efficiency η; (

**b**) power coefficient c

_{p}; (

**c**) heave amplitude ΔY

_{max}/c; (

**d**) pitch amplitude Δθ

_{max}; (

**e**) with k

_{h}* = 2.0, hydraulic efficiency η

_{2}calculated by the larger value of the maximum heaving amplitude in the leading-edge point or trailing-edge point as sweep width.

**Figure 16.**Foil performance parameters as a function of the damping coefficient (ζ) for various values of the frequency ratio in heave motion ω

_{h}*. (

**a**) Efficiency η; (

**b**) power coefficient c

_{p}; (

**c**) heave amplitude ΔY

_{max}/c; (

**d**) pitch amplitude Δθ

_{max}.

**Figure 17.**Reduced frequency characteristics of double-elastic-constrained flapping hydrofoil in all working conditions involved in this paper.

Symbol | Units | Definition |
---|---|---|

V_{∞} | m/s | Freestream velocity |

c_{EA} | m | Distance between the elastic axis and the leading edge of the airfoil |

y | m | Instantaneous displacement in heaving motion (upward positive in Figure 1) |

θ | rad | Pitch angle (clockwise negative in Figure 1) |

${m}_{h}$ | Kg | Mass of all the components undergoing the heaving motion (including foil) |

${D}_{h}$ | Ns/m | Total heave damping |

${k}_{h}$ | N/m | Heave spring stiffness |

${I}_{\theta}$ | Kg m^{2} | Moment of inertia about the pitching axis |

${D}_{\theta}$ | Nm s/rad | Total pitch damping |

${k}_{\theta}$ | Nm/rad | Pitch spring stiffness |

$S$ | Kg m | m_{θ}x_{θ}Static moment (mass of the components only undergoing the pitch motion times x _{θ}) |

${m}_{\theta}$ | Kg | Mass of all the components undergoing the pitch motion (including foil) |

${x}_{\theta}$ | m | Distance between the pitch axis and the center of pitching mass (defined positive when the pitch axis is upstream of the center of mass) |

${F}_{y}$ | N | Hydrodynamic force component in the heave (y) direction |

${M}_{z}$ | N m | Hydrodynamic moment about the pitch axis |

b | m | Blade span length |

Non-Dimensional Parameters | Definition |
---|---|

${k}_{h}^{*}=\frac{{k}_{h}}{\rho {V}_{\infty}^{2}b}$ | Heave spring stiffness coefficient, represents the magnitude of heave amplitude |

${k}_{\theta}^{*}=\frac{{k}_{\theta}}{\rho {V}_{\infty}^{2}b{c}^{2}}$ | Pitch spring stiffness coefficient, represents the magnitude of pitch amplitude |

${\omega}_{h}^{*}=\frac{c}{{V}_{\infty}}\sqrt{\frac{{k}_{h}}{{m}_{h}}}$ | Frequency ratio in the heave direction, represents the natural frequency in the heave direction |

${\omega}_{\theta}^{*}=\frac{c}{{V}_{\infty}}\sqrt{\frac{{k}_{\theta}}{{I}_{\theta}}}$ | Frequency ratio in the pitch direction, represents the natural frequency in the pitch direction |

${\zeta}_{h}=\frac{{D}_{h}}{2\sqrt{{m}_{h}{k}_{h}}}$ | Total linear heave damping coefficient |

${\zeta}_{\theta}=\frac{{D}_{\theta}}{2\sqrt{{I}_{\theta}{k}_{\theta}}}$ | Total linear pitch damping coefficient |

${S}^{*}=\frac{S}{{m}_{h}c}$ | Mathematically static imbalance arm length |

${c}_{EA}^{*}=\frac{{c}_{EA}}{c}$ | Position of pitch axis |

Airfoil | Dimension | ${\mathit{k}}_{\mathit{h}}^{*}$ | ${\mathit{k}}_{\mathit{\theta}}^{*}$ | ${\mathit{\omega}}_{\mathit{h}}^{*}$ | ${\mathit{\omega}}_{\mathit{\theta}}^{*}$ | ${\mathit{\zeta}}_{\mathit{h}}$ | ${\mathit{\zeta}}_{\mathit{\theta}}$ | ${\mathit{S}}^{*}$ | ${\mathit{c}}_{\mathit{E}\mathit{A}}^{*}$ | b |
---|---|---|---|---|---|---|---|---|---|---|

NACA0012 | 2D | 2.0 | 0.08 | 0.707 | 0.894 | 0.0~6.0 | 0.0 | 0.0 | 0.35 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, J.; Yan, W.; Mei, L.; Cong, L.; Shi, W.
Principal Parameters Analysis of the Double-Elastic-Constrained Flapping Hydrofoil for Tidal Current Energy Extraction. *J. Mar. Sci. Eng.* **2022**, *10*, 855.
https://doi.org/10.3390/jmse10070855

**AMA Style**

Zhou J, Yan W, Mei L, Cong L, Shi W.
Principal Parameters Analysis of the Double-Elastic-Constrained Flapping Hydrofoil for Tidal Current Energy Extraction. *Journal of Marine Science and Engineering*. 2022; 10(7):855.
https://doi.org/10.3390/jmse10070855

**Chicago/Turabian Style**

Zhou, Junwei, Wenhui Yan, Lei Mei, Lixin Cong, and Weichao Shi.
2022. "Principal Parameters Analysis of the Double-Elastic-Constrained Flapping Hydrofoil for Tidal Current Energy Extraction" *Journal of Marine Science and Engineering* 10, no. 7: 855.
https://doi.org/10.3390/jmse10070855