1. Introduction
The wide variety of applications of oscillating structures in a fluid flow has drawn a lot of interest in the past decades. Numerous academic and industrial investigations have led to considerable advances in fluid dynamics and moving deformable structures [
1]. Oscillating structures in a fluid flow can perform in two distinct modes: Power extraction and propulsion [
2]. While the later has been widely investigated, among others, for aerodynamics or propulsion of flying and aquatic species [
3], the former remains a recent application.
Recent interest in wind and tidal energies has led to various researches in order to understand the dynamics of tidal and wind turbines, and underlying physics [
4,
5,
6,
7]. Indeed, heaving and pitching movements of submerged structures have been the subject of numerical [
4,
8,
9] and experimental analysis [
3,
5,
10]. These works were actively interested in optimizing the hydrodynamic performance of these structures, such as minimizing drag or maximizing their thrust or lift, depending on their use [
2,
11,
12]. However, most of these studies are based on 2D modeling of immersed rigid solids. Therefore they do not take into account the Fluid-Structure Interactions (FSI) effects [
1]. More recently, with the advances in computing resources and numerical coupling methods in FSI, realistic and efficient full-scale simulations are now possible [
7].
Structure flexibility can enhance thrust and drag efficiency of a moving foil [
6,
13]. This behavior has been observed in natural phenomena and since then, studied for propulsion and energy extraction. Flexible flukes, for instance, boost the thrust efficiency of bottlenose dolphins by around 20% [
14]. Studies have shown that deformations of flexible structures must be precisely anticipated since they have a significant influence on structural performance [
15]. Therefore, studies on the flexibility of blade materials have been carried out to improve the efficiency of wind and water turbines. For both technologies, flexible blades have been shown to act as a passive pitch control mechanism by making a dynamic adjustment and can prevent the stall effect [
16]. Moreover, the flexibility of the blades provides higher thrust, lower normal forces, and minimal torque as well as reduced turbine vibrations, leading to higher efficiency and a significant increase of the self-starting capability [
6,
15,
16].
Monolithic and partitioned approaches are two of the most well-known numerical methods for handling FSI problems [
17,
18,
19]. The monolithic method is more accurate than partitioned approaches and unconditionally stable [
20]. It is, however, more difficult to implement. Therefore, it is often employed to solve simple FSI problems. Otherwise, partitioned methods are widely adopted in practice due to their ease of implementation. Indeed, by coupling with a dedicated interface, the conventional Computational Fluid Dynamics (CFD) and Computational Solid Dynamics (CSD) programs, they can be utilized for this purpose. In order to ensure the stability and accuracy of this approach for a wide range of applications, various coupling schemes have been developed. In addition, partitioned approaches allow the use of simplified models for the fluid or the structure, independently of one another. For instance, the slender structure can be modeled with beam theory [
21,
22].
While FSI numerical approaches for analyzing the dynamic response of embedded deformable structures are commonly employed [
23,
24], taking into account, in addition to deformations, large free or forced displacements of the structure remains an unexplored field of investigation in the literature. Indeed, multiple sophisticated models are needed to acquire a reasonable description of the fluid and structural dynamics, as well as the dynamic mesh [
19,
25]. For any of these reasons, experimental investigations are more common in this situation. They are, however, still complex and expensive to set up [
3,
5,
11,
26].
As previously shown, the efficiency of rotating wind and tidal turbines can be improved by using flexible blades. While numerous numerical and experimental investigations on rigid oscillating structures have been carried out in order to enhance their hydrodynamic performances, only few of them have investigated the fluid-induced deformations effects on these systems [
27]. In addition, most of these works focused on the propulsive performances of the structure and not on the power extraction performances. For example, Alben [
28] employed an analytical model to establish that the thrust power of a flexible and pitching structure has a series of resonant peaks as a function of its flexibility. David et al. [
29] present an experimental work on the study of the thrust generation of a pitching and rigid foil with flexible flaps connected to its trailing edge. It has been established that the flexible flap significantly improves the thrust performance.
However, there is a lack of knowledge in the literature on the hydrodynamic performance of deformable heaving and pitching structures for energy extraction regime. For example, Yin and Luo [
30] investigated the effects of the wing inertia for a range of wing stiffness on lift and drag performance and on energy consumption. It has been found that both inertia-induced deformations and flow-induced deformations can improve the lift of the structure. Furthermore, flow-induced deformations, as in the case of a low-mass wing, creates less drag and leads to greater aerodynamic power efficiency. Tian et al. [
31] studied the effects of the flexibility of a flapping plate flow energy harvester. They present a strategy to enhance the energy extraction capability of the hydrofoil based on flapping plate active control.
Finally, while the fluid-induced deformations are considered in these investigation, the internal stresses and deformations of the structure are not resolved. Lately, Manjunathan and Bhardwaj [
27] have conducted a numerical analysis in which the plate’s internal stresses are resolved. This study showed that the propulsion is optimal for a specific pitching frequency. However, rigid plates outperform the flexible ones for large pitching frequency. The objective of this work is to carry out a numerical analysis of the FSI effects on a 2D flexible hydrofoil, subjected to a forced oscillating movement. A study of the impact of structure flexibility on hydrodynamic forces and on the efficiency of the hydrofoil are achieved.
In this paper, we first present the mathematical formulation of the problem in
Section 2.
Section 3 outlines the numerical methods applied to solve the problem. Then, the numerical validations and the obtained results are discussed in
Section 4. Lastly, an overall conclusion is drawn in
Section 5.
4. Results and Discussion
4.1. Analysis of the Flow in the Wake of the Hydrofoil
Two characteristic behaviors of the flow can be observed with respect to the pitching amplitude of the hydrofoil. Indeed, for this specific oscillating frequency (
), large pitching amplitude cases generate some dynamic-stall vortex shedding. Conversely, small pitching amplitude cases do not generate vortex. Both behaviors are illustrated by the instantaneous vorticity fields (Equation (
17)) for the cases
° and
° respectively, represented in
Figure 5 and Figure 7 for the hydrofoils 1, 2, and 3.
The snapshots are made at characteristic times
, defined as
where
and
T corresponds to the period number and the period value respectively:
For a small pitching amplitude (
°), no vortex is generated in the flow by the hydrofoil dynamics, as seen in
Figure 5. This applies to all flexible materials of the hydrofoil. For a non-deformable hydrofoil under the same conditions, the similar flow behavior has already been observed and discussed in the literature [
2]. Indeed, it is shown that this is typical for a low oscillation frequency (in our case,
) and a low pitching amplitude.
For this case, the flow around the hydrofoil is slightly affected by the flexibility of the hydrofoil materials. However, small differences do appear and can be seen in the vorticity fields (
Figure 5). These differences are imperceptible along the leading edge of the structure up to about
of its chord. This is because the first third of the structure is non-deformable from the leading edge to its center of rotation, located at
of the chord (
Figure 2). On the other hand, notable differences appear from the last third of the hydrofoil, especially up to the trailing edge, and propagate into the wake. For
, a positive vorticity is generated at the trailing edge of the hydrofoils 2 and 3 cases. The generated vorticity is more important for the more flexible hydrofoil (hydrofoil 3), which is the most flexible hydrofoil. However, there is no vortex generation in the less flexible hydrofoil case (hydrofoil 1). The opposite phenomenon is observed for
, where a negative vorticity is generated at the trailing edge. This flow disturbance is then propagated into the wake and amplified according to the hydrofoil material flexibility. It is then more important in the case of hydrofoil 3.
Dynamic-stall vortex shedding appears when the pitching amplitudes increases. The vorticity amplitude increases with the hydrofoil pitching amplitudes, as shown in
Figure 6. For example, at
, a leading edge vortex is generated along the hydrofoil upper surface for
°. In addition, for
°, a trailing edge vortex is generated along the hydrofoil upper surface.
For the large pitching amplitude (
°), the flow along the hydrofoil surface, from the leading edge up to the
of its chord, does not show any variation according to the three materials (
Figure 7). This is also observed for the case of
° and for all times. This is because of the rigidity of this part of the hydrofoil (see
Figure 2).
As shown previously, the material flexibility in the case of ° has a small impact on the flow dynamics. Conversely, for °, this has a significant impact. First, large vortices are now observed, due to the higher value of the angle of attack. At the time , the flow along the upper surface of hydrofoil stalls earlier for the less flexible flexible hydrofoil (hydrofoil 1). Conversely, it never stalls for the most flexible hydrofoil (hydrofoil 3). On the lower surface, the opposite phenomenon occurs to a lesser extent. In fact, the flow almost stalls at the trailing edge for hydrofoil 3 while it never stalls for hydrofoil 2 and therefore for = hydrofoil 1. At , a similar phenomenon, but symmetrical with respect to the chord, appears. In this case, the stall is observed along the lower surface of hydrofoils 1 and 2 and along the upper surface of hydrofoil 3.
Moreover, for , a positive vortex is generated at the leading edge. The more flexible the hydrofoil material is, the faster the vortex is released. Indeed, at the same time, the vortex is completely detached from hydrofoil 3, while it is about to be released from hydrofoil 2 and is still attached to hydrofoil 1. It is then advected by the flow. At , the same phenomenon is observed on the extrados but a positive vortex becomes negative, and vice versa.
4.2. Flexibility Influence on the Hydrodynamic Forces
The qualitative fluid flow analysis in the previous section shows that the wake of the structure becomes increasingly disturbed as the flexibility of the hydrofoil increases. These flow disturbances has a significant impact on the hydrodynamic forces and therefore on the power extraction efficiency. The hydrodynamic coefficients
(Equation (
18)),
(Equation (
15)), and
(Equation (
16)) are analyzed in this section. The pitching moment coefficient
is calculated with respect to the structure center of rotation:
Figure 8 shows the time variations of hydrodynamic loads for the three hydrofoil materials and pitching amplitude ranging from
° to
°. The hydrodynamic loads are almost periodic for all pitching amplitudes. Thus, a single oscillating period
T is represented.
Firstly, for low pitching amplitude (
°), the coefficient
remains relatively low (
Figure 8). Nevertheless, the amplitude of
increases with the flexibility of the structure. A similar behavior is observed for the coefficient
. The average value of these two coefficients is almost zero for all materials. This can be explained by the fact that the heaving and the pitching motions are symmetrical. On the other hand, the mean value of the coefficient
is equal to
for all the hydrofoils. However, the amplitude of this coefficient increases with the flexibility of the structure. This is probably due to the increase of the hydrofoil projected area in the plane perpendicular to the fluid flow direction. Finally, additional oscillations can be observed on the coefficient curve. This is due to the deformation of the structure.
For °, the hydrodynamic coefficients of hydrofoils 1 and 2 are almost the same. However, the coefficient amplitudes for the most flexible hydrofoil (3) are larger than for hydrofoils 1 and 2. For this and a higher pitching amplitude, flexibility has less impact than for °. However, for high pitching amplitudes, a phase advance of the coefficients is generated. For example, the maximum value of for ° is always obtained earlier for the most flexible materials.
There is a significant increase in the hydrodynamic forces for the pitching amplitude
° compared to
° (
Figure 8). The mean value of the coefficient
is also significantly higher for these cases. This is consistent with the vortex generation and the pitching amplitude increase.
Finally, for the pitching amplitude
°, quite significant vibrations appear for all coefficients. This seems consistent with the appearance of vortices (
Figure 7). These vibrations are maximal for
and
, when the angle of attack is decreasing, from its maximum value (10° to 40°) to its mean value (0°). It also occurs when the leading edge vortices break away from the structure. On the other hand, these disturbances tend to disappear for
and
, when the angle of attack increases. Indeed, for
and
, the vortices have been advected and are therefore already in the wake of the structure. These remarks are valid for all hydrofoils.
The horizontal force coefficient
is not taken into account for the calculation of the energy extraction efficiency. However, mitigating the horizontal force component would reduce interactions with the hydrofoil support structures of the hydrofoil, and therefore enables more reliable and cheaper tidal turbines. The fluctuations of the horizontal forces have already been discussed (
Figure 8). Here, the mean horizontal force coefficient
is plotted in
Figure 9 for all pitching amplitudes
. It can be seen that it increases with both the pitching amplitude
and the flexibility of the hydrofoil. Indeed, the more flexible the material, the larger the mean horizontal forces. This is true for all pitching amplitudes. However, for low pitching amplitudes (
°), the flexibility has very little effect on mean horizontal forces.
Conversely, for a higher , the flexibilty effects are more visible and tend to increase the gap between the three hydrofoils. Finally, increases linearly up to ° for hydrofoils 1 and 2, while it increases linearly up to ° for hydrofoil 3.
4.3. Flexibility Influence on the Power Extraction Efficiency of the Hydrofoil
As shown in
Figure 10, the instantaneous hydrodynamic power coefficient exhibits two distinct behaviors with respect to the pitching amplitudes. Firstly, for small pitching amplitude, such as
° and
°, the power coefficient oscillates quasi-sinusoidally around its mean value. The higher the pitching amplitude, the higher the mean power coefficient value. Indeed, it is almost zero for
° and around
for
° (
Figure 10). For a very low pitching amplitude, such as
°, the power coefficient amplitudes increase with the flexibility of the structure. Conversely, for a medium pitching amplitude (
°), the power coefficient is less impacted by the flexibility. However, the flexibility tends to generate a phase lag, which is especially observable for hydrofoil 3. This is also observed for a higher pitching amplitude such as
°. For these large pitching amplitudes, the instantaneous power coefficient remains periodic but no longer oscillates sinusoidally. For hydrofoil 1 and 2, two peaks are observed at
and
, corresponding to a maximum angle of attack.
Table 5 summarizes the power extraction efficiency for each hydrofoil and pitching amplitude. It can be seen that the efficiency increases with the pitching amplitude for all hydrofoils. Indeed, its value is very low (<1%) for
° and reaches a peak of 20.45% for hydrofoil 3 at
°.
Moreover, the hydrofoil flexibility has a significant impact on its power extraction efficiency. Overall, flexibility tends to increase the efficiency. However, two behaviors are noticeable. Indeed, the efficiency increases with the flexibility of the hydrofoil for the lowest and the highest pitching amplitudes. For example, it can be seen that for °, the efficiency shows a relative improvement of 44% between hydrofoils 1 and 3, while it shows a relative improvement of 46% for °. For these pitching amplitudes, hydrofoil 2 generates a higher efficiency than hydrofoil 1 and a lower one than hydrofoil 3.
On the other hand, for moderate pitching amplitudes 20° °, flexibility seems to have less impact on the efficiency. For example, there is no increase in power extraction efficiency between hydrofoils 2 and 3 for ° and between hydrofoils 1 and 2 for °.
4.4. Fluid Pressure and Vorticity Fields Analysis
As discussed previously, the power extraction coefficient
and the horizontal force component are highly dependent on the hydrofoil material flexibility. It has been shown that the well-known leading edge vortex (LEV) phenomenon, which is frequently investigated when studying an oscillating structure in a fluid flow, can significantly increase the performances of oscillating foil [
2]. In the present study, LEV phenomenon is observed for a pitching amplitude
greater or equal to 25° and higher amplitudes. This is illustrated in
Figure 6 and
Figure 11. Since the hydrofoil heaving displacements are symmetrical, it is expected to observe the same
coefficient values during downward displacements (
) as during upward ones (
). Furthermore, it can be seen in
Figure 10 that flexibility mainly influences the power coefficient in the first halves of the downward (
) and upward (
) heaving phase.
In
Figure 11, the instantaneous vorticity and pressure fields are presented at four times, as marked in
Figure 10. First, it is recalled that the vortex structures are similar in shapes for all the materials and for a specific pitching amplitude. This was already discussed in
Section 4.1. Therefore, it is not surprising to observe similar pressure distributions around the hydrofoils. The pressure and vorticity around hydrofoil 1 (
Figure 11a) give an example to understand the relationship between these flow fields and the power coefficient
. As the hydrofoil is pitching and heaving downward, a LEV is generated (
) along the downstream surface of the hydrofoil, along with a low pressure region. As the LEV is thickening, the power coefficient is increasing. Then, as it is convected by the fluid flow (
), the power coefficient starts decreasing.
The greater deformation of the more flexible hydrofoils (
Figure 11b,c) causes a lower pressure load on the downstream surface and higher pressure load on the upstream surface of the hydrofoil. In addition, the LEV is generated and detached earlier. This phenomenon was also observed by Tian et al. [
31]. This causes a higher maximum value of the power coefficient (
Figure 10) for the more flexible structure. Furthermore, as the LEV appears earlier, the maximum value of
is generated earlier. It is close to
and
for hydrofoils 2 and 3, respectively.
5. Conclusions
A numerical investigation of a deformable hydrofoil in forced heaving and pitching motion for energy production is presented in this study. The effects of both the material flexibility and the pitching amplitude on the hydrodynamic performances are investigated for one oscillation frequency and one heaving amplitude. The FSI effects are considered using a partitioned implicit coupling approach.
Thus, for this specific oscillation frequency, it has been shown that the efficiency is mostly improved with the flexibility of the structure. For a low pitching amplitude, no LEV has been observed. However, for higher pitching amplitudes, LEV occurs and has a significant impact on the hydrofoil performance. Indeed, the relationship between LEV, the low pressure region, and power extraction coefficient have been presented. Flexibility tends to lower the pressure drop and therefore to increase the power extraction coefficient. In addition, LEV stalls earlier for flexible material, which is coherent with the fact that the maximum value is earlier for the most flexible material. A maximum relative improvement of 46% between the most and least flexible hydrofoil has been obtained for %.
Finally, this study shows that the consideration of the hydrofoil deformations is necessary since it can have a huge benefit on its performances. This work should be completed by a complete parametric study taking into account different heaving amplitudes and different oscillation frequencies.