Experimental investigation of the hydrodynamic characteristics of heave plates using forced oscillation

Ocean Engineering 66 (2013) 82-91

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Ocean Engineering

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Experimental investigation of the hydrodynamic characteristics of

heave plates using forced oscillation

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Jinxuan Li"*'*, Shuxue Liu", M i n Zhao'''^ Bin Teng"

"state Key Laboratory of Coastal and Offsliore Engineering, Dalian University of Technology. Dalian 116024, China CCCC Third Harbor Consultants Co., Ltd. Shanghai 200032, China

A R T I C L E I N F O

Article history:

Received 26 July 2012 Accepted 6 April 2013 Available online 10 May 2013

Keywords:

Heave plate Forced oscillation

Hydrodynamic characteristics

A B S T R A C T

The heave plate is the key component of a Spar platform as it can effectively improve the heave response of the platform system by providing additional damping and added mass. This paper investigates the hydrodynamic coefficients of heave plates by using forced oscillation model tests. The effects of variables such as the Keulegan-Carpenter (/fC) number, frequency of oscillation, plate depth, thickness-to-width ratio, shape of the edge, perforation ratio and hole size on the hydrodynamic coefficients were analyzed. Experiments using a group of three solid or perforated heave plates were also carried out and the experimental results were compared with those for a single plate. The relationship between the spacing of the heave plates and hydrodynamic coefficients was studied.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The Spar platform has become one o f the main types of platform used in deepwater oil and gas exploration due to its excellent motion performance and relatively low cost. However, it may experience heave resonant oscillation which often causes damage to the risers and mooring systems under severe sea states. So it is essential to reduce the amplitude of the heave response to a safe range. Heave plates are horizontal plates fitted beneath the Spar platform. These plates provide the added mass which can increase the natural heave period of the platform and move it outside the wave frequency range. In addition, the sharp edges of the heave plates enhance the flow separation and the vortex shedding process that help to provide extra viscous damping. All these factors can effectively reduce the heave motion and improve the stability of the platform.

Currently, most studies on heave plates are focused on their geometrical features such as the thickness-to-width ratio, the opening area and plate spacing, as well as motion features which include the amplitude and frequency of the oscillations. Prislin et al. (1998) investigated the hydrodynamic coefficients of single plates and multi-plates w i t h different spacings using free decay tests. The results showed that the drag coefficient is only depen-dent on the Keulegan-Carpenter (KC) number (see Section 3) at values of the Reynolds number beyond 105. Molin (2001) studied arrays of porous disks i n oscillatory flow perpendicular to their

"Corresponding author. Tel.: +86 411 84708520; fax: -1-86 411 84708526. E-mail address: lijx@dlut.edu.cn ü- Li).

0029-8018/$-see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.oceaneng.2013.04.012

planes using potential flow theory. He concluded that no extra damping can be gained by making the disk porous when the KC number is larger than 1.0, but as the KC number becomes smaller, the heave damping of a porous disk is obviously larger than that of a solid disk. In addition, the added mass is sensitive to the amplitude of the motion. That is important i n the vertical resonance of a spar platform. But the effects of the shape of the edge of the disks were not included in the potential flow approach used by Molin (2001). Tao and Thiagarajan (2003a, 2003b) used a numerical calculation to obtain the hydrodynamics of a cylinder associated w i t h plates. The effect of the KC number, the thickness-to-diameter ratio and the diameter ratio (of the plate diameter and cylinder parameter) on the hydrodynamics was investigated. Chua et al. (2005) carried out a series of forced oscillation tests on a solid and porous plate w i t h a large central opening. They found that the greater the perforation ratio, the lower the value of the added mass coefficient and the higher the value of the drag coefficient. Tao and Dray (2008) studied the hydrodynamic per-formance of solid and porous disks. They presented the hydro-dynamic coefficients of the solid or porous disk and examined the sensitivities of the damping and added mass coefficients to the porosity of the disks.

Experimental research is the main means to study the hydro-dynamic characteristics of heave plates, but due to the different ranges of test parameters or the limitations of the test conditions, the conclusions f r o m different researchers were not totally con-sistent w i t h each other. Chua et al. (2005) reported that the drag coefficient decreases w i t h increasing oscillation frequency while the experiments by Tao and Dray (2008) revealed that the hydrodynamic damping is effectively independent o f the

J. Li et al. / Ocean Engineering 66 (2013) 82-91 83

frequency of the oscillation, although their parameters were similar, w i t h the KC numbers in the range 0.2-1.2. Therefore, dedicated experiments and further research are necessary to ascertain the factors that affect the hydrodynamic characteristics of heave plates. The present worl< investigates the hydrodynamic coefficients of a single plate or multi-plates by using forced oscillation tests. All the experiments were conducted in quiescent water and the heave plates were oscillating harmonically using KC numbers in the range 0.2-1.2 and at frequencies i n the range 0.2¬ 1.0 Hz. The effects of/CC number, oscillation frequency, plate depth, thickness-to-width ratio, shape of the edge, perforation ratio, hole size and the spacing of the plates were analyzed. These results can be used in practical engineering design and numerical calculations.

2. Experimental setup

All experiments were conducted in the nonlinear wave tank at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. The wave tank is 60.0 m long, 4.0 m wide and 2.5 m deep. Fig. 1 shows the experimental setup. The heave plates were forced to oscillate harmonically in a vertical direction in quiescent water and the heave motions were generated by a vertically mounted servomotor capable of produ-cing periodic and sinusoidal motion. The model heave plates were attached to the slide block via two steel screw rods which had a diameter of 20 mm. The slide block was able to oscillate periodi-cally along the guide rail controlled by the servomotor. A load cell connected the two rods. It was used to record the heave force on the plate system during the tests. In addition, a linear variable displacement transducer (LVDT) was fitted on the slide block to record the displacement of the heave plates since the plates were rigidly connected to the slide block. It was used to verify the oscillation of the plate w i t h the given displacement for the forced oscillating system. The outputs f r o m the load cell and LVDT were amplified and then sampled and acquired by a data acquisition board (DAQ) w i t h a sample rate of 100 Hz.

In order to investigate the effect of the thickness-to-width ratio, shape of the edge, perforation ratio and hole size on the hydrodynamics of the heave plate, the forced tests were carried out on nine different model heave plates. All the model plates were flat square plates w i t h dimensions of 400 m m x 400 m m . Detailed information on the plates used is given i n Table 1. Fig. 2 gives the specific dimensions of the perforated plates (Plates 6-9),

where the center hole, used to connect the plate, is kept constant at <j)20mm. The dimensions of other holes changed. The three different shapes (rectangular, triangular and semicircular) are shown in Fig. 3.

For the plates described above, the experiments were mainly divided into five parts:

Part 1: Tests on Plate 1 at varying depths, oscilladon ampli-tudes and frequencies to check the effect of the immersed depth of the plate (plate depth) on the results and to investi-gate the hydrodynamics of a single plate.

Part 2: Tests on Plates 1-3 to study the infiuence of the thickness-to-width ratio on the hydrodynamics of the plates. Part 3: Tests on Plates 1, 4 and 5 to study the influence of the shape of the edge on the hydrodynamics of the plates. Part 4: Tests on Plates 6-9 to study the influence of the perforation ratio and hole size on the hydrodynamics of the plates.

Part 5: Tests on multi-plates consisting of three plates (all three plates being either Plate 1 or Plate 7) w i t h different spacing to investigate the effect of the spacing on the hydrodynamics of the mulri-plates.

The experimental parameters for the five part tests are listed in Table 2. The test water was 2.0 m in depth. For tests on m u l t i -plates, the intermediate plate depth was kept constant at 1.0 m and the other plate depths were determined by a given plate

Table 1

Dimensions of model plates.

Model Thickness Shape of Perforation Hole diameter

plate (mm) edge ratio (mm)

Plate 1 t = 5 Rectangular 0 (solid) 0 (solid) (see Fig. 3a)

Plate 2 t = 8 Rectangular 0 (solid) 0 (solid) Plate 3 t = 1 0 Rectangular 0 (solid) 0 (solid) Plate 4 t = 5 Triangular 0 (solid) 0 (solid)

(see Fig. 3b)

Plate 5 t = 5 Semicircular 0 (solid) 0 (solid) (see Fig. 3c)

Plate 6 t = 5 Rectangular 1% 0 = 2 2 Plate 7 t = 5 Rectangular 5% 0 = 2 2 Plate 8 t = 5 Rectangular 5% 0 = 5 0 Plate 9 t = 5 Rectangular 10% 0 = 22

84 ]. Li et al. / Ocean Engineering 66 (2013) 82-91

a

c

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b

O O CD

O

© O ;

-! © © O

© © - © - ^ - ®

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Fig. 2. Sizes of holes used in perforated plates, (a) Perforation rado 1% (Plate 6), (b) perforation ratio 5% (Plate 7), (c) perforadon rado 5% (Plate 8), (d) perforation ratio 10% (Plate 9).

Fig. 3. Model of heave plates with different edge shapes, (a) Rectangular, (b) triangular, (c) semicircular.

Table 2

Expenmental parameters.

Test Depth of plate (m) Amplitude of oscilladon (mm) Frequency of oscilladon (Hz) Relative spacing of plates Part-1 0.4, 0.6, 0.8,1.0,1.2 a = 13, 25, 32, 38, 51, 64, 76 /=0.2, 0.4. 0.5, 0.6, 0.8,1.0 _

Part-2 1.0 0 = 13, 25, 32, 38, 51, 64 7=0.2, 0.5 _

Part-3 1.0 a = 13, 25, 32, 38, 51. 64 / = 0 . 2 , 0.5

-Part-4 1.0 a = 13, 25, 32, 38, 51, 64 /=0.2, 0.5

J. Li et at / Ocean Engineering 66 (2013) 82-91 85

spacing. Eacii vibration test lasted 15-20 cycles and was repeated twice to verify the repeatability of the experimental results.

3. Data processing and analysis

According to the Morison equation (Sarpkaya and Isaacson, 1981), the governing equadon of motion of the plate system can be represented by

(M + pVCa)X + \pACi\X\X = F (1)

where M is the mass of the system, p is the density of water, V is the immersed volume of the plate including the connecting rod, A is the area of the plate, Q is the added mass coefficient and Q is the drag coefficient. X, X and X are the displacement, velocity and acceleration, respectively, of the forced motion. F is the total force on the plate system. In the present study of thin plates forced to oscillate in water, V is defined as the volume of the "equivalent" cube, i.e. V = A x L where L represents the w i d t h of the plate.

During the tests, the load cell recorded the time history of the force, while the LVDT recorded the time history of the displace-ment of the model plate. The velocity and acceleration time series can be calculated by differentiating the displacement, once and twice, respectively. So there are only two unknown variables, i.e.

• Displacemcnt(mcasurcnicnt) • Displacenicnt(filtercd) ~ Force(nicasurcmcnO • Forcc(filtcrcd) /=0.5Hz,KC=\.0, Perforation ratio 5% Q Time (s)

Fig. 4. Typical time tiistories of displacement and force.

Ca and Cd in Eq. (1). They can be obtained by using the least

squares method according to the known force, displacement, velocity and acceleration time series based on Eq. (1).

Due to the high frequency noise f r o m the servomotor con-troller, the outputs f r o m the instruments were filtered using the Fourier transform method. Fig. 4 gives the typical measured and filtered time histories of the displacement and the force. It shows that this filtering method could effectively remove the noise.

The hydrodynamic forces on the oscillating plate in still water depend on two fundamental non-dimensional parameters that are defined as

KC = 2na

P =

(2)

(3)

where L is the w i d t h of the plate, a and ƒ are the amplitude and frequency of the oscillation, respectively, and u is the kinematic viscosity of the fluid.

4. Results and discussion

The hydrodynamic coefficients, including the added mass coefficient Q and the drag coefficient Cj, can be obtained by the least squares method for all test cases. The effects of the different parameters and the results are summarized below.

4.Ï. Influence of the plate depth, KC number and fisquency

The test water was 2.0 m deep and the plate was 0.4 m wide. In order to examine the influence of the depth of the plate on the experimental results and the validity of the experimental results, experiments on Plate 1 using different depths were carried out.

As the w i d t h of the plate, L, remains constant, the KC number represents the amplitude of the heave oscillation. According to Eq. (2) and Table 2, the tests on Plate 1 were carried out using ICC numbers in the range 0.2-1.2, which is typical of the vertical oscillations of deepwater offshore structures. The test oscillation

0.75 p 0.70 0.65 0.60 0.55 0.50 0.45 0.40 -0.35 0.75 0.70 0.65 -0.60 0.55 0.50 -0.45 0.40 0.35 • plate dcpth^0.4tii plalc depth^O.óm • plalc depth=0.8m plate dcptli=1.0m plate depths 1.2m Plalc l,/,=0.2Hz 0,0 0,6 KC M plate deptli=0.4m

—O— plate depth^0.6m

A plate deptli-O.Sm -^Z""- plate depth^l.Om — p l a t e dcpth^l.2m Plate l,/3=0.8Hz 0,0 0,6 KC 0,8 0.75 0.70 0.65 —•— plate depth=0.4m plate deptli=0.6ni . A • plate dcpth-0.8m - — p l a t e depth-1 .Om —^—plate dcptli=l .2m 0.60 r ü° 0.55 - ^ g f P ^ ^ 0.50 - i f 0.45 0.40 0.35 l . l . Plate l,/,=0.5Hz' 0.2 0.6 KC D.75 0.70 0,65 0.60 0.55 0.50 0,45 0.40 0,35 —•— plate deptli=0.4m —e— plate deptli=0.6m " A' • plate dcpth-O.Sm — ' plate dcplh=I,Om plate dcpth=1.2iTi 1.0 1.2 Plate \,l\=\mz 0,2 0.6 KC 1.0 1.2

86 i i et al. I Ocean Engineering 66 (2013) 82-91

frequencies varied w i t h i n the range 0.2-1,0 Hz, corresponding to /?=29,090-160,000.

Fig. 5 gives the added mass coefficients of Plate 1 at different depths. In addition the values of Q at frequency 1.0 Hz, the values of Ca analyzed for different depths appear to be approximately the same. This means that the effect of the plate depth on the added mass coefficient is sufficiendy small that it can be neglected when the plate depth is around 1 m.

Fig. 6 shows the drag coefficient of Plate 1 at different depths. It can be seen that the values of Q for the same /CC value also appear to be idendcal to each other when the plate depth is greater than 0.4 m. When the plate depth is small, some small waves were observed in the water surface when the oscilladon frequency was larger, which may affect the experimental results. However, the experimental results show that the influence of the plate depth is negligible when the plate depth varies w i t h i n the range 0.6-1.2 m. So, the plate depth was set to 1.0 m for the following tests on a single plate i n order to avoid including the effect of the plate depth on the experimental results.

After verifying the influence of the plate depth on the results, and keeping the depth of the plate at 1.0 m, the experiments on Plate 1 at different oscilladng frequencies were systematically conducted. Fig. 7 gives the variadon of the analyzed hydrodynamic coefficients of Plate 1 at different oscillating frequencies w i t h the /CC number. Fig. 7 shows that the added mass coefficient Ca

increases w i t h increasing KC number and there is a nearly linear reladonship between them. As the frequency increases, the value of Ca increases slightly, but it is not obvious. Thus, the oscillating frequencies have only a slight influence on the added mass coefficient of a plate.

In contrast to the added mass coefficient, the drag coefficient C^ decreases w i t h increasing KC number, as shown in Fig. 7. The values of Cd fall rapidly when KC < 0.6 and then tend to become constant for larger values of KC. Moreover, i t is obvious that the drag coefficient is independent of the oscilladng frequencies. However, the effects of the oscilladng frequencies on the drag coefficient are small and can be neglected. This observadon of the effect of the frequencies on is consistent w i t h that given by Tao and Dray (2008). The reason may be that w i t h i n a KC range of 0.2¬ 1.2, convecdon is the dominant feature in the flow generated by an oscilladng plate and is only slightly related to the frequency.

Therefore, the main effect factor for the hydrodynamic coeffi-cients, Ca and Cd, is the KC number. The reason may be that the KC number characterizes the amplitude of fluid motion relative to the diameter of the disk, or in the present cases of forced oscilladon, the amplitude of plate motion. In fact, vortex shedding around the plate occurs when the plate is oscillating. They w i l l evidendy affect the hydrodynamic coefficients. Tao and Thiagarajan (2003a) observed three vortex shedding modes, i.e. independent, inter-active and asymmetric unidirectional vortex shedding, w i t h

10 9 8 6 5 10 9 8 6 5 ' le l,/|=0.2Hz - • - plale deplh=0.4m f < - plate depth=0.6ni plate depth=0.8iTi ' plate depth=1.0m plate depth=1.2in 6 5 0.0 0.2 0.4 0.6 0. KC Plate l,/3=0.8Hz 1.0 1.2 plate depth=0.4in plate depth=0.6m plate depth=0.8m •7 - plate depth=1.0iii <i— plate deptli=1.2in

0.0 0.2 0.4 0.6 0.8 1.0 KC

Plate l,/,=0.5Hz - • - plate depth=0.4m • plate depth=0.6iTi -é^ plale depth=0.8in —-7- plale depth=1.0ra plate depth=1.2iTi 0.0 0.2 0.4 0.6 O.f KC

Plate l,/4=1.0Hz - I plate depth=0.4m plate depth=0.6m plale deptli=0.8m plale depth=1.0in plale deplh=1.2m

Fig. 6. Vanation of with KC for Plate 1 oscilladng at different depths.

0.75 0.70 0.65 0.60 U= 0.55 0.50 0.45 0.40 0.35 0.0 0.2 0.4 0.6 KC 0.8 10 r-9 8 7 6 5 Plate 1, depth ofplate=I.Om - • - y ; = o . 2 H z - • ~ / , = 0 . 4 H z -A,-/3=0.5Hz -'.7-/^=0.6Hz -'*-/5=0.8Hz - > - / , = 1 . 0 H z 0.2 0.4 0,6 KC 0,8 Fig. 7. Variation of C„ and with KC for Plate 1 oscilladng at different frequencies.

J. Li et ai / Ocean Engineering 66 (2013) 82-91 87

numerical investigation on an oscillating cylinder w i t h a single disk of different thickness. The occurrence of the vortex shedding modes was shown to primarily depend on KC number and the dimension of the plate.

It should be noted that the minimum frequency used i n this paper is 0.2 Hz. However, as Tao and Dray (2008) mentioned, when the KC number is very low, the flow around the edges of the plate is dominated by diffusion. The results w i l l be dependent on the frequencies of the oscillations because the dominance of the diffusion is related to the frequency. He (2003) showed experi-mentally that, for solid disks, the damping coefficient varies w i t h the frequency when KC<OA.

4.2. Influence of thickness-to-width ratio

He (2003) carried out experimental studies on the effects of the thickness-to-diameter ratio on the hydrodynamics for three cir-cular disks w i t h a thickness-to-diameter rado of 1/87.5,1/37 and 1 /25, respectively. He used KC numbers in the range 0-1.1 and an oscilladng frequency of 4 Hz. His results show that the thickness-to-diameter ratio affects the damping coefficients dramadcally at a small KC number, while there is a slight dependence of the added mass on the thickness-to-diameter rado.

To study the influence of the thickness-to-width ratio on the hydrodynamics of the square plate, experiments on three plates w i t h thickness of 5, 8 and 10 mm, corresponding to a thickness-to-w i d t h rado of 1/80,1/50 and 1/40, respectively, thickness-to-were conducted at oscilladng frequencies of 0.2 Hz and 0.5 Hz. The KC numbers were in the range 0.2-1.0. Fig. 8 gives the variadon of the added mass coefficients of the three plates w i t h KC number. As can be seen in Fig. 8, similar to the results given by He (2003), the thickness-to-w i d t h rado has litde influence on the added mass of the plate, although there is a slight decrease in Ca as the thickness-to-width rado increases.

The variation of the drag coefficients of the three plates w i t h KC number is shown i n Fig. 9. It is evident that the drag coefficient decreases as the thickness-to-width ratio increases, especially at

low KC numbers. Fig. 9 shows that, at a KC number of 0.2, the drag coefficients for the thickness-to-width ratios of 1/80, 1/50 and 1/40 are 9.14, 8.63 and 6.97 at frequency 0.2 Hz, while they are 9.61, 8.03 and 5.91 at frequency 0.5 Hz, respectively. All these observations are in good agreement w i t h the results given by He (2003) who reported that the damping of the plate decreases significandy when the thickness-to-diameter ratio is less than 1/ 50. However, the influence of the thickness-to-width ratio on the drag coefficient is dependent on the KC number. As the KC number increases, such influence appears to be somewhat weak. A t

KC > 0.4, the value of Q for thickness-to-width ratios of 1 /40 is

no longer reducing as dramatically as that at KC=0.2.

Fig. 10 shows the variation of the added mass and drag coefficients of the three plates w i t h KC number for different frequencies w i t h the same thickness-to-width ratios. This clearly demonstrates the effect of the frequencies on the hydrodynamic coefficients. The frequency has litde effect on the added mass coefficients of the three plates. This is consistent w i t h the results described in Section 4.2. In contrast, the frequency appears to affect the drag coefficients for plates w i t h a thickness-to-width rado of 1/50 and 1/40 at KC<0.4. It should be noted that the dominant flow generated by the oscillating plate changes from convection to diffusion as the thickness increases. Since the dominance of the diffusion is related to the frequency, the results for the drag coefficient depend on the frequency.

These results are basically identical to the results given by Tao and Thiagarajan (2003a). As mentioned above, they reported that the thickness of a disk and the KC number significantly affects the vortex shedding behaviors around the plate. So i t is reasonable that the influence o f t h e thickness-to-width ratio for a square plate on the drag coefficients is dependent on the KC number.

4.3. Influence of the shape of the edge

In practical engineering, the sharp edges of marine structures may become blunted in the hostile marine environment. In the present study, experiments on t w o plates w i t h triangular and

Fig. 8. Variation of C„ witli KC for three plates with different thickness-to-width ratios.

J. Li et aL / Ocean Engineering 66 (2013) 82-91

J. Li et aL / Ocean Engineering 66 (2013) 82-91 89

semicircular edges (see Fig. 3) were also carried out and the results were compared with those of Plate 1, which has a rectangular edge, to examine the influence of the shape of the edge on the hydrodynamics. The added mass coefficients of the three plates are shown in Fig. 11. It can be seen that the plate w i t h a rectangular edge yields the greatest added mass while the values of Co for the other two plates are nearly identical to each other. This means that the added mass reduces when the sharp edge is cut or become round. Fig. 12 gives the variation of the drag coefficients of the three plates w i t h KC number. It can be seen that the values of Cd for the three plates w i t h different edges are quite similar to each other. That indicates that the drag coefficient is basically independent of the shape of the edge.

4.4. Influence of perforation ratio and hole size

The heave plates have perforated openings that fit the vertical risers and these openings play an important role in tuning the hydrodynamics of the heave plates. The present study concerns experiments on three plates w i t h perforation ratios of 1%, 5% and 10%, respectively. The diameter of the holes in these plates is 22 m m (see Fig. 2 for Plates 6, 7 and 9). The results are shown in Figs. 13 and 14 and the hydrodynamic coefficients of the solid plate

(Plate 1) are also given in those figures for comparison. Because the fluid can pass through the hole, i t is reasonable that the added mass coefficient decreases w i t h increasing perforation ratio. As can be seen f r o m Fig. 14, the values of Cd increase w i t h increasing perforation ratio at low KC numbers. As the KC number increases, such influence appears to diminish. The drag coefficients of these plates w i t h different perforation ratios are nearly identical to each other at /CC=1.0, which is quite consistent w i t h the results of Molin (2001) who reported that no extra damping can be gained by making the plate porous when the KC number is larger than 1.0. Therefore, a perforated plate could gain more damping than a solid plate at /fC<1.0 and the greater the perforated ratio, the greater the drag coefficient. This phenomenon may be also attributed to the vortex shedding around the plate. It can be deduced that the hole on the plate can influence the vortex shedding development. The greater the perforated ratio, the bigger its effect on the vortex shedding patterns.

The hole size or the number of holes may influence the hydrodynamics of a perforated plate. Experiments on another perforated plate (Plate 8) w i t h a perforation ratio of 5% were conducted to investigate the effect of the size of the hole. The perforated ratio of Plate 7 is the same as that of Plate 8, but the diameter of the hole in Plate 8 is 50 m m while the diameter of

0.8 0.7 0.6 U° 0.5 0.4 0,3 0.2 Plate \.P0=0% « ~ Plate 6, Pi=l% Plate 7, P2-S% Pli ƒ = 0.2Hz 0.0 0.2 0.4 0.6 KC 0.8 1.0 0.8 0.7 0.6 0 ° 0.5 0.4 0.3 0.2 - • - Plale 1 , P 0 = 0 % - m - Phte6,P]=\% - A - Plate 7, P 2 = 5 % —V— Plate f-0.5Hz 0.0 0.2 0.4 0.6 KC 0.8 1.0

Fig. 13. Variation of Ca with KC for plates with different perforated ratios.

ƒ = 0.2Hz • — P l a t e 1,/>0=0% • • - P l a t e 6 , P l = l % A ^ P l a t e 7 , / > 2 = 5 % •V— Plate 9 , P 3 = I 0 % ƒ = 0 . 5 H z I — Plate 1, P 0 = 0 % i - P l a l e 6 , P l = l % . - Plale 7, P2=5% — Plale 9, « = 1 0 % 0.2 0.4 0.6 KC 1.0 0,0 0,2 0,4 0.6 KC

Fig. 14. Variation of Cj with KC for plates with different perforated rados.

0.8 0.7 0.6 0 ° 0.5 0,4 0.3 0,2 ƒ = 0.2112 • Plate 7,d\=22mm • Plate 8, rf2=50mm 0.4 0.6 0.8 KC 0.8 0.7 0.6 0 ° 0.5 0.4 0.3 0.2 / = 0 . 5 1 1 z -m- P l a t e 7 , c / l = 2 2 m m — P l a t e 8, d 2 = 5 0 m m 0.4 0.6 O.S KC

90 _{J. Li et al. / Ocean Engineering 66 (2013) 82-91}

the hole in Plate 7 is 22 mm. The added mass coefficient and the drag coefficient of the two plates are shown in Figs. 15 and 16, respectively. It can be seen from these figures that the hole size has little influence on the hydrodynamics of the perforated plates. The hydrodynamic coefficients of the two plates are almost the same. So the heave plate could have any suitably sized hole which fits the risers when the perforation ratio is fixed. The size of the hole cannot change the hydrodynamics of the plate.

4.5. Influence of plate spacing

In order to investigate the relationship between the hydro-dynamic coefficients and the plate spacing for multi-plates, the tests on multi-plates (consisting of three plates each of Plates 1 or 7) w i t h different spacing were carried out. Figs. 17 and 18 show the added mass coefficients of one solid plate and one perforated plate in multi-plates w i t h different plate spacing, respectively. The

15 12 9 6 -/ = 0 , 2 H z Plate 7, </l=22mm 18 Plale 8, rf2=50mm IS 12 0.2 0.4 0.6 0.8 KC / = 0.5Hz - • — Plate?, rfl=22mm - O - Plate 8, rf2=50mm 0.0 0.2 0.4 0.6 0.8 1.0 KC Fig. 16. Variation of Q with KC for perforated plates with different sizes of hole.

Fig. 17. Variation of C„ with KC for one solid heave plate in muld-plates with different spacings.

J. Li et aL / Ocean Engineering 66 (2013) 82-91 91

results for a single plate are also given for comparison. In Figs. 17 and 18, H is the plate spacing. It should be emphasized that the so-called added mass coefficient of a single plate means that the value is the results for the multi-plates divided by the number of plates. In addition, the maximum spacing is 0.6 m and the immersed depth of the upper plate is 0.4 m. This depth is a litde smaller, compared w i t h Figs. 5 and 6, the effect of the plate depth is not as significant as that of the spacing. It can be seen f r o m Figs. 17 and 18 that the added mass coefficient per solid plate and perforated plate increases as the spacing increases. The value of Q for each solid plate tends to be the same as the value of Ca for a single solid plate at H/L=1.5. However, for a perforated plate w i t h a perfora-don ratio of 5%, the value of Ca for each perforated plate is close to the result for a single perforated plate at H/L=1.2. This means that the effect of the spacing on the added mass coefficient is dependent on the perforadon rado.

Figs. 19 and 20 show the drag coefficients of one solid plate and one perforated plate in muld-plates w i t h different plate spacings, respectively. Similarly, the drag coefficient per solid plate or perforated plate increases as the spacing increases and tends to the result for a single plate atH/L=1.5.

In fact, Tao et al. (2007) had investigated the span-wise length effects on the hydrodynamic properties based on a heaving verdcal cylinder w i t h two heave plates in the form of two circular disks attached by numencal method. Their results revealed that the relative spacing not only affects vortex shedding development in the way that the vortices developed w i t h i n the two disks are suppressed, but also changes the vortex shedding regime due to the stronger interaction between the vortex shedding processes of the t w o disks as the spacing continues to decrease. Contranly, the critical spacing exists for a definite plate, beyond which the hydrodynamic coefficients are independent on the relative spa-cing. So, i n engineering design practice, as Tao et al. (2007) recommended, the heave plates should be arranged in the inde-pendent region in order to achieve maximum benefit of increasing added mass and drag damping to reduce the verdcal oscillation.

5. Conclusions

The hydrodynamic characteristics o f a heave plate have been studied experimentally in this paper using forced oscillation. The effects of different variables on the hydrodynamic coefficients were analyzed. The following conclusions can be drawn. The added mass coefficient increases as the KC number increases. The frequency has little influence on i t when the KC number in the range 0.2-1.2.

The dependence of the KC number and the frequency of the drag coefficients of an oscillating plate are found to be sensitive to the thickness-to-width ratio. For plates w i t h a thickness-to-width ratio of 1/80 and 1/50, the drag coefficient decreases as the KC

number increases. The drag coefficient is independent of the frequency for a plate w i t h a thickness-to-width ratio of 1/80. However, the frequency appears to influence the drag coefficient at low KC numbers for a thickness-to-width ratio of less than 1 /SO, the greater the frequency, the lower the drag coefficient. But the thickness-to-width rado has a little influence on the added mass.

The added mass of the two plates with a triangular edge and a semicircular edge are nearly idendcal to each other, and the plate w i t h a rectangular edge yields the greatest added mass. In contrast, the drag coefficient is almost independent of the shape of the edge.

The added mass coefficient decreases w i t h increasing tion ratio while drag coefficient increases w i t h increasing perfora-tion ratio at low KC numbers. However, the drag coefficients for plates w i t h different perforation ratios are nearly identical to each other at /CC=1.0. The size of the hole has littie influence on the hydrodynamics of the perforated plates.

Both the added mass coefficient and drag coefficient increase as the plate spacing increases. For a solid plate, the added mass coefficient and the drag coefficient of one plate in multi-plates tend to be those of a single plate at the relative space H/L= 1.5. But for a perforated plate w i t h a perforation ratio of 5%, the added mass coefficient of one plate in multi-plates tends to be that of a single plate at H/L=1.2 while the drag coefficient at H/f.= 1.5.

Acloiowledgements

This work was supported financially by the National Natural Science Foundation of China (No. 51221961), National Basic Research Program of China (No. 2011CB013703) and National Science and Technology Major Project (No. 2008ZX05026-02-02).

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