Analytical and computational approach for studying the interaction between waves and cylindrical wave energy converters oscillating in two modes

ANALYTICAL AND COMPUTATIONAL APPROACH FOR STUDYING THE INTERACTION BETWEEN WAVES AND CYLINDRICAL WAVE ENERGY CONVERTERS OSCILLATING IN TWO MODES HEIDI K. HEIKKINEN*, MARKKU J. LAMPINEN* AND JARI M. BÖLING† * Aalto University School of Engineering Department of Energy Technology Applied Thermodynamics Sähkömiehentie 4J, 02150 Espoo, Finland e-mail: [email protected], [email protected], web page: http://www.aalto.fi † Åbo Akademi Department of Chemical Engineering Process Control Laboratory Piispankatu 8, 20500 Turku, Finland e-mail: jbolin[email protected], web page: http://www.abo.fi Key words: Wave Energy Converter, Cylinder, Horizontal and Vertical Oscillations, Phase Shift. Summary. Ocean wave energy may be recovered by oscillating wave energy converters. The energy converter studied in this work is a horizontally orientated cylinder which may be placed at different depths in the sea. The cylinder can oscillate in horizontal and vertical directions and transfer mechanical energy forward by hydraulic cylinders. To study the interaction between the waves and the converter, we have used potential flow theory separately for both the waves and the oscillating cylinder, and then combined these potential functions by using the principle of superposition. Combined potential flow fields, together with Euler’s equations, enable us to obtain the pressure distribution around the cylinder. When knowing the pressure distribution, both the force upon the cylinder, and the net mechanical power transferred from the waves to the moving cylinder, can be calculated. With this model we have analyzed several interesting topics which affect the efficiency of the wave energy converter. The phase shift is the most important parameter - with the phase shift π/2 the best efficiency 0.5 was achieved. To achieve the right phase shift for different waves is essential due to the power capture. Furthermore it is shown that feedback control is necessary for keeping the phase shift constant. Also the cylinder radius has a great effect on the efficiency. The other important parameters studied in this work were the wave height and the wave period. International Conference on Computational Methods in Marine Engineering MARINE 2011 L.Eça, E. Oñate, J. García, T. Kvamsdal and P. Bergan (Eds) Analytical and Computational Approach for Studying the Interaction between Waves and Cylindrical Wave Energy Converters Oscillating in Two Modes 413 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 2 1 INTRODUCTION To define the exact interaction between the waves and the converter is a complex task, and there are no simple and unequivocal models for calculating the actual process. The issue has been approached before using linear hydrodynamic numerical models among others for flap configurations1. In this study, we take a different approach and perform an analytical and computational model using a cylindrical wave energy converter. There are earlier calculations of wave forces on marine structures2,3 and especially on fixed vertical piles. This thought is utilized here; the most significant difference and renewal is the horizontally aligned cylinder oscillating in two modes. Letting the cylinder oscillate in both horizontal and vertical directions enables higher power capture from the waves and improves the efficiency of the converter. 2 SYSTEM DESCRIPTION The energy converter studied in this work is a horizontally orientated cylinder which may be placed at different depths in the sea. The cylinder can oscillate in horizontal and vertical directions and transfer mechanical energy forward by hydraulic cylinders situated at the both ends of the converter. With the aid of hydraulic cylinders, salt water is pressurized and then feed to a reverse osmosis desalination plant which is directly coupled to the wave energy converter. One part of the pressurized water flows through the reverse osmosis membranes and becomes unsalted, even drinking water if desired, while the other part of the pressurized water can be used for power production for example by turbines. Operation of the wave energy converter in waves is shown in Fig. 1. The dimensions of the system are also included in the figure: H is the wave height, L the wavelength, d the water depth, a the radius of the cylinder and h the distance between the center of the cylinder and still water level. Figure 1. A wave energy converter oscillating in ocean waves. 414 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 3 3 INTERACTION BETWEEN WAVES AND OSCILLATING WAVE ENERGY CONVERTERS To study the interaction between the waves and the wave energy converter, we will use potential flow theory separately both for the waves and the oscillating cylinder, and then combine these flow fields by using the principle of superposition. Combined potential flow fields together with Euler’s equations enable us to obtain the pressure distribution around the cylinder. Knowing the pressure distribution, the force upon the cylinder and also the net mechanical power transferred from the waves to the moving cylinder can be solved. We define the potential function Φ(x, y, t) as follows u x    (1) vy    (2) where u and w are the fluid velocity components in horizontal and vertical directions. When the fluid density is constant, the continuity equation is 22 2 22 0 uv xyx y            (3) The velocity flow field defined by Eqs. (1) and (2) satisfies the two-dimensional irrotationality condition uv yx      (4) The Euler equations of the flow in x-y plane are 1uuu p uv t xy x       (5) 1vvv p uv g t xy y        (6) where ρ is the density, p the pressure and g the acceleration of gravity. Using the Eqs.(1), (2) and (4), the Euler equations may be rewritten as 22 0 2 uv p xt          (7) 415 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 4 22 0 2 uv p gy yt            (8) Integration of these equations with respect to x and y gives 22 (,) 2 uv p F yt t       (9) 22 (,) 2 uv p g y Gxt t        (10) Subtracting Eq. (10) from Eq. (9) we obtain (,) (,)F y t Gxt gy   (11) From Eq. (11) we see that by choosing ( ,) ()Gxt Gt  , Eqs. (9) and (10) are identical and can be rewritten as 22 () 2 uv p g y Gt t        (12) which is the Bernoulli equation. Equation (3) may be rewritten in polar coordinates as 22 2 2 22 11 0 r rr r          (13) and the velocity components ur and uθ can be defined as r ur    (14) 1 ur      (15) which gives 22 22 r uvuu   and hence from Eq. (12) 22 ( , , ) ( sin ) ( ) 2 r uu p r t gh r Gt t               (16) where siny hr    and 0h is the depth of the midpoint of the cylinder. Horizontal flow around the cylinder is shown in Fig. 2. 416 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 5 Figure 2. Potential flow around a circular cylinder. In the potential flow theory different velocity potentials may be summed up owing to the principle of superposition. Here we divide the motion of the cylinder and water particles into horizontal and vertical parts and build up the velocity potentials respectively. The procedure of forming the Bernoulli equation and the horizontal velocity potential is quite similar to that used by Dean and Dalrymple3. The most important difference is the oscillating and horizontally aligned cylinder used in this model. An approximative solution for the velocity potential which satisfies the Laplace equation and the boundary conditions on the surface of the cylinder can be expressed as 22 cos ( ) cos sin ( ) sin HV HV aa ur u U vr v V rr               (17) where u is the horizontal velocity of the water particle, U the horizontal velocity of the cylinder, v is the vertical velocity of the water particle, V the vertical velocity of the cylinder, a the radius of the cylinder, r the distance from the midpoint of the cylinder and θ the angle between the horizontal axis and the point. Both functions ΦH and ΦV satisfy Eq. (13) for any time dependent velocities u(t), U(t), v(t) and V(t), and approximately for velocities u and v which are given by Eqs. (21) and (22) and used in the calculations of Figs.3-11. Assuming the flow to be slow around the cylinder, the term 22 ( )2 r uu   in Eq. (16) can be neglected2 and the pressure distribution on the cylinder may be obtained by using Equations (16) and (17): ( , , ) 2 cos 2 sin ( sin ) ( ) uU vV p at a a g h a Gt tt tt                             (18) Integration of the pressure around the cylinder gives the horizontal and vertical forces on the cylinder (per unit length): 417 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 6    2 '2 0 , cos 2 H uU F pa a d a tt               (19)    2 '2 0 , sin 2 V vV Fpaad a g tt                (20) According the linear wave theory, the horizontal and vertical wave velocity components are4   2 cosh 22 cos 2 2cosh yd L H gT x t ud L LT L                 (21)   2 sinh 22 sin 2 2cosh yd L H gT x t vd L LT L                 (22) where H is the wave height, T the wave period, L the wavelength and d the water depth. Because of the relative velocity between the cylinder and wave used in the velocity potential, we can set the values 0x and 0yh  (the distance between still water level and the midpoint of the cylinder). The horizontal and vertical velocity components of an oscillating cylinder are 0 2 cos t UU T       (23) 0 2 sin t VV T        (24) where U0 and V0 are the maximum velocity components of the cylinder and φ the phase shift between the wave and the cylinder. The wave power on the cylinder is the product of force and velocity: converter H V P FU FV  (25) The mean energy flux of the wave energy converter can now be calculated 418 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 7 0 0022 1 2( ) 2( ) cosh sinh sin 2 cosh T converter converter E P dt T hd hd UV g aH LL d L L                   (26) The mean energy flux or power transmission per wave2 is 224 114 8 82 sinh g wave d H gc Hg L L Ed T L                       (27) The efficiency of the wave energy converter is converter wave E E    (28) 4 THE EFFECTS OF PHASE SHIFT AND DIFFERENT SIZE WAVES ON WAVE ENERGY CONVERTERS With this model we have analyzed several interesting topics which affect the efficiency of the wave energy converter. The phase shift is the most important parameter - with the optimum phase shift π/2 the best efficiency 0.5 is achieved when both horizontal and vertical oscillations are allowed. The other important parameters studied in this work are the wave height and the wave period. Also the cylinder radius has a great impact on the efficiency. Two different size cylinders are modeled: the smaller has a radius of two meters and is situated in three meter depth from still water level, while the bigger has radius of three meters and the depth four meters. In these calculations, the wave energy converter is located in the area where the water depth is 12 meters which means transitional water. Figure 3 presents instantaneous velocity potentials ΦH and stream function curves ψH around the cylinder at 0t when the phase shift is optimal, π/2. The impact of the cylinder on the stream is clear; the farthest parts of the stream are almost parallel. The water surface is also drawn in the figure. Instantaneous pressure distribution at 0t  caused by the wave and hydrostatic pressure on the cylinder is presented in Fig. 4. 419 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 8 Figure 3. Constant velocity potentials ΦH and corresponding stream function curves ψH around the cylinder (t = 0, phase shift = π/2). Figure 4. Instantaneous pressure distribution on the cylinder caused by the wave and hydrostatic pressure (t = 0). The charts in figures 5-7 describe the interaction between one wave and wave energy converter. The cylinder has a radius of two meters and the depth three meters from the still water level. The wave height is 1.34 meters, the period ten seconds and the power 19 kW per wave crest meter. In all figures, H refers to horizontal movement and V to vertical movement. The phase shift between the cylinder velocity and the water particle velocity is well shown in Fig. 5. Figure 6 presents the wave forces faced by the oscillating wave energy converter. Figure 7 presents the power captured from the wave; the advantage of the two-dimensional path of the cylinder compared to the only horizontal path of the cylinder appears clearly in the graph. With optimal phase shift, power can be captured during whole wave period. 420 Heidi K. Heikkinen, Markku J. Lampinen and Jari M. Böling 9 -1.00 -0.50 0.00 0.50 1.00 0 5 10 Time [s] Velocities [m/s] u: Water particle U: Cylinder v: Water particle V: Cylinder Figure 5. The velocities of the cylinder and water particles with optimum phase shift π/2 when cylinder radius is 3 m, wave height 1.34 m and wave period 10 s. Figure 6. The force components on the cylinder when phase shift is optimal π/2, cylinder radius 3 m, wave height 1.34 m and wave period 10 s. Buoyancy is about 300 kN/m and therefore left from the chart. Figure 7. The power capture when phase shift is optimal π/2, cylinder radius 3 m, wave height 1.34 m and wave period 10 s. 421