Viscous effect for heaving cylindrical point absorbers controlled by a latching control system and a novel approach to viscous force

Abstract

The modelling of nonlinear viscous effects for a wave energy converter (WEC) under control systems represents a difficult task. The control system aims to maximize the power capture of WECs. Consequently, large velocities of these devices observe. In addition, the linear potential theory is unable to show a realistic prediction of power production by neglecting viscous effects. A common way of modelling the viscous drag force of WECs is based on the viscous term of the Morison equation. In the literature, this approach shows that drag coefficient has a high variability that depends mainly on the relative velocity of the device. In the paper, two models analyze the nonlinear viscous effects on three cylindrical point absorbers. They have different diameter (D) to draft (T) ratios and are controlled by latching control. The first model approaches the viscous force according to the Morison equation. The second model is a heuristic model proposed by the present study. This new approach calculates the viscous force based on two parameters. Computational fluid dynamics (CFD) evaluates the fidelity of both models and the power production of the three-point absorbers. The proposed model shows good concordance with the CFD results at low and high velocities of point absorbers. The power production of the three-point absorbers controlled by latching shows that the power conversion increase is proportional to the D/T ratio.

Appendix

1.1 Mesh sensitivity study

The mesh sensitivity study of the buoy b107 uses the free decay test developed by Jin et al. (2018), see Fig. 13. The experimental test shows that after 3.5 s the buoy suffered a slight mechanical friction, which prevented its free movement. This explains the difference between the CFD and experimental results in the last amplitude.

Fig. 13

Mesh sensitivity study, buoy b107

The decay test simulation uses two symmetrical planes that pass through the vertical axis of each buoy. The symmetrical planes avoid high computational cost. The mesh sensitivity study focused on the free surface around the buoy. Different sizes of elements are tested before obtaining the first numerical result close to the experimental result, Mesh 1. From Mesh1, a sensitivity analysis is developed focused on the refinement and dimensions of the NWT, Mesh 2 and Mesh 3. The element size is 0.0124 m for Mesh 1, and 0.0062 m for Mesh 2 and Mesh 3. Mesh 1 and Mesh 2 considered the width of the NWT shown in Fig. 3. Mesh 3 considered the dimension of NWT with the same width of the experimental tank in Jin et al. (2018), 6 m. Mesh 2 and Mesh 3 show good agreement with the experimental result (amplitude and period). Therefore, for the free surface around each buoy the element size is 0.0062 m. Considering the Mesh 3, Fig. 14 shows the displacement and velocity of the buoy b107 in the decay test.

Fig. 14

Displacement and velocity of the decay test, buoy b107

1.2 Laminar model

In this research a laminar solution is assumed. Different analysis are presented below that justify that a laminar solution is enough in the present work:

• First analysis, the oscillating flow around the buoy. Since the hydrodynamics of cylindrical structures in oscillatory flows, the Keulegan–Carpenter (KC) number and the Reynolds number (Re), are defined by

  $$\text{Re}=\frac{\rho {U}_{\text{m}}D}{\mu }$$

  (18)
  $$\text{KC}=\frac{{U}_{\text{m}}{T}_{\text{w}}}{D}$$

  (19)

  where $\mu$ is the dynamic viscosity of the water, ${U}_{\text{m}}$ is the amplitude of the horizontal flow velocity, ${T}_{\text{w}}$ is the wave period, and $D$ is the diameter of the cylinder. The values of KC and Re numbers are equal to 0.85 and 4.5 × 10⁴, respectively. For the calculated KC, the orbital motion of the water particles around the body wall does not show a clear turbulent behavior according to Sumer and Fredsøe (2006). Similar values of KC and Re numbers are considered in Devolder et al. (2018), where the results of laminar CFD simulation showed good agreement with experimental results.

• Second analysis, the flow of the buoy wall is analyzed as a flat plate. Applying the latching control system without PTO (wave height = 0.08 m), the maximum relative velocity recorded between flow and body is 0.78 m/s, buoy b107. Considering a characteristic length as the sum of the draft (0.28 m) plus maximum amplitude of motion (0.126 m) of buoy b107, the Re on the wall of the body is 3 × 10⁵ and is less than critical Reynolds for a flat plate, 5 × 10⁵ (steady flow). Considering the latching control system and laminar CFD solutions, Giorgi and Ringwood (2016) and Giorgi et al. (2016) show good results for a heaving-buoy wave energy converter. Laminar CFD solutions are also observed in Devolder et al. (2016, 2018), Giorgi et al. (2016) and Anbarsooz et al. (2014).

• Third analysis, behavior of the vortices on the sharp edge of the cylinder. As expected, the numerical results show vortices in the lower part of the buoy, see Figs. 15 and 16. They are of low intensity and with little interaction with the body wall. The vorticity field shows that there is no beginning of a chaotic flow behavior as in turbulent flow with eddies of different length scales. The vortices generated are well-defined structures. They do not show signs of breakdown, because the injection of kinetic energy is not enough and is not constant. The latching duration contributes to decreasing in vortex intensity when the body begins its movement.

  Fig. 15

  

  Free surface and vorticity field (in the plane of symmetry) around b107 downwards, latching duration 0.2 s, wave direction (from left to right), (a) before maximum amplitude, (b) at maximum amplitude

  Fig. 16

  

  Free surface and vorticity field (in the plane of symmetry) around b107 upwards, latching duration 0.2 s, wave direction (from left to right), (a) before maximum amplitude, (b) at maximum amplitude

• Fourth analysis, experimental decay test. Laminar solution reproduced the experimental decay test of buoy b107 satisfactorily, see Fig. 14. This numerical result shows a maximum velocity equal to 0.87 m/s, higher than the maximum relative velocity obtained using the latching control system, 0.78 m/s (without PTO).

• Finally, from the point of view of the numerical solution stability, the laminar flow solution always showed stability, indicating that there is no evidence of a possible transition from laminar to turbulent flow. In view of the previous discussion, the laminar CFD solution is justified in this work. It is important to mention that when PTO is applied the amplitude and velocity of each buoy are reduced. Based on numerical simulation, the maximum velocity of the buoy obtained with PTO is 0.47 m/s, which corresponds to the buoy b010 and a wave height of 0.1 m.

The vorticity field of Figs. 15 and 16 is described in more detail, only the vortices generated in the water are analyzed. When the buoy goes down, Fig. 15a, vorticity detaches from the lower body. When the body reaches the maximum amplitude, Fig. 15b, vorticity detaches from the wall. Finally, both generated vortices show destructive interaction, because they have opposite rotation. When the buoy goes up, Fig. 16a, the vorticity detaches from the wall. In addition, at the maximum amplitude, Fig. 16b, the vorticity detaches from the lower body. Again, an adverse interaction occurs between the generated vortices. The latching duration contributes to decreasing in vortex intensity when the body begins its movement. In Figs. 15a and 16a, the movement of the water particles of the incoming waves affects the vortices generated, stretching their structure. When the PTO system is applied, the intensity of vortices decreases, because the buoy velocity also decreases.

1.3 Latching duration

The latching duration represents the parameter that defines the efficiency of the latching control system. Based on CFD, the approximation of the optimum constant latching duration for each buoy is calculated by testing different latching durations for a regular wave of height and period of 0.08 m and 1.66 s, respectively. According to Fig. 17 the optimum latching duration for buoy b060, b107 and b154 are 0.04 s, 0.20 s and 0.18 s, respectively. PTO is not considered in this analysis.

Fig. 17

Buoy amplitude versus latching duration, PTO is not considered