Numerical study on active wave devouring propulsion
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Abstract
The possibility of extracting energy from gravity waves for marine propulsion was numerically studied by a two-dimensional oscillating hydrofoil in this study. The commercially available computational fluid dynamics software FLUENT was used for the unstructured grid based on the Reynolds-average Navier–Stokes equation. The free surface waves and motion of the flapping foil were implemented by customizing the FLUENT solver using a user-defined function technique. In addition, dynamic mesh technology and post processing capabilities were fully utilized. The validation of the model was carried out using experimental data for an oscillation hydrofoil under the waves. The results of the simulation were investigated in detail in order to explain the increase of propeller efficiency in gravity waves. Eight design parameters were identified and it was found that some of them greatly affected the performance of wave energy extraction by the active oscillating hydrofoil. Finally, the overall results suggested that when the design parameters are correctly maintained, the present approach can increase the performance of the oscillating hydrofoil by absorbing energy from sea waves.
Keywords
Wave devouring propulsion Flapping foil Numerical simulation1 Introduction
Rising energy bills and intensifying pressure to reduce CO_{2} emission have caused increasing demand for alternative energy sources for many sectors including marine propulsion. The ocean waves have long been considered as a substantial source of energy for marine propulsion. However, utilization of wave energy has not been studied extensively because of the complexity of nonlinear oscillatory hydrodynamics.
Several experiments have been performed to examine the possibility of utilizing the energy from ocean waves for marine propulsion (Jakobsen [1], Isshiki et al. [2], Terao [3]). In their concept, one or two hydrofoils were fixed to the ship through a spring system, in a manner that the angle of the hydrofoils could be adjusted as the direction of incoming water is changed. In the meantime, the ship and the hydrofoils are heaving and pitching with incoming waves and produce forward thrust on the hydrofoil.
The corresponding problem with the moving foil oscillating in an unbounded fluid has been studied for several decades. Swimming of slender fish has been treated by Lighthill [4], and the waving motion of a two-dimensional flexible plate has been calculated by Wu [5]. Later, based on the potential flow approach, a series of papers by Wu [6, 7, 8] optimized the oscillation motion and shape parameters for two-dimensional flat plates. Furthermore, several theories of an oscillating foil have been developed. Bose [9] has developed a time-domain panel method for analysis of an oscillating foil in unsteady motion. Kubota et al. [10] and Kudo et al. [11] have developed two dimensional linear and nonlinear theories which can estimate propulsive performance of partly flexible as well as rigid oscillating propulsors. They also showed that an aft-half elastic foil reveals 8 % higher efficiency than a rigid foil. Yamaguchi and Bose [12] extended the work of Kubota et al. and Kudo et al. to design rigid and aft-half elastic oscillating foils for a large-scale ship. Their results showed that both oscillating foils can give higher propulsive efficiency than an optimal screw propeller, and the elastic foil gives 5–7 % higher propulsive efficiency than the screw propeller. Early hydrodynamics models were restricted to potential flow assumption. But with the advancement of computers, more sophisticated numerical models have been introduced to analyze the performance of an oscillating foil. Pedro et al. [13] and Guglielmini and Blondeaux [14] have investigated the performance of a low Reynolds number oscillating foil based on a computational fluid dynamics (CFD) approach and promising results have been reported. Other studies (Lai et al. [15], Anderson et al. [16] and Gopalkrishnan et al. [17]) have addressed the thrust producing capability of an oscillating hydrofoil by experimental work. They have shown that the potential efficiency of the oscillation hydrofoil propulsor can compete with that of a conventional rotating propeller. However, an oscillating hydrofoil has not been considered as a practical replacement because of the mechanical complexity even with the improvement of efficiency up to some extent. Because of that, some of them extended their studies on oscillating foils to consider propulsion by using wave energy as described in the following paragraphs.
For the first time in history Wu [18] introduced the theory for extracting energy from surrounding flows by a two-dimensional hydrofoil oscillating through gravity waves in water. According to his theory, it has been found that the energy extraction is impossible if the flow is uniform, and only feasible when the primary flow contains a wave component which has vertical velocity normal to the mean free stream and the wing span. Finally, he was able to obtain the best mode of heave and pitching for extraction of wave energy by passive type wave devouring propulsor.
Later, Isshiki [19] employed Wu’s theory of an oscillating hydrofoil and extended it by introducing a free surface effect for investigating the possibility of wave devouring propulsion by a passive type oscillating hydrofoil. Further, not only theoretically, but also experimentally, Isshiki and Murakami [20] studied the basic concept of passive type wave devouring capability of an oscillating hydrofoil.
In addition, to illustrate the unsteady foil motions and wave devouring capabilities, Grue et al. [21] developed a theory for a two-dimensional flat plate near the free surface using a frequency-domain integral equation approach. The theory in both head and following waves was in good agreement with the experiment conducted by Isshiki and Murakami [20]; however, with lower wave numbers there were systematic discrepancies between the theory and the experimental results as nonlinear effects and free surface effects were not fully accounted for in the theory.
Despite these inviting results by several independent studies, a significant commercial success is yet to be seen. The drawbacks of the concept have added resistance in calm seas as well as mechanical complexity. Although the oscillating foil propulsor is mechanically complex, if the gain by recovering the wave energy is considerable, it may be more attractive than the conventional screw propeller. In the present study, therefore, we propose a new concept of wave energy recovering through a powered oscillating foil propulsor, which is designed to replace the conventional screw propeller. The objectives are to develop a numerical model which can predict the oscillating foil performance in a wave field and to elucidate the physical mechanisms.
2 Problem description and performance indices
In the above equation g is gravitational acceleration.
3 Modeling of oscillating foil under the wave
The CFD code, FLUENT 6.3 used in this study employs the cell-centered finite volume method that allows the use of computational elements with arbitrary shapes. Free surface deformation was captured using the Volume of Fraction (VOF) method. Reynolds averaging approach with two equations model of k-ω SST model was used for modeling the turbulence. Convective terms were discretized using the second-order accurate upwind scheme, while diffusive terms were discretized using the second-order accurate central differentiation scheme and the volume fraction was discretized with modified high resolution interface capturing (HRIC) option. The velocity pressure coupling and overall solution procedure were based on a SIMPLE type segregated algorithm adapted to structured and unstructured grids. Finally the discretized equations were solved by using point wise Gauss–Seidel iteration.
Boundary conditions were set to simulate the oscillation of the foil under a wave: on the inlet boundary, horizontal and vertical velocity components for generating waves, u and v (Eqs. 14, 15), were imposed with a turbulence intensity of 1 % and pressure extrapolated from inside. On the foil surface, a no-slip condition was imposed, i.e., zero relative velocity with extrapolated pressure. On the outer boundary, the pressure was set to a hydrostatic pressure as a function of depth from the free surface, while all other variables were extrapolated. The first order accurate scheme was used for the extrapolation.
The above wave generation equations and foil motions (in Eqs. 4, 5) are implemented in the FLUENT solver by programming a user defined function (UDF).
Towards the end of the computational domain, an artificial damping zone was applied, so that the wave energy is gradually dissipated in the direction of wave propagation to prevent the wave reflection. The profile and magnitude of the artificial damping have to be designed to minimize the possible wave reflection at entrance of the damping zone and maximize the wave energy dissipation. After comprehensive tests, the length of the damping zone was determined as to be at least three wavelengths. The damping zone was designed by introducing numerical source term into the momentum equation specified by UDF.
u, v, are the velocities in the Cartesian coordinate system, U is the mean flow velocity and, X1 and X2 are the x coordinates of starting and ending points of the damping zone. The performance and efficiency of the artificial damping coefficient C_{a} were numerically tested and confirmed.
4 Grid dependency study on an oscillating foil under a wave
Verification refers to an estimation of the numerical errors and uncertainties in the process of iteration and grid refinement, which are inevitable issues in the numerical computations. To evaluate the grid dependency, four grids were generated with the same meshing strategy (as described in Sect. 3), but with systematically decreasing the element size on the foil surface and near the free surface. In those four test cases non-dimensional parameters (describe in Sect. 6) were set as, wave phase angle −90°, phase difference of heave and pitch 40°, Froude number 0.87, Reynolds number 5 × 10^{7}, non-dimensional heave amplitude 0.5, extended feathering parameter 0.2 and non-dimensional submergence depth 1.28, and further, wave amplitude 1 m and the wave encounter frequency and foil oscillation frequency were kept equal at 0.156 Hz.
Grid convergence study for thrust coefficient
Grid no. | Grid | Number of cells | Average thrust coefficient (C_{x}) | ε_{i}% of difference in thrust coefficient from the finer grid result |
---|---|---|---|---|
1 | Fine | 195704 | 0.29712 | – |
2 | Medium-1 | 123506 | 0.29742 | 0.1 % |
3 | Medium-2 | 78274 | 0.29792 | 0.17 % |
4 | Coarse | 58591 | 0.29846 | 0.18 % |
Grid studies were conducted using four grids, which enables two separate three-grid studies to be performed and compared. Grid study 1 (GS1) gives estimates for grid errors and uncertainties on grid 1 using the three finer grids 1–3 while grid study 2 (GS2) gives estimates for grid errors and uncertainties on grid 2 using the three coarser grids 2–4.
Verification of average thrust coefficient
Study | R | p | C | δ_{RE} (%) | U_{G} (%) |
---|---|---|---|---|---|
GS1 (grids 1–3) | 0.61 | 1.43 | 0.64 | 0.16 | 0.1 |
GS2 (grids 2–4) | 0.90 | 0.31 | 0.11 | 1.44 | 0.17 |
Comparing the estimated grid error through uncertainty U_{G}, the grid uncertainty is less for GS1 than GS2 and the values (0.1 and 0.17 % grid 1, respectively) are reasonable in consideration of the overall number of grid points used. Therefore, for the rest of the numerical study medium-1 grid is used.
5 Comparison to the experiment
Parameters of experimental setup
Hydrofoil profile | NACA0015 |
Chord length | 40 cm |
Pitching center from LE | 12 cm |
Water depth | 71 cm |
Mean angle of attack | −8° |
Wave amplitude | 4.8 cm |
Even though the experiment was conducted with passive type oscillation foil, the simulations were carried out by using active type oscillation foil. However, the simulations motion condition of heave amplitude, pitch amplitude, phase differences and oscillation frequencies are kept identical with the experiment, and the oscillation frequency of the foil and the wave encounter frequency are set to be equal. Further, the heave and pitching motions in the free running test were assumed with corresponding sine function and also average advanced velocity of the free running test was assumed for the advance velocity of the active oscillation foil in simulations. The amplitude of the waves observed in the experiment has shown a considerable scattering; therefore, the mean value of the amplitude was chosen for comparison. The size of the numerical domain was the same as the experimental tank and the same meshing strategy as described in Sect. 3 was used for modeling the numerical domain.
6 Discussions
Foil and wave motion condition: basic case for parametric study
Foil chord length (c) | 7 m |
Advancing speed (U) | 7.2 m/s |
Wave length (λ) | 140.2 m |
Wave angular frequency (ω) | 0.6629 rad/s |
Encounter frequency (ω_{0}) | 0.9855 rad/s |
Wave height (2a) | 2 m |
Heave amplitude (h_{0}) | 4.2 m |
Pitch amplitude (θ_{0}) | 15.2° |
Submergence (h_{1}) | 9 m |
Phase difference of heave and pitching (φ) | 105° |
Phase difference of wave and motion (ψ) | −90° |
6.1 Wave encounter frequency effect
Wu [18] investigated the effect of frequency on extraction of wave energy by his theoretical study. It has been confirmed that when the wave encounter frequency and foil oscillation frequency are equal, the energy gain is maximized. In other words, the energy gain is always accompanied by the increase of the leading-edge suction, suggesting a tendency towards leading edge stall. The above argument has been further investigated numerically by the present study. The non-dimensionalized wave encounter frequency (ω_{0}/2πf) was changed from 0.5 to 2, keeping the foil oscillation frequency (f) constant and changing the wave encounter frequency (ω_{0}) accordingly. The wave length was changed according to the wave encounter frequency and rest of the parameters were kept constant as Froude number 0.87, wave phase angle −90°, phase difference between heave and pitch 40°, non-dimensional heave amplitude 0.6 and non-dimensional submergence depth 1.28.
6.2 Effect of phase difference between wave and foil motion (ψ)
Phase difference between wave and foil motion is discussed in this section. Nine simulations of different phase angles (ψ) were performed varying from −180° to +180°. The following parameters were set to be constant as Froude number 0.87, phase difference between heave and pitch 105°, non-dimensional heave amplitude 0.6, extended feathering parameter 0.43 and non-dimensional submergence depth 1.28. The physical interpretation of wave phase difference ψ is defined as when the wave frame of reference has been fixed at the wave crest (the vertical component of the wave orbital velocity is about to go downwards), then the wave phase difference is at 0° and 180°, the foil is at the mean position (submergence depth = h_{1}) and about to go upwards and downwards with respect to the wave frame of reference respectively. And also when the wave phase difference is −90° and +90°, the foil is at its bottommost position and uppermost position respectively.
6.3 Phase angle effect between heave and pitching motion (φ)
The previous studies on infinite water oscillating foils suggested that the phase angle between heave and pitching motion is a very sensitive parameter for the efficiency and thrust generation mechanism. Therefore, the thrust generation mechanism of an oscillating foil propulsor under the waves has been investigated in different phase angles. Sixteen simulations were carried out with varying phase angles φ from 0 to 180° while keeping the other design parameters constant as Froude number 0.87, wave phase angle −90°, non-dimensional heave amplitude 0.6, extended feathering parameter 0.43 and non-dimensional submergence depth 1.28. Varying the phase from 90° causes the hydrofoil to have a non-zero pitch angle at the top and bottom positions. If the phase angle is greater than 90° at the lowest position of the heaving motion, the hydrofoil will pitch upwards and, if less than 90° at the same position, the hydrofoil will pitch downwards.
In Fig. 14, maximum resultant forces are applied to the foil due to the increase in effective angle of attack. Horizontal components of those resultant forces are much higher and thereby increase the thrust. Also, the vertical component of force is high which results in high instantaneous power supply to the propulsor. As shown in Fig. 15, at the phase angle 60°, the force contribution to the thrust is much higher and the magnitude of instantaneous lift is the smallest. Because of that reason maximum efficiency is achieved near 60°. In Fig. 16, the vertical component of instantaneous force is dominant at phase angle 110°, thus making the higher power input to the foil. Also, the horizontal component gets the minimum values for thrust.
6.4 Froude number effect
The Froude number provides a significant contribution to the wave devouring propulsion because the wave encounter frequencies are highly dependent on the advancing velocity of oscillating foil propulsor. In this section, wave devouring performance is numerically investigated by changing the Froude number based on advancing speed of the oscillating foil. The Froude number is changed from 0.6 to 2. In order to change the Froude number, advancing speed and wave length have to change accordingly. Even though the wave length has changed by the simulation, the wave encounter frequency and foil oscillation frequency are kept constant at 0.23 Hz. To keep the extended feathering parameter at 0.2 the pitching amplitude has to change accordingly. Moreover, the other parameters are kept constants as non-dimensional heave amplitude 0.5, wave phase angle −90°, phase difference between heave and pitch 40° and non-dimensional submergence depth 1.28.
6.5 Feathering parameter
The extended feathering parameter provides a measure of the relative magnitude of pitch and heave velocities with included vertical wave velocity (Eq. 26). Physically, the feathering parameter is denoted as the ratio of the foil slope to the slope of the path traveled by the pitching axis of the oscillating foil with respect to the space fixed incoming wave profile. In the computation the feathering parameter ε is varied from 0 to 1 while changing the pitching amplitude 0–36° accordingly. The same variation of fathering parameter could be achieved by varying the heave amplitude instead of pitching amplitude. But if we changed the heave amplitude instead of pitching amplitude to match the varying fathering parameter, the foil will operate in different influence of wave orbital velocity range. Because the wave effect (wave orbital velocity) decreases exponentially from the free surface and the most effective height exists in the range of free surface to 80 % of the wave length. Therefore we have kept the heave amplitude as a constant and varied the pitch amplitude to change the feathering parameter. The other parameters that kept constant are, Froude number 0.87, non-dimensional heave amplitude 0.6, wave phase angle −90°, phase difference of heave and pitch 40° and non-dimensional submergence depth 1.71.
6.6 Wave amplitude effect
Tabulated data for capability of devouring wave energy with respect to wave amplitude
a (m) | θ_{0} (°) | U (m/s) | ω_{0} (rad/s) | \( \frac{{\omega_{ 0}^{2} c}}{g} \) | \( \frac{{h_{0} }}{c} \) | ε | \( \frac{{h_{1} }}{c} \) | Fr | φ (°) | ψ (°) | |
---|---|---|---|---|---|---|---|---|---|---|---|
W1 | 0.5 | 9.30 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
W2 | 1 | 9.63 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
W3 | 1.5 | 9.96 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
W4 | 2 | 10.29 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
W5 | 2.5 | 10.62 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
W6 | 3 | 10.95 | 6.63 | 1.0588 | 0.8 | 0.7 | 0.2 | 1.71 | 0.8 | 40 | −90 |
Even though previous investigations were carried out with non-dimensional heave amplitude of 1.28, the present computations were done with non-dimensional submergence depth of 1.71. To capture the increment of wave amplitude, we had to increase the size of the non-deformed finer structured mesh around the free surface which leads to a decrease in the space between the foil and free surface mesh (mesh generation is described in Sect. 3). Therefore, to avoid the collision between the foil and free surface mesh, the submergence depth had to be increased up to 1.71. However, a series of computations were performed to check the effect of the new submergence depth of 1.71 on best oscillation parameters obtained by previous computations. Surprisingly, the new submergence did not affect the best values of motion parameters obtained previously as wave phase angle −90°, phase difference of heave and pitch 40°, Froude number 0.8, non-dimensional heave amplitude 0.7 and extended feathering parameter 0.2. But, as we predicted, the thrust and efficiencies were bit lower than the previous submergence at 1.28 cases.
On the other hand, to compare the wave amplitude effect on oscillation foil, a series of simulations were carried out without incoming waves to have a bottom line conditions for wave energy capturing.
Result of wave energy recovery
Wave amplitude (m) a | Input wave power (W/m) P_{w} | Recovered wave power (W/m) P_{r} | Percentage of recovery (%) |
---|---|---|---|
0.5 | 16521 | 15354 | 93 |
1 | 66086 | 46063 | 70 |
1.5 | 148693 | 69094 | 46 |
2 | 264344 | 78307 | 30 |
2.5 | 413038 | 101338 | 25 |
3 | 594775 | 122835 | 21 |
7 Conclusions
The work presented is an effort towards a systematic understanding of the influence of various motion parameters on thrust generation from an active type oscillation foil in a wave field. The parameters that have been investigated are phase difference between wave and foil motion ψ, phase difference between heave and pitching motion φ, submergence depth of foil h_{1}/c, non-dimensional frequency of oscillation ω_{0}^{2}c/g, heave amplitude h_{0}/c, non-dimensional wave encounter frequency ω_{0}/2πf, Froude number Fr and extended feathering parameter ε.
The result of computational verification and validation for an oscillation foil under a wave has been presented. An unstructured grid based Reynolds-averaged Navier–Stokes method was used. First, the effects of oscillating foil frequency and wave encounter frequency were investigated. When the hydrofoil oscillates at the wave encounter frequency, the two motions are correlated and give the peaks of both thrust and efficiency. It was also found that small variation in oscillation frequencies considerably affects the efficiency and thrust. The effects of phase angles between heave and pitch motion and between wave and foil motion were also studied. It was found that when the wave has a −90° phase difference with foil heave motion, the efficiency and thrust reached the maximum due to the high utilization of wave orbital velocity. When selecting the phase angle between heave and pitching motion a design trade-off was found between the efficiency and generated thrust. Then without forfeiting the efficiency or thrust, a value of 40° was found appropriate as the phase angle between heave and pitching motion. Finally, it was observed that increasing wave amplitude increases the efficiency of the propulsor and decreases the percentage of wave energy recovery. It can be seen that 18 % of efficiency increment can be achieved, when the ratio of wave amplitude to foil chord length is 3/7. And when the wave amplitude to foil chord length is less than 1/7, more than 70 % wave energy could be recovered as useful propulsion energy.
The overall results suggest that the present approach (active wave oscillation foil in wavy flow) has the possibility to recover the wave energy for marine propulsion. However, analyzing the interaction between an oscillating propulsor and hull, and developing the hull form suitable for oscillating propulsion could be considered as opportunities for further research.
Notes
Acknowledgments
The authors express their sincere gratitude to Dr. Takafumi Kawamura, who worked together with the authors in the beginning of this work. Further, their gratitude is extended to Professor Hidetoshi Sueoka and Professor Hiromichi Akimoto for their advice during the progress of the project. The authors also deeply thank the editor and anonymous reviewers for their valuable comments and advice.
Open Access
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