# Experimental study of the heave added mass and damping of solid and perforated disks close to the free surface

Bernard Molin, Fabien Remy, Thierry Rippol

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Bernard Molin, Fabien Remy, Thierry Rippol. Experimental study of the heave added mass and damping of solid and perforated disks close to the free surface. Proceedings of the 12th International Maritime Association of the Mediterranean (IMAM) 2007, Sep 2007, Varna, Bulgaria. hal-00456812

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Experimental study of the heave added mass and damping of solid and
perforated disks close to the free surface

Bernard Molin¹, Fabien Remy¹ & Thierry Rippol²

¹ Ecole Centrale Marseille, 13 451 Marseille cedex 20

² Océanide, BP 63, 83 502 la Seyne sur mer

Forced heave experiments are performed with solid and perforated disks of large diameters (60 cm) and very small thicknesses (1mm). The submergence of the disks varies from 5 to 25 cm, while the waterdepth is set at 50 cm. Added mass and damping coefficients, as obtained from the measured vertical force, are compared with numerical values from a linearized potential flow approach. The importance of flow separation from the outer rim of the disks is emphasized.

# 1 INTRODUCTION

There are many examples, in the offshore field, of relatively flat structures with small thicknesses, in some cases perforated. For instance the truss spars are equipped with square horizontal plates that have the main purposes of enhancing the vertical added mass and inducing viscous damping; it has been proposed that perforating the plates would improve their hydrodynamic performances (Downie *et al.* 2000). Another case is the hatch covers used as protection structures for submarine equipment such as wellheads. As shown in figure 1 these hatch covers are tubular frames with open-area ratios in the range 0.15-0.50 (Sandvik *et al.* 2006).

<figure><img src="image_2.png" /><figcaption>Figure 1: Hatch cover model.</figcaption></figure>

To be mentioned as well are the mudmat foundations, for which perforations offer some advantages, reducing the installation and pull-out loads (White *et al.* 2005).

A problem associated with these hatch covers or mudmats is their installation procedures, and the loads they

withstand when being lowered through the splash zone, and then all the way down to the sea-floor. In this respect it is important to have a good knowledge of their hydrodynamic characteristics, i.e. vertical added mass and damping coefficients. Model tests were conducted by Norsk Hydro, some of them reported in Sandvik *et al.* (2006). The investigations reported in Downie *et al.* (2000) are also purely experimental.

On the theoretical viewpoint, it has been proposed by the first author to apply potential flow theory, making the assumption that the perforations induce head losses proportional to the square of the traversing velocities. Moreover the perforations are assimilated with a constant (or slowly-varying) porosity distributed over the surface. First applications concerned circular and square cylinder shapes, used as stabilizers in some designs of compliant towers. A striking result was that added mass and damping coefficients depended, to a large extent, on the motion amplitude. These results were confirmed through dedicated model tests (Molin & Legras 1990). Later on Molin (2001) considered the case of a periodic array of circular disks, in unbounded fluid. Recently Molin & Nielsen (2004) considered the case of an isolated disk below the free surface. Some comparisons were presented with model test results, with reasonable agreement, but without free surface effects, the model being deeply submerged. Moreover in those tests, the models were not circular, but rectangular as shown in figure 1.

In this paper we report on some recent experiments, carried out in the BGO-First offshore wave tank, with horizontal disks in forced heave motion. From the force measurements added mass and damping coefficients are derived and compared with numerical predictions.

The paper is divided as follows: section 2 presents the numerical model, which is the same as described in Molin & Nielsen (2004). Section 3 presents the experimental setup and the considered cases. Finally section 4 presents the
comparisons between measurements and calculations.

## 2 NUMERICAL MODEL

The fluid domain is bounded by the bottom ($z = 0$) and the free surface ($z = h$). The disk radius is $a$ and it is located at a distance $d$ from the bottom. The fluid is divided into 3 sub-domains: an outer domain 1 ($R \ge a$, $0 \le z \le h$), a domain 2 below the disk ($R \le a$, $0 \le z \le d$), and a domain 3 above the disk ($R \le a$, $d \le z \le h$) (see figure 2).

<figure><img src="image_3.png" /><figcaption>Figure 2: Geometry.</figcaption></figure>

We take $W = A\omega \cos\omega t$ as the imposed vertical velocity and we use potential flow theory. For a solid disk this is an acceptable assumption as long as the Keulegan-Carpenter number is sufficiently small. For a perforated disk we must also make sure that the flow separates through the openings and that the rotational wakes remain confined within a short distance of the disk. This means that the openings must be small (and numerous enough to be idealized as uniform porosity). Then it makes some sense to use potential flow theory to describe the outer flow.

As in Molin (2001) the discharge equation is taken under the form

$$p_2 - p_3 = \rho \frac{1 - \tau}{2\mu\tau^2} (\Phi_z - A\omega \cos\omega t) |\Phi_z - A\omega \cos\omega t|. \quad (1)$$

Here $\tau$ is the porosity ratio (open area divided by total area) and $\mu$ a discharge coefficient (usually in-between 0.5 and 1).

Equation (1) is partly linearized into

$$\varphi_2 - \varphi_3 = -\frac{i}{\omega} \frac{8}{3\pi} \frac{1-\tau}{2\mu\tau^2} (\varphi_z - A\omega) \|\varphi_z - A\omega\| \quad (2)$$

where $\|\|$ stands for the modulus of the complex number and $\Phi(R, z, t) = \Re\{\varphi(R, z) \exp(-i\omega t)\}$.

The boundary value problem to be solved is then:

$$\Delta\varphi = 0 \quad \text{in the fluid} \quad (3)$$

$$g\varphi_z - \omega^2\varphi = 0 \quad \text{at the mean free surface } z = h \quad (4)$$

$$\varphi_2 - \varphi_3 = -\frac{i}{\omega} \frac{8}{3\pi} \frac{1-\tau}{2\mu\tau^2} (\varphi_z - A\omega) \|\varphi_z - A\omega\| \quad \text{on the disk } z=d \ 0 \le R \le a \quad (5)$$

$$\varphi_z = 0 \quad \text{on the bottom } z=0 \quad (6)$$

$$\text{radiation condition} \quad R \to \infty \quad (7)$$

At the outer edge of the disk the pressure differential is nil. Hence the traversing velocity is also zero. We take advantage of this feature to expand $\varphi_z(R, d)$ as

$$\varphi_z(R, d) = A \omega \left[ 1 + \sum_{i=1}^{\infty} D_i J_0(\nu_i R) \right] \quad (8)$$

where the "wave numbers" $\nu_i$ are the roots of $J_0(\nu_i a) = 0$. This provides an orthogonal basis over the disk.

The BVP is solved through eigen-function expansions, with the velocity potential being developed as:

### Sub-domain 1

$$\varphi_1(R, z) = A\omega a \left\{ A_0 \frac{\cosh k_0 z}{\cosh k_0 h} \frac{H_0(k_0 R)}{H_0(k_0 a)} + \sum_{n=1}^{N_1} A_n \cos k_n z \frac{K_0(k_n R)}{K_0(k_n a)} \right\} \quad (9)$$

### Sub-domain 2

$$\varphi_2(R, z) = \frac{A\omega}{2d} \left(z^2 - \frac{R^2}{2}\right) + A\omega a \left\{ B_0 + \sum_{n=1}^{N_2} B_n \cos \lambda_n z \frac{I_0(\lambda_n R)}{I_0(\lambda_n a)} \right\} + A\omega \sum_{i=1}^{N_4} \frac{D_i}{\nu_i} \frac{\cosh \nu_i z}{\sinh \nu_i d} J_0(\nu_i R) \quad (10)$$

### Sub-domain 3

$$\varphi_3(R, z) = A\omega \left[ z - h + \frac{g}{\omega^2} \right] + A\omega a C_0 \frac{\cosh \mu_0 (z-d)}{\cosh \mu_0 (h-d)} J_0(\mu_0 R) + A\omega a \sum_{n=1}^{N_3} C_n \cos \mu_n (z-d) \frac{I_0(\mu_n R)}{I_0(\mu_n a)} + A\omega \sum_{i=1}^{N_4} \frac{D_i}{\nu_i} \left( \delta_i \cosh \nu_i (z-d) + \sinh \nu_i (z-d) \right) J_0(\nu_i R) \quad (11)$$

where

$$\begin{aligned} \omega^2 &= g k_0 \tanh k_0 h = -g k_n \tan k_n h \\ &= g \mu_0 \tanh \mu_0 (h-d) = -g \mu_n \tan \mu_n (h-d) \end{aligned} \quad (12)$$

$$\delta_i = \frac{\omega^2 \tanh \nu_i (h-d) - g \nu_i}{g \nu_i \tanh \nu_i (h-d) - \omega^2} \quad (13)$$
and $J_0$, $I_0$, $K_0$, $H_0 = J_0 + iY_0$ are the standard notations for Bessel and Hankel functions.

The conditions that remain to be fulfilled are the matching equations for $\varphi$ and $\varphi_R$ at $R = a$, and the discharge equation. The difficulty arising from its non-linearity is overcome through an iterative procedure whereby equation (5) is written as

$$ \sum_j \left[ -\frac{8i}{3\pi} \tilde{K}_C f(R) - \frac{\coth \nu_j d - \delta_j}{\nu_j a} \right] D_j J_0(\nu_j R) \\ = \frac{1}{2ad} \left( d^2 - \frac{R^2}{2} \right) - \frac{1}{a} \left( d - h + \frac{g}{\omega_2} \right) \\ + B_0 + \sum_n (-1)^n B_n \frac{I_0(\lambda_n R)}{I_0(\lambda_n a)} \\ - C_0 \frac{J_0(\mu_0 R)}{\cosh \mu_0 (h-d)} - \sum_n C_n \frac{I_0(\mu_n R)}{I_0(\mu_n a)} \quad (14) $$

where $\tilde{K}_C$ is the "porous Keulegan-Carpenter number", defined as (cf Molin, 2001):

$$ \tilde{K}_C = \frac{1 - \tau}{2\mu\tau^2} \frac{A}{a} \quad (15) $$

and $f(R) = \| \sum_i D_i J_0(\nu_i R) \|$ is evaluated from the solutions obtained at the previous iterations.

Multiplying both sides of equation (14) with $R J_0(\nu_i R)$ and integrating in $R$ from 0 to $a$ for $i = 1, \dots, N_4$ gives the linear system

$$ \mathbf{M}^{(i)} \cdot \vec{D}^{(i)} = \vec{E}_D^{(i)} + \mathbf{DB} \cdot \vec{B}^{(i)} + \mathbf{DC} \cdot \vec{C}^{(i)} \quad (16) $$

Similarly the matching conditions at $R = a$ result into the vectorial equations (applying a procedure à la Garrett, 1971):

$$ \vec{B}^{(i)} = \vec{E}_{21} + \mathbf{BA} \cdot \vec{A}^{(i)} \quad (17) $$

$$ \vec{C}^{(i)} = \vec{E}_{31} + \mathbf{CA} \cdot \vec{A}^{(i)} \quad (18) $$

$$ \vec{A}^{(i)} = \vec{E}_1 + \mathbf{AB} \cdot \vec{B}^{(i)} + \mathbf{AC} \cdot \vec{C}^{(i)} + \mathbf{AD} \cdot \vec{D}^{(i)} \quad (19) $$

Eliminating $\vec{A}^{(i)}, \vec{B}^{(i)}$ and $\vec{C}^{(i)}$ in-between equations (16) through (19) gives a linear system in $\vec{D}^{(i)}$ which is solved by a standard Gauss routine. Convergence is reached within a few iterations, starting from the solid case. The complex added mass coefficient is then obtained by

$$ \frac{M_{33}}{\rho a^3} = C_a + i C_b = 2\pi \left[ \begin{aligned} & \frac{2h-d}{4a} - \frac{a}{16d} - \frac{g}{2a\omega^2} \\ & + \frac{B_0}{2} + \sum_{n=1}^{N_2} (-1)^n \frac{\beta_n B_n}{\lambda_n a} \\ & - \frac{J_1(\mu_0 a) C_0}{\mu_0 a \cosh \mu_0 (h-d)} - \sum_{n=1}^{N_3} \frac{\gamma_n C_n}{\mu_n a} \\ & + \sum_{i=1}^{N_4} (\coth \nu_i d - \delta_i) \frac{D_i}{\nu_i^2 a^2} J_1(\nu_i a) \end{aligned} \right] \quad (20) $$

where $\beta_n = I_1(\lambda_n a)/I_0(\lambda_n a)$ and $\gamma_n = I_1(\mu_n a)/I_0(\mu_n a)$.

The main deficiency in our potential idealization is that it does not account for flow separation at the outer edge of the disk. An empirical correction, inspired from the Morison equation, consists in adding up a supplementary drag term, expressed as

$$ F_v = -\frac{1}{2} \rho C_D \pi a^2 V |V| \quad (21) $$

where $V$ is the disk velocity ($A \omega \cos \omega t$) minus the relative fluid velocity through the disk, averaged over the disk:

$$ \begin{aligned} V &= \Re \left\{ A \omega \left[ 1 + 2 \sum_{i=1}^{N_4} \frac{D_i}{\nu_i a} J_1(\nu_i a) \right] e^{-i \omega t} \right\} \\ &= \Re \left\{ A \omega Q e^{-i \omega t} \right\} \end{aligned} \quad (22) $$

In the end this provides the following correction to the added mass and damping coefficients:

$$ \Delta C_a + i \Delta C_b = \frac{4}{3} i \frac{A}{a} C_D Q \|Q\| \quad (23) $$

Following Graham (1980) and as in Sandvik et al. (2006), we shall relate the drag coefficient $C_D$ to the Keulegan-Carpenter number $KC = \pi A/a$ by

$$ C_D = \alpha KC^{-1/3} \quad (24) $$

where $\alpha$ is to be adjusted.

### 3 DESCRIPTION OF THE EXPERIMENTS

Experiments were performed in the BGO-First facilities, in la Seyne sur mer (near Toulon). This tank has a width of 16 m, with a total length, from the wavemakers to the opposite wall, of about 40 m. During the tests its false bottom was raised to a depth of 50 cm from the free surface.

Three disk models were tested, with the same diameters: 60 cm, and thicknesses of only 1 mm. Rigidity was ensured by a radial stiffening, as can be seen in the photographs (figures 3 and 4). Rigidity checks were performed by locating two accelerometers on the disks, one at the base of the shaft, the second one at the rim (in between two stiffeners): they yielded nearly identical records, in good agreement with the value derived from the imposed motion amplitude and frequency.

One disk was solid, while the perforated ones had circular holes with the same arrangement, the diameters of the holes being 4 mm and 6 mm. Given that some holes were obstructed by the radial stiffening, the overall open-area ratios were, respectively, 8.5 and 19.2 %. For simplicity they will be referenced as 10 and 20 % in the text.

The vertical load was measured by a force sensor located at the base of the shaft connecting the disk to the hydraulic jack. Five wave gauges, of resistive type, measured free surface elevations every 10 cm from the axisymmetry axis along a radial line. The first three ones
<figure><img src="image_4.png" /><figcaption>Figure 3: Solid disk in the tank.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Perforated (20 % porosity) disk in the tank.</figcaption></figure>

were removed at small disk submergence. The sampling frequency was 400 Hz for all signals while the periods of the imposed motion ranged from 0.6 to 2 seconds.

The measured vertical force was processed through Fourier analysis over sliding windows 3 periods long. In most tests steady state was reached after initial transients and before contamination by reflection from the tank walls. Figure 5 shows a typical time series of the recorded force, and the amplitude of its first harmonic obtained by the Fourier analysis.

## TEST CASES

Two series of tests were performed.

The first series focused on the effect of motion amplitude, the oscillation period taking three different values (0.8, 1.2 and 1.6 s). The solid and perforated disks were studied at 5, 10 and 25 cm of submergence. The perforated ones were also tested at 40 cm submergence (5 cm from the tank bottom). Amplitudes ranged from 5 mm up to 10 cm.

In the second series of tests the solid disk only was tested

<figure><img src="image_6.png" /><figcaption>Figure 5: Time trace and harmonic analysis of the vertical force.</figcaption></figure>

at the rather shallow submergence of 5 cm, while the oscillation period varied from 0.6 up to 2 s, by 0.1 s increment, and the oscillation amplitude took the 4 values of 3, 5, 10 and 15 mm.

Altogether over 200 test cases were covered.

# 4 COMPARISONS BETWEEN CALCULATIONS AND MEASUREMENTS

## 4.1 Solid disk at 5 cm immersion

We start with this case where the sensitivity of the hydrodynamic coefficients to the oscillation period was extensively investigated. The added mass and damping coefficients, as obtained numerically and as derived from the measurements, are shown in figures 6 and 7. Figure 8 gives the Response Amplitude Operators (RAOs) of the free surface elevation at gauge S5, located at 50 cm from the disk axis.

<figure><img src="image_7.png" /><figcaption>Figure 6: Solid disk. Immersion 5 cm. Added mass coefficient. Calculated (solid lines) and measured values (symbols).</figcaption></figure>

In figure 6 it can be observed that the theoretical added mass coefficient is negative over the period range [0.8 s -
<figure><img src="image_8.png" /><figcaption>Figure 7: Solid disk. Immersion 5 cm. Damping coefficient. Calculated (solid lines) and measured values (symbols).</figcaption></figure>

1.3 s] (roughly) and that it attains the very high value of about 10 at a period of 1.6 s. It can be reminded here that in unbounded fluid the added mass of a disk is $8/3ρa^3$, meaning a $C_a$ value of 8/3. Likewise the theoretical damping coefficient attains values as high as almost 12 at the oscillation period of 1.4 s. It is quite remarkable that the experimental added mass and damping coefficients agree well with the numerical predictions at oscillation periods from 0.6 up to 1.2 s and that the agreement deteriorates at higher periods, with the experimental values shifting away from the theoretical ones quickly as the oscillation amplitude increases. Even at the smallest amplitude of only 3 mm the shift is appreciable. A kink in the experimental coefficients appear around 1.8 s, which can also be seen in figure 8, showing the RAO of the free surface elevation at gauge S5.

<figure><img src="image_9.png" /><figcaption>Figure 8: Solid disk. Immersion 5 cm. RAO gauge S5. Modulus.</figcaption></figure>

The reason why nonlinear effects come into play at oscillation periods larger than 1.2 s, and not below, is probably associated with time variation of the thickness of the water column above the disk. Figure 10 shows the calculated flux of water in-between domain 3 (above the disk) and domain 1 (outer domain, see figure 1). It is made non

<figure><img src="image_10.png" /><figcaption>Figure 9: Solid disk. Immersion 5 cm. RAO gauge S5. Phase.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Solid disk. Immersion 5 cm. Non-dimensional flux.</figcaption></figure>

dimensional by dividing it by $\pi A \omega a^2$. This quantity is also, from mass conservation, the RAO of the variation of the (averaged) water layer thickness above the disk. At periods less than 1 s, the plotted value is less than 0.5: for a motion amplitude of 10 mm, the averaged thickness varies from 45 to 55 mm. At a period of 1.4 s, a value as high as 3 is attained: with the same motion amplitude the thickness varies from 20 to 80 mm. This affects the phase velocity of the waves traveling over the disk and creates time lags, visible in figure 9.

This is but an interpretation which would need to be confirmed numerically.

## 4.2 Solid disk at 25 cm immersion

In this configuration the disk is midway in between the free surface and the false bottom of the tank. Figure 11 shows the theoretical added mass and damping coefficients, versus the oscillation period. Due to the immersion the damping coefficient is small and the added mass coefficient varies little.

Tests were performed at three oscillation periods: 0.8, 1.2 and 1.6 s, while the amplitudes ranged from 1 cm up to 8 cm. Figures 12 and 13 show the experimental added mass and damping coefficients, vs. the amplitude. The val-
<figure><img src="image_12.png" /><figcaption>Figure 11: Solid disk. Immersion 25 cm. Calculated added mass and damping coefficients.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 12: Solid disk. Immersion 25 cm. Measured added mass coefficient.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 13: Solid disk. Immersion 25 cm. Measured damping coefficients.</figcaption></figure>

uues plotted at zero amplitude are the potential values, from figure 11. It is remarkable that, from an amplitude of 6 cm, that is a KC number of 0.6, the damping coefficient ex-

ceeds the added mass coefficient (for the 1.2 and 1.6 s oscillation period).

In the right-hand-side figure, referring to the damping coefficients, we have drawn three curves, obtained by adding up, to the potential flow value, a viscous term, the viscous force being expressed as

$$F_v = -\frac{1}{2} \rho C_D \pi a^2 V |V| \approx -\frac{4}{3} \rho C_D a^2 A^2 \omega^2 \cos \omega t$$

meaning a supplementary damping coefficient

$$\Delta C_b = \frac{4}{3} C_D \frac{A}{a}$$

The drag coefficient $C_D$ is taken as

$$C_D = \alpha KC^{-1/3}$$

with the Keulegan-Carpenter number $KC = \pi A/a$.

With a well chosen $\alpha$ value a nearly perfect fit with the experimental damping coefficients is achieved, but the $\alpha$ value depends on the oscillation period. This suggests that the free surface is playing some role. As a matter of fact the experimental RAOs at gauges 4 and 5 increase with the motion amplitude at the 1.2 and 1.6 s periods, while they slightly decrease at the 0.8 s period.

It should be noticed that our $\alpha$ values imply drag coefficients much higher than reported in previous papers. For instance Lake et al. (2000) perform experiments on a much smaller (and thicker) disk (diameter 18.64 cm; thickness 7.75 mm) and obtain a drag coefficient in between 3 and 3.5 when KC is comprised between 0.2 and 0.4. With $\alpha \sim 10$, in this KC range our drag coefficient is around 15! Presumably this is due to the different value of the diameter over thickness ratio (24 in Lake et al.'s experiments, 600 in ours). This ratio has been known to be an important parameter at small Keulegan-Carpenter numbers (e.g. see Tao & Thiagarajan 2003).

## 4.3 Perforated disks at 25 cm immersion

We now address the perforated disk case, first at the intermediate immersion of 25 cm. We focus on the tests performed at the 1.2 s oscillation period, where a large number of amplitudes was covered.

Figures 14 and 15 show the experimental added mass and damping coefficients, vs. the amplitude, for the two porosity ratios of 10 and 20 %, and for the solid case. The values plotted at zero amplitude, for the solid case, are the potential flow calculated values. Strong differences are observed for the added mass coefficients, and much less, surprisingly, for the damping coefficients: they are nearly identical except at very small amplitudes where the 10 % porosity yields higher values, and at the highest amplitudes where the 20 % porosity values are a bit lower.

In figures 16 and 17, referring to the 20 % porosity case, we show the measured and calculated hydrodynamic coefficients. In figure 16 calculated values are shown with $\alpha = 0$ (no viscous correction), and with $\alpha = 9$, which gave the best fit for the solid disk. The discharge coefficient $\mu$ in
<figure><img src="image_15.png" /><figcaption>Figure 14: Immersion 25 cm. Period 1.2 s. Measured added mass coefficients.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure 15: Immersion 25 cm. Period 1.2 s. Measured damping coefficients.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 16: Porosity 20 %. Immersion 25 cm. Period 1.2 s. Measured and calculated added mass and damping coefficients.</figcaption></figure>

equation (1) has been taken equal to 0.5. In figure 17 the coefficient $\alpha$ has been decreased to 6. A better fit between numerical and experimental values is then obtained. It is un-

<figure><img src="image_18.png" /><figcaption>Figure 17: Porosity 20 %. Immersion 25 cm. Period 1.2 s. Measured and calculated added mass and damping coefficients.</figcaption></figure>

derstandable that, since some flow can take place through the disk, the vortex shedding at the outer edge be weaker in the perforated than in the solid case. Only asymptotically, when the motion amplitude increases, can the solid $\alpha$ value be recovered. This means that, in the perforated case, the viscous correction should be given a different dependence on the $KC$ number than in the solid case.

It is remarkable that the viscous correction modifies the added mass coefficient as well, whereas only the damping coefficient is modified in the solid case. This correction is negative: the added mass coefficient decreases.

## 4.4 Perforated disks at 5 cm immersion

Finally we present results for the 20 % porosity disk at the shallower immersion of 5 cm. Results at 0.8 and 1.6 s excitation period are given in figures 18 and 19.

<figure><img src="image_19.png" /><figcaption>Figure 18: Porosity 20 %. Immersion 5 cm. Measured and calculated added mass and damping coefficients. Period 0.8 s.</figcaption></figure>

At 0.8 s a good fit between experimental and numerical hydrodynamic coefficients is obtained with an $\alpha$ value taken equal to zero: probably the proximity of the free surface inhibits or lowers flow separation. It must also be commented that our expression of the drag force is based on an
<figure><img src="image_20.png" /><figcaption>Figure 19: Porosity 20 %. Immersion 5 cm. Measured and calculated added mass and damping coefficients. Period 1.6 s.</figcaption></figure>

averaged velocity through the disk (equation (22)), considered to be representative of the velocity at the edge of the disk. This is correct in unbounded fluid but not close to the free surface where the flow velocity varies along the radius, at a scale related to the wavelength. Probably an alternative formulation of the drag force could be devised.

At the 1.6 s period, experimental and numerical values quickly deviate from each other as the motion amplitude increases: increasing the motion amplitude is alike decreasing porosity and similar effects as in the solid case are obtained.

## 5 FINAL REMARKS

Some illustrative results have been presented from an extensive experimental campaign on forced heave tests of solid and porous disks. This campaign differs from literature-reported experiments by the size of the model (60 cm diameter), its large diameter over thickness ratio (600), and by the proximities of the free surface and bottom. The *KC* range covered is from 0 to 1.

At mid immersion, where the confinement effects are the lowest, very large damping coefficients have been derived from the force measurements, notably higher than the damping values reported in the literature. This feature is likely associated with our large diameter over thickness ratio. As a consequence, it is only for very small *KC* values (less than 0.1!) that porosity yields larger damping coefficients than in the solid case. The effect is more notable for the added mass coefficient, suggesting that rendering porous the heave plates of truss spars would allow to decrease the structural loads while keeping the damping value the same.

When oscillated close to the free surface, the hydrodynamic coefficients of the solid disk were found to be in good agreement with linearized potential flow calculations at the smaller periods, while nonlinear effects quickly came into play at the larger ones. This feature was associated with variations of the thickness of the water layer above the disk. This contention would deserve some confirmation. One way would be to couple a nonlinear shallow water model above the disk with a linear model in the outer

domain, in the manner used by Grue (1992) in the 2D case. Another approach, which we are considering, would be to use a CFD model.

As for the perforated disks, the numerical model of Molin & Nielsen (2004) was found to produce added mass and damping coefficients in good concordance with the measured ones only at small oscillation amplitudes. As the amplitude increases, effects associated with flow separation at the outer edge quickly come into play. An empirical viscous correction has been proposed to account for the associated drag force. As observed experimentally, it predicts an increased damping coefficient and a decreased added mass coefficient, but no perfect fit with the experimental values could be achieved.

## ACKNOWLEDGMENTS

The model tests at BGO-First were carried out within the GIS-HYDRO organization, with financial support from Conseil Général du Var. Financial support from Norsk Hydro for the analysis is gratefully acknowledged.

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