Numerical study of the propulsive performance of two-dimensional pitching foils at very high frequencies: scaling laws and turbulence effects

Purpose This paper aims to analyze the propulsive performance of small-amplitude pitching foils at very high frequencies with double objectives: to find out scaling laws for the time-averaged thrust and propulsive efficiency at very high frequencies; and to characterize the Strouhal number above which the effect of turbulence on the mean values cannot be neglected. Design/methodology/approach The thrust force and propulsive efficiency of a pitching NACA0012 foil at high reduced frequencies (k) and a Reynolds number Re = 16 000 are analyzed using accurate numerical simulations, both assuming laminar flow and using a transition turbulence model. The time-averaged results are validated with available experimental data for k up to about 12 (Strouhal number, St, up to 0.6). This study also compares the present numerical results with the predictions of theoretical models and existing numerical results. For a foil pitching about its quarter chord with amplitude α0 = 8o, the reduced frequency is varied here up to k = 30 (St up to 2), much higher than in any previous numerical or experimental work. Findings For this pitch amplitude, turbulence effects are found negligible for St ≲ 0.8, and affecting less than 10% to the time-averaged thrust coefficient CT¯ for larger St Linear potential theory fails for very large k, even for the small pitch amplitude considered, particularly for the power coefficient, and therefore for the propulsive efficiency. It is found that CT¯ ∼ St2 for large St, in agreement with recent models, and the propulsive efficiency decays as 1/k, in disagreement with the linear potential theory. Originality/value Pitching foils are increasingly studied as efficient propellers and energy harvesting devices. Their performance at very high reduced frequencies has not been sufficiently analyzed before. The authors provide accurate numerical simulations to discern when turbulence is relevant for the computation of the time-averaged thrust and efficiency and how their scaling with the reduced frequency is affected in relation to the laminar-flow predictions. This is relevant because some small-amplitude theoretical models predict high propulsive efficiency of pitching foils at very high frequencies over certain ranges of the structural parameters, and only very accurate numerical simulations may decide on these predictions.

Study of Motion Performance of a Floating System With Four Moonpools and a VAWT

In this paper the hydrodynamic performance of a barge-type floating foundation installed with four moonpools and a VAWT was investigated through model tests and theoretical calculations. The characteristics of wave-induced motion responses and tether tensions and the effects of turbine rotations were examined. Physical model tests were conducted in a wave tank using regular waves with the wave period ranging from 0.6 to 1.6 seconds and 0.01 or 0.02 meters in amplitude. A 2-MW-class VAWT was modelled with a scale ratio of 1/100 in the experiments. By varying the mass and the rotational speed of the turbine, gyroscopic moment effects were studied. In addition, numerical calculations based on the linear potential theory and Green function method were carried out to estimate motion responses and tether tensions. The present results indicate that the gyroscopic effect due to turbine rotations can be profound. It was found that the first-order motions of the floating system were substantially reduced by the gyroscopic effect, while the second-order motions and tether tensions may be significantly increased. Moreover, the viscous damping of water motions in moonpools was found not negligible. As a result, theoretical models based on linear potential theory should be used with care in hydrodynamic analysis with regard to the floating systems with VAWT rotations. In addition, the present in-house program code was validated against WAMIT through comparing hydrodynamic predictions of a floating foundation with four moonpools, with reasonable agreement.

Hydrodynamic Efficiency of a Wave Energy Converter in Front of an Orthogonal Breakwater

In the present study, the hydrodynamic efficiency of a cylindrical wave energy converter (WEC) of vertical symmetry axis and arranged in front of a reflecting orthogonal breakwater is explored. The idea is based on exploiting the anticipated amplification of the scattered and the reflected wave fields originating from the presence of the vertical walls, towards increasing the WEC’s wave power absorption due to the walls’ wave reflections. Two types of converters are examined, namely the heaving device and the oscillating water column (OWC) device, assuming linear potential theory. The associated diffraction-, motion-, and pressure-radiation problems are solved using axisymmetric eigenfunction expansions for the velocity potential around the WECs by properly accounting for the wave field’s modification due to the walls’ presence. To this end, a theoretical formulation dealing with the evaluation of the converter’s performance is presented accounting for the coupling between the WEC and the reflecting vertical walls. The results depict that the amount of the harvested wave power by the WEC in front of an orthogonal wall is amplified compared to the absorbed wave power by the same WEC in the open sea.

Hydrodynamic Analysis and Motions of Ship with Forward Speed via a Three-Dimensional Time-Domain Panel Method

A new three-dimensional (3D) time-domain panel method is developed to solve the ship hydrodynamic problem and motions. For an advancing ship with a constant forward speed in regular waves, the ship’s hull can be discretized and processed into a number of quadrilateral panels. Based on Green’s theorem, an analytical expression for Froude–Krylov (F–K) forces evaluation on the quadrilateral panels is derived without accuracy loss. Within the linear potential theory, the transient free surface Green function (TFSGF) is applied to solve the boundary value problem. To improve the efficiency and numerical stability of TFSGF evaluation, a precise integration method with variable parameters setting for extended identity matrix is developed to compute the TFSGF in the computation domain. Then, radiation and diffraction forces can be evaluated by means of the impulse response function method. The Wigley I hull form is taken as a study case, and the computed hydrodynamic coefficients, wave exciting forces, and motions by the present method are compared with previous literature experimental data and prior published results. It manifests that the three-dimensional time-domain panel method proposed in this paper has good accuracy.

Non-Linear Motion Characteristics of a Shallow Draft Cylindrical Barge Type Floater for a FOWT in Waves

A shallow draft cylindrical barge type floater with footing close to the water surface was experimentally evaluated in waves to investigate non-linear motion characteristics. The floater was designed to be used as an option for FOWT—floating offshore wind turbines. The non-linear mechanism can be promoted due to the viscous force that acts on the footing edges and the footing interaction with the free surface. In general, the observed non-linear viscous damping is modeled as a force proportional to the square of the relative velocity between the floater and the water. Therefore, the viscous damping levels is expected to increase, and the response in waves, to decrease. However, an increase in motion responses was observed for a broad range of wave periods. An attempt was made to clarify the hydrodynamic mechanism by comparing wave tank experiments, numerical calculations by CFD—computational fluid dynamics codes, and linear potential theory codes. Regular wave tests for three different wave height conditions were carried out, including free decay tests in still waters. For CFD simulations, the OpenFOAM code was selected. For potential theory simulations, the WAMIT code was chosen. As a result of the research, three points could be highlighted and discussed: first, the hydrodynamic phenomenon that contributed to the non-linear motion of the floater was identified; second, the increase and coupling of the motions response of heave and pitch motions; and finally, the phenomenon that the footing held water mass and lifted it to the water surface. The CFD calculations were able to get good qualitative results compared with the experiments and confirmed the use of CFD as a useful tool to capture the non-linear hydrodynamic phenomenon. The linear potential theory was not able to capture the phenomenon discussed herein.

Performance of an Array of Oscillating Water Column Devices in Front of a Fixed Vertical Breakwater

The present study explores the performance of an array of cylindrical oscillating water column (OWC) devices, having a vertical symmetry axis, placed in front of a bottom seated, surface piercing, vertical breakwater. The main goal of this study is the investigation of a possible increase in the power efficiency of an OWC array by applying, in the vicinity of the array, a barrier to the wave propagation, aiming at amplifying the scattered and reflected waves originating from the presence of the devices and the wall. To cope with the set goal, a theoretical analysis is presented in the framework of linear potential theory, based on the solution of the proper diffraction, and pressure-radiation problems in the frequency domain, using the image theory, the matched axisymmetric eigenfunction expansion formulation, and the multiple scattering approach. Numerical results are presented and discussed in terms of the expected power absorption by the OWCs comparing different array’s characteristics i.e.,: (a) angle of incidence of the incoming wave train; (b) distances from the breakwater; and (c) geometric characteristics of the different arrangements. The results show that compared to the isolated OWC array (i.e., no presence of the wall), the power efficiency of the OWCs in front of a breakwater is amplified at specific frequency ranges.

Theoretical Analysis of a Vertical Cylindrical Floater in Front of an Orthogonal Breakwater

This study investigates the effect of an orthogonal-shaped reflecting breakwater on the hydrodynamic characteristics of a vertical cylindrical body. The reflecting walls are placed behind the body, which can be conceived as a floater for wave energy absorption. Linear potential theory is assumed, and the associated diffraction and motion radiation problems are solved in the frequency domain. Axisymmetric eigenfunction expansions of the velocity potential are introduced into properly defined ring-shaped fluid regions surrounding the floater. The hydrodynamic interaction phenomena between the body and the adjacent breakwaters are exactly taken into account by using the method of images. Results are presented and discussed concerning the exciting wave forces on the floater and its hydrodynamic coefficients, concluding that the hydrodynamics of a vertical cylindrical body in front of an orthogonally shaped breakwater differ from those in unbounded waters.

Effects of Four Moon Pools on a Floating System Installed With Twin-VAWTs

This paper describes characteristics of motion responses and tether tensions of a floating structure with four moon pools, on which one or two vertical axis wind turbine models are installed. Effects of several moon pools founded in a floating structure on motion characteristics have been unclear. In this study, the authors proposed a twin-VAWT installed floating system, which was a pontoon based structure. However four moon pools were set on. The study conducted model experiments in a wave tank using regular waves with 0.6 to 2.0 seconds in wave periods and 0.02 and 0.04 m in wave height. The model had four moon pools and was installed with one or two vertical axis turbine models. From it, gyroscopic moment effects were investigated. Besides, the study performed numerical calculations with the linear potential theory based method which were a Green function method. As a results, responses of the twin-turbine model are not affected by gyroscopic moment. The study discusses motion responses and tether tensions with nonlinear behaviours from mainly the experimental results. Also the effect of moon pools were investigated from the calculations. From comparisons of motion results on calculation models with same displacement but different draft, the results suggested that not only heave motion but also roll motion could be reduced because of the moon pools if the size of the moon pools were optimized.